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A waterbody number , waterbody index number or waterbody ID is used for the hydrographic classification of waterbodies . Where classification only covers bodies of flowing water such as rivers, it may be called a watercourse number .
Many different systems are used in limnology to number waterbodies such as lakes, rivers and streams, ranging from basic index numbers representing their location in the river system , such as serial numbers for the major catchment areas of the rivers, or numbers indicating their hierarchical position in the system. On the basis of such numbers the aim is to give waterbodies, river sections or catchment areas a clear identification number . As a result of the systems used in various specialist fields there may even be different national and regional waterbody index numbers for the same body of water.
A general geographical and hydrographical classification uses Strahler numbers to determine the:
Hydrological-ecological classifications:
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https://en.wikipedia.org/wiki/Waterbody_number
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The Waterborne Disease and Outbreak Surveillance System (WBDOSS) is a national surveillance system maintained by the U.S. Centers for Disease Control and Prevention (CDC) . The WBDOSS receives data about waterborne disease outbreaks and single cases of waterborne diseases of public health importance (for example, Primary Amebic Meningoencephalitis (PAM) ) in the United States and then disseminates information about these diseases, outbreaks, and their causes. WBDOSS was initiated in 1971 by CDC, the Council of State and Territorial Epidemiologists (CSTE), and the Environmental Protection Agency (EPA) . Data are reported by public health departments in individual states, territories, and the Freely Associated States (composed of the Republic of the Marshall Islands, the Federated States of Micronesia and the Republic of Palau; formerly parts of the U.S.-administered Trust Territories of the Pacific Islands). Although initially designed to collect data about drinking water outbreaks in the United States, WBDOSS now includes outbreaks associated with recreational water, as well as outbreaks associated with water that is not intended for drinking (non-recreational) and water for which the intended use is unknown. [ 1 ] [ 2 ]
Waterborne disease outbreaks may be associated with recreational water, water intended for drinking, water not intended for drinking (non-recreational water, for example, from cooling towers or ornamental fountains) and water of unknown intent. In order for a waterborne disease outbreak to be included in WBDOSS there must be an epidemiologic link between two or more persons that includes a location of water exposure, a clearly defined time period for the water exposure, and one or more waterborne illnesses caused by pathogens such as bacteria, parasites and viruses, or by chemicals/toxins. Common routes of exposure to waterborne pathogens include swallowing contaminated water, inhaling water droplets or airborne chemicals from the water, and direct physical contact with contaminated water. Epidemiologic evidence must implicate water or volatile compounds from the water that have entered the air as the probable source of the illness. WBDOSS outbreaks are further evaluated and classified based on the strength of evidence in the outbreak report that implicates water as the source of the outbreak. Waterborne disease outbreaks that have both strong epidemiologic data and comprehensive water-quality testing data are assigned a higher class than outbreaks with weak epidemiologic data and little or no water-quality testing data. [ 1 ] [ 2 ]
Public health departments investigate waterborne disease outbreaks in states, territories, and Freely Associated States and are essential contributors to the WBDOSS. The primary reporting tool for WBDOSS prior to 2009 was the CDC 52.12 waterborne disease outbreak reporting form. Beginning in 2009, this form was replaced by the electronic National Outbreak Reporting System (NORS) . Secondary data sources include case reports of water-associated cases of PAM caused by Naegleria fowleri infections, case reports for chemical/toxin poisoning and wound infections (reported sporadically), data about recreational water-associated Vibrio cases from the Cholera and Other Vibrio Surveillance System, and case reports for pool chemical-related health events not associated with recreational water (reported sporadically. [ 1 ] [ 2 ]
CDC has published WBDOSS surveillance summaries on an annual or biennial basis since 1971. Summary statistics and descriptions of waterborne disease outbreaks were published in CDC reports until 1984 and have been published in the Morbidity and Mortality Weekly Report (MMWR) since 1985. Public health researchers and policy makers use the data to understand and reduce waterborne disease and outbreaks. WBDOSS data are available to support EPA efforts to improve drinking water quality and to provide direction for CDC’s recreational water activities, such as the Healthy Swimming program. [ citation needed ]
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https://en.wikipedia.org/wiki/Waterborne_Disease_and_Outbreak_Reporting_System
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Waterborne resins are sometimes called water-based resins. They are resins or polymeric resins that use water as the carrying medium as opposed to solvent or solvent-less. Resins are used in the production of coatings , adhesives , sealants , elastomers and composite materials . [ 1 ] [ 2 ] When the phrase waterborne resin is used, it usually describes all resins which have water as the main carrying solvent. The resin could be water-soluble, [ 3 ] water reducible or water dispersed. [ 4 ]
Most coatings have four basic components. These are the resin , solvent , pigment and additive systems [ 5 ] but the resin or binder is the key ingredient. Continuing environmental legislation in many countries along with geopolitics such as oil production are ensuring that chemists are increasingly turning to waterborne technology for paint/coatings and since resins or binders are the most important part of a coating, more of them are being developed and designed waterborne and there is a constantly increasing use by coating formulators. The use of waterborne coatings and hence waterborne resins really started to grow in the 1960s led by the United States and was driven by: a) the need to reduce flammability; b) environmental legislation aimed at reducing the amount of solvent vapor (VOC - Volatile organic compound ) discharged into the atmosphere; c) cost; d) political factors i.e. security of supply. [ 6 ] All these factors helped the desire to reduce the reliance on oil derived solvents. The use of water as the carrying solvent for coatings and hence resins has been increasing ever since. The same holds true for adhesives. Water is generally a low cost (but not free) commodity in plentiful supply with no toxicity problems so there has always been a desire to produce paints, inks, adhesives and textile sizes etc. with water as the carrying solvent. This has required the production of waterborne resins designed for these systems. In recent years legislative pressure has ensured that waterborne systems and hence waterborne resins are coming increasingly to the fore. [ 7 ] [ 8 ] [ 9 ]
An epoxy resin system generally consists of a curing agent and an epoxy resin. Both the curing agent and the epoxy resin can be made waterborne. Solid epoxy resin (molecular weight >1000) dispersions are available and consist of an epoxy resin dispersed in water sometimes with the aid of co-solvents and surfactants . The resin backbone is often modified to ensure water dispersibility. These resins dry in their own right by water/co-solvent evaporation and the particles coalescence. [ 10 ] To cure the resin and crosslink it, an amine-based curing agent is usually added. This produces a two-component system. An alternative is to use standard medium viscosity liquid epoxy resins and emulsify them in a water-soluble polyamine or polyaminoamide hardener resin which also gives a two-component system. Polyaminoamides (or polyamidoamines) are made by reacting ethylene amines with dimerized fatty acids to give a species with amide links but still having amine functionality. Water is liberated during the condensation reaction. These resins can then be made water-soluble by reacting further with glacial organic acids or formaldehyde . Resins like these are usually left with yet further amine functionality on the polymer backbone to enable them to cure and crosslink an epoxy resin. [ 11 ] Paints may then be made from them by pigmenting either the epoxy or the amine hardener portion or even both. [ 12 ] [ 13 ] Polyamine curing resins as opposed to polyaminoamide resins are generally made by partially adducting polyfunctional amines with an epoxy resin and/or epoxy diluent and leaving the species with residual amine functionality. This adduct can then be dissolved in water and used to emulsify more epoxy resin and again either portion or both may be pigmented. The advantage with these systems is that they do not need glacial organic acids to solubilize them. This is an advantage if the coating is to be used over a highly alkaline substrate such as fresh concrete, as the alkali from the cement will neutralise the acid and cause instability on repeated dipping of a brush into the can. [ 14 ] Even though water is present and is a fuel for corrosion , water-based metal coatings based on waterborne epoxy can also be formulated. [ 15 ] Other research is investigating the benefits of combining graphene technology with waterborne epoxy. [ 16 ]
Research continues and many patents and journal papers continue to be published with novel ways of converting epoxy systems to their waterborne counterparts. One such method is to take a molecule that already is intrinsically partially hydrophilic such as a diol with a polypropylene oxide backbone, and then reacting it with epichlorohydrin and then dehydrochlorinated with sodium hydroxide. This produces a diepoxy terminated polypropylene glycol molecule. This can now be reacted with an ethyleneamine such as triethylenetetramine (TETA) to produce an amine terminated moiety that is intrinsically hydrophilic and able to cure an epoxy resin. [ 17 ] [ 18 ] These waterbased wpoxy coatings when used with the right choice of pigments, can be used to coat the inside of oil tanks. [ 19 ]
Water reducible alkyds are basically conventional alkyd resins (i.e., polyesters based on saturated or unsaturated oils or fatty acids, polybasic acids and alcohols) modified to confer water miscibility. Typical components are vegetable oils or fatty acids such as linseed , soybean, castor, dehydrated castor, safflower, tung, coconut and tall oil. Acids include isophthalic, terephthalic, adipic, benzoic, succinic acids and phthalic, maleic and trimellitic anhydride. Polyols include glycerol , pentaerythritol , Trimethylolpropane , ethylene glycol , propylene glycol , diethylene glycol , neopentyl glycol , 1,6-hexanediol and 1,4-butanediol . [ 20 ] Typical methods for introducing varying degrees of water miscibility are similar to other resin systems. Methods basically involve introducing hydrophilic centres such as acid groups that can then be neutralised to form a salt. [ 21 ] Introducing polar groups onto the backbone is another method. With alkyds typical methods include maleinazation of unsaturated fatty acids with maleic anhydride . This involves making a Diels-Alder adduct near the double bond sites. The acid groups introduced can then be further reacted with polyols. A Diels-Alder reaction only occurs where there is a conjugated double bond system. Simple addition occurs if not conjugated. Other techniques include synthesizing the resin with hydroxyl functional oligomers e.g. containing ethylene glycol then adding specific acid or hydroxyl containing substances towards the end of the reaction. Another technique is making an acrylic functional alkyd with an acrylic monomer blend rich in carboxylic acid groups. Synthesis techniques have been studied and published for acrylic modified water-reducible alkyds. [ 22 ]
Late twentieth century technology allowed the production of alkyd emulsions. [ 23 ] The technology continues to evolve including production of DTM (Direct To Metal) finishes. [ 24 ] The biggest issue has been getting VOC content below 250g/L. Poor corrosion resistance has also been an issue. Alkyd emulsion technology uses a reactive surfactant that has double bonds and thus oxidative drying properties like a conventional alkyd. The material is then put under shear and water added slowly. Initially a water in oil emulsion is formed but continued water addition and shear results in inversion and a stable oil in water emulsion is formed. [ 25 ] [ 26 ] Sustainability and other market factors mean a number of companies are entering the market. [ 27 ] As well as patents, doctoral theses are being done at universities on the subject. [ 28 ]
Saturated polyester resins contain many of the materials used in conventional alkyd resins but without the oil or fatty acid components. Typical components for these resins are poly carboxylic and polyhydroxyl components. The more commonly used polyacids are phthalic, isophthalic, terephthalic and adipic acid. Phthalic and trimellitic anhydrides may also be used. Polyols tend to be neopentyl glycol, 1,6-hexanediol and trimethylolpropane . To make them waterborne organic acids or anhydrides are added in a two-stage process but there are other methods too. [ 29 ] [ 30 ]
Polyurethanes resins are available waterborne. The single component versions are usually referred to as Polyurethane dispersions (PUD).
They are available in anionic, cationic and nonionic versions though anionic moieties are the most readily available commercially. [ 31 ] The use of an anionic or cationic center or indeed a hydrophilic non-ionic manufacturing technique tends to result in a permanent inbuilt water resistance weakness. Research is being conducted and techniques developed to combat this weakness. [ 32 ] Cationic PUD also introduce hydrophilic components when synthesized, but techniques have and are being researched to improve the performance and water resistance properties by various techniques. This includes introducing star-branched polydimethylsiloxane. [ 33 ]
Waterborne polyurethanes are also available in 2 component versions. [ 34 ] As a 2 component polyurethane consists of polyol (s) and an isocyanate and isocyanates react with water this requires special formulating and production techniques. [ 35 ] [ 36 ] The polyisocyanate that is water-dispersible maybe modified with sulfonate [ 37 ] for example. PUDs are not usually synthesised with plant based polyols because they don't have other performance enhancing functional groups. Recent work (2021) reports modification to achieve this and enable even greener versions. [ 38 ] Work is also ongoing to get the performance of 1 component waterborne polyurethanes to match that of 2 component versions. [ 39 ] Self-healing versions of two-component waterborne polyurethanes are being researched. [ 40 ] Research has shown that modification of these resin systems with polyaniline improves a number of properties including corrosion resistance. [ 41 ] Higher solid content is also desired for economics of not transporting water. [ 42 ]
Ionic centers are usually introduced with waterborne PUDs, and so the water resistance in the resultant film has been studied. The nature of the polyol and the level of COOH groups and hydrophobic modification with other moieties can improve the hydrophilicity. Polyester polyols give the biggest improvements. [ 43 ] [ 44 ] Polycarbonate polyols also enhance properties, [ 45 ] [ 46 ] especially if the polycarbonate is also fluorinated. [ 47 ]
Silicone modification of the resin makes the species much more hydrophobic and water resistant. [ 48 ] [ 49 ] [ 50 ]
As the world attempts to move towards a low-carbon economy , carbon capture by using carbon dioxide from the atmosphere is gaining attention and research being done. Using carbon dioxide in PUD production is being researched. [ 51 ]
A latex is a stable dispersion ( emulsion ) of polymer in water . Synthetic lattices are usually made by polymerizing a monomer such as vinyl acetate that has been emulsified with surfactants dispersed in water . [ 52 ] The overall technique is called Emulsion polymerization . Other techniques including inversion from water in oil to oil in water emulsions are available. [ 53 ] Particular emphasis in recent years has been the production of self-crosslinking versions especially acrylic emulsions. As an example, these may be produced by modifying with divinyl silane. [ 54 ] Some examples include vinyl acetate based latices, acrylics and styrene-butadiene versions. They may be used to produce waterborne direct to metal coatings. [ 55 ] Waterborne acrylic resins are also used frequently in water-based paints. [ 56 ]
Acrylic latices prepared by emulsion polymerization are often improved by copolymerizing other functional monomers. [ 57 ] Glycidyl methacrylate is one such monomer used which then incorporates oxirane functionality into the polymer. This would then improve the properties (such as scrub resistance) of the paint formulated from this resin. DMAEMA (dimethylaminoethyl methacrylate) is another such species. [ 58 ] Other innovative techniques for improving acrylic latices include incorporating a biocide with acrylic functionality as the modifying monomer. This allows the binder for a waterborne paint to be inherently anti-biocidal. [ 59 ] Techniques exist to speed up the cure of waterborne acrylics. [ 60 ] Waterborne acrylic latices and polyurethane acrylates that are UV curable have also been produced. [ 61 ]
Polymeric and oligomeric aziridines are one of the moieties used to crosslink waterborne resins. They usually react with the carboxyl groups present on these species. Potlife is usually improved along with other properties. [ 62 ]
Emulsion polymerization : Polymerization whereby monomer(s), initiator, dispersion medium, and possibly colloid stabilizer constitute initially an inhomogeneous system resulting in particles of colloidal dimensions containing the formed polymer.
Note : With the exception of mini-emulsion polymerization , the term “emulsion polymerization” does not mean that polymerization occurs in the droplets of a monomer emulsion. [ 63 ]
Batch emulsion polymerization : Emulsion polymerization in which all the ingredients are placed in a reactor prior to reaction. [ 64 ]
see article Electrophoretic deposition
The resins used for electrodeposition are usually epoxy, acrylic or phenolic resin types. They are formulated with functional groups which when neutralised form ionic groups on the polymer backbone. These confer water solubility on the polymer. They are available as anodic versions which deposit on the cathode of an electrochemical cell or cathodic which deposit on the cathode. [ 65 ] Cathodic electrodeposition resins dominate and they have revolutionised corrosion protection in the automotive industry. Ceramics as well as metals may be coated this way. [ 66 ] They are applied as OEM (Original Equipment Manufacture) rather than as a refinishing system. Cathodic resins contain amines on the polymer backbone which are neutralized by acids groups such as acetic acid to give a stable aqueous dispersion. When an electric current is passed through a car body that is dipped in a bath containing a paint based on a cathodic electrodeposition resin, the hydroxyl ions formed near the cathode deposit the paint on the car body. The electric current needed for this is determined by the number of ionic centers.
Dispersions of waterborne resins for electrocoating usually contain some co-solvents such as butyl glycol and isopropanol and are usually very low in solids content i.e. 15%. They usually have molecular weights in the region of 3000–4000. Paints based on them tend to have PVCs of less than 10 i.e. a very high binder to pigment ratio.
Cathodic electrophoretic deposition coatings can be made that are self-healing even at room temperature. The base polymer used for this synthesis is, a waterborne Polyurethane Dispersion (PUD) that is cationic rather than anionic. [ 67 ]
Many resins are available waterborne but can be hybrids or blends. An example would be polyurethane dispersions blended or hybridized with acrylic resins, [ 68 ] [ 69 ] which are commonly used in automotive paint . Such systems can be made by using acrylic monomers and a polyurethane dispersion which will polymerise simultaneously to give an interpenetrating polymer network , without the need for NMP as a cosolvent. This combines the lower cost of acrylic with the high performance of a polyurethane. [ 70 ] Waterborne epoxy resins may be modified with acrylate and then further modified with side chains having many fluorine atoms on them. [ 71 ] Waterborne resins are also available that use both water and renewable raw materials. [ 72 ] Another example is to combine alkyd resins with acrylics to make them waterborne. Using hyperbranched alkyds and modifying them with acrylic monomers and using mini emulsion polymerization, suitable hybrids maybe formed. [ 73 ] As well as hybridization of the resins, a combination of techniques maybe employed. As an example, ultraviolet curing coatings that can be electrodeposited and are waterborne hybrids of epoxy and acrylic resins maybe produced. [ 74 ] [ 75 ]
Hybrid resins include among others, PUDs that are both waterborne and UV curable. They are being researched and many papers published. [ 76 ] [ 77 ] [ 78 ] [ 79 ] [ 80 ] [ 81 ] PUD- acrylics using epoxidized soybean oil have been produced that are UV curable. [ 82 ] The structure and type of acrylate will affect the properties. [ 83 ] Hybrid resins used in coatings that are vegetable based, waterborne and UV curable are considered very green and have also been investigated. [ 84 ] [ 85 ] Similarly, UV-curable waterborne fluorinated polyurethane-acrylate resins can be designed and used in coatings. [ 86 ] As well as acrylic PUD hybridization, further modification with silane monomers can be undertaken. [ 87 ]
Other examples of hybridization include modifying waterborne epoxy with latex dispersions. The latex-modified epoxy aqueous dispersions are treated by evaporation techniques. Nitrile latices were used in the study. [ 88 ]
Modification of soybean oil that has been epoxidized and then reacted with acrylic acid will produce waterborne epoxy acrylates that are also based on some renewable content. The corrosion resistance properties are improved using this technique. [ 89 ]
Alkyds can likewise be hybridised and made water reducible. This may be achieved by acrylic modification. [ 90 ] Waterborne epoxy resins may also be acrylated and hybridized and much research has gone into these systems. [ 91 ] [ 92 ] [ 93 ] [ 94 ] Research is also taking place using waterborne alkyd resins hybridized with styrene-acrylic emulsions. These then find use in waterborne exterior decorative and architectural paints. [ 95 ]
High bio-based content or renewability of materials is highly prized as there is a trend in some parts of the world to a low-carbon economy. [ 96 ] Waterborne resins are already perceived as environmentally friendly but work is ongoing to improve this further by using non-petroleum based raw materials where possible. [ 97 ] Waterborne epoxies are one such area of research. [ 98 ] Since waterborne resins are usually considered green and environmentally friendly, techniques are being researched that include capturing carbon dioxide from the atmosphere to make the raw materials and then further synthesis. [ 99 ]
Water is in some ways an unusual chemical. It is a very powerful and universal solvent. Most liquids decrease in volume on freezing, but water expands. It occurs naturally on earth in all three states of solid (ice), liquid (water) and gas(water vapour and steam). At 273.16 K or 0.16 °C (known as the triple point ) it can coexist in all three states simultaneously. It has a very low molecular weight of 18 and yet a relatively high boiling point of 100 0 C. This is due to inter molecular forces and in particular hydrogen bonding . The surface tension is also high at 72 dynes/cm (mN/metre) which affects its ability to wet certain surfaces. It evaporates (latent heat of evaporation 2260 kJ per kg) very slowly in comparison to some solvents and hardly at all when the relative humidity is very high. It has a very high specific heat capacity (4.184 kJ/kg/K ) and that is why it is used in central heating systems in the United Kingdom and Europe . These factors have to be borne in mind when formulating waterborne resins and other water based systems such as adhesives and coatings. [ 100 ] [ 101 ]
Waterborne resins find use in Coatings , Adhesives , Sealants and Elastomers and other applications. Specifically they find use in textile coatings, [ 102 ] industrial coatings, [ 103 ] UV coatings, [ 104 ] floor coatings, [ 105 ] hygiene coatings, [ 106 ] wood coatings, [ 107 ] adhesives, [ 108 ] concrete coatings, [ 109 ] automotive coatings, [ 110 ] [ 111 ] clear coatings [ 112 ] and anticorrosive applications including waterborne epoxy based anticorrosive primers [ 113 ] [ 114 ] [ 115 ] They are also used in the design and manufacture of medical devices such as the polyurethane dressing, a liquid bandage based on polyurethane dispersion. [ 116 ] Over the years they have also been used in polymer modified cements and repair mortars [ 117 ] They have also found use in general textile applications including coating nonwovens. [ 118 ] Recent (post 2020) innovations have included producing a waterborne polyurethane that has embedded silver particles to combat COVID . [ 119 ] Waterborne polyurethane dispersions with antimicrobial properties have also been developed. [ 120 ]
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A water cut meter measures the water content (cut) of crude oil and hydrocarbons as they flow through a pipeline. While the title "Water cut" has been traditionally used, the current API naming is Water Cut Analyser or WCA as OWD or On-Line Water Determination is trademarked. The API and ISO committees have not yet come out with a standard for these devices. Though the API is currently in late stages of balloting a draft. There are however standards in place for fiscal automatic sampling of crude oil namely API 8.2 and ISO 3171. [ 1 ]
Water cut meters are typically used in the petroleum industry to measure the water cut of oil flowing from a well, produced oil from a separator , crude oil transfer in pipelines and in loading tankers.
There are several technologies used. The main technologies are dielectric measurements using radio or microwave frequency and NIR measurements and less common are gamma ray based instruments.
The water cut is the ratio of water produced compared to the volume of total liquids produced from an oil well. The water cut in waterdrive reservoirs can reach very high values. [ 2 ]
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https://en.wikipedia.org/wiki/Watercut_meter
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The waterfall model is a breakdown of developmental activities into linear sequential phases, meaning that each phase is passed down onto each other, where each phase depends on the deliverables of the previous one and corresponds to a specialization of tasks. [ 1 ] This approach is typical for certain areas of engineering design . In software development , [ 1 ] it tends to be among the less iterative and flexible approaches, as progress flows in largely one direction (downwards like a waterfall ) through the phases of conception, initiation, analysis , design , construction , testing , deployment , and maintenance . [ 2 ] The waterfall model is the earliest systems development life cycle ( SDLC ) approach used in software development. [ 3 ] When it was first adopted, there were no recognized alternatives for knowledge-based creative work. [ 4 ]
The first known presentation describing the use of such phases in software engineering was held by Herbert D. Benington at the Symposium on Advanced Programming Methods for Digital Computers on 29 June 1956. [ 5 ] This presentation was about the development of software for SAGE . In 1983, Benington republished his paper with a foreword explaining that the phases were on purpose organized according to the specialization of tasks, and pointing out that the process was not in fact performed in a strict top-down fashion, but depended on a prototype. [ 6 ] [ better source needed ]
Although the term "waterfall" is not used in the paper, the first formal detailed diagram of the process later known as the "waterfall model" is often [ 7 ] cited as coming from a 1970 article by Winston W. Royce . [ 8 ] [ 9 ] [ 10 ] However, he commented that it had major flaws stemming from how testing only happened at the end of the process, which he described as being "risky and [inviting] failure". [ 8 ] The rest of his paper introduced five steps which he felt were necessary to "eliminate most of the development risks" associated with the unaltered waterfall approach. [ 8 ]
Royce's five additional steps (which included writing complete documentation at various stages of development) never took mainstream hold, but his diagram of what he considered a flawed process became the starting point when describing a "waterfall" approach. [ 11 ] [ 12 ]
The earliest use of the term "waterfall" may have been in a 1976 paper by Bell and Thayer. [ 13 ] [ better source needed ]
In 1985, the United States Department of Defense adopted the waterfall model in the DOD-STD-2167 standard for working with software development contractors. This standard referred for iterations of a software development [ 14 ] to " the sequential phases of a software development cycle " and stated that " the contractor shall implement a software development cycle that includes the following six phases: Software Requirement Analysis, Preliminary Design, Detailed Design, Coding and Unit Testing, Integration, and Testing ". [ 14 ] [ 15 ]
Although Royce never recommended nor described a waterfall model, [ 16 ] rigid adherence to the following phases are criticized by him:
Thus, the waterfall model maintains that one should move to a phase only when its preceding phase is reviewed and verified.
Various modified waterfall models (including Royce's final model), however, can include slight or major variations on this process. [ 8 ] These variations include returning to the previous cycle after flaws are found downstream, or returning to the design phase if downstream phases are deemed insufficient.
Time spent early in the software production cycle can reduce costs at later stages. For example, a problem found in the early stages (such as requirements specification) is cheaper to fix than the same bug found later on in the process (by a factor of 50 to 200). [ 17 ]
In common practice, waterfall methodologies result in a project schedule with 20–40% of the time invested for the first two phases, 30–40% of the time to coding, and the rest dedicated to testing and implementation. With the project organization needing to be highly structured, most medium and large projects will include a detailed set of procedures and controls, which regulate every process on the project. [ 18 ] [ failed verification ]
A further argument supporting the waterfall model is that it places emphasis on documentation (such as requirements documents and design documents) as well as source code . [ citation needed ] In less thoroughly designed and documented methodologies, knowledge is lost if team members leave before the project is completed, and it may be difficult for a project to recover from the loss. If a fully working design document is present (as is the intent of big design up front and the waterfall model), new team members and new teams should be able to familiarise themselves to the project by reading the documents. [ 19 ]
The waterfall model provides a structured approach; the model itself progresses linearly through discrete, easily understandable and explainable phases and thus is easy to understand. It also provides easily identifiable milestones in the development process, often being used as a beginning example of a development model in many software engineering texts and courses. [ 20 ]
Similarly, simulation can play a valuable role within the waterfall model. [ 21 ] By creating computerized or mathematical simulations of the system being developed, teams can gain insights into how the system will perform before proceeding to the next phase. Simulations allow for testing and refining the design, identifying potential issues or bottlenecks, and making informed decisions about the system's functionality and performance.
Clients may not know the exact requirements before they see working software and thus change their requirements further on, leading to redesign, redevelopment, and retesting, and increased costs. [ 22 ]
Designers may not be aware of future difficulties when designing a new software product or feature, in which case revising the design initially can increase efficiency in comparison to a design not built to account for newly discovered constraints, requirements, or problems. [ 23 ]
Organisations may attempt to deal with a lack of concrete requirements from clients by employing systems analysts to examine existing manual systems and analyse what they do and how they might be replaced. However, in practice, it is difficult to sustain a strict separation between systems analysis and programming, [ 24 ] as implementing any non-trivial system will often expose issues and edge cases that the systems analyst did not consider.
Some organisations, such as the United States Department of Defense, now have a stated preference against waterfall-type methodologies, starting with MIL-STD-498 released in 1994, which encourages evolutionary acquisition and iterative and incremental development . [ 25 ]
In response to the perceived problems with the "pure" waterfall model, many 'modified waterfall models' have been introduced. These models may address some or all of the criticisms of the "pure" waterfall model.
These include the rapid development models that Steve McConnell calls "modified waterfalls": [ 17 ] Peter DeGrace's "sashimi model" (waterfall with overlapping phases), waterfall with subprojects, and waterfall with risk reduction. Other software development model combinations such as "incremental waterfall model" also exist. [ 26 ]
Winston W. Royce 's final model, his intended improvement upon his initial "waterfall model", illustrated that feedback could (should, and often would) lead from code testing to design (as testing of code uncovered flaws in the design) and from design back to requirements specification (as design problems may necessitate the removal of conflicting or otherwise unsatisfiable/undesignable requirements). [ citation needed ] In the same paper Royce also advocated large quantities of documentation, doing the job "twice if possible" (a sentiment similar to that of Fred Brooks , famous for writing the Mythical Man Month — an influential book in software project management — who advocated planning to "throw one away"), and involving the customer as much as possible (a sentiment similar to that of extreme programming ).
Royce notes on the final model are the following:
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Waterfall plots are often used to show how two-dimensional phenomena change over time. [ 1 ] A three-dimensional spectral waterfall plot is a plot in which multiple curves of data, typically spectra , are displayed simultaneously. Typically the curves are staggered both across the screen and vertically, with "nearer" curves masking the ones behind. The result is a series of "mountain" shapes that appear to be side by side. The waterfall plot is often used to show how two-dimensional information changes over time or some other variable such as rotational speed . Waterfall plots are also often used to depict spectrograms or cumulative spectral decay [ definition needed ] (CSD).
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The Waterloopkundig Laboratorium ( Hydraulic Research Laboratory ) was an independent Dutch scientific institute specialising in hydraulics and hydraulic engineering . The laboratory was established in Delft from 1927, moving to a new location in the city in 1973. The institute later became known as WL | Delft Hydraulics . In 2008, the laboratory was incorporated into the international nonprofit Deltares institute. [ 1 ]
The Hydraulic Laboratory was classified by the Dutch Government as a major technological institute [ nl ] and was tasked with acquiring, generating, and disseminating knowledge on hydraulics and hydraulic engineering. [ 2 ]
The laboratory conducted research into the causes of changes in the course of rivers , estuaries , and coasts , and the possible influences on them due to hydraulic engineering activities, along with a range of studies on topics such as dredging , wave action and coastal morphodynamics . The laboratory played a significant advisory role in the conception, design, and implementation of the Zuiderzee Works and the Delta Works , [ 3 ] along with several international projects. [ 4 ] [ 5 ] [ 6 ] [ 7 ]
The laboratory was established in 1927 by Rijkswaterstaat , under the directorship of Professor ir. J.Th. Thijsse (1893-1984). It was initially located in the basement of the Civil Engineering Department building at Delft University of Technology . [ 8 ]
Thijsse's role on the Zuiderzee State Commission had introduced him to hydrodynamic model research, an innovative approach to understanding the dynamics of water. In 1927, both Rijkswaterstaat and Delft University of Technology began incorporating this research methodology, prompting the establishment of the laboratory. [ 5 ]
The impetus for the formation of the laboratory began in the 1920s, and lay in the design of the sluices for the Afsluitdijk , a significant project requiring extensive research and experimentation. The task was initially assigned to Professor Theodor Rehbock at the Flussbaulaboratorium (river construction laboratory) at the Technical University of Karlsruhe , a major institute in the field of hydraulic engineering research at the time. The results of this investigation were documented in a report which was published in 1931. [ 9 ]
This report was subject to review by Thijsse, who advised the Dutch authorities on the need for additional research of this type, not just for the Zuiderzee Works, but also for other projects across the Netherlands. This recommendation precipitated the decision to establish a laboratory similar to that in Karlsruhe, to serve the Netherlands. Thijsse spearheaded the initial research at the newly formed laboratory and documented the findings in a follow-up report to Rehbock's original study. [ 10 ]
To facilitate third-party contract research, such as work for Rijkswaterstaat and international schemes, it was decided that the laboratory would operate independently from the Delft University of Technology, and be established as a financially autonomous foundation, with its board appointed from university staff, major consultants, and representatives from Rijkswaterstaat. [ 2 ] [ 11 ]
Experiments into the behaviour of irregular waves had been undertaken in the Netherlands since 1920, with initial experiments on irregular wave behaviour in wind tunnels . This pioneering research, including investigations into wave run-up , led to the construction of a specialised wind wave flume at the laboratory in 1933. Unprecedented at the time of construction, the flume boasted dimensions of 25 metres in length, 4 metres in width, and a maximum water depth of 0.45 metres. [ 12 ]
Subsequently, in order to better satisfy the necessary conditions for wave height and period, the flume was extended to 50 metres in length, and fitted with a monochromatic wave generator. These enhancements enabled a wider variety of research projects, including studies on wave overtopping , the stability of rubble-mound breakwaters , wave impact forces, and the stability of floating structures. By the time of World War II , research had extended into model investigations of wave generation, with outcomes corroborating prototype data collected by Harald Sverdrup and Walter Munk . [ 13 ] [ 14 ]
In 1969, new wave flumes with typical widths of 8 metres were installed in the laboratory in order to permit modelling and testing of breakwaters and dikes whilst simulating arbitrary angles of wave attack. The previously available flume widths of 4 metres had proved too small for this purpose, and the new flumes therefore provided the laboratory with the ability to model and test the performance of significant coastal and river engineering structures. [ 15 ] [ 16 ]
In 1973, the laboratory moved from its location in the centre of Delft to a new location at the most southern end of the Delft Technological University campus, becoming known locally as the Thijsse-erf (Thijsse yard) . [ 17 ] [ 18 ]
Throughout its history, the laboratory undertook national and international research on numerous civil and hydraulic engineering subjects including dredging technology, [ 19 ] density issues, pumps , and detailed structural studies on locks and weirs . [ 20 ] [ 21 ] International projects included the Belgian Port of Zeebrugge (1933–36), the cut-off of the Abidjan lagoon (1933–46), [ 4 ] and flood prevention works in Nottingham (1946–51). [ 11 ] [ 22 ] [ 23 ]
From 1951 to 1996, a second location known as the Waterloopkundig Laboratorium de Voorst (WLV) (Hydraulic Laboratory "de Voorst" ) was located in Noordoostpolder , between Marknesse , Kraggenburg , and Vollenhove . The establishment of a second laboratory at de Voorst was prompted by the lack of space in Delft for large outdoor models. Utilising land on the outskirts of Delft was not feasible due to the damp peat soil, which made it difficult to construct large models without soil settlement . In an environment where water levels are measured on a millimetric scale, even minute settlements were unacceptable. [ 24 ] [ 2 ]
Additional benefits of the de Voorst location included its location within a low-lying polder , eliminating the need for an additional pumping system, and its availability due to the heavy boulder clay composition of the soil making it unsuitable for farming . Since the land was government-owned, no financial acquisition was required. [ 25 ] From 1951, the Waterloopkundig Laboratorium therefore operated two facilities: an indoor modelling laboratory in Delft, and an outdoor model facility in De Voorst. In the 1970s, indoor laboratory facilities were added to the de Voorst location. [ 26 ]
A significant advantage of the de Voorst location was the ability to construct large-scale models of estuaries and ports , enabling model tests to predict the influence of hydraulic works on the watercourses , making use of the large differences in water levels from the surrounding surface water. These models were pivotal during the planning and construction phase of the Delta Works in Zeeland , and also allowed research works to be undertaken for international projects such as the reconstruction of the Port of Lagos . [ 4 ] [ 27 ]
Other international projects where research was carried out at the laboratory to inform the design and construction included the construction of the Eider Barrage , and works at the mouth of the Volta River in Ghana . [ 4 ] [ 6 ]
The scale of the physical models in the laboratory were often substantial, with many being large enough to permit model ships which necessitated pilotage by helmsmen , an example being the model created for Jo Thijsse's design for the junction of the Amsterdam–Rhine Canal and the Lek , a large structure which came to be known as De Eieren van Thijsse (Thijsse's eggs). [ 28 ]
From the 1980s, computer-assisted mathematical modelling began to be useful in mapping potential water flows, reducing the need for very large-scale physical water models. [ 29 ] Consequently, the decision was made in 1995 to concentrate activities at the Delft location and close the de Voorst facilities. The site was purchased by Natuurmonumenten and renamed the Waterloopbos , where visitors can view the models and associated infrastructure via a walking route through the woods. [ 30 ] [ 31 ] [ 32 ]
By 2008, the Delft laboratory had become known by the English name WL | Delft Hydraulics , and in an effort to consolidate knowledge with similar institutes, it was merged with other research institutes and sections of Rijkswaterstaat to form the Deltares Institute . [ 33 ] The laboratory continues to operate today as part of Deltares. [ 2 ]
The following people were directors of the laboratory from its foundation in 1927 until it merged with Deltares in 2008. [ 2 ]
The following personnel served as Heads of the de Voorst facility.
Significant engineering figures who undertook research or served in senior positions with the Waterloopkundig Laboratorium included Eco Bijker (various roles including head of department, head of the de Voorst Laboratory, and deputy director), [ 34 ] [ 35 ] Pieter Abraham van de Velde , [ 36 ] Frank Spaargaren (interim general director, 1995–1997), [ 37 ] [ 38 ] Krystian Pilarczyk (research engineer, 1966–1968), [ 39 ] [ 40 ] and PJ Wemelsfelder , who undertook research at the facility and served as head of the Hydrometric Department . [ 41 ]
A similar institution known as the Waterbouwkundig Laboratorium (Hydraulic Engineering Research Laboratory) is located in Borgerhout , Belgium. It was established in 1933. [ 42 ]
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A watermaker is a device used to obtain potable water by reverse osmosis of seawater. In boating and yachting circles, desalinators are often referred to as "watermakers".
The devices can be expensive to acquire and maintain, but are quite valuable because they reduce the need for large water tanks for a long passage.
The term watermaker may also refer to an atmospheric water generator , a machine that extracts potable water from the humidity in air using a refrigeration or a desiccant .
Many versions are used by long-distance ocean cruisers.
Depending on the design, watermakers can be powered by electricity from the battery bank, an engine, an AC generator or hand operated. There is a portable, towed, water-powered watermaker available which converts to hand operation in an emergency.
There is great variation in the amount of water consumed.
At home in the United States , each person uses about 55 U.S. gallons (210 liters) of water per day on average. [ 1 ] Where supplies are limited, and in emergencies, much less may be used.
Typical cruising yachts use from 4 to 20 liters (1.1 to 5.3 U.S. gallons) per person per day, the average probably being about 6 liters (1.6 U.S. gallons). The minimum water intake required to maintain body hydration is 1.5 liters (0.40 U.S. gallons) per day. The amount of water that is required for a person to consume is dependent on different factors. Some of these factors include weight, height and gender. Men on average needs a greater amount of water than women do. [ 2 ] [ citation needed ]
Popular brands of yacht watermakers typically make from 2 to 150 litres per hour of operation (0.53 to 41 gallons) depending on the model.
There are strong opinions among small boat cruisers about the usefulness of these devices. The arguments may be summarised as:
Some manufacturers of electrically powered watermakers have energy recovery systems in their design which reduce the power consumption; however, these are typically some 50% more expensive for any similar size due to their additional complexity. As a guideline, assuming a 12V DC system, the energy recovery incorporated in those watermakers have the effect of reducing the electric current used from perhaps typically 20A to about 8A. Like any piece of equipment, it is bound to fail at some time and cause expense/anxiety.
All watermakers designed for small boats and yachts rely on essentially the same technology, exploiting the principle of "reverse osmosis": a high pressure pump forcing seawater through a membrane that allows water but not salt to pass.
The common comparison is that of a filter; however, as the holes in the membrane are smaller than molecules of sodium chloride (salt) and indeed smaller than bacteria, and pressures in the nature of 45-50 bar are required, the process is much more complex than the common water filter or the oil filter found in automobile engines.
An atmospheric water generator is a machine that extracts potable water from the humidity in air using a refrigeration or a desiccant . Condensing moisture by refrigeration requires a minimum ambient temperature of about 10–15 °C (50–59 °F), while desiccant adsorbers have no such restriction. Either method is suitable for a desert climate, where water production is dependent on ambient humidity. The Negev desert in Israel, for example, has a significant average relative humidity of 64%. [ 3 ]
Contrary to some online sources, [ 4 ] a 1922 article in Popular Science cites an average relative humidity of 30% for the Sahara Desert, about half the humidity in an air-conditioned home. [ 5 ] Moreover, the effect of the dew point causes early mornings to have higher humidity, so that atmospheric water generation is possible even in the harshest climates. [ citation needed ]
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Waterproofing is the process of making an object, person or structure waterproof or water-resistant so that it remains relatively unaffected by water or resists the ingress of water under specified conditions. Such items may be used in wet environments or underwater to specified depths.
Water-resistant and waterproof often refer to resistance to penetration of water in its liquid state and possibly under pressure, whereas damp proof refers to resistance to humidity or dampness. Permeation of water vapour through a material or structure is reported as a moisture vapor transmission rate (MVTR).
The hulls of boats and ships were once waterproofed by applying tar or pitch . Modern items may be waterproofed by applying water-repellent coatings or by sealing seams with gaskets or o-rings .
Waterproofing is used in reference to building structures (such as basements , decks, or wet areas), watercraft, canvas, clothing ( raincoats or waders ), electronic devices and paper packaging (such as cartons for liquids).
In construction, a building or structure is waterproofed with the use of membranes and coatings to protect contents and structural integrity. The waterproofing of the building envelope in construction specifications is listed under 07 - Thermal and Moisture Protection within MasterFormat 2004, by the Construction Specifications Institute , and includes roofing and waterproofing materials. [ citation needed ]
In building construction , waterproofing is a fundamental aspect of creating a building envelope , which is a controlled environment. The roof covering materials, siding , foundations, and all of the various penetrations through these surfaces must be water-resistant and sometimes waterproof. Roofing materials are generally designed to be water-resistant and shed water from a sloping roof, but in some conditions, such as ice damming and on flat roofs , the roofing must be waterproof. Many types of waterproof membrane systems are available, including felt paper or tar paper with asphalt or tar to make a built-up roof, other bituminous waterproofing , ethylene propylene diene monomer EPDM rubber , hypalon , polyvinyl chloride , liquid roofing , and more.
Walls are not subjected to standing water, and the water-resistant membranes used as housewraps are designed to be porous enough to let moisture escape. Walls also have vapor barriers or air barriers . Damp proofing is another aspect of waterproofing. Masonry walls are built with a damp-proof course to prevent rising damp , and the concrete in foundations needs to be damp-proofed or waterproofed with a liquid coating, basement waterproofing membrane (even under the concrete slab floor where polyethylene sheeting is commonly used), or an additive to the concrete.
Within the waterproofing industry, below-ground waterproofing is generally divided into two areas:
In buildings using earth sheltering , too much humidity can be a potential problem, so waterproofing is critical. Water seepage can lead to mold growth, causing significant damage and air quality issues. Properly waterproofing foundation walls is required to prevent deterioration and seepage.
Another specialized area of waterproofing is rooftop decks and balconies. Waterproofing systems have become quite sophisticated and are a very specialized area. Failed waterproof decks, whether made of polymer or tile, are one of the leading causes of water damage to building structures and personal injury when they fail. Major problems occur in the construction industry when improper products are used for the wrong application. While the term waterproof is used for many products, each of them has a very specific area of application, and when manufacturer specifications and installation procedures are not followed, the consequences can be severe. Another factor is the impact of expansion and contraction on waterproofing systems for decks. Decks constantly move with changes in temperatures, putting stress on the waterproofing systems. One of the leading causes of waterproof deck system failures is the movement of underlying substrates (plywood) that causes too much stress on the membranes, failing the system. While beyond the scope of this reference document, waterproofing of decks and balconies is a complex of many complimentary elements. These include the waterproofing membrane used, adequate slope-drainage, proper flashing details, and proper construction materials.
The penetrations through a building envelope must be built in a way such that water does not enter the building, such as using flashing and special fittings for pipes, vents, wires, etc. Some caulkings are durable, but many are unreliable for waterproofing.
Also, many types of geomembranes are available to control water, gases, or pollution.
From the late 1990s to the 2010s, the construction industry has had technological advances in waterproofing materials, including integral waterproofing systems and more advanced membrane materials. Integral systems such as hycrete work within the matrix of a concrete structure, giving the concrete itself a waterproof quality. There are two main types of integral waterproofing systems: the hydrophilic and the hydrophobic systems. A hydrophilic system typically uses a crystallization technology that replaces the water in the concrete with insoluble crystals. Various brands available in the market claim similar properties, but not all can react with a wide range of cement hydration by-products and thus require caution. Hydrophobic systems use concrete sealers or even fatty acids to block pores within the concrete, preventing water passage.
Sometimes, the same materials used to keep water out of buildings are used to keep water in, such as a pool or pond liners .
New membrane materials seek to overcome shortcomings in older methods like polyvinyl chloride (PVC) and high-density polyethylene (HDPE). Generally, new technology in waterproof membranes relies on polymer -based materials that are very adhesive to create a seamless barrier around the outside of a structure.
Waterproofing should not be confused with roofing , since roofing cannot necessarily withstand hydrostatic head while waterproofing can.
The standards for waterproofing bathrooms in domestic construction have improved over the years, due in large part to the general tightening of building codes.
Some garments , and tents , are designed to give greater or lesser protection against rain. For urban use, raincoats and jackets are used; for outdoor activities in rough weather, there is a range of hiking apparel . Typical descriptions are "showerproof", "water resistant", and "waterproof". [ 1 ] These terms are not precisely defined. A showerproof garment will usually be treated with a water-resisting coating but is not rated to resist a specific hydrostatic head . This is suitable for protection against light rain, but after a short time, water will penetrate. A water-resistant garment is similar, perhaps slightly more resistant to waste,r but also not rated to resist a specific hydrostatic head. A garment described as waterproof will have a water-repellent coating, with the seams also taped to prevent water ingress there. Better waterproof garments have a membrane lining designed to keep water out but allow trapped moisture to escape (" breathability ")—a totally waterproof garment would retain body sweat and become clammy. Waterproof garments specify their hydrostatic rating, ranging from 1,500 for light rain to 20,000 for heavy rain.
Waterproof garments are intended for use in weather conditions which are often windy as well as wet and are usually also wind resistant.
Footwear can also be made waterproof by using a variety of methods, including but not limited to, the application of beeswax, waterproofing spray, or mink oil . [ 2 ]
Waterproofing methods have been implemented in many types of objects, including paper packaging, cosmetics, and, more recently, consumer electronics. Electronic devices used in military and severe commercial environments are routinely conformally coated in accordance with IPC-CC-830 to resist moisture and corrosion, but encapsulation is needed to become truly waterproof. Even though it is possible to find waterproof wrapping or other types of protective cases for electronic devices, a new technology enabled the release of diverse waterproof smartphones and tablets in 2013. [ 3 ] This method is based on a special nanotechnology coating a thousand times thinner than a human hair which protects electronic equipment from damage due to the penetration of water. Several manufacturers use the nano coating method on their smartphones, tablets, and digital cameras.
A 2013 study found that nanotextured surfaces using cone forms produce highly water-repellent surfaces. These nanocone textures are superhydrophobic (extremely water-hating). [ 4 ] [ 5 ]
Waterproof packaging or other types of protective cases for electronic devices can be found. A new technology enabled the release of various waterproof smartphones and tablets in 2013. [ 6 ] A study from 2013 found that nano-textured surfaces using cone shapes produce highly water-repellent surfaces. These "nanocone" textures are superhydrophobic. [ 7 ] [ 8 ]
Media related to Waterproofing at Wikimedia Commons
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For watershed segmentation , a technique for image processing, see Watershed (image processing) .
Watershed delineation is the process of identifying the boundary of a watershed, also referred to as a catchment, drainage basin , or river basin. It is an important step in many areas of environmental science, engineering, and management, for example to study flooding, aquatic habitat, or water pollution.
The activity of watershed delineation is typically performed by geographers, scientists, and engineers. Historically, watershed delineation was done by hand on paper topographic maps , sometimes supplemented with field research. In the 1980s, automated methods were developed for watershed delineation with computers and electronic data, and these are now in widespread use.
Computerized methods for watershed delineation use digital elevation models (DEMs), datasets that represent the height of the Earth's land surface. Computerized watershed delineation may be done using specialized hydrologic modeling software such as WMS , geographic information system software like ArcGIS or QGIS , or with programming languages like Python or R .
Watersheds are a fundamental geographic unit in hydrology , the science concerned with the movement, distribution, and management of water on Earth. Delineating watersheds may be considered an application of hydrography , the branch of applied sciences which deals with the measurement and description of the physical features of oceans, seas, coastal areas, lakes and rivers. It is also related to geomorphometry , the quantitative science of analyzing land surfaces. Watershed delineation continues to be an active area of research, with scientists and programmers developing new algorithms and methods, and making use of increasingly high-resolution data from aerial or satellite remote sensing .
The conventional method of finding a watershed boundary is to draw it by hand on a paper topographic map , or on a transparent overlay. The watershed area can then be estimated using a planimeter , by overlaying graph paper and counting grid cells, or the result can be digitized for use with mapping software. The same process can be done on a computer, sketching the watershed boundary (with a mouse or stylus) over a digital copy of a topographic map. [ 1 ] This is referred to as "heads up digitizing" or "on-screen digitizing." [ 2 ]
For "manual" watershed delination, one must know how to read and interpret a topographic map, for example to identify ridges, valleys, and the direction of steepest slope. [ 3 ] Even in the computer era, manual watershed delineation is still a useful skill, in order to check whether watersheds generated with software are correct. [ 1 ]
Instructions for manual watershed delineation can be found in some textbooks in geography or environmental management, in government pamphlets, [ 4 ] [ 5 ] or in online video tutorials. [ 6 ]
According to the US Geological Survey, there are 5 steps to manual watershed delineation: [ 6 ]
General Rules:
One disadvantage to manual watershed delineation is that it is subject to errors and the individual judgment of the analyst. The Illinois Environmental Protection Agency wrote, "bear in mind that delineating a watershed is an inexact science. Any two people, even if both are experts, will come up with slightly different boundaries." [ 5 ]
Especially for smaller watersheds and when accurate results are important, field reconnaissance may be needed to find features that are not shown on maps. "Going out into the field allows you to identify human alterations, such as road ditches, storm sewers and culverts that could change the direction of waters flow and thus change the watershed boundaries." [ 5 ]
Using computer software to delineate watersheds can be much faster than manual methods. It may also be more consistent, as it removes analyst's subjectivity. Automatic methods of watershed delineation have been in use since the 1980s, and are now in widespread use in the science and engineering communities. Researchers have even used computer methods to delineate watersheds on Mars. [ 7 ] [ 8 ]
Automated watershed delineation methods use digital data of the earth's elevation, a Digital Elevation Model , or DEM. Typically, algorithms use the method of "steepest slope" to calculate the flow direction from a grid cell (or pixel) to one of its neighbors. [ 9 ]
It is possible to use DEMs in different formats for watershed delineation, such as a Triangular Irregular Network (TIN), [ 10 ] or Hexagonal tiling [ 11 ] however most contemporary algorithms make use of a regular rectangular grid. [ 12 ] In the 1980s and 1990s, digital elevation models were often obtained by scanning and digitizing the contours on paper topographic maps, which were then converted to a TIN or a gridded DEM. [ 13 ] More recently, the DEM is obtained by aerial or satellite remote sensing , using stereophotogrammetry , lidar , or radar . [ 14 ]
To use a rectangular grid DEM for watershed delineation, it must first be processed or "conditioned" in order to return realistic results. [ 9 ] The result is sometimes referred to as a "hydro-enforced" DEM or a "HydroDEM." Most of the software packages listed below can perform these functions on a "raw" DEM, or analysts can download hydrologically-conditioned DEMs such as the near-global HydroSHEDS, [ 15 ] MERIT-Hydro, [ 16 ] or EDNA [ 17 ] for the continental United States. The usual steps for hydrologic conditioning of a DEM are:
Additionally, some methods allow for "fencing ridgelines" and burning in flow pathways through lakes. [ 18 ] Some methods also enforce a small slope onto flat areas so that flow will continue to move toward the outlet. [ 19 ] The step of "burning in" stream channels involves artificially deepening the channel, by subtracting a large elevation value from pixels that represent the channel. This ensures that once flow has entered the channel, it will stay there rather than jumping out and flowing overland or into another channel. Some algorithms infer the location of channels automatically from the DEM. Better results are usually obtained by burning in mapped stream channels, or channels derived from satellite or aerial imagery. [ 20 ]
There are several different algorithms available for calculating flow direction from a DEM. The first method, introduced by Australian geographers O'Callaghan and Mark in 1984, is referred to as D8. [ 12 ] Water flows from a pixel to one of 8 possible directions to a neighboring cell (including diagonally), based on the direction of steepest slope. There are disadvantages to this method as water flow is limited to 8 directions, separated by 45°, which may result in unrealistic flow patterns. Also, because all of the flow is routed in one direction, the D8 method is unable to model situations where the flow diverges, such as on convex hillsides, in a river delta , or in branched or braided rivers . Alternative algorithms have been proposed and implemented to overcome this limitation, such as D∞. [ 21 ] Nevertheless, the D8 algorithm remains in widespread use, and has been used to create important datasets such as HydroBasins [ 15 ] and MERIT-Basins. [ 16 ]
Computerized watershed delineation is not always correct. Some errors stem from incorrectly placing the watershed outlet on the digital river network, or "snapping the pour point." [ 22 ] Another class of errors stems from inaccuracies in the digital terrain data, or where its resolution is too coarse to capture flow pathways. [ 2 ] In general, DEMs with higher spatial resolution can more realistically describe topography of the land surface and flow direction. However, there is a tradeoff, as a finer grid with more pixels increases computing time. [ 16 ] Nevertheless, even high-resolution data may not adequately capture flow pathways in complex environments like cities and suburbs, where flow is directed by curbs, culverts, and storm drains. [ 23 ] Finally, some errors can result from the algorithm or the choice of parameters. [ 24 ]
Because errors are common, some authorities insist that the results of automated delineation must be carefully checked. The US Geological Survey's standards for the US Watershed Boundary Dataset allow the use of software "to generate intermediate or “draft” boundary lines," which then must be verified by the analyst by overlaying them on a computer display over basemaps (scanned topographic maps, aerial photographs) to verify their accuracy. [ 1 ]
Some of the first watershed delineation software was written in FORTRAN, such as CATCH [ 25 ] and DEDNM. [ 19 ] Watershed delineation tools are a part of several Geographic Information System software packages such as ArcGIS , QGIS , and GRASS GIS . There are standalone programs for watershed delineation such as TauDEM. Watershed delineation tools are also incorporated into some hydrologic modeling software packages.
Software developers have also published libraries or modules in several languages (see list below). Many of these packages are free and open source, which means they can be expanded or adapted by those willing and able to write or modify code. Finally, there are web applications for delineating watersheds. Some of these web apps have extra features for science and engineering like calculating flow statistics or watershed land cover types (e.g.: StreamStats, Model My Watershed).
There are a number of vector datasets representing watersheds as polygons that can be displayed and analyzed with GIS or other software. In these datasets, the entire land surface is divided into "subwatersheds" or "unit catchments." Individual unit watersheds can be combined or merged to find larger watersheds. The unit catchments have linked hydrological code data or similar metadata to create a flow network , so flow pathways and connections can be determined via network analysis. [ 34 ]
This list is non-exhaustive, as many organizations and territories have produced their own watershed map data and have published via the web. Notable datasets include:
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Watershed management is the study of the relevant characteristics of a watershed aimed at the sustainable distribution of its resources and the process of creating and implementing plans, programs and projects to sustain and enhance watershed functions that affect the plant , animal , and human communities within the watershed boundary. [ 1 ] Features of a watershed that agencies seek to manage to include water supply , water quality , drainage , stormwater runoff , water rights and the overall planning and utilization of watersheds. Landowners , land use agencies, stormwater management experts, environmental specialists, water use surveyors and communities all play an integral part in watershed management.
In agricultural systems, common practices include the use of buffer strips , grassed waterways, the re-establishment of wetlands , and forms of sustainable agriculture practices such as conservation tillage , crop rotation and inter-cropping . After certain practices are installed, it is important to continuously monitor these systems to ensure that they are working properly in terms of improving environmental quality. [ 2 ]
In urban settings, managing areas to prevent soil loss and control stormwater flow are a few of the areas that receive attention. A few practices that are used to manage stormwater before it reaches a channel are retention ponds , filtering systems and wetlands. It is important that storm-water is given an opportunity to infiltrate so that the soil and vegetation can act as a "filter" before the water reaches nearby streams or lakes. In the case of soil erosion prevention, a few common practices include the use of silt fences, landscape fabric with grass seed and hydroseeding . The main objective in all cases is to slow water movement to prevent soil transport.
The 2nd World Water Forum held in The Hague in March 2000 raised some controversies that exposed the multilateral nature and imbalance the demand and supply management of freshwater . While donor organizations, private and government institutions backed by the World Bank , believe that freshwater should be governed as an economic good by appropriate pricing, NGOs however, held that freshwater resources should be seen as a social good . [ 3 ] The concept of network governance where all stakeholders form partnerships and voluntarily share ideas towards forging a common vision can be used to resolve this clash of opinion in freshwater management. Also, the implementation of any common vision presents a new role for NGOs because of their unique capabilities in local community coordination, thus making them a valuable partner in network governance . [ 4 ]
Watersheds replicate this multilateral terrain with private industries and local communities interconnected by a common watershed. Although these groups share a common ecological space that could transcend state borders, their interests, knowledge and use of resources within the watershed are mostly disproportionate and divergent, resulting to the activities of a specific group adversely impacting on other groups. Examples being the Minamata Bay poisoning that occurred from 1932 to 1968, killing over 1,784 individuals and the Wabigoon River incidence of 1962. Furthermore, while some knowledgeable groups are shifting from efficient water resource exploitation to efficient utilization, net gain for the watershed ecology could be lost when other groups seize the opportunity to exploit more resources .
Moreover, the need to create partnerships between donor organizations, private and government institutions and community representatives like NGOs in watersheds is to enhance an "organizational society" among stakeholders. [ 5 ]
Several riparian states have adopted this concept in managing the increasingly scarce resources of watersheds. These include the nine Rhine states, with a common vision of pollution control , [ 6 ] the Lake Chad and river Nile Basins, whose common vision is to ensure environmental sustainability . [ 7 ] As a partner in the commonly shared vision, NGOs has adopted a new role in operationalizing the implementation of regional watershed management policies at the local level. For instance, essential local coordination and education are areas where the services of NGOs have been effective. [ 8 ] This makes NGOs the "nuclei" for successful watershed management. [ 4 ] Recently, artificial Intelligence techniques such as neural networks have been utilized to address the problem of watershed management. [ 9 ]
Environmental laws often dictate the planning and actions that agencies take to manage watersheds. Some laws require that planning be done, others can be used to make a plan legally enforceable and others set out the ground rules for what can and cannot be done in development and planning. Most countries and states have their own laws regarding watershed management.
Those concerned about aquatic habitat protection have a right to participate in the laws and planning processes that affect aquatic habitats. By having a clear understanding of whom to speak to and how to present the case for keeping our waterways clean a member of the public can become an effective watershed protection advocate.
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The water–cement ratio ( w/c ratio , or water-to-cement ratio , sometimes also called the Water-Cement Factor , f ) is the ratio of the mass of water ( w ) to the mass of cement ( c ) used in a concrete mix:
The typical values of this ratio f = w ⁄ c are generally comprised in the interval 0.40 and 0.60.
The water-cement ratio of the fresh concrete mix is one of the main, if not the most important, factors determining the quality and properties of hardened concrete, as it directly affects the concrete porosity, and a good concrete is always a concrete as compact and as dense as possible. A good concrete must be therefore prepared with as little water as possible, but with enough water to hydrate the cement minerals and to properly handle it.
A lower ratio leads to higher strength and durability , but may make the mix more difficult to work with and form. Workability can be resolved with the use of plasticizers or super-plasticizers . A higher ratio gives a too fluid concrete mix resulting in a too porous hardened concrete of poor quality.
Often, the concept also refers to the ratio of water to cementitious materials, w/cm. Cementitious materials include cement and supplementary cementitious materials such as ground granulated blast-furnace slag (GGBFS), fly ash (FA), silica fume (SF), rice husk ash (RHA), metakaolin (MK), and natural pozzolans . Most of supplementary cementitious materials (SCM) are byproducts of other industries presenting interesting hydraulic binding properties. After reaction with alkalis (GGBFS activation) and portlandite ( Ca(OH) 2 ), they also form calcium silicate hydrates (C-S-H), the "gluing phase" present in the hardened cement paste. These additional C-S-H are filling the concrete porosity and thus contribute to strengthen concrete. SCMs also help reducing the clinker content in concrete and therefore saving energy and minimizing costs, while recycling industrial wastes otherwise aimed to landfill .
The effect of the water-to-cement (w/c) ratio onto the mechanical strength of concrete was first studied by René Féret (1892) in France, and then by Duff A. Abrams (1918) (inventor of the concrete slump test ) in the USA, and by Jean Bolomey (1929) in Switzerland.
The 1997 Uniform Building Code specifies a maximum of 0.5 w/c ratio when concrete is exposed to freezing and thawing in moist conditions or to de-icing salts , and a maximum of 0.45 w/c ratio for concrete in severe, or very severe, sulfate conditions.
Concrete hardens as a result of the chemical reaction between cement and water (known as hydration and producing heat ). For every mass ( kilogram , pound , or any unit of weight ) of cement (c), about 0.35 mass of water (w) is needed to fully complete the hydration reactions. [ 1 ]
However, a fresh concrete with a w/c ratio of 0.35 may not mix thoroughly, and may not flow well enough to be correctly placed and to fill all the voids in the forms, especially in the case of a dense steel reinforcement . More water is therefore used than is chemically and physically necessary to react with cement. Water–cement ratios in the range of 0.40 to 0.60 are typically used. For higher-strength concrete, lower w/c ratios are necessary, along with a plasticizer to increase flowability.
A w/c ratio higher than 0.60 is not acceptable as fresh concrete becomes "soup" [ 2 ] and leads to a higher porosity and to very poor quality hardened concrete as publicly stated by Prof. Gustave Magnel (1889-1955, Ghent University , Belgium) during an official address to American building contractors at the occasion of one of his visits in the United States in the 1950s to build the first prestressed concrete girder bridge in the USA: the Walnut Lane Memorial Bridge in Philadelphia open to traffic in 1951. [ 3 ] [ 4 ] [ 5 ] [ 6 ] The famous sentence of Gustave Magnel, facing reluctance from a contractor, when he was requiring a very low w/c ratio, zero-slump , concrete for casting the girders of this bridge remains in many memories: "American makes soup, not concrete" . [ 7 ]
When the excess water added to improve the workability of fresh concrete, and not consumed by the hydration reactions, leaves concrete as it hardens and dries, it results in an increased concrete porosity only filled by air . A higher porosity reduces the final strength of concrete because the air present in the pores is compressible and concrete microstructure can be more easily " crushed ".
Moreover, a higher porosity also increases the hydraulic conductivity ( K , m/s) of concrete and the effective diffusion coefficients ( D e , m 2 /s) of solutes and dissolved gases in the concrete matrix. This increases water ingress into concrete, accelerates its dissolution ( calcium leaching ), favors harmful expansive chemical reactions ( ASR , DEF), and facilitates the transport of aggressive chemical species such as chlorides ( pitting corrosion of reinforced bars ) and sulfates (internal and external sulfate attacks, ISA and ESA, of concrete) inside the concrete porosity.
When cementitious materials are used to encapsulate toxic heavy metals or radionuclides , a lower w/c ratio is required to decrease the matrix porosity and the effective diffusion coefficients of the immobilized elements in the cementitious matrix. A lower w/c ratio also contributes to minimize the leaching of the toxic elements out of the immobilization material.
A higher porosity also facilitates the diffusion of gases into the concrete microstructure . A faster diffusion of atmospheric CO 2 increases the concrete carbonation rate . When the carbonation front reaches the steel reinforcements (rebar), the pH of the concrete pore water at the steel surface decreases. At a pH value lower than 10.5, the carbon steel is no longuer passivated by an alkaline pH and starts to corrode ( general corrosion ). A faster diffusion of oxygen ( O 2 ) into the concrete microstructure also accelerates the rebar corrosion.
Moreover, on the long term, a concrete mix with too much water will experience more creep and drying shrinkage as excess water leaves the concrete porosity, resulting in internal cracks and visible fractures (particularly around inside corners), which again will reduce the concrete mechanical strength.
Finally, water added in excess also facilitates the segregation of fine and coarse aggregates ( sand and gravels ) from the fresh cement paste and causes the formation of honeycombs (pockets of gravels without hardened cement paste) in concrete walls and around rebar. It also causes water bleeding at the surface of concrete slabs or rafts (with a dusty surface left after water evaporation).
For all the afore mentioned reasons, it is strictly forbidden to add extra water to a ready-mix concrete truck when the delivery time is exceeded, and the concrete becomes difficult to pour because it starts to set. Such diluted concrete immediately loses any official certification and the responsibility of the contractor accepting such a deleterious practice is also engaged. In the worst case, an addition of superplasticizer can be made to increase again the concrete workability and to salvage the content of a ready-mix concrete truck when the maximum concrete delivery time is not exceeded.
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https://en.wikipedia.org/wiki/Water–cement_ratio
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The water–gas shift reaction (WGSR) describes the reaction of carbon monoxide and water vapor to form carbon dioxide and hydrogen :
The water gas shift reaction was discovered by Italian physicist Felice Fontana in 1780. [ 1 ] [ 2 ] It was not until much later that the industrial value of this reaction was realized. Before the early 20th century, hydrogen was obtained by reacting steam under high pressure with iron to produce iron oxide and hydrogen. With the development of industrial processes that required hydrogen, such as the Haber–Bosch ammonia synthesis, a less expensive and more efficient method of hydrogen production was needed. As a resolution to this problem, the WGSR was combined with the gasification of coal to produce hydrogen.
The WGSR is a highly valuable industrial reaction that is used in the manufacture of ammonia, hydrocarbons , methanol , and hydrogen . Its most important application is in conjunction with the conversion of carbon monoxide from steam reforming of methane or other hydrocarbons in the production of hydrogen. [ 3 ] In the Fischer–Tropsch process , the WGSR is one of the most important reactions used to balance the H 2 /CO ratio. It provides a source of hydrogen at the expense of carbon monoxide, which is important for the production of high purity hydrogen for use in ammonia synthesis.
The water–gas shift reaction may be an undesired side reaction in processes involving water and carbon monoxide, e.g. the rhodium-based Monsanto process . The iridium-based Cativa process uses less water, which suppresses this reaction.
The WGSR can aid in the efficiency of fuel cells by increasing hydrogen production. The WGSR is considered a critical component in the reduction of carbon monoxide concentrations in cells that are susceptible to carbon monoxide poisoning such as the proton-exchange membrane (PEM) fuel cell . [ 4 ] The benefits of this application are two-fold: not only would the water gas shift reaction effectively reduce the concentration of carbon monoxide, but it would also increase the efficiency of the fuel cells by increasing hydrogen production. [ 4 ] Unfortunately, current commercial catalysts that are used in industrial water gas shift processes are not compatible with fuel cell applications. [ 5 ] With the high demand for clean fuel and the critical role of the water gas shift reaction in hydrogen fuel cells, the development of water gas shift catalysts for the application in fuel cell technology is an area of current research interest.
Catalysts for fuel cell application would need to operate at low temperatures. Since the WGSR is slow at lower temperatures where equilibrium favors hydrogen production, WGS reactors require large amounts of catalysts, which increases their cost and size beyond practical application. [ 4 ] The commercial LTS catalyst used in large scale industrial plants is also pyrophoric in its inactive state and therefore presents safety concerns for consumer applications. [ 5 ] Developing a catalyst that can overcome these limitations is relevant to implementation of a hydrogen economy.
The WGS reaction is used in combination with the solid adsorption of CO 2 in the sorption enhanced water gas shift (SEWGS) in order to produce a high pressure hydrogen stream from syngas . [ 6 ]
The equilibrium of this reaction shows a significant temperature dependence and the equilibrium constant decreases with an increase in temperature, that is, higher hydrogen formation is observed at lower temperatures.
With increasing temperature, the reaction rate increases, but hydrogen production becomes less favorable thermodynamically [ 7 ] since the water gas shift reaction is moderately exothermic ; this shift in chemical equilibrium can be explained according to Le Chatelier's principle . Over the temperature range of 600–2000 K, the equilibrium constant for the WGSR has the following relationship: [ 5 ]
In the range of 600-1200 K, the simpler expression derived by Moe [ 8 ] can be used:
Alternatively, the equilibrium constant for the WGSR directly derived from thermodynamic quantities leads to: [ 9 ]
In order to take advantage of both the thermodynamics and kinetics of the reaction, the industrial scale water gas shift reaction is conducted in multiple adiabatic stages consisting of a high temperature shift (HTS) followed by a low temperature shift (LTS) with intersystem cooling. [ 10 ] The initial HTS takes advantage of the high reaction rates, but results in incomplete conversion of carbon monoxide. A subsequent low temperature shift reactor lowers the carbon monoxide content to <1%. Commercial HTS catalysts are based on iron oxide – chromium oxide and the LTS catalyst is a copper-based. The copper catalyst is susceptible to poisoning by sulfur . Sulfur compounds are removed prior to the LTS reactor by a guard bed. An important limitation for the HTS is the H 2 O/CO ratio where low ratios may lead to side reactions such as the formation of metallic iron, methanation , carbon deposition, and the Fischer–Tropsch reaction.
The typical composition of commercial HTS catalyst has been reported as 74.2% Fe 2 O 3 , 10.0% Cr 2 O 3 , 0.2% MgO (remaining percentage attributed to volatile components). [ 11 ] The chromium acts to stabilize the iron oxide and prevents sintering . The operation of HTS catalysts occurs within the temperature range of 310 °C to 450 °C. The temperature increases along the length of the reactor due to the exothermic nature of the reaction. As such, the inlet temperature is maintained at 350 °C to prevent the exit temperature from exceeding 550 °C. Industrial reactors operate at a range from atmospheric pressure to 8375 kPa (82.7 atm). [ 11 ] The search for high performance HT WGS catalysts remains an intensive topic of research in fields of chemistry and materials science. Activation energy is a key criteria for the assessment of catalytic performance in WGS reactions. To date, some of the lowest activation energy values have been found for catalysts consisting of copper nanoparticles on ceria support materials, [ 12 ] with values as low as Ea = 34 kJ/mol reported relative to hydrogen generation.
Catalysts for the lower temperature WGS reaction are commonly based on copper or copper oxide loaded ceramic phases, While the most common supports include alumina or alumina with zinc oxide, other supports may include rare earth oxides, spinels or perovskites. [ 13 ] A typical composition of a commercial LTS catalyst has been reported as 32-33% CuO, 34-53% ZnO, 15-33% Al 2 O 3 . [ 5 ] The active catalytic species is CuO. The function of ZnO is to provide structural support as well as prevent the poisoning of copper by sulfur. The Al 2 O 3 prevents dispersion and pellet shrinkage. The LTS shift reactor operates at a range of 200–250 °C. The upper temperature limit is due to the susceptibility of copper to thermal sintering. These lower temperatures also reduce the occurrence of side reactions that are observed in the case of the HTS. Noble metals such as platinum, supported on ceria, have also been used for LTS. [ 14 ]
The WGSR has been extensively studied for over a hundred years. The kinetically relevant mechanism depends on the catalyst composition and the temperature. [ 10 ] [ 18 ] Two mechanisms have been proposed: an associative Langmuir–Hinshelwood mechanism and a redox mechanism. The redox mechanism is generally regarded as kinetically relevant during the high-temperature WGSR (> 350 °C) over the industrial iron-chromia catalyst. [ 7 ] Historically, there has been much more controversy surrounding the mechanism at low temperatures. Recent experimental studies confirm that the associative carboxyl mechanism is the predominant low temperature pathway on metal-oxide-supported transition metal catalysts. [ 19 ] [ 17 ]
In 1920 Armstrong and Hilditch first proposed the associative mechanism. In this mechanism CO and H 2 O are adsorbed onto the surface of the catalyst, followed by formation of an intermediate and the desorption of H 2 and CO 2 . In general, H 2 O dissociates onto the catalyst to yield adsorbed OH and H. The dissociated water reacts with CO to form a carboxyl or formate intermediate. The intermediate subsequently dehydrogenates to yield CO 2 and adsorbed H. Two adsorbed H atoms recombine to form H 2 .
There has been significant controversy surrounding the kinetically relevant intermediate during the associative mechanism. Experimental studies indicate that both intermediates contribute to the reaction rate over metal oxide supported transition metal catalysts. [ 19 ] [ 17 ] However, the carboxyl pathway accounts for about 90% of the total rate owing to the thermodynamic stability of adsorbed formate on the oxide support. The active site for carboxyl formation consists of a metal atom adjacent to an adsorbed hydroxyl. This ensemble is readily formed at the metal-oxide interface and explains the much higher activity of oxide-supported transition metals relative to extended metal surfaces. [ 17 ] The turn-over-frequency for the WGSR is proportional to the equilibrium constant of hydroxyl formation, which rationalizes why reducible oxide supports (e.g. CeO 2 ) are more active than irreducible supports (e.g. SiO 2 ) and extended metal surfaces (e.g. Pt). In contrast to the active site for carboxyl formation, formate formation occurs on extended metal surfaces. The formate intermediate can be eliminated during the WGSR by using oxide-supported atomically dispersed transition metal catalysts, further confirming the kinetic dominance of the carboxyl pathway. [ 20 ]
The redox mechanism involves a change in the oxidation state of the catalytic material. In this mechanism, CO is oxidized by an O-atom intrinsically belonging to the catalytic material to form CO 2 . A water molecule undergoes dissociative adsorption at the newly formed O-vacancy to yield two hydroxyls. The hydroxyls disproportionate to yield H 2 and return the catalytic surface back to its pre-reaction state.
The mechanism entails nucleophilic attack of water or hydroxide on a M-CO center, generating a metallacarboxylic acid . [ 4 ] [ 21 ]
The WGSR is exergonic , with the following thermodynamic parameters at room temperature (298 K) [ 22 ]
In aqueous solution, the reaction is less exergonic. [ 23 ]
In the conversion of carbon dioxide to useful materials, the water–gas shift reaction is used to produce carbon monoxide from hydrogen and carbon dioxide. This is sometimes called the reverse water–gas shift reaction . [ 24 ]
Water gas is defined as a fuel gas consisting mainly of carbon monoxide (CO) and hydrogen (H 2 ). The term 'shift' in water–gas shift means changing the water gas composition (CO:H 2 ) ratio. The ratio can be increased by adding CO 2 or reduced by adding steam to the reactor.
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The Watt steam engine design was an invention of James Watt that became synonymous with steam engines during the Industrial Revolution , and it was many years before significantly new designs began to replace the basic Watt design.
The first steam engines , introduced by Thomas Newcomen in 1712, were of the "atmospheric" design. At the end of the power stroke , the weight of the object being moved by the engine pulled the piston to the top of the cylinder as steam was introduced. Then the cylinder was cooled by a spray of water, which caused the steam to condense, forming a partial vacuum in the cylinder. Atmospheric pressure on the top of the piston pushed it down, lifting the work object. James Watt noticed that it required significant amounts of heat to warm the cylinder back up to the point where steam could enter the cylinder without immediately condensing. When the cylinder was warm enough that it became filled with steam the next power stroke could commence.
Watt realised that the heat needed to warm the cylinder could be saved by adding a separate condensing cylinder. After the power cylinder was filled with steam, a valve was opened to the secondary cylinder, allowing the steam to flow into it and be condensed, which drew the steam from the main cylinder causing the power stroke. The condensing cylinder was water cooled to keep the steam condensing. At the end of the power stroke, the valve was closed so the power cylinder could be filled with steam as the piston moved to the top. The result was the same cycle as Newcomen's design, but without any cooling of the power cylinder which was immediately ready for another stroke.
Watt worked on the design over a period of several years, introducing the condenser, and introducing improvements to practically every part of the design. Notably, Watt performed a lengthy series of trials on ways to seal the piston in the cylinder, which considerably reduced leakage during the power stroke, preventing power loss. All of these changes produced a more reliable design which used half as much coal to produce the same amount of power. [ 1 ]
The new design was introduced commercially in 1776, with the first example sold to the Carron Company ironworks. About the same time, Watt encountered a business problem that led him to introduce a new unit of measurement of power, or the rate at which work is done: the horsepower . His earlier business agreements framed his earnings in how much coal the customer of the steam engine saved, but when discussing installing a steam engine for a London brewer, that business did not use coal - it used horses to drive the mills. [ 2 ]
Watt continued working to improve the engine, and in 1781 introduced a system using a sun and planet gear to turn the linear motion of the engines into rotary motion. This made it useful not only in the original pumping role, but also as a direct replacement in roles where a water wheel would have been used previously. This was a key moment in the industrial revolution, since power sources could now be located anywhere instead of, as previously, needing a suitable water source and topography . Watt's partner Matthew Boulton began developing a multitude of machines that made use of this rotary power, developing the first modern industrialized factory, the Soho Foundry , which in turn produced new steam engine designs. Watt's early engines were like the original Newcomen designs in that they used low-pressure steam, and all of the power was produced by atmospheric pressure. When, in the early 1800s, other companies introduced high-pressure steam engines, Watt was reluctant to follow suit due to safety concerns. [ 3 ] Wanting to improve on the performance of his engines, Watt began considering the use of higher-pressure steam, as well as designs using multiple cylinders in both the double-acting concept and the multiple-expansion concept. These double-acting engines required the invention of the parallel motion , which allowed the piston rods of the individual cylinders to move in straight lines, keeping the piston true in the cylinder, while the walking beam end moved through an arc, somewhat analogous to a crosshead in later steam engines.
In 1698, the English mechanical designer Thomas Savery invented a pumping appliance that used steam to draw water directly from a well by means of a vacuum created by condensing steam. The appliance was also proposed for draining mines , but it could only draw fluid up approximately 25 feet (7.5 m), meaning it had to be located within this distance of the mine floor being drained. As mines became deeper, this was often impractical. It also consumed a large amount of fuel compared with later engines. [ 4 ]
The solution to draining deep mines was found by Thomas Newcomen who developed an "atmospheric" engine that also worked on the vacuum principle. It employed a cylinder containing a movable piston connected by a chain to one end of a rocking beam that worked a mechanical lift pump from its opposite end. At the bottom of each stroke, steam was allowed to enter the cylinder below the piston. As the piston rose within the cylinder, drawn upward by a counterbalance, it drew in steam at atmospheric pressure. At the top of the stroke the steam valve was closed, and cold water was briefly injected into the cylinder as a means of cooling the steam. This water condensed the steam and created a partial vacuum below the piston. The atmospheric pressure outside the engine was then greater than the pressure within the cylinder, thereby pushing the piston into the cylinder. The piston, attached to a chain and in turn attached to one end of the "rocking beam", pulled down the end of the beam, lifting the opposite end of the beam. Hence, the pump deep in the mine attached to opposite end of the beam via ropes and chains was driven. The pump pushed, rather than pulled the column of water upward, hence it could lift water any distance. Once the piston was at the bottom, the cycle repeated. [ 4 ]
The Newcomen engine was more powerful than the Savery engine. For the first time water could be raised from a depth of over 300 feet (90 m). [ 5 ] The first example from 1712 was able to replace a team of 500 horses that had been used to pump out the mine. Seventy-five Newcomen pumping engines were installed at mines in Britain, France, Holland, Sweden and Russia. In the next fifty years only a few small changes were made to the engine design.
While Newcomen engines brought practical benefits, they were inefficient in terms of the use of energy to power them. The system of alternately sending jets of steam, then cold water into the cylinder meant that the walls of the cylinder were alternately heated, then cooled with each stroke. Each charge of steam introduced would continue condensing until the cylinder approached working temperature once again. So at each stroke part of the potential of the steam was lost.
In 1763, James Watt was working as instrument maker at the University of Glasgow when he was assigned the job of repairing a model Newcomen engine and noted how inefficient it was. [ 6 ]
In 1765, Watt conceived the idea of equipping the engine with a separate condensation chamber, which he called a "condenser" . Because the condenser and the working cylinder were separate, condensation occurred without significant loss of heat from the cylinder. The condenser remained cold and below atmospheric pressure at all times, while the cylinder remained hot at all times.
Steam was drawn from the boiler to the cylinder under the piston . When the piston reached the top of the cylinder, the steam inlet valve closed and the valve controlling the passage to the condenser opened. The condenser being at a lower pressure, drew the steam from the cylinder into the condenser where it cooled and condensed from water vapour to liquid water, maintaining a partial vacuum in the condenser that was communicated to the space of the cylinder by the connecting passage. External atmospheric pressure then pushed the piston down the cylinder.
The separation of the cylinder and condenser eliminated the loss of heat that occurred when steam was condensed in the working cylinder of a Newcomen engine. This gave the Watt engine greater efficiency than the Newcomen engine, reducing the amount of coal consumed while doing the same amount of work as a Newcomen engine.
In Watt's design, the cold water was injected only into the condensation chamber. This type of condenser is known as a jet condenser . The condenser is located in a cold water bath below the cylinder. The volume of water entering the condenser as spray absorbed the latent heat of the steam, and was determined as seven times the volume of the condensed steam. The condensate and the injected water was then removed by the air pump, and the surrounding cold water served to absorb the remaining thermal energy to retain a condenser temperature of 30 to 45 °C (85 to 115 °F) and the equivalent pressure of 0.04 to 0.1 bars (4.0 to 10.0 kPa; 0.6 to 1.5 psi). [ 7 ]
At each stroke the warm condensate was drawn off from the condenser and sent to a hot well by a vacuum pump, which also helped to evacuate the steam from under the power cylinder. The still-warm condensate was recycled as feedwater for the boiler.
Watt's next improvement to the Newcomen design was to seal the top of the cylinder and surround the cylinder with a jacket. Steam was passed through the jacket before being admitted below the piston, keeping the piston and cylinder warm to prevent condensation within it. The second improvement was the utilisation of steam expansion against the vacuum on the other side of the piston. The steam supply was cut during the stroke, and the steam expanded against the vacuum on the other side. This increased the efficiency of the engine, but also created a variable torque on the shaft which was undesirable for many applications, in particular pumping. Watt therefore limited the expansion to a ratio of 1:2 (i.e. the steam supply was cut at half stroke). This increased the theoretical efficiency from 6.4% to 10.6%, with only a small variation in piston pressure. [ 7 ] Watt did not use high pressure steam because of safety concerns. [ 3 ] : 85
These improvements led to the fully developed version of 1776 that actually went into production. [ 8 ]
The separate condenser showed dramatic potential for improvements on the Newcomen engine but Watt was still discouraged by seemingly insurmountable problems before a marketable engine could be perfected. It was only after entering into partnership with Matthew Boulton that such became reality. Watt told Boulton about his ideas on improving the engine, and Boulton, an avid entrepreneur, agreed to fund development of a test engine at Soho , near Birmingham . At last Watt had access to facilities and the practical experience of craftsmen who were soon able to get the first engine working. As fully developed, it used about 75% less fuel than a similar Newcomen one.
In 1775, Watt designed two large engines: one for the Bloomfield Colliery at Tipton , completed in March 1776, and one for John Wilkinson 's ironworks at Broseley in Shropshire , which was at work the following month. A third engine, at Stratford-le-Bow in east London, was also working that summer. [ 9 ]
Watt had tried unsuccessfully for several years to obtain an accurately bored cylinder for his steam engines, and was forced to use hammered iron, which was out of round and caused leakage past the piston. Joseph Wickham Roe stated in 1916: "When [John] Smeaton saw the first engine he reported to the Society of Engineers that 'Neither the tools nor the workmen existed who could manufacture such a complex machine with sufficient precision ' ". [ 10 ]
In 1774, John Wilkinson invented a boring machine in which the shaft that held the cutting tool was supported on both ends and extended through the cylinder, unlike the cantilevered borers then in use. Boulton wrote in 1776 that "Mr. Wilkinson has bored us several cylinders almost without error; that of 50 inches diameter, which we have put up at Tipton, does not err on the thickness of an old shilling in any part". [ 10 ]
Boulton and Watt 's practice was to help mine-owners and other customers to build engines, supplying men to erect them and some specialised parts. However, their main profit from their patent was derived from charging a licence fee to the engine owners, based on the cost of the fuel they saved. The greater fuel efficiency of their engines meant that they were most attractive in areas where fuel was expensive, particularly Cornwall , for which three engines were ordered in 1777, for the Wheal Busy , Ting Tang , and Chacewater mines. [ 11 ]
The first Watt engines were atmospheric pressure engines, like the Newcomen engine but with the condensation taking place separate from the cylinder. Driving the engines using both low pressure steam and a partial vacuum raised the possibility of reciprocating engine development. [ 12 ] An arrangement of valves could alternately admit low pressure steam to the cylinder and then connect with the condenser. Consequently, the direction of the power stroke might be reversed, making it easier to obtain rotary motion. Additional benefits of the double acting engine were increased efficiency, higher speed (greater power) and more regular motion.
Before the development of the double acting piston, the linkage to the beam and the piston rod had been by means of a chain, which meant that power could only be applied in one direction, by pulling. This was effective in engines that were used for pumping water, but the double action of the piston meant that it could push as well as pull. This was not possible as long as the beam and the rod were connected by a chain. Furthermore, it was not possible to connect the piston rod of the sealed cylinder directly to the beam, because while the rod moved vertically in a straight line, the beam was pivoted at its centre, with each side inscribing an arc. To bridge the conflicting actions of the beam and the piston, Watt developed his parallel motion . This device used a four bar linkage coupled with a pantograph to produce the required straight line motion much more cheaply than if he had used a slider type of linkage. He was very proud of his solution.
Having the beam connected to the piston shaft by a means that applied force alternately in both directions also meant that it was possible to use the motion of the beam to turn a wheel. The simplest solution to transforming the action of the beam into a rotating motion was to connect the beam to a wheel by a crank , but because another party had patent rights on the use of the crank, Watt was obliged to come up with another solution. [ 14 ] He adopted the epicyclic sun and planet gear system suggested by an employee William Murdoch , only later reverting, once the patent rights had expired, to the more familiar crank seen on most engines today. [ 15 ] The main wheel attached to the crank was large and heavy, serving as a flywheel which, once set in motion, by its momentum maintained a constant power and smoothed the action of the alternating strokes. To its rotating central shaft, belts and gears could be attached to drive a great variety of machinery.
Because factory machinery needed to operate at a constant speed, Watt linked a steam regulator valve to a centrifugal governor which he adapted from those used to automatically control the speed of windmills. [ 16 ] The centrifugal was not a true speed controller because it could not hold a set speed in response to a change in load. [ 17 ]
These improvements allowed the steam engine to replace the water wheel and horses as the main sources of power for British industry, thereby freeing it from geographical constraints and becoming one of the main drivers in the Industrial Revolution .
Watt was also concerned with fundamental research on the functioning of the steam engine. His most notable measuring device, still in use today, is the Watt indicator incorporating a manometer to measure steam pressure within the cylinder according to the position of the piston, enabling a diagram to be produced representing the pressure of the steam as a function of its volume throughout the cycle.
The oldest surviving Watt engine is Old Bess of 1777, now in the Science Museum, London .
The oldest working engine in the world is the Smethwick Engine , brought into service in May 1779 and now at Thinktank in Birmingham (formerly at the now defunct Museum of Science and Industry, Birmingham ).
The oldest still in its original engine house and still capable of doing the job for which it was installed is the 1812 Boulton and Watt engine at the Crofton Pumping Station in Wiltshire . This was used to pump water for the Kennet and Avon Canal ; on certain weekends throughout the year the modern pumps are switched off and the two steam engines at Crofton still perform this function.
The oldest extant rotative steam engine, the Whitbread Engine (from 1785, the third rotative engine ever built), is located in the Powerhouse Museum in Sydney, Australia.
A Boulton-Watt engine of 1788 may be found in the Science Museum, London , [ 18 ] while an 1817 blowing engine , formerly used at the Netherton ironworks of M W Grazebrook now decorates Dartmouth Circus , a traffic island at the start of the A38(M) motorway in Birmingham.
The Henry Ford Museum in Dearborn, Michigan houses a replica of a 1788 Watt rotative engine. It is a full-scale working model of a Boulton-Watt engine. The American industrialist Henry Ford commissioned the replica engine from the English manufacturer Charles Summerfield in 1932. [ 19 ] The museum also holds an original Boulton and Watt atmospheric pump engine, originally used for canal pumping in Birmingham, [ 20 ] illustrated below, and in use in situ at the Bowyer Street pumping station, [ 21 ] [ 22 ] from 1796 until 1854, and afterwards removed to Dearborn in 1929.
Another one is preserved at Fumel factory, France.
In the 1880s, Hathorn Davey and Co / Leeds produced a 1 hp / 125 rpm atmospheric engine with external condenser but without steam expansion. It has been argued that this was probably the last commercial atmospheric engine to be manufactured. As an atmospheric engine, it did not have a pressurised boiler. It was intended for small businesses. [ 23 ]
Watt's Expansion Engine is generally considered as of historic interest only. There are however some recent developments which may lead to a renaissance of the technology. Today, there is an enormous amount of waste steam and waste heat with temperatures between 100 and 150 °C (210 and 300 °F) generated by industry. In addition, solarthermal collectors, geothermal energy sources and biomass reactors produce heat in this temperature range. There are technologies to utilise this energy, in particular the Organic Rankine Cycle (ORC). In principle, these are steam turbines which do not use water but a fluid (a refrigerant) which evaporates at temperatures below 100 °C (212 °F). Such systems are however fairly complex. They work with pressures of 6 to 20 bars (600 to 2,000 kPa; 87 to 290 psi), so that the whole system has to be completely sealed.
The Expansion Engine can offer significant advantages here, in particular for lower power ratings of 2 to 100 kW: with expansion ratios of 1:5, the theoretical efficiency reaches 15%, which is in the range of ORC systems. The Expansion Engine uses water as working fluid which is simple, cheap, non-toxic, non-flammable and non-corrosive. It works at pressure near and below atmospheric, so that sealing is not a problem. And it is a simple machine, implying cost effectiveness. Researchers from the University of Southampton / UK are currently developing a modern version of Watt's engine in order to generate energy from waste steam and waste heat. They improved the theory, demonstrating that theoretical efficiencies of up to 17.4% (and actual efficiencies of 11%) are possible. [ 24 ]
In order to demonstrate the principle, a 25 watt experimental model engine was built and tested. The engine incorporates steam expansion as well as new features such as electronic control. The picture shows the model built and tested in 2016. [ 25 ] Currently, a project to build and test a scaled-up 2 kW engine is under preparation. [ 26 ]
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Wattle and daub is a composite building method in which a woven lattice of wooden strips called " wattle " is "daubed" with a sticky material usually made of some combination of wet soil, clay, sand, and straw. Wattle and daub has been used for at least 6,000 years and is still an important construction method in many parts of the world. Many historic buildings include wattle and daub construction.
The wattle and daub technique has been used since the Neolithic period. It was common for houses of Linear pottery and Rössen cultures of middle Europe, but is also found in Western Asia ( Çatalhöyük , Shillourokambos ) as well as in North America ( Mississippian culture ) and South America ( Brazil ). In Africa it is common in the architecture of traditional houses such as those of the Ashanti people . Its usage dates back at least 6,000 years. There are suggestions that construction techniques such as lath and plaster and even cob may have evolved from wattle and daub. Fragments from prehistoric wattle and daub buildings have been found in Africa, Europe, Mesoamerica and North America. [ 1 ]
Evidence for wattle and daub (or "wattle and reed") fire pits, storage bins, and buildings shows up in Egyptian archaeological sites such as Merimda and El Omari, dating back to the 5th millennium BCE, predating the use of mud brick and continuing to be the preferred building material until about the start of the First Dynasty (3100BCE). It continued to flourish well into the New Kingdom and beyond. [ 2 ] Vitruvius refers to it as being employed in Rome . [ 3 ] A review of English architecture especially reveals that the sophistication of this craft is dependent on the various styles of timber frame housing. [ 4 ]
The wattle and plaster process has been replaced in modern architecture by brick and mortar or by lath and plaster , a common building material for wall and ceiling surfaces, in which a series of nailed wooden strips are covered with plaster smoothed into a flat surface. In many regions this building method has itself been overtaken by drywall construction using plasterboard sheets.
The wattle is made by weaving thin branches (either whole, or more usually split) or slats between upright stakes. The wattle may be made as loose panels, slotted between timber framing to make infill panels, or made in place to form the whole of a wall. In different regions, the material of wattle can be different. For example, at the Mitchell Site on the northern outskirts of the city of Mitchell, South Dakota, willow has been found as the wattle material of the walls of the house. [ 5 ] Reeds and vines can also be used as wattle material. [ 6 ] [ 7 ] The origin of the term wattle describing a group of acacias in Australia, is derived from the common use of acacias as wattle in early Australian European settlements. [ 8 ]
Daub is usually created from a mixture of ingredients from three categories: binders , aggregates and reinforcement. Binders hold the mix together and can include clay, lime , chalk dust and limestone dust. Aggregates give the mix its bulk and dimensional stability through materials such as mud, sand, crushed chalk and crushed stone. Reinforcement is provided by straw, hair, hay or other fibrous materials, and helps to hold the mix together as well as to control shrinkage and provide flexibility. [ 9 ] The daub may be mixed by hand, or by treading – either by humans or livestock . It is then applied to the wattle and allowed to dry, and often then whitewashed to increase its resistance to rain. Sometimes there can be more than one layer of daub. At the Mitchell Site, the anterior of the house had double layers of burned daub. [ 10 ]
There were two popular choices for wattle and daub infill paneling: close-studded paneling and square paneling.
Close-studding panels create a much narrower space between the timbers: anywhere from 7 to 16 inches (18 to 40 cm). For this style of panel, weaving is too difficult, so the wattles run horizontally and are known as ledgers. The ledgers are sprung into each upright timber (stud) through a system of augered holes on one side and short chiseled grooves along the other. The holes (along with holes of square paneling) are drilled at a slight angle towards the outer face of each stud. This allows room for upright hazels to be tied to ledgers from the inside of the building. The horizontal ledgers are placed every two to three feet (0.6 to 0.9 metres) with whole hazel rods positioned upright top to bottom and lashed to the ledgers. These hazel rods are generally tied a finger-width apart with 6–8 rods each with a 16-inch (40 cm) width. Gaps allow key formation for drying. [ 11 ]
Square panels are large, wide panels typical of some later timber-frame houses. These panels may be square in shape, or sometimes triangular to accommodate arched or decorative bracing. This style requires the wattles to be woven for better support of the daub.
To insert wattles in a square panel several steps are required. First, a series of evenly spaced holes are drilled along the middle of the inner face of each upper timber. Next, a continuous groove is cut along the middle of each inner face of the lower timber in each panel. Vertical slender timbers, known as staves, are then inserted and these hold the whole panel within the timber frame. The staves are positioned into the holes and then sprung into the grooves. They must be placed with sufficient gaps to weave the flexible horizontal wattles.
In some places or cultures, the technique of wattle and daub was used with different materials and thus has different names.
In the early days of the colonisation of South Australia , in areas where substantial timber was unavailable, pioneers' cottages and other small buildings were frequently constructed with light vertical timbers, which may have been "native pine" ( Callitris or Casuarina spp. ), driven into the ground, the gaps being stopped with pug (kneaded clay and grass mixture). Another term for this construction is palisade and pug . [ 12 ]
"Mud and stud" is a similar process to wattle and daub, with a simple frame consisting only of upright studs joined by cross rails at the tops and bottoms. Thin staves of ash were attached, then daubed with a mixture of mud, straw, hair and dung. The style of building was once common in Lincolnshire . [ 13 ]
Pierrotage is the infilling material used in French Vernacular architecture of the Southern United States to infill between half-timbering with diagonal braces, which is similar to daub. It is usually made of lime mortar clay mixed with small stones. It is also called bousillage or bouzillage, especially in French Vernacular architecture of Louisiana of the early 1700s. The materials of bousillage are Spanish moss or clay and grass. Bousillage also refers to the type of brick molded with the same materials and used as infilling between posts. Columbage refers to the timber-framed construction with diagonal bracing of the framework. Pierratage or bousillage is the material filled into the structural timbers. [ 14 ]
Bajarreque is a wall constructed with the technique of wattle and daub. The wattle here is made of bagasse , and the daub is the mix of clay and straw. [ 15 ]
Jacal can refer to a type of crude house whose wall is built with wattle and daub in southwestern US. Closely spaced upright sticks or poles driven into the ground with small branches (wattle) interwoven between them make the structural frame of the wall. Mud or an adobe clay (daub) is covered outside. To provide additional weather protection, the wall is usually plastered. [ 16 ]
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Watts & Co. was a British publishing house which aimed to promote rationalism and secular education and "publish free thought books at affordable prices". [ 1 ] The firm had a close relationship with the Rationalist Press Association (later known as the Rationalist Association) and many of books it published were imprinted "Issued for the Rationalist Press Association Limited". [ 2 ]
Watts & Co. was founded in 1864 by the English secularist Charles Watts and his elder brother John Watts (1834-1866). [ 3 ] [ 4 ] In 1880 the firm's name, Watts & Co., was changed to C. Watts and in 1882 changed to C. A. Watts & Co., [ 5 ] but it was still commonly referred to, including on the title pages of many of the published books, as Watts & Co. Its premises were initially located at 17 Johnson's Court (just off Fleet Street ), London. [ 6 ]
In the first ten years of its operation the firm competed with its main business rival, the secularist Charles Bradlaugh 's Freethought Publishing Company , for market share. To do that it published the works of William Stewart Ross (who penned essays using his nom de plume , "Saladin"), who was a strong critic of Bradlaugh on certain aspects of doctrine and social policy. [ 7 ]
When Charles Watts migrated to America in 1882 [ 8 ] he handed over the publishing the firm to his son, Charles Watts Jr. (also referred to as C. A. Watts).
In 1885 the firm launched Watts's Literary Guide , a long-lasting periodical which changed its title three times: in 1884 to The Literary Guide and Rationalist Review , in 1956 to the Humanist and in 1972 to the New Humanist . [ 9 ] Under the latest title it still being published in 2024.
After publishing Joseph McCabe 's The Religion of the Twentieth Century in 1899 to modest acclaim, the firm had its first major success in 1900 with Ernst Haeckel 's The Riddle of the Universe which had sold 100,000 copies by 1905. In 1902 the firm launched its Cheap Reprints series, which Watts viewed a vehicle to make great works of free thought and "rationalist thought available to people ... of modest means and limited leisure". [ 10 ] Authors in this series included Charles Darwin , Thomas Huxley and John Stuart Mill . Each volume was sold for sixpence, a price shunned by mainstream publishers as inviting bankruptcy but which in fact ended up selling "in the region of 4,000,000 copies". [ 11 ]
Watts & Co. launched the Thinker's Library series in 1929 under the editorship of Charles Watts Jr.'s son Fredrick. Running for 22 years, it comprised 140 titles as hardbacks bound in red cloth, with the selling price set in the beginning at no more than one shilling per volume. [ 12 ] The series "sold several million copies of philosophical works over the years". [ 13 ]
Edited by: A.C. Grayling, Naomi Goulder, and Andrew Pyle
Over the years C. A. Watts & Company Limited maintained a strong link with the Rationalist Press Association. In 1953 the organizations agreed that "the publishing policy of CA Watts & Co would be decided by the Rationalist Press Association Board". [ 14 ]
In 1960 Watts & Co. was sold to Sir Isaac Pitman and Sons. Henceforth the Pemberton Publishing Company, a subsidiary of the Rationalist Press Association, would handle "all the publishing affairs of the ... Association". [ 15 ] The last Pemberton title was published in 1989 and from that time all titles were directly published by the Rationalist Press Association itself. [ 15 ]
Watts & Co. successfully published modestly-priced books on "rationalist and humanist" [ 19 ] themes which were read by millions of people throughout the English-speaking world. This was done in the face of hostility from certain church leaders, booksellers, judges and politicians who objected to books which "challenged the traditional religious outlook" [ 20 ] and in the face of skeptical mainstream publishers who believed that Watts' publishing program would lead to their bankruptcy.
Watts & Co. "provided a vehicle for unusual talents" such as Joseph McCabe, Hector Hawton and Nicolas Walter "to thrive". [ 21 ]
The Rationalist Press Association/Watts & Co. partnership operated "one of the first book clubs " where "members of the association subscribing five shillings and upwards" and in return "received books to the value of their subscriptions". [ 20 ]
Bill Cooke has argued that the Watts & Co. was "the first systematic venture to publish affordable non-fiction in paperback form", [ 22 ] with its Cheap Reprints series appearing before the launch of Ernest Benn Limited 's Sixpenny Library , [ 23 ] J. M. Dent 's Everyman's Library and Penguin 's Pelican Books .
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The reflected binary code ( RBC ), also known as reflected binary ( RB ) or Gray code after Frank Gray , is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For example, the representation of the decimal value "1" in binary would normally be " 001 ", and "2" would be " 010 ". In Gray code, these values are represented as " 001 " and " 011 ". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.
Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. [ 3 ]
Many devices indicate position by closing and opening switches. If that device uses natural binary codes , positions 3 and 4 are next to each other but all three bits of the binary representation differ:
The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce , the transition might look like 011 — 001 — 101 — 100 . When the switches appear to be in position 001 , the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic , then the sequential system may store a false value.
This problem can be solved by changing only one switch at a time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers , or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as unit-distance , [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] single-distance , single-step , monostrophic [ 9 ] [ 10 ] [ 7 ] [ 8 ] or syncopic codes , [ 9 ] in reference to the Hamming distance of 1 between adjacent codes.
In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code , or BRGC . Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. [ 11 ] [ 12 ] [ 13 ] Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". [ 14 ] He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".
In the standard encoding of the Gray code the least significant bit follows a repetitive pattern of 2 on, 2 off (... 11001100 ...); the next digit a pattern of 4 on, 4 off; the i -th least significant bit a pattern of 2 i on 2 i off. The most significant digit is an exception to this: for an n -bit Gray code, the most significant digit follows the pattern 2 n −1 on, 2 n −1 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2 n −2 places. The four-bit version of this is shown below:
For decimal 15 the code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code. [ 15 ]
In modern digital communications , Gray codes play an important role in error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Despite the fact that Stibitz described this code [ 11 ] [ 12 ] [ 13 ] before Gray, the reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; [ 16 ] [ 17 ] one of those also lists "minimum error code" and "cyclic permutation code" among the names. [ 17 ] A 1954 patent application refers to "the Bell Telephone Gray code". [ 18 ] Other names include "cyclic binary code", [ 12 ] "cyclic progression code", [ 19 ] [ 12 ] "cyclic permuting binary" [ 20 ] or "cyclic permuted binary" (CPB). [ 21 ] [ 22 ]
The Gray code is sometimes misattributed to 19th century electrical device inventor Elisha Gray . [ 13 ] [ 23 ] [ 24 ] [ 25 ]
Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle , a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872. [ 26 ] [ 13 ]
It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. [ 27 ] [ 28 ] [ 29 ] [ 30 ] Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes. [ 31 ]
Martin Gardner wrote a popular account of the Gray code in his August 1972 "Mathematical Games" column in Scientific American . [ 32 ]
The code also forms a Hamiltonian cycle on a hypercube , where each bit is seen as one dimension.
When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 [ 33 ] or 1876, [ 34 ] [ 35 ] he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, [ 36 ] [ 37 ] [ 38 ] and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code. [ 13 ] This code became known as Baudot code [ 39 ] and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932. [ 40 ] [ 41 ] [ 38 ]
About the same time, the German-Austrian Otto Schäffler [ de ] [ 42 ] demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874. [ 43 ] [ 13 ]
Frank Gray , who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube -based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, [ 14 ] and the name of Gray stuck to the codes. The " PCM tube " apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code. [ 44 ]
Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose.
Gray codes are used in linear and rotary position encoders ( absolute encoders and quadrature encoders ) in preference to weighted binary encoding. This avoids the possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others.
For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in the form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.
Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking.
In contrast, the Gray code used by position encoders ensures that the codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position.
Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms . [ 15 ] They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties.
Gray codes are also used in labelling the axes of Karnaugh maps since 1953 [ 45 ] [ 46 ] [ 47 ] as well as in Händler circle graphs since 1958, [ 48 ] [ 49 ] [ 50 ] [ 51 ] both graphical methods for logic circuit minimization .
In modern digital communications , 1D- and 2D-Gray codes play an important role in error prevention before applying an error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies.
If a system has to cycle sequentially through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves.
A balanced Gray code can be constructed, [ 52 ] that flips every bit equally often. Since bit-flips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit.
George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already. [ 11 ] [ 12 ] [ 13 ]
A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains. [ 53 ] The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.
Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, [ nb 1 ] it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous. [ 54 ]
To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, [ 55 ] add one to it with a standard binary adder, and then convert the result back to Gray code. [ 56 ] Other methods of counting in Gray code are discussed in a report by Robert W. Doran , including taking the output from the first latches of the master-slave flip flops in a binary ripple counter. [ 57 ]
As the execution of program code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs. [ 58 ] [ 59 ]
The binary-reflected Gray code list for n bits can be generated recursively from the list for n − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary 0 , prefixing the entries in the reflected list with a binary 1 , and then concatenating the original list with the reversed list. [ 13 ] For example, generating the n = 3 list from the n = 2 list:
The one-bit Gray code is G 1 = ( 0,1 ). This can be thought of as built recursively as above from a zero-bit Gray code G 0 = ( Λ ) consisting of a single entry of zero length. This iterative process of generating G n +1 from G n makes the following properties of the standard reflecting code clear:
These characteristics suggest a simple and fast method of translating a binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the n th Gray code is obtained by computing n ⊕ ⌊ n 2 ⌋ {\displaystyle n\oplus \left\lfloor {\tfrac {n}{2}}\right\rfloor } . Prepending a 0 bit leaves the order of the code words unchanged, prepending a 1 bit reverses the order of the code words. If the bits at position i {\displaystyle i} of codewords are inverted, the order of neighbouring blocks of 2 i {\displaystyle 2^{i}} codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence, the order of two neighbouring codewords is reversed
If bit 1 is inverted, blocks of 2 codewords change order:
If bit 2 is inverted, blocks of 4 codewords reverse order:
Thus, performing an exclusive or on a bit b i {\displaystyle b_{i}} at position i {\displaystyle i} with the bit b i + 1 {\displaystyle b_{i+1}} at position i + 1 {\displaystyle i+1} leaves the order of codewords intact if b i + 1 = 0 {\displaystyle b_{i+1}={\mathtt {0}}} , and reverses the order of blocks of 2 i + 1 {\displaystyle 2^{i+1}} codewords if b i + 1 = 1 {\displaystyle b_{i+1}={\mathtt {1}}} . Now, this is exactly the same operation as the reflect-and-prefix method to generate the Gray code.
A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming g i {\displaystyle g_{i}} is the i {\displaystyle i} th Gray-coded bit ( g 0 {\displaystyle g_{0}} being the most significant bit), and b i {\displaystyle b_{i}} is the i {\displaystyle i} th binary-coded bit ( b 0 {\displaystyle b_{0}} being the most-significant bit), the reverse translation can be given recursively: b 0 = g 0 {\displaystyle b_{0}=g_{0}} , and b i = g i ⊕ b i − 1 {\displaystyle b_{i}=g_{i}\oplus b_{i-1}} . Alternatively, decoding a Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.
To construct the binary-reflected Gray code iteratively, at step 0 start with the c o d e 0 = 0 {\displaystyle \mathrm {code} _{0}={\mathtt {0}}} , and at step i > 0 {\displaystyle i>0} find the bit position of the least significant 1 in the binary representation of i {\displaystyle i} and flip the bit at that position in the previous code c o d e i − 1 {\displaystyle \mathrm {code} _{i-1}} to get the next code c o d e i {\displaystyle \mathrm {code} _{i}} . The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ... [ nb 2 ] See find first set for efficient algorithms to compute these values.
The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist. [ 60 ] [ 55 ] [ nb 1 ]
On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set . If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation.
In practice, "Gray code" almost always refers to a binary-reflected Gray code (BRGC). However, mathematicians have discovered other kinds of Gray codes. Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word).
It is possible to construct binary Gray codes with n bits with a length of less than 2 n , if the length is even. One possibility is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. [ 61 ] OEIS sequence A290772 [ 62 ] gives the number of possible Gray sequences of length 2 n that include zero and use the minimum number of bits.
0 → 000 1 → 001 2 → 002 10 → 012 11 → 011 12 → 010 20 → 020 21 → 021 22 → 022 100 → 122 101 → 121 102 → 120 110 → 110 111 → 111 112 → 112 120 → 102 121 → 101 122 → 100 200 → 200 201 → 201 202 → 202 210 → 212 211 → 211 212 → 210 220 → 220 221 → 221
There are many specialized types of Gray codes other than the binary-reflected Gray code. One such type of Gray code is the n -ary Gray code , also known as a non-Boolean Gray code . As the name implies, this type of Gray code uses non- Boolean values in its encodings.
For example, a 3-ary ( ternary ) Gray code would use the values 0,1,2. [ 31 ] The ( n , k )- Gray code is the n -ary Gray code with k digits. [ 63 ] The sequence of elements in the (3, 2)-Gray code is: 00,01,02,12,11,10,20,21,22. The ( n , k )-Gray code may be constructed recursively, as the BRGC, or may be constructed iteratively . An algorithm to iteratively generate the ( N , k )-Gray code is presented (in C ):
There are other Gray code algorithms for ( n , k )-Gray codes. The ( n , k )-Gray code produced by the above algorithm is always cyclical; some algorithms, such as that by Guan, [ 63 ] lack this property when k is odd. On the other hand, while only one digit at a time changes with this method, it can change by wrapping (looping from n − 1 to 0). In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.
Gray codes are not uniquely defined, because a permutation of the columns of such a code is a Gray code too. The above procedure produces a code in which the lower the significance of a digit, the more often it changes, making it similar to normal counting methods.
See also Skew binary number system , a variant ternary number system where at most two digits change on each increment, as each increment can be done with at most one digit carry operation.
Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity". [ 52 ] In balanced Gray codes , the number of changes in different coordinate positions are as close as possible. To make this more precise, let G be an R -ary complete Gray cycle having transition sequence ( δ k ) {\displaystyle (\delta _{k})} ; the transition counts ( spectrum ) of G are the collection of integers defined by
λ k = | { j ∈ Z R n : δ j = k } | , for k ∈ Z n {\displaystyle \lambda _{k}=|\{j\in \mathbb {Z} _{R^{n}}:\delta _{j}=k\}|\,,{\text{ for }}k\in \mathbb {Z} _{n}}
A Gray code is uniform or uniformly balanced if its transition counts are all equal, in which case we have λ k = R n n {\displaystyle \lambda _{k}={\tfrac {R^{n}}{n}}} for all k . Clearly, when R = 2 {\displaystyle R=2} , such codes exist only if n is a power of 2. [ 64 ] If n is not a power of 2, it is possible to construct well-balanced binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either 2 ⌊ 2 n 2 n ⌋ {\displaystyle 2\left\lfloor {\tfrac {2^{n}}{2n}}\right\rfloor } or 2 ⌈ 2 n 2 n ⌉ {\displaystyle 2\left\lceil {\tfrac {2^{n}}{2n}}\right\rceil } . [ 52 ] Gray codes can also be exponentially balanced if all of their transition counts are adjacent powers of two, and such codes exist for every power of two. [ 65 ]
For example, a balanced 4-bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced: [ 52 ]
whereas a balanced 5-bit Gray code has a total of 32 transitions, which cannot be evenly distributed among the positions. In this example, four positions have six transitions each, and one has eight: [ 52 ]
We will now show a construction [ 66 ] and implementation [ 67 ] for well-balanced binary Gray codes which allows us to generate an n -digit balanced Gray code for every n . The main principle is to inductively construct an ( n + 2)-digit Gray code G ′ {\displaystyle G'} given an n -digit Gray code G in such a way that the balanced property is preserved. To do this, we consider partitions of G = g 0 , … , g 2 n − 1 {\displaystyle G=g_{0},\ldots ,g_{2^{n}-1}} into an even number L of non-empty blocks of the form
{ g 0 } , { g 1 , … , g k 2 } , { g k 2 + 1 , … , g k 3 } , … , { g k L − 2 + 1 , … , g − 2 } , { g − 1 } {\displaystyle \left\{g_{0}\right\},\left\{g_{1},\ldots ,g_{k_{2}}\right\},\left\{g_{k_{2}+1},\ldots ,g_{k_{3}}\right\},\ldots ,\left\{g_{k_{L-2}+1},\ldots ,g_{-2}\right\},\left\{g_{-1}\right\}}
where k 1 = 0 {\displaystyle k_{1}=0} , k L − 1 = − 2 {\displaystyle k_{L-1}=-2} , and k L ≡ − 1 ( mod 2 n ) {\displaystyle k_{L}\equiv -1{\pmod {2^{n}}}} ). This partition induces an ( n + 2 ) {\displaystyle (n+2)} -digit Gray code given by
If we define the transition multiplicities
m i = | { j : δ k j = i , 1 ≤ j ≤ L } | {\displaystyle m_{i}=\left|\left\{j:\delta _{k_{j}}=i,1\leq j\leq L\right\}\right|}
to be the number of times the digit in position i changes between consecutive blocks in a partition, then for the ( n + 2)-digit Gray code induced by this partition the transition spectrum λ i ′ {\displaystyle \lambda '_{i}} is
λ i ′ = { 4 λ i − 2 m i , if 0 ≤ i < n L , otherwise {\displaystyle \lambda '_{i}={\begin{cases}4\lambda _{i}-2m_{i},&{\text{if }}0\leq i<n\\L,&{\text{ otherwise }}\end{cases}}}
The delicate part of this construction is to find an adequate partitioning of a balanced n -digit Gray code such that the code induced by it remains balanced, but for this only the transition multiplicities matter; joining two consecutive blocks over a digit i {\displaystyle i} transition and splitting another block at another digit i {\displaystyle i} transition produces a different Gray code with exactly the same transition spectrum λ i ′ {\displaystyle \lambda '_{i}} , so one may for example [ 65 ] designate the first m i {\displaystyle m_{i}} transitions at digit i {\displaystyle i} as those that fall between two blocks. Uniform codes can be found when R ≡ 0 ( mod 4 ) {\displaystyle R\equiv 0{\pmod {4}}} and R n ≡ 0 ( mod n ) {\displaystyle R^{n}\equiv 0{\pmod {n}}} , and this construction can be extended to the R -ary case as well. [ 66 ]
Long run (or maximum gap ) Gray codes maximize the distance between consecutive changes of digits in the same position. That is, the minimum run-length of any bit remains unchanged for as long as possible. [ 68 ]
Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors. [ 69 ] If we define the weight of a binary string to be the number of 1s in the string, then although we clearly cannot have a Gray code with strictly increasing weight, we may want to approximate this by having the code run through two adjacent weights before reaching the next one.
We can formalize the concept of monotone Gray codes as follows: consider the partition of the hypercube Q n = ( V n , E n ) {\displaystyle Q_{n}=(V_{n},E_{n})} into levels of vertices that have equal weight, i.e.
V n ( i ) = { v ∈ V n : v has weight i } {\displaystyle V_{n}(i)=\{v\in V_{n}:v{\text{ has weight }}i\}}
for 0 ≤ i ≤ n {\displaystyle 0\leq i\leq n} . These levels satisfy | V n ( i ) | = ( n i ) {\displaystyle |V_{n}(i)|=\textstyle {\binom {n}{i}}} . Let Q n ( i ) {\displaystyle Q_{n}(i)} be the subgraph of Q n {\displaystyle Q_{n}} induced by V n ( i ) ∪ V n ( i + 1 ) {\displaystyle V_{n}(i)\cup V_{n}(i+1)} , and let E n ( i ) {\displaystyle E_{n}(i)} be the edges in Q n ( i ) {\displaystyle Q_{n}(i)} . A monotonic Gray code is then a Hamiltonian path in Q n {\displaystyle Q_{n}} such that whenever δ 1 ∈ E n ( i ) {\displaystyle \delta _{1}\in E_{n}(i)} comes before δ 2 ∈ E n ( j ) {\displaystyle \delta _{2}\in E_{n}(j)} in the path, then i ≤ j {\displaystyle i\leq j} .
An elegant construction of monotonic n -digit Gray codes for any n is based on the idea of recursively building subpaths P n , j {\displaystyle P_{n,j}} of length 2 ( n j ) {\displaystyle 2\textstyle {\binom {n}{j}}} having edges in E n ( j ) {\displaystyle E_{n}(j)} . [ 69 ] We define P 1 , 0 = ( 0 , 1 ) {\displaystyle P_{1,0}=({\mathtt {0}},{\mathtt {1}})} , P n , j = ∅ {\displaystyle P_{n,j}=\emptyset } whenever j < 0 {\displaystyle j<0} or j ≥ n {\displaystyle j\geq n} , and
P n + 1 , j = 1 P n , j − 1 π n , 0 P n , j {\displaystyle P_{n+1,j}={\mathtt {1}}P_{n,j-1}^{\pi _{n}},{\mathtt {0}}P_{n,j}}
otherwise. Here, π n {\displaystyle \pi _{n}} is a suitably defined permutation and P π {\displaystyle P^{\pi }} refers to the path P with its coordinates permuted by π {\displaystyle \pi } . These paths give rise to two monotonic n -digit Gray codes G n ( 1 ) {\displaystyle G_{n}^{(1)}} and G n ( 2 ) {\displaystyle G_{n}^{(2)}} given by
G n ( 1 ) = P n , 0 P n , 1 R P n , 2 P n , 3 R ⋯ and G n ( 2 ) = P n , 0 R P n , 1 P n , 2 R P n , 3 ⋯ {\displaystyle G_{n}^{(1)}=P_{n,0}P_{n,1}^{R}P_{n,2}P_{n,3}^{R}\cdots {\text{ and }}G_{n}^{(2)}=P_{n,0}^{R}P_{n,1}P_{n,2}^{R}P_{n,3}\cdots }
The choice of π n {\displaystyle \pi _{n}} which ensures that these codes are indeed Gray codes turns out to be π n = E − 1 ( π n − 1 2 ) {\displaystyle \pi _{n}=E^{-1}\left(\pi _{n-1}^{2}\right)} . The first few values of P n , j {\displaystyle P_{n,j}} are shown in the table below.
These monotonic Gray codes can be efficiently implemented in such a way that each subsequent element can be generated in O ( n ) time. The algorithm is most easily described using coroutines .
Monotonic codes have an interesting connection to the Lovász conjecture , which states that every connected vertex-transitive graph contains a Hamiltonian path. The "middle-level" subgraph Q 2 n + 1 ( n ) {\displaystyle Q_{2n+1}(n)} is vertex-transitive (that is, its automorphism group is transitive, so that each vertex has the same "local environment" and cannot be differentiated from the others, since we can relabel the coordinates as well as the binary digits to obtain an automorphism ) and the problem of finding a Hamiltonian path in this subgraph is called the "middle-levels problem", which can provide insights into the more general conjecture. The question has been answered affirmatively for n ≤ 15 {\displaystyle n\leq 15} , and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839 N , where N is the number of vertices in the middle-level subgraph. [ 70 ]
Another type of Gray code, the Beckett–Gray code , is named for Irish playwright Samuel Beckett , who was interested in symmetry . His play " Quad " features four actors and is divided into sixteen time periods. Each period ends with one of the four actors entering or leaving the stage. The play begins and ends with an empty stage, and Beckett wanted each subset of actors to appear on stage exactly once. [ 71 ] Clearly the set of actors currently on stage can be represented by a 4-bit binary Gray code. Beckett, however, placed an additional restriction on the script: he wished the actors to enter and exit so that the actor who had been on stage the longest would always be the one to exit. The actors could then be represented by a first in, first out queue , so that (of the actors onstage) the actor being dequeued is always the one who was enqueued first. [ 71 ] Beckett was unable to find a Beckett–Gray code for his play, and indeed, an exhaustive listing of all possible sequences reveals that no such code exists for n = 4. It is known today that such codes do exist for n = 2, 5, 6, 7, and 8, and do not exist for n = 3 or 4. An example of an 8-bit Beckett–Gray code can be found in Donald Knuth 's Art of Computer Programming . [ 13 ] According to Sawada and Wong, the search space for n = 6 can be explored in 15 hours, and more than 9500 solutions for the case n = 7 have been found. [ 72 ]
Snake-in-the-box codes, or snakes , are the sequences of nodes of induced paths in an n -dimensional hypercube graph , and coil-in-the-box codes, [ 73 ] or coils , are the sequences of nodes of induced cycles in a hypercube. Viewed as Gray codes, these sequences have the property of being able to detect any single-bit coding error. Codes of this type were first described by William H. Kautz in the late 1950s; [ 5 ] since then, there has been much research on finding the code with the largest possible number of codewords for a given hypercube dimension.
Yet another kind of Gray code is the single-track Gray code (STGC) developed by Norman B. Spedding [ 74 ] [ 75 ] and refined by Hiltgen, Paterson and Brandestini in Single-track Gray Codes (1996). [ 76 ] [ 77 ] The STGC is a cyclical list of P unique binary encodings of length n such that two consecutive words differ in exactly one position, and when the list is examined as a P × n matrix , each column is a cyclic shift of the first column. [ 78 ]
The name comes from their use with rotary encoders , where a number of tracks are being sensed by contacts, resulting for each in an output of 0 or 1 . To reduce noise due to different contacts not switching at exactly the same moment in time, one preferably sets up the tracks so that the data output by the contacts are in Gray code. To get high angular accuracy, one needs lots of contacts; in order to achieve at least 1° accuracy, one needs at least 360 distinct positions per revolution, which requires a minimum of 9 bits of data, and thus the same number of contacts.
If all contacts are placed at the same angular position, then 9 tracks are needed to get a standard BRGC with at least 1° accuracy. However, if the manufacturer moves a contact to a different angular position (but at the same distance from the center shaft), then the corresponding "ring pattern" needs to be rotated the same angle to give the same output. If the most significant bit (the inner ring in Figure 1) is rotated enough, it exactly matches the next ring out. Since both rings are then identical, the inner ring can be cut out, and the sensor for that ring moved to the remaining, identical ring (but offset at that angle from the other sensor on that ring). Those two sensors on a single ring make a quadrature encoder. That reduces the number of tracks for a "1° resolution" angular encoder to 8 tracks. Reducing the number of tracks still further cannot be done with BRGC.
For many years, Torsten Sillke [ 79 ] and other mathematicians believed that it was impossible to encode position on a single track such that consecutive positions differed at only a single sensor, except for the 2-sensor, 1-track quadrature encoder. So for applications where 8 tracks were too bulky, people used single-track incremental encoders (quadrature encoders) or 2-track "quadrature encoder + reference notch" encoders.
Norman B. Spedding, however, registered a patent in 1994 with several examples showing that it was possible. [ 74 ] Although it is not possible to distinguish 2 n positions with n sensors on a single track, it is possible to distinguish close to that many. Etzion and Paterson conjecture that when n is itself a power of 2, n sensors can distinguish at most 2 n − 2 n positions and that for prime n the limit is 2 n − 2 positions. [ 80 ] The authors went on to generate a 504-position single track code of length 9 which they believe is optimal. Since this number is larger than 2 8 = 256, more than 8 sensors are required by any code, although a BRGC could distinguish 512 positions with 9 sensors.
An STGC for P = 30 and n = 5 is reproduced here:
Each column is a cyclic shift of the first column, and from any row to the next row only one bit changes. [ 81 ] The single-track nature (like a code chain) is useful in the fabrication of these wheels (compared to BRGC), as only one track is needed, thus reducing their cost and size.
The Gray code nature is useful (compared to chain codes , also called De Bruijn sequences ), as only one sensor will change at any one time, so the uncertainty during a transition between two discrete states will only be plus or minus one unit of angular measurement the device is capable of resolving. [ 82 ]
Since this 30 degree example was added, there has been a lot of interest in examples with higher angular resolution. In 2008, Gary Williams, [ 83 ] [ user-generated source? ] based on previous work, [ 80 ] discovered a 9-bit single track Gray code that gives a 1 degree resolution. This Gray code was used to design an actual device which was published on the site Thingiverse . This device [ 84 ] was designed by etzenseep (Florian Bauer) in September 2022.
An STGC for P = 360 and n = 9 is reproduced here:
Two-dimensional Gray codes are used in communication to minimize the number of bit errors in quadrature amplitude modulation (QAM) adjacent points in the constellation . In a typical encoding the horizontal and vertical adjacent constellation points differ by a single bit, and diagonal adjacent points differ by 2 bits. [ 85 ]
Two-dimensional Gray codes also have uses in location identifications schemes, where the code would be applied to area maps such as a Mercator projection of the earth's surface and an appropriate cyclic two-dimensional distance function such as the Mannheim metric be used to calculate the distance between two encoded locations, thereby combining the characteristics of the Hamming distance with the cyclic continuation of a Mercator projection. [ 86 ]
If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4-bit Gray code, the resulting code will be an "excess Gray code". This code shows the property of counting backwards in those extracted bits if the original value is further increased. Reason for this is that Gray-encoded values do not show the behaviour of overflow, known from classic binary encoding, when increasing past the "highest" value.
Example: The highest 3-bit Gray code, 7, is encoded as (0)100. Adding 1 results in number 8, encoded in Gray as 1100. The last 3 bits do not overflow and count backwards if you further increase the original 4 bit code.
When working with sensors that output multiple, Gray-encoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single Gray code or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.
The bijective mapping { 0 ↔ 00 , 1 ↔ 01 , 2 ↔ 11 , 3 ↔ 10 } establishes an isometry between the metric space over the finite field Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} with the metric given by the Hamming distance and the metric space over the finite ring Z 4 {\displaystyle \mathbb {Z} _{4}} (the usual modular arithmetic ) with the metric given by the Lee distance . The mapping is suitably extended to an isometry of the Hamming spaces Z 2 2 m {\displaystyle \mathbb {Z} _{2}^{2m}} and Z 4 m {\displaystyle \mathbb {Z} _{4}^{m}} . Its importance lies in establishing a correspondence between various "good" but not necessarily linear codes as Gray-map images in Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} of ring-linear codes from Z 4 {\displaystyle \mathbb {Z} _{4}} . [ 87 ] [ 88 ]
There are a number of binary codes similar to Gray codes, including:
The following binary-coded decimal (BCD) codes are Gray code variants as well:
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In physics , mathematics , engineering , and related fields, a wave is a propagating dynamic disturbance (change from equilibrium ) of one or more quantities . Periodic waves oscillate repeatedly about an equilibrium (resting) value at some frequency . When the entire waveform moves in one direction, it is said to be a travelling wave ; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave . In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero.
There are two types of waves that are most commonly studied in classical physics : mechanical waves and electromagnetic waves . In a mechanical wave, stress and strain fields oscillate about a mechanical equilibrium . A mechanical wave is a local deformation (strain) in some physical medium that propagates from particle to particle by creating local stresses that cause strain in neighboring particles too. For example, sound waves are variations of the local pressure and particle motion that propagate through the medium. Other examples of mechanical waves are seismic waves , gravity waves , surface waves and string vibrations . In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according to Maxwell's equations . Electromagnetic waves can travel through a vacuum and through some dielectric media (at wavelengths where they are considered transparent ). Electromagnetic waves, as determined by their frequencies (or wavelengths ), have more specific designations including radio waves , infrared radiation , terahertz waves , visible light , ultraviolet radiation , X-rays and gamma rays .
Other types of waves include gravitational waves , which are disturbances in spacetime that propagate according to general relativity ; heat diffusion waves ; plasma waves that combine mechanical deformations and electromagnetic fields; reaction–diffusion waves , such as in the Belousov–Zhabotinsky reaction ; and many more. Mechanical and electromagnetic waves transfer energy , [ 1 ] momentum , and information , but they do not transfer particles in the medium. In mathematics and electronics waves are studied as signals . [ 2 ] On the other hand, some waves have envelopes which do not move at all such as standing waves (which are fundamental to music) and hydraulic jumps .
A physical wave field is almost always confined to some finite region of space, called its domain . For example, the seismic waves generated by earthquakes are significant only in the interior and surface of the planet, so they can be ignored outside it. However, waves with infinite domain, that extend over the whole space, are commonly studied in mathematics, and are very valuable tools for understanding physical waves in finite domains.
A plane wave is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel. Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency. In linear media, complicated waves can generally be decomposed as the sum of many sinusoidal plane waves having different directions of propagation and/or different frequencies . A plane wave is classified as a transverse wave if the field disturbance at each point is described by a vector perpendicular to the direction of propagation (also the direction of energy transfer); or longitudinal wave if those vectors are aligned with the propagation direction. Mechanical waves include both transverse and longitudinal waves; on the other hand electromagnetic plane waves are strictly transverse while sound waves in fluids (such as air) can only be longitudinal. That physical direction of an oscillating field relative to the propagation direction is also referred to as the wave's polarization , which can be an important attribute.
A wave can be described just like a field, namely as a function F ( x , t ) {\displaystyle F(x,t)} where x {\displaystyle x} is a position and t {\displaystyle t} is a time.
The value of x {\displaystyle x} is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . However, in many cases one can ignore one dimension, and let x {\displaystyle x} be a point of the Cartesian plane R 2 {\displaystyle \mathbb {R} ^{2}} . This is the case, for example, when studying vibrations of a drum skin. One may even restrict x {\displaystyle x} to a point of the Cartesian line R {\displaystyle \mathbb {R} } – that is, the set of real numbers . This is the case, for example, when studying vibrations in a violin string or recorder . The time t {\displaystyle t} , on the other hand, is always assumed to be a scalar ; that is, a real number.
The value of F ( x , t ) {\displaystyle F(x,t)} can be any physical quantity of interest assigned to the point x {\displaystyle x} that may vary with time. For example, if F {\displaystyle F} represents the vibrations inside an elastic solid, the value of F ( x , t ) {\displaystyle F(x,t)} is usually a vector that gives the current displacement from x {\displaystyle x} of the material particles that would be at the point x {\displaystyle x} in the absence of vibration. For an electromagnetic wave, the value of F {\displaystyle F} can be the electric field vector E {\displaystyle E} , or the magnetic field vector H {\displaystyle H} , or any related quantity, such as the Poynting vector E × H {\displaystyle E\times H} . In fluid dynamics , the value of F ( x , t ) {\displaystyle F(x,t)} could be the velocity vector of the fluid at the point x {\displaystyle x} , or any scalar property like pressure , temperature , or density . In a chemical reaction, F ( x , t ) {\displaystyle F(x,t)} could be the concentration of some substance in the neighborhood of point x {\displaystyle x} of the reaction medium.
For any dimension d {\displaystyle d} (1, 2, or 3), the wave's domain is then a subset D {\displaystyle D} of R d {\displaystyle \mathbb {R} ^{d}} , such that the function value F ( x , t ) {\displaystyle F(x,t)} is defined for any point x {\displaystyle x} in D {\displaystyle D} . For example, when describing the motion of a drum skin , one can consider D {\displaystyle D} to be a disk (circle) on the plane R 2 {\displaystyle \mathbb {R} ^{2}} with center at the origin ( 0 , 0 ) {\displaystyle (0,0)} , and let F ( x , t ) {\displaystyle F(x,t)} be the vertical displacement of the skin at the point x {\displaystyle x} of D {\displaystyle D} and at time t {\displaystyle t} .
Waves of the same type are often superposed and encountered simultaneously at a given point in space and time. The properties at that point are the sum of the properties of each component wave at that point. In general, the velocities are not the same, so the wave form will change over time and space.
Sometimes one is interested in a single specific wave. More often, however, one needs to understand large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a drum stick , or all the possible radar echoes one could get from an airplane that may be approaching an airport .
In some of those situations, one may describe such a family of waves by a function F ( A , B , … ; x , t ) {\displaystyle F(A,B,\ldots ;x,t)} that depends on certain parameters A , B , … {\displaystyle A,B,\ldots } , besides x {\displaystyle x} and t {\displaystyle t} . Then one can obtain different waves – that is, different functions of x {\displaystyle x} and t {\displaystyle t} – by choosing different values for those parameters.
For example, the sound pressure inside a recorder that is playing a "pure" note is typically a standing wave , that can be written as
The parameter A {\displaystyle A} defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); c {\displaystyle c} is the speed of sound; L {\displaystyle L} is the length of the bore; and n {\displaystyle n} is a positive integer (1,2,3,...) that specifies the number of nodes in the standing wave. (The position x {\displaystyle x} should be measured from the mouthpiece , and the time t {\displaystyle t} from any moment at which the pressure at the mouthpiece is maximum. The quantity λ = 4 L / ( 2 n − 1 ) {\displaystyle \lambda =4L/(2n-1)} is the wavelength of the emitted note, and f = c / λ {\displaystyle f=c/\lambda } is its frequency .) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters.
As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance r {\displaystyle r} from the center of the skin to the strike point, and on the strength s {\displaystyle s} of the strike. Then the vibration for all possible strikes can be described by a function F ( r , s ; x , t ) {\displaystyle F(r,s;x,t)} .
Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to the temperature in a metal bar when it is initially heated at various temperatures at different points along its length, and then allowed to cool by itself in vacuum. In that case, instead of a scalar or vector, the parameter would have to be a function h {\displaystyle h} such that h ( x ) {\displaystyle h(x)} is the initial temperature at each point x {\displaystyle x} of the bar. Then the temperatures at later times can be expressed by a function F {\displaystyle F} that depends on the function h {\displaystyle h} (that is, a functional operator ), so that the temperature at a later time is F ( h ; x , t ) {\displaystyle F(h;x,t)}
Another way to describe and study a family of waves is to give a mathematical equation that, instead of explicitly giving the value of F ( x , t ) {\displaystyle F(x,t)} , only constrains how those values can change with time. Then the family of waves in question consists of all functions F {\displaystyle F} that satisfy those constraints – that is, all solutions of the equation.
This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For example, if F ( x , t ) {\displaystyle F(x,t)} is the temperature inside a block of some homogeneous and isotropic solid material, its evolution is constrained by the partial differential equation
where Q ( p , f ) {\displaystyle Q(p,f)} is the heat that is being generated per unit of volume and time in the neighborhood of x {\displaystyle x} at time t {\displaystyle t} (for example, by chemical reactions happening there); x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} are the Cartesian coordinates of the point x {\displaystyle x} ; ∂ F / ∂ t {\displaystyle \partial F/\partial t} is the (first) derivative of F {\displaystyle F} with respect to t {\displaystyle t} ; and ∂ 2 F / ∂ x i 2 {\displaystyle \partial ^{2}F/\partial x_{i}^{2}} is the second derivative of F {\displaystyle F} relative to x i {\displaystyle x_{i}} . (The symbol " ∂ {\displaystyle \partial } " is meant to signify that, in the derivative with respect to some variable, all other variables must be considered fixed.)
This equation can be derived from the laws of physics that govern the diffusion of heat in solid media. For that reason, it is called the heat equation in mathematics, even though it applies to many other physical quantities besides temperatures.
For another example, we can describe all possible sounds echoing within a container of gas by a function F ( x , t ) {\displaystyle F(x,t)} that gives the pressure at a point x {\displaystyle x} and time t {\displaystyle t} within that container. If the gas was initially at uniform temperature and composition, the evolution of F {\displaystyle F} is constrained by the formula
Here P ( x , t ) {\displaystyle P(x,t)} is some extra compression force that is being applied to the gas near x {\displaystyle x} by some external process, such as a loudspeaker or piston right next to p {\displaystyle p} .
This same differential equation describes the behavior of mechanical vibrations and electromagnetic fields in a homogeneous isotropic non-conducting solid. Note that this equation differs from that of heat flow only in that the left-hand side is ∂ 2 F / ∂ t 2 {\displaystyle \partial ^{2}F/\partial t^{2}} , the second derivative of F {\displaystyle F} with respect to time, rather than the first derivative ∂ F / ∂ t {\displaystyle \partial F/\partial t} . Yet this small change makes a huge difference on the set of solutions F {\displaystyle F} . This differential equation is called "the" wave equation in mathematics, even though it describes only one very special kind of waves.
Consider a traveling transverse wave (which may be a pulse ) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling
This wave can then be described by the two-dimensional functions
or, more generally, by d'Alembert's formula : [ 6 ] u ( x , t ) = F ( x − v t ) + G ( x + v t ) . {\displaystyle u(x,t)=F(x-vt)+G(x+vt).} representing two component waveforms F {\displaystyle F} and G {\displaystyle G} traveling through the medium in opposite directions. A generalized representation of this wave can be obtained [ 7 ] as the partial differential equation 1 v 2 ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 . {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}.}
General solutions are based upon Duhamel's principle . [ 8 ]
The form or shape of F in d'Alembert's formula involves the argument x − vt . Constant values of this argument correspond to constant values of F , and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x -direction at velocity v (and G will propagate at the same speed in the negative x -direction). [ 9 ]
In the case of a periodic function F with period λ , that is, F ( x + λ − vt ) = F ( x − vt ), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F ( x − v ( t + T )) = F ( x − vt ) provided vT = λ , so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ / v . [ 10 ]
The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave ), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form: [ 11 ] [ 12 ] [ 13 ] u ( x , t ) = A ( x , t ) sin ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x,t)\sin \left(kx-\omega t+\phi \right),} where A ( x , t ) {\displaystyle A(x,\ t)} is the amplitude envelope of the wave, k {\displaystyle k} is the wavenumber and ϕ {\displaystyle \phi } is the phase . If the group velocity v g {\displaystyle v_{g}} (see below) is wavelength-independent, this equation can be simplified as: [ 14 ] u ( x , t ) = A ( x − v g t ) sin ( k x − ω t + ϕ ) , {\displaystyle u(x,t)=A(x-v_{g}t)\sin \left(kx-\omega t+\phi \right),} showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation . [ 14 ] [ 15 ]
There are two velocities that are associated with waves, the phase velocity and the group velocity .
Phase velocity is the rate at which the phase of the wave propagates in space : any given phase of the wave (for example, the crest ) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as v p = λ T . {\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}
Group velocity is a property of waves that have a defined envelope, measuring propagation through space (that is, phase velocity) of the overall shape of the waves' amplitudes—modulation or envelope of the wave.
A sine wave , sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function . In mechanics , as a linear motion over time, this is simple harmonic motion ; as rotation , it corresponds to uniform circular motion . Sine waves occur often in physics , including wind waves , sound waves, and light waves, such as monochromatic radiation . In engineering , signal processing , and mathematics , Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.
A plane wave is a kind of wave whose value varies only in one spatial direction. That is, its value is constant on a plane that is perpendicular to that direction. Plane waves can be specified by a vector of unit length n ^ {\displaystyle {\hat {n}}} indicating the direction that the wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction ( n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {x}}} ) and time ( t {\displaystyle t} ). Since the wave profile only depends on the position x → {\displaystyle {\vec {x}}} in the combination n ^ ⋅ x → {\displaystyle {\hat {n}}\cdot {\vec {x}}} , any displacement in directions perpendicular to n ^ {\displaystyle {\hat {n}}} cannot affect the value of the field.
Plane waves are often used to model electromagnetic waves far from a source. For electromagnetic plane waves, the electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other.
A standing wave, also known as a stationary wave , is a wave whose envelope remains in a constant position. This phenomenon arises as a result of interference between two waves traveling in opposite directions.
The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave . Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut , where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node . Halfway between two nodes there is an antinode , where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.
A soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.
Wave propagation is any of the ways in which waves travel. With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves .
Electromagnetic waves propagate in vacuum as well as in material media. Propagation of other wave types such as sound may occur only in a transmission medium .
The propagation and reflection of plane waves—e.g. Pressure waves ( P wave ) or Shear waves (SH or SV-waves) are phenomena that were first characterized within the field of classical seismology , and are now considered fundamental concepts in modern seismic tomography . The analytical solution to this problem exists and is well known. The frequency domain solution can be obtained by first finding the Helmholtz decomposition of the displacement field, which is then substituted into the wave equation . From here, the plane wave eigenmodes can be calculated. [ citation needed ] [ clarification needed ]
The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of the reflected SV wave is identical to the incidence wave, while the angle of the reflected P wave is greater than the SV wave. For the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture. [ 16 ]
Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different. [ clarification needed ]
Wave velocity is a general concept, of various kinds of wave velocities, for a wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as: v p = ω k , {\displaystyle v_{\rm {p}}={\frac {\omega }{k}},} where:
The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω cannot be chosen independently from the wavenumber k , but both are related through the dispersion relationship : ω = Ω ( k ) . {\displaystyle \omega =\Omega (k).}
In the special case Ω( k ) = ck , with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c . For instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of the dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the waves propagate and on the type of waves (for instance electromagnetic , sound or water waves).
The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation : v g = ∂ ω ∂ k {\displaystyle v_{\rm {g}}={\frac {\partial \omega }{\partial k}}}
In almost all cases, a wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium.
Waves exhibit common behaviors under a number of standard situations, for example:
Waves normally move in a straight line (that is, rectilinearly) through a transmission medium . Such media can be classified into one or more of the following categories:
Waves are usually defined in media which allow most or all of a wave's energy to propagate without loss . However materials may be characterized as "lossy" if they remove energy from a wave, usually converting it into heat. This is termed "absorption." A material which absorbs a wave's energy, either in transmission or reflection, is characterized by a refractive index which is complex . The amount of absorption will generally depend on the frequency (wavelength) of the wave, which, for instance, explains why objects may appear colored.
When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.
Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law .
A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.
When waves in a linear medium (the usual case) cross each other in a region of space, they do not actually interact with each other, but continue on as if the other one were not present. However at any point in that region the field quantities describing those waves add according to the superposition principle . If the waves are of the same frequency in a fixed phase relationship, then there will generally be positions at which the two waves are in phase and their amplitudes add , and other positions where they are out of phase and their amplitudes (partially or fully) cancel . This is called an interference pattern .
The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization . A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter .
Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.
Dispersion is the frequency dependence of the refractive index , a consequence of the atomic nature of materials. [ 17 ] : 67 A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is seen by letting white light pass through a prism , the result of which is to produce the spectrum of colors of the rainbow. Isaac Newton was the first to recognize that this meant that white light was a mixture of light of different colors. [ 17 ] : 190
The Doppler effect or Doppler shift is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. [ 18 ] It is named after the Austrian physicist Christian Doppler , who described the phenomenon in 1842.
A mechanical wave is an oscillation of matter , and therefore transfers energy through a medium . [ 19 ] While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefore, the oscillating material does not move far from its initial position. Mechanical waves can be produced only in media which possess elasticity and inertia . There are three types of mechanical waves: transverse waves , longitudinal waves , and surface waves .
The transverse vibration of a string is a function of tension and inertia, and is constrained by the length of the string as the ends are fixed. This constraint limits the steady state modes that are possible, and thereby the frequencies.
The speed of a transverse wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension of the string ( T ) over the linear mass density ( μ ):
where the linear density μ is the mass per unit length of the string.
Acoustic or sound waves are compression waves which travel as body waves at the speed given by:
or the square root of the adiabatic bulk modulus divided by the ambient density of the medium (see speed of sound ).
Body waves travel through the interior of the medium along paths controlled by the material properties in terms of density and modulus (stiffness). The density and modulus, in turn, vary according to temperature, composition, and material phase. This effect resembles the refraction of light waves. Two types of particle motion result in two types of body waves: Primary and Secondary waves.
Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy. They include body waves—the primary ( P waves ) and secondary waves ( S waves )—and surface waves, such as Rayleigh waves , Love waves , and Stoneley waves .
A shock wave is a type of propagating disturbance. When a wave moves faster than the local speed of sound in a fluid , it is a shock wave. Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in pressure , temperature and density of the medium. [ 20 ]
Shear waves are body waves due to shear rigidity and inertia. They can only be transmitted through solids and to a lesser extent through liquids with a sufficiently high viscosity.
An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum , the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light . From this emerged the idea that light is an electromagnetic wave. The unification of light and electromagnetic waves was experimentally confirmed by Hertz in the end of the 1880s. Electromagnetic waves can have different frequencies (and thus wavelengths), and are classified accordingly in wavebands, such as radio waves , microwaves , infrared , visible light , ultraviolet , X-rays , and gamma rays . The range of frequencies in each of these bands is continuous, and the limits of each band are mostly arbitrary, with the exception of visible light, which must be visible to the normal human eye.
The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics . Solutions of this equation are wave functions which can be used to describe the probability density of a particle.
The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin- 1 ⁄ 2 particles.
Louis de Broglie postulated that all particles with momentum have a wavelength
where h is the Planck constant , and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics . Nowadays, this wavelength is called the de Broglie wavelength . For example, the electrons in a CRT display have a de Broglie wavelength of about 10 −13 m.
A wave representing such a particle traveling in the k -direction is expressed by the wave function as follows:
where the wavelength is determined by the wave vector k as:
and the momentum by:
However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet , [ 24 ] a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.
In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet . [ 25 ] [ 26 ] [ 27 ] Gaussian wave packets also are used to analyze water waves. [ 28 ]
For example, a Gaussian wavefunction ψ might take the form: [ 29 ]
at some initial time t = 0, where the central wavelength is related to the central wave vector k 0 as λ 0 = 2π / k 0 . It is well known from the theory of Fourier analysis , [ 30 ] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian. [ 31 ] Given the Gaussian:
the Fourier transform is:
The Gaussian in space therefore is made up of waves:
that is, a number of waves of wavelengths λ such that kλ = 2 π.
The parameter σ decides the spatial spread of the Gaussian along the x -axis, while the Fourier transform shows a spread in wave vector k determined by 1/ σ . That is, the smaller the extent in space, the larger the extent in k , and hence in λ = 2π/ k .
Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy works to restore equilibrium. Surface waves on water are the most familiar example.
Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016. [ 32 ] Gravitational waves are disturbances in the curvature of spacetime , predicted by Einstein's theory of general relativity .
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Wave-making resistance or wave resistance is a form of drag that affects any object moving on a surface of a fluid, such as boats and ships moving on the surface of water, and reflects the energy required to push the water out of the way of that body. For example, the hull of a moving watercraft creates waves (a wake ) which carry energy away and resist the motion of the watercraft. Wave resistance is only one of the components of the total resistance or drag experienced by a body moving on a surface of a fluid, other being viscous drag and pressure drag .
For small displacement hulls , such as sailboats or rowboats, wave-making resistance is the major source of the marine vessel drag .
A salient property of water waves is dispersiveness; i.e., the greater the wavelength, the faster it moves. Waves generated by a ship are affected by her geometry and speed, and most of the energy given by the ship for making waves is transferred to water through the bow and stern parts. Simply speaking, these two wave systems, i.e., bow and stern waves, interact with each other, and the resulting waves are responsible for the resistance. If the resulting wave is large, it carries much energy away from the ship, delivering it to the shore or wherever else the wave ends up or just dissipating it in the water, and that energy must be supplied by the ship's propulsion (or momentum), so that the ship experiences it as drag. Conversely, if the resulting wave is small, the drag experienced is small.
The amount and direction (additive or subtractive) of the interference depends upon the phase difference between the bow and stern waves (which have the same wavelength and phase speed), and that is a function of the length of the ship at the waterline. For a given ship speed, the phase difference between the bow wave and stern wave is proportional to the length of the ship at the waterline. For example, if the ship takes three seconds to travel its own length, then at some point the ship passes, a stern wave is initiated three seconds after a bow wave, which implies a specific phase difference between those two waves. Thus, the waterline length of the ship directly affects the magnitude of the wave-making resistance.
For a given waterline length, the phase difference depends upon the phase speed and wavelength of the waves, and those depend directly upon the speed of the ship. For a deepwater wave, the phase speed is the same as the propagation speed and is proportional to the square root of the wavelength . That wavelength is dependent upon the speed of the ship.
Thus, the magnitude of the wave-making resistance is a function of the speed of the ship in relation to its length at the waterline.
A simple way of considering wave-making resistance is to look at the hull in relation to bow and stern waves. If the length of a ship is half the length of the waves generated, the resulting wave will be very small due to cancellation, and if the length is the same as the wavelength, the wave will be large due to enhancement.
The phase speed c {\displaystyle c} of waves is given by the following formula:
c = g 2 π l {\displaystyle c={\sqrt {{\frac {g}{2\pi }}l}}}
where l {\displaystyle l} is the length of the wave and g {\displaystyle g} the gravitational acceleration. Substituting in the appropriate value for g {\displaystyle g} yields the equation:
c in knots ≈ 1.341 × length in ft ≈ 4 3 × length in ft {\displaystyle {\mbox{c in knots}}\approx 1.341\times {\sqrt {\mbox{length in ft}}}\approx {\frac {4}{3}}\times {\sqrt {\mbox{length in ft}}}}
or, in metric units:
c in knots ≈ 2.429 × length in m ≈ 6 × length in m ≈ 2.5 × length in m {\displaystyle {\mbox{c in knots}}\approx 2.429\times {\sqrt {\mbox{length in m}}}\approx {\sqrt {6\times {\mbox{length in m}}}}\approx 2.5\times {\sqrt {\mbox{length in m}}}}
These values, 1.34, 2.5 and very easy 6, are often used in the hull speed rule of thumb used to compare potential speeds of displacement hulls, and this relationship is also fundamental to the Froude number , used in the comparison of different scales of watercraft.
When the vessel exceeds a " speed–length ratio " (speed in knots divided by square root of length in feet) of 0.94, it starts to outrun most of its bow wave , the hull actually settles slightly in the water as it is now only supported by two wave peaks. As the vessel exceeds a speed-length ratio of 1.34, the wavelength is now longer than the hull, and the stern is no longer supported by the wake, causing the stern to squat, and the bow to rise. The hull is now starting to climb its own bow wave, and resistance begins to increase at a very high rate. While it is possible to drive a displacement hull faster than a speed-length ratio of 1.34, it is prohibitively expensive to do so. Most large vessels operate at speed-length ratios well below that level, at speed-length ratios of under 1.0.
Since wave-making resistance is based on the energy required to push the water out of the way of the hull, there are a number of ways that this can be minimized.
Reducing the displacement of the craft, by eliminating excess weight, is the most straightforward way to reduce the wave making drag. Another way is to shape the hull so as to generate lift as it moves through the water. Semi-displacement hulls and planing hulls do this, and they are able to break through the hull speed barrier and transition into a realm where drag increases at a much lower rate. The disadvantage of this is that planing is only practical on smaller vessels, with high power-to-weight ratios, such as motorboats. It is not a practical solution for a large vessel such as a supertanker .
A hull with a blunt bow has to push the water away very quickly to pass through, and this high acceleration requires large amounts of energy. By using a fine bow, with a sharper angle that pushes the water out of the way more gradually, the amount of energy required to displace the water will be less. A modern variation is the wave-piercing design. The total amount of water to be displaced by a moving hull, and thus causing wave making drag, is the cross sectional area of the hull times distance the hull travels, and will not remain the same when prismatic coefficient is increased for the same lwl and same displacement and same speed.
A special type of bow, called a bulbous bow , is often used on large power vessels to reduce wave-making drag. The bulb alters the waves generated by the hull, by changing the pressure distribution ahead of the bow. Because of the nature of its destructive interference with the bow wave, there is a limited range of vessel speeds over which it is effective. A bulbous bow must be properly designed to mitigate the wave-making resistance of a particular hull over a particular range of speeds. A bulb that works for one vessel's hull shape and one range of speeds could be detrimental to a different hull shape or a different speed range. Proper design and knowledge of a ship's intended operating speeds and conditions is therefore necessary when designing a bulbous bow.
If the hull is designed to operate at speeds substantially lower than hull speed then it is possible to refine the hull shape along its length to reduce wave resistance at one speed. This is practical only where the block coefficient of the hull is not a significant issue.
Since semi-displacement and planing hulls generate a significant amount of lift in operation, they are capable of breaking the barrier of the wave propagation speed and operating in realms of much lower drag, but to do this they must be capable of first pushing past that speed, which requires significant power. This stage is called the transition stage and at this stage the rate of wave-making resistance is the highest. Once the hull gets over the hump of the bow wave, the rate of increase of the wave drag will start to reduce significantly. [ 1 ] The planing hull will rise up clearing its stern off the water and its trim will be high. Underwater part of the planing hull will be small during the planing regime. [ 2 ]
A qualitative interpretation of the wave resistance plot is that a displacement hull resonates with a wave that has a crest near its bow and a trough near its stern, because the water is pushed away at the bow and pulled back at the stern. A planing hull simply pushed down on the water under it, so it resonates with a wave that has a trough under it. If it has about twice the length it will therefore have only square root (2) or 1.4 times the speed. In practice most planing hulls usually move much faster than that. At four times hull speed the wavelength is already 16 times longer than the hull.
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In mathematical analysis , more precisely in microlocal analysis , the wave front (set) WF( f ) characterizes the singularities of a generalized function f , not only in space , but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970.
In more familiar terms, WF( f ) tells not only where the function f is singular (which is already described by its singular support ), but also how or why it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimensions at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is microlocal smoothness .
Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction tangent to the curve, the function remains smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction v , one can try to smooth the function out by averaging in directions perpendicular to v . If the resulting function is smooth, then we regard ƒ to be smooth in the direction of v . Otherwise, v is in the wavefront set.
Formally, in Euclidean space , the wave front set of ƒ is defined as the complement of the set of all pairs ( x 0 , v ) such that there exists a test function ϕ ∈ C c ∞ {\displaystyle \phi \in C_{c}^{\infty }} with ϕ {\displaystyle \phi } ( x 0 ) ≠ 0 and an open cone Γ containing v such that the estimate
holds for all positive integers N . Here ( ϕ f ) ∧ {\displaystyle (\phi f)^{\wedge }} denotes the Fourier transform. Observe that the wavefront set is conical in the sense that if ( x , v ) ∈ Wf(ƒ), then ( x ,λ v ) ∈ Wf(ƒ) for all λ > 0. In the example discussed in the previous paragraph, the wavefront set is the set-theoretic complement of the image of the tangent bundle of the curve inside the tangent bundle of the plane.
Because the definition involves cutoff by a compactly supported function, the notion of a wave front set can be transported to any differentiable manifold X . In this more general situation, the wave front set is a closed conical subset of the cotangent bundle T * ( X ), since the ξ variable naturally localizes to a covector rather than a vector. The wave front set is defined such that its projection on X is equal to the singular support of the function.
In Euclidean space, the wave front set of a distribution ƒ is defined as
where Σ x ( f ) {\displaystyle \Sigma _{x}(f)} is the singular fibre of ƒ at x . The singular fibre is defined to be the complement of all directions ξ {\displaystyle \xi } such that the Fourier transform of f , localized at x , is sufficiently regular when restricted to an open cone containing ξ {\displaystyle \xi } . More precisely, a direction v is in the complement of Σ x ( f ) {\displaystyle \Sigma _{x}(f)} if there is a compactly supported smooth function φ with φ( x ) ≠ 0 and an open cone Γ containing v such that the following estimate holds for each positive integer N :
Once such an estimate holds for a particular cutoff function φ at x , it also holds for all cutoff functions with smaller support, possibly for a different open cone containing v .
On a differentiable manifold M , using local coordinates x , ξ {\displaystyle x,\xi } on the cotangent bundle , the wave front set WF( f )
of a distribution ƒ can be defined in the following general way:
where the singular fibre Σ x ( f ) {\displaystyle \Sigma _{x}(f)} is again the complement of all directions ξ {\displaystyle \xi } such that the Fourier transform of f , localized at x , is sufficiently regular when restricted to a conical neighbourhood of ξ {\displaystyle \xi } . The problem of regularity is local, so it can be checked in the local coordinate system, using the Fourier transform on the x variables. The required regularity estimate transforms well under diffeomorphism , and so the notion of regularity is independent of the choice of local coordinates.
The notion of a wave front set can be adapted to accommodate other notions of regularity of a function. Localized can here be expressed by saying that f is truncated by some smooth cutoff function not vanishing at x . (The localization process could be done in a more elegant fashion, using germs .)
More concretely, this can be expressed as
where
Typically, sections of O are required to satisfy some growth (or decrease) condition at infinity, e.g. such that ( 1 + | ξ | ) s v ( ξ ) {\displaystyle (1+|\xi |)^{s}v(\xi )} belong to some L p space .
This definition makes sense, because the Fourier transform becomes more
regular (in terms of growth at infinity) when f is truncated with the smooth cutoff ϕ {\displaystyle \phi } .
The most difficult "problem", from a theoretical point of view,
is finding the adequate sheaf O characterizing functions belonging to a given subsheaf E of the space G of generalized functions.
If we take G = D ′ the space of Schwartz distributions and want to characterize distributions which are locally C ∞ {\displaystyle C^{\infty }} functions, we must take for O (Ω) the classical function spaces called O ′ M (Ω) in the literature.
Then the projection on the first component of a distribution's wave front set is nothing else than its classical singular support , i.e. the complement of the set on which its restriction would be a smooth function .
The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators . The propagation of singularities theorem characterizes the wave front set.
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In mathematical physics, the wave maps equation is a geometric wave equation that solves
where D {\displaystyle D} is a connection . [ 1 ] [ 2 ]
It can be considered a natural extension of the wave equation for Riemannian manifolds . [ 3 ]
This applied mathematics –related article is a stub . You can help Wikipedia by expanding it .
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In fluid dynamics , the wave method (WM) , or wave characteristic method (WCM) , is a model describing unsteady flow of fluids in conduits (pipes).
The wave method is based on the physically accurate concept that transient pipe flow occurs as a result of pressure waves generated and propagated from a disturbance in the pipe system (valve closure, pump trip, etc.) This method was developed and first described by Don J. Wood in 1966. [ 1 ] A pressure wave, which represents a rapid pressure and associated flow change, travels at sonic velocity for the liquid pipe medium, and the wave is partially transmitted and reflected at all discontinuities in the pipe system (pipe junctions, pumps, open or closed ends, surge tanks, etc.) A pressure wave can also be modified by pipe wall resistance. This description is one that closely represents the actual mechanism of transient pipe flow. [ 1 ]
The WM has the very significant advantage [ according to whom? ] that computations need be made only at nodes in the piping system. Other techniques such as the method of characteristics (MOC) require calculations at equally spaced interior points in a pipeline. This requirement can easily increase the number of calculations by a factor of 10 or more. However, virtually identical solutions are obtained by the WM and the MOC. [ 2 ]
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Wave overtopping is the time-averaged amount of water that is discharged (in liters per second) per structure length (in meters) by waves over a structure such as a breakwater , revetment or dike which has a crest height above still water level.
When waves break over a dike, it causes water to flow onto the land behind it. Excessive overtopping is undesirable because it can compromise the integrity of the structure or result in a safety hazard, particularly when the structure is in an area where people, infrastructure or vehicles are present, such as in the case of a dike fronting an esplanade or densely populated area.
Wave overtopping typically transpires during extreme weather events, such as intense storms, which often elevate water levels beyond average due to wind setup . These effects may be further intensified when the storm coincides with a high spring tide .
Excessive overtopping may cause damage to the inner slope of the dike, potentially leading to failure and inundation of the land behind the dike, or create water-related issues on the inside of the dike due to excess water pressure and inadequate drainage . The process is highly stochastic , and the amount of overtopping depends on factors including the freeboard, wave height , wave period , the geometry of the structure, and slope of the dike. [ 1 ]
Overtopping can transpire through various combinations of water levels and wave heights, wherein a low water level accompanied by high waves may yield an equivalent overtopping outcome to that of a higher water level with lower waves. This phenomenon is inconsequential when water levels and wave heights exhibit correlation; however, it poses difficulties in river systems where these factors are uncorrelated. In such instances, a probabilistic calculation is necessary.
The freeboard is the height of the dike's crest above the still water level, which usually corresponds to the determining storm surge level or river water level. Overtopping is typically expressed in litres per second per metre of dike length (L/s/m), as an average value. Overtopping follows the cyclical nature of waves, resulting in a large amount of water flowing over a structure, followed by a period with no water. The official website of the EurOtop Manual , which is widely used in the design of coastal engineering structures, features a number of visualisations of wave overtopping. [ 2 ] [ 3 ] [ 4 ] [ 5 ]
In the case of overtopping at rubble-mound breakwaters, recent research using numerical models indicates that overtopping is strongly dependent on the slope angle. [ 6 ] Since present design guidelines for non-breaking waves do not include the effect of the slope angle, modified guidelines have also been proposed. Whilst these observed slope effects are too large to be ignored, they still need to be verified by tests using physical models . [ 6 ] [ 7 ]
Overtopping behaviour is also influenced by the geometry and layout of different coastal structures. For example, seawalls (which are typically vertical, or near-vertical, as opposed to sloping breakwaters or revetments), are often situated behind natural beaches . Scour at the base of these structures during storms can have a direct impact on wave energy dissipation along their frontage, thus influencing wave overtopping. This phenomenon assumes critical importance when storms occur in such quick succession that the beach doesn't have sufficient time for sediments removed by the storm to be re-established. Experimental results show that, for near-vertical structures at the back of a beach, there is an increase in wave overtopping volume for a storm that starts from an eroded beach configuration, rather than a simple slope. [ 8 ]
Wave overtopping predominantly depends on the respective heights of individual waves compared to the crest level of the coastal structure involved. This overtopping doesn't occur continuously; rather, it's a sporadic event that takes place when particularly high waves within a storm impact the structure. [ 3 ] [ 11 ]
The extent of wave overtopping is quantified by the volume of water that overflows onto the adjacent land. This can be measured either as the volume of water per wave for each unit length of the seawall, or as the average rate of overtopped water volume per unit length during the storm wave period. [ 11 ]
Much research into overtopping has been carried out, ranging from laboratory experiments to full-scale testing and the use of simulators. [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ] In 1971, Jurjen Battjes developed a theoretically accurate equation for determining the average overtopping. [ 19 ] [ 20 ] However, the formula's complexity, involving error functions , has limited its widespread adoption in practical applications. Consequently, an alternative empirical relationship has been established:
in which Q {\displaystyle Q} is the dimensionless overtopping, and R {\displaystyle R} is the dimensionless freeboard:
Q = q g H s 2 h / L 0 tan α {\displaystyle Q={\frac {q}{\sqrt {gH_{s}^{2}}}}{\sqrt {\frac {h/L_{0}}{\tan \alpha }}}}
in which:
The values of a {\displaystyle a} and b {\displaystyle b} depend on the type of breaking wave , as shown in the table below:
The resistance term γ {\displaystyle \gamma } has a value between approximately 0.5 (for two layers of loosely dumped armourstone ) and 1.0 (for a smooth slope). The effect of a berm and obliquely incident waves is also taken into account through the resistance term. This is determined in the same way as when calculating wave run-up. Special revetment blocks that reduce wave run-up (e.g., Hillblock, Quattroblock) also reduce wave overtopping. [ 21 ] [ 22 ] Since the governing overtopping is the boundary condition, this means that the use of such elements allows for a slightly lower flood barrier. [ 23 ]
Research for the EurOtop manual has provided much additional data, and based on this, the formula has been slightly modified to:
with a maximum of:
It turns out that this formula is also a perfect rational approximation of the original Battjes formula.
In certain applications, it may also be necessary to calculate individual overtopping quantities, i.e. the overtopping per wave. The volumes of individual overtopping waves are Weibull distributed . The overtopping volume per wave for a given probability of exceedance is given by:
in which P v {\displaystyle P_{v}} is the probability of exceedance of the calculated volume, P o v {\displaystyle P_{ov}} is the probability of overtopping waves, and h c {\displaystyle h_{c}} is the crest height. [ 24 ] [ 25 ]
In terms of revetments, the overtopping discussed in the EurOtop manual refers to the overtopping measured at the seaward edge of the revetment crest. [ 4 ] The formulas above describe the wave overtopping occurring at the sea-side edge of the crest. In scenarios where the crest is impermeable (for example, a road surface or a clay layer), the volume of water overtopping the inland side of the crest would roughly equal that on the seaside. However, in the case of a rock armour breakwater with a more permeable crest, a large part of the overtopping water will seep into the crest, thus providing less overtopping on the inside of it. To analyse this effect, reduction coefficient γ {\displaystyle \gamma } can be used. This factor can be multiplied by 0.5 for a standard crest, with a width of about three rocks. This can result in a significant reduction in overtopping, and thus in the required crest height. If, behind the crest at a lower level, a permeable rock armour layer is installed with width x {\displaystyle x} , the amount of overtopping on the landside of this layer decreases still further. In that case, the reduction term γ {\displaystyle \gamma } (not to be confused with the reduction co-efficient γ b {\displaystyle \gamma _{b}} ) can be multiplied by − 0.142 x B + 0.577 {\displaystyle -0.142{\frac {x}{B}}+0.577} , in which B {\displaystyle B} is the crest width. [ 26 ] [ 27 ] [ 28 ]
The circumstances surrounding overtopping at berm-type breakwaters differ slightly from those of dikes. Minor wave overtopping may occur as splashes from waves striking individual rocks. However, significant overtopping typically results in a horizontal flow across the crest, similar to what happens with dikes. The primary distinction lies in the wave heights used for designing these structures. Dikes rarely face wave heights exceeding 3 metres, while berm breakwaters are often designed to withstand wave heights of around 5 metres. This difference impacts the overtopping behaviour when dealing with smaller overtopping discharges. [ 29 ]
An understanding wave overtopping involves a combination of empirical data , physical modelling , and numerical simulations to predict and mitigate its impacts on coastal structures and safety. [ 3 ] Traditionally, permissible average overtopping discharge has been utilised as a standard for designing coastal structures. It is necessary to restrict the average overtopping discharge to guarantee both the structural integrity of the structure, as well as the protection of individuals, vehicles, and properties situated behind it. Design handbooks often stipulate the thresholds for the maximum individual overtopping volumes, necessitating the examination of wave overtopping on a wave-per-wave basis. Often, to ensure a more dependable level of safety for pedestrians and vehicles, or to evaluate the stability of the inner slope of a revetment, it is necessary to consider the peak velocity and thickness of the overtopping flow. [ 30 ]
The tolerable overtopping is the overtopping which the design accepts may occur during a design storm condition. It is dependent on a number of factors including the intended use of the dike or coastal structure, and the quality of the revetment. Tolerable overtopping volumes are site-specific and depend on various factors, including the size and usage of the receiving area, the dimensions and capacity of drainage ditches, damage versus inundation curves, and return period. For coastal defences safeguarding the lives and well-being of residents, workers, and recreational users, designers and overseeing authorities must also address the direct hazards posed by overtopping. This necessitates evaluating the level of hazard and its likelihood of occurrence, thereby enabling the development of suitable action plans to mitigate risks associated with overtopping events. [ 4 ]
For rubble mound breakwaters (e.g., in harbour breakwaters) and a significant wave height H m 0 {\displaystyle H_{m0}} greater than 5m on the outside, a heavy rubble mound revetment on the inside is required for overtopping of 10-30 L/s per metre. For overtopping of 5-20 L/s per metre, there is a high risk of damage to the crest.
For regular grass, an average overtopping of 5 L/s per metre of dike is considered permissible. For very good grass cover, without special elements or street furniture such as stairs, sign poles, or fences, 10 L/s per metre is allowed. Overtopping tests with a wave overtopping simulator have shown that for an undamaged grass cover, without special elements, 50L/s per metre often causes no damage. The problem is not so much the strength of the grass cover, but the presence of other elements such as gates, stairs and fences. It should be considered that, for example, 5 L/s per metre can occur due to high waves and a high freeboard, or low waves with a low freeboard. In the first case, there are not many overtopping waves, but when one overtops, it creates a high flow velocity on the inner slope. In the second case, there are many overtopping waves, but they create relatively low flow velocities. As a result, the requirements for overtopping over river dikes are different from those for sea dikes. [ 26 ]
A good sea dike with a continuous grass cover can easily handle 10 L/s per metre without problems, assuming good drainage is provided at the foot of the inner slope. Without adequate drainage, the amount of water that could potentially enter properties at the foot of the inner slope would be unacceptable, which is why such dikes are designed for a lower overtopping amount. Since it has been found that a grass cover does not fail due to the average overtopping, but rather due to the frequent occurrence of high flow velocities, coastal authorities such as Rijkswaterstaat in the Netherlands have decided (since 2015) to no longer test grass slopes on the inner side of the dike for average overtopping discharge, but rather for the frequency of high flow velocities during overtopping. [ 31 ] [ 32 ]
Research has shown that grass roots can contribute to improving the shear strength of soil used in dike construction, providing that the grass is properly maintained. [ 33 ] Developing a grass cover takes time and requires a suitable substrate, such as lean and reasonably compacted clay . Firmly compacted clay soil is initially unsuitable for colonisation by grass plants. However, after a frost or winter period, the top layer of such a compacted clay layer is sufficiently open for the establishment of grass. To function properly, grass cover formation must begin well before winter. [ 34 ]
Research in The Netherlands has found that dikes with a well-compacted and flat clay lining can withstand a limited wave height or limited wave overtopping, such as in the majority of river areas, during the first winter after construction even without a grass cover, for many days without significant damage. If the wave load in the river area is higher, no damage that threatens safety will occur if the clay lining is thick enough (0.8 metres or more) and adequately compacted throughout its entire thickness. An immature grass cover can be temporarily protected against hydraulic loads with stapled geotextile mats. [ 35 ]
For damage to ships in harbours or marinas, the following figures can be used:
These values provide guidance on the expected impact of overtopping on ships in marinas or harbours, on nearby buildings and other infrastructure, depending on the significant wave height H m 0 {\displaystyle H{m0}} and overtopping rate (in L/s per metre). This information then helps to inform the appropriate design, the required protection measures, and response plans for different scenarios. [ 24 ]
When there is water on both sides of a barrier (such as in the case of a harbour dam, breakwater or closure dam), wave overtopping over the dam will also generate waves on the other side of the dam. This is called wave transmission. To determine the amount of wave transmission, it is not necessary to determine the amount of overtopping. The transmission depends only on the wave height on the outer side, the freeboard, and the roughness of the slope. For a smooth slope, the transmission coefficient (the relationship between the wave on the inside of the dam and the incoming wave) is:
In which ξ 0p is the Iribarren number based on the peak period of the waves, and β is the angle of incidence of the waves. [ 36 ] [ 24 ]
In order to assess the safety and resilience of dikes, as well as the robustness of the grass lining on their crests and landward slopes, a wave overtopping simulator can be employed. The most onerous wave conditions for which a dike is designed occur relatively rarely, so using a wave overtopping simulator enables in-situ replication of anticipated conditions on the dike itself. This allows the responsible organisation overseeing the structure to evaluate its capacity to withstand predicted wave overtopping during specific extreme scenarios. [ 37 ]
During these tests, the wave overtopping simulator is positioned on the dike's crest and continuously filled with water. The device features valves at its base that can be opened to release varying volumes of water, thereby simulating a wide range of wave overtopping events. This approach helps ensure that the dike's integrity is accurately and effectively assessed. [ 38 ]
In the case of dikes with grass slopes, another test method is to use a sod puller to determine the tensile strength of the sod, which can then be translated into strength under the load caused by wave overtopping. In addition to simulating wave overtopping, the simulation of wave impacts and wave run-up is possible with a specially developed generator and simulator. [ 39 ] [ 40 ] [ 24 ]
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https://en.wikipedia.org/wiki/Wave_overtopping
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In physics , a wave packet (also known as a wave train or wave group ) is a short burst of localized wave action that travels as a unit, outlined by an envelope . A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of component sinusoidal waves of different wavenumbers , with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. [ 1 ] Any signal of a limited width in time or space requires many frequency components around a center frequency within a bandwidth inversely proportional to that width; even a gaussian function is considered a wave packet because its Fourier transform is a "packet" of waves of frequencies clustered around a central frequency. [ 2 ] Each component wave function , and hence the wave packet, are solutions of a wave equation . Depending on the wave equation, the wave packet's profile may remain constant (no dispersion ) or it may change ( dispersion ) while propagating.
Ideas related to wave packets – modulation , carrier waves , phase velocity , and group velocity – date from the mid-1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877. [ 3 ]
Erwin Schrödinger introduced the idea of wave packets just after publishing his famous wave equation . [ 4 ] He solved his wave equation for a quantum harmonic oscillator , introduced the superposition principle , and used it to show that a compact state could persist. While this work did result in the important concept of coherent states , the wave packet concept did not endure. The year after Schrödinger's paper, Werner Heisenberg published his paper on the uncertainty principle , showing in the process, that Schrödinger's results only applied to quantum harmonic oscillators , not for example to Coulomb potential characteristic of atoms. [ 4 ] : 829
The following year, 1927, Charles Galton Darwin explored Schrödinger's equation for an unbound electron in free space, assuming an initial Gaussian wave packet . [ 5 ] Darwin showed that at time t {\displaystyle t} later the position x {\displaystyle x} of the packet traveling at velocity v {\displaystyle v} would be
x 0 + v t ± σ 2 + ( h t / 2 π σ m ) 2 {\displaystyle x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}}
where σ {\displaystyle \sigma } is the uncertainty in the initial position.
Later in 1927 Paul Ehrenfest showed that the time, T {\displaystyle T} for a matter wave packet of width Δ x {\displaystyle \Delta x} and mass m {\displaystyle m} to spread by a factor of 2 was T ≈ m Δ x 2 / ℏ {\textstyle T\approx m{\Delta x}^{2}/\hbar } . Since ℏ {\displaystyle \hbar } is so small, wave packets on the scale of macroscopic objects, with large width and mass, double only at cosmic time scales. [ 6 ] : 49
Quantum mechanics describes the nature of atomic and subatomic systems using Schrödinger's wave equation . The classical limit of quantum mechanics and many formulations of quantum scattering use wave packets formed from various solutions to this equation.
Quantum wave packet profiles change while propagating; they show dispersion. Physicists have concluded that "wave packets would not do as representations of subatomic particles". [ 4 ] : 829
Schrodinger developed wave packets in hopes of interpreting quantum wave solutions as locally compact wave groups. [ 4 ] Such packets tradeoff position localization for spreading momentum. In the coordinate representation of the wave (such as the Cartesian coordinate system ), the position of the particle's localized probability is specified by the position of the packet solution. The narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the momentum of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle.
One kind of optimal tradeoff minimizes the product of position uncertainty Δ x {\displaystyle \Delta x} and momentum uncertainty Δ p x {\displaystyle \Delta p_{x}} . [ 7 ] : 60 If we place such a packet at rest it stays at rest: the average value of the position and momentum match a classical particle. However it spreads out in all directions with a velocity given by the optimal momentum uncertainty Δ p x {\displaystyle \Delta p_{x}} . The spread is so fast that in the distance of once around an atom the wave packet is unrecognizable.
Particle interactions are called scattering in physics; wave packet mathematics play an important role in quantum scattering approaches . A monochromatic (single momentum) source produces convergence difficulties in the scattering models. [ 8 ] : 150 Scattering problems also have classical limits. Whenever the scattering target (for example an atom) has a size much smaller than wave packet, the center of the wave packet follows scattering classical trajectories. In other cases, the wave packet distorts and scatters as it interacts with the target. [ 9 ] : 295
Without dispersion the wave packet maintains its shape as it propagates.
As an example of propagation without dispersion , consider wave solutions to the following wave equation from classical physics ∂ 2 u ∂ t 2 = c 2 ∇ 2 u , {\displaystyle {\partial ^{2}u \over \partial t^{2}}=c^{2}\,\nabla ^{2}u,}
where c is the speed of the wave's propagation in a given medium.
Using the physics time convention, e − iωt , the wave equation has plane-wave solutions u ( x , t ) = e i ( k ⋅ x − ω ( k ) t ) , {\displaystyle u(\mathbf {x} ,t)=e^{i{(\mathbf {k\cdot x} }-\omega (\mathbf {k} )t)},}
where the relation between the angular frequency ω and angular wave vector k is given by the dispersion relation : ω ( k ) = ± | k | c = ± 2 π c λ , {\displaystyle \omega (\mathbf {k} )=\pm |\mathbf {k} |c=\pm {\frac {2\pi c}{\lambda }},} such that ω 2 / | k | 2 = c 2 {\displaystyle \omega ^{2}/|\mathbf {k} |^{2}=c^{2}} . This relation should be valid so that the plane wave is a solution to the wave equation. As the relation is linear , the wave equation is said to be non-dispersive .
To simplify, consider the one-dimensional wave equation with ω(k) = ±kc . Then the general solution is u ( x , t ) = A e i k ( x − c t ) + B e i k ( x + c t ) , {\displaystyle u(x,t)=Ae^{ik(x-ct)}+Be^{ik(x+ct)},} where the first and second term represent a wave propagating in the positive respectively negative x -direction .
A wave packet is a localized disturbance that results from the sum of many different wave forms . If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. [ 10 ] From the basic one-dimensional plane-wave solutions, a general form of a wave packet can be expressed as u ( x , t ) = 1 2 π ∫ − ∞ ∞ A ( k ) e i ( k x − ω ( k ) t ) d k . {\displaystyle u(x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }A(k)~e^{i(kx-\omega (k)t)}\,dk.} where the amplitude A ( k ) , containing the coefficients of the wave superposition , follows from taking the inverse Fourier transform of a " sufficiently nice "
initial wave u ( x , t ) evaluated at t = 0 : A ( k ) = 1 2 π ∫ − ∞ ∞ u ( x , 0 ) e − i k x d x . {\displaystyle A(k)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\,\infty }u(x,0)~e^{-ikx}\,dx.} and 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}} comes from Fourier transform conventions .
For example, choosing u ( x , 0 ) = e − x 2 + i k 0 x , {\displaystyle u(x,0)=e^{-x^{2}+ik_{0}x},}
we obtain A ( k ) = 1 2 e − ( k − k 0 ) 2 4 , {\displaystyle A(k)={\frac {1}{\sqrt {2}}}e^{-{\frac {(k-k_{0})^{2}}{4}}},}
and finally u ( x , t ) = e − ( x − c t ) 2 + i k 0 ( x − c t ) = e − ( x − c t ) 2 [ cos ( 2 π x − c t λ ) + i sin ( 2 π x − c t λ ) ] . {\displaystyle {\begin{aligned}u(x,t)&=e^{-(x-ct)^{2}+ik_{0}(x-ct)}\\&=e^{-(x-ct)^{2}}\left[\cos \left(2\pi {\frac {x-ct}{\lambda }}\right)+i\sin \left(2\pi {\frac {x-ct}{\lambda }}\right)\right].\end{aligned}}}
The nondispersive propagation of the real or imaginary part of this wave packet is presented in the above animation.
By contrast, in the case of dispersion, a wave changes shape during propagation. For example, the free Schrödinger equation , i ℏ ∂ ψ ∂ t = − ℏ 2 2 m ∇ 2 ψ , {\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi ,} has plane-wave solutions of the form: ψ ( r , t ) = A e i [ k ⋅ r − ω ( k ) t ] , {\displaystyle \psi (\mathbf {r} ,t)=Ae^{i{[\mathbf {k\cdot r} }-\omega (\mathbf {k} )t]},} where A {\displaystyle A} is a constant and the dispersion relation satisfies [ 11 ] [ 12 ] ω ( k ) = ℏ k 2 2 m = ℏ 2 m ( k x 2 + k y 2 + k z 2 ) , {\displaystyle \omega (\mathbf {k} )={\frac {\hbar \mathbf {k} ^{2}}{2m}}={\frac {\hbar }{2m}}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}),} with the subscripts denoting unit vector notation . As the dispersion relation is non-linear, the free Schrödinger equation is dispersive .
In this case, the wave packet is given by: ψ ( r , t ) = 1 ( 2 π ) 3 / 2 ∫ g ( k ) e i [ k ⋅ r − ω ( k ) t ] d 3 k {\displaystyle \psi (\mathbf {r} ,t)={\frac {1}{(2\pi )^{3/2}}}\int g(\mathbf {k} )e^{i{[\mathbf {k\cdot r} }-\omega (\mathbf {k} )t]}d^{3}k} where once again g ( k ) {\displaystyle g(\mathbf {k} )} is simply the Fourier transform of ψ ( k , 0 ) {\displaystyle \psi (\mathbf {k} ,0)} . If ψ ( k , 0 ) {\displaystyle \psi (\mathbf {k} ,0)} (and therefore g ( k ) {\displaystyle g(\mathbf {k} )} ) is a Gaussian function , the wave packet is called a Gaussian wave packet . [ 13 ]
For example, the solution to the one-dimensional free Schrödinger equation (with 2Δ x , m , and ħ set equal to one) satisfying the initial condition ψ ( x , 0 ) = 2 / π 4 exp ( − x 2 + i k 0 x ) , {\displaystyle \psi (x,0)={\sqrt[{4}]{2/\pi }}\exp \left({-x^{2}+ik_{0}x}\right),} representing a wave packet localized in space at the origin as a Gaussian function, is seen to be ψ ( x , t ) = 2 / π 4 1 + 2 i t e − 1 4 k 0 2 e − 1 1 + 2 i t ( x − i k 0 2 ) 2 = 2 / π 4 1 + 2 i t e − 1 1 + 4 t 2 ( x − k 0 t ) 2 e i 1 1 + 4 t 2 ( ( k 0 + 2 t x ) x − 1 2 t k 0 2 ) . {\displaystyle {\begin{aligned}\psi (x,t)&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{4}}k_{0}^{2}}~e^{-{\frac {1}{1+2it}}\left(x-{\frac {ik_{0}}{2}}\right)^{2}}\\&={\frac {\sqrt[{4}]{2/\pi }}{\sqrt {1+2it}}}e^{-{\frac {1}{1+4t^{2}}}(x-k_{0}t)^{2}}~e^{i{\frac {1}{1+4t^{2}}}\left((k_{0}+2tx)x-{\frac {1}{2}}tk_{0}^{2}\right)}~.\end{aligned}}}
An impression of the dispersive behavior of this wave packet is obtained by looking at the probability density: | ψ ( x , t ) | 2 = 2 / π 1 + 4 t 2 e − 2 ( x − k 0 t ) 2 1 + 4 t 2 . {\displaystyle |\psi (x,t)|^{2}={\frac {\sqrt {2/\pi }}{\sqrt {1+4t^{2}}}}~e^{-{\frac {2(x-k_{0}t)^{2}}{1+4t^{2}}}}~.} It is evident that this dispersive wave packet, while moving with constant group velocity k o , is delocalizing rapidly: it has a width increasing with time as √ 1 + 4 t 2 → 2 t , so eventually it diffuses to an unlimited region of space.
The above dispersive Gaussian wave packet, unnormalized and just centered at the origin, instead, at t =0, can now be written in 3D, now in standard units: [ 14 ] [ 15 ] ψ ( r , 0 ) = e − r ⋅ r / 2 a , {\displaystyle \psi (\mathbf {r} ,0)=e^{-\mathbf {r} \cdot \mathbf {r} /2a},} The Fourier transform is also a Gaussian in terms of the wavenumber, the k -vector, ψ ( k , 0 ) = ( 2 π a ) 3 / 2 e − a k ⋅ k / 2 . {\displaystyle \psi (\mathbf {k} ,0)=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}.} With a and its inverse adhering to the uncertainty relation Δ x Δ p x = ℏ / 2 , {\displaystyle \Delta x\Delta p_{x}=\hbar /2,} such that a = 2 ⟨ r ⋅ r ⟩ / 3 ⟨ 1 ⟩ = 2 ( Δ x ) 2 , {\displaystyle a=2\langle \mathbf {r} \cdot \mathbf {r} \rangle /3\langle 1\rangle =2(\Delta x)^{2},} can be considered the square of the width of the wave packet , whereas its inverse can be written as 1 / a = 2 ⟨ k ⋅ k ⟩ / 3 ⟨ 1 ⟩ = 2 ( Δ p x / ℏ ) 2 . {\displaystyle 1/a=2\langle \mathbf {k} \cdot \mathbf {k} \rangle /3\langle 1\rangle =2(\Delta p_{x}/\hbar )^{2}.}
Each separate wave only phase-rotates in time, so that the time dependent Fourier-transformed solution is
Ψ ( k , t ) = ( 2 π a ) 3 / 2 e − a k ⋅ k / 2 e − i E t / ℏ = ( 2 π a ) 3 / 2 e − a k ⋅ k / 2 − i ( ℏ 2 k ⋅ k / 2 m ) t / ℏ = ( 2 π a ) 3 / 2 e − ( a + i ℏ t / m ) k ⋅ k / 2 . {\displaystyle {\begin{aligned}\Psi (\mathbf {k} ,t)&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2}e^{-iEt/\hbar }\\&=(2\pi a)^{3/2}e^{-a\mathbf {k} \cdot \mathbf {k} /2-i(\hbar ^{2}\mathbf {k} \cdot \mathbf {k} /2m)t/\hbar }\\&=(2\pi a)^{3/2}e^{-(a+i\hbar t/m)\mathbf {k} \cdot \mathbf {k} /2}.\end{aligned}}}
The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.
Ψ ( r , t ) = ( a a + i ℏ t / m ) 3 / 2 e − r ⋅ r 2 ( a + i ℏ t / m ) . {\displaystyle \Psi (\mathbf {r} ,t)=\left({a \over a+i\hbar t/m}\right)^{3/2}e^{-{\mathbf {r} \cdot \mathbf {r} \over 2(a+i\hbar t/m)}}.}
The integral of Ψ over all space is invariant, because it is the inner product of Ψ with the state of zero energy, which is a wave with infinite wavelength, a constant function of space. For any energy eigenstate η ( x ) , the inner product, ⟨ η | ψ ⟩ = ∫ η ( r ) ψ ( r ) d 3 r , {\displaystyle \langle \eta |\psi \rangle =\int \eta (\mathbf {r} )\psi (\mathbf {r} )d^{3}\mathbf {r} ,} only changes in time in a simple way: its phase rotates with a frequency determined by the energy of η . When η has zero energy, like the infinite wavelength wave, it doesn't change at all.
For a given t {\displaystyle t} , the phase of the wave function varies with position as ℏ t / m 2 ( a 2 + ( ℏ t / m ) 2 ) ‖ r ‖ 2 {\displaystyle {\frac {\hbar t/m}{2(a^{2}+(\hbar t/m)^{2})}}\|\mathbf {r} \|^{2}} . It varies quadratically with position, which means that it is different from multiplication by a linear phase factor e i k ⋅ r {\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }} as is the case of imparting a constant momentum to the wave packet. In general, the phase of a gaussian wave packet has both a linear term and a quadratic term. The coefficient of the quadratic term begins by increasing from − ∞ {\displaystyle -\infty } towards 0 {\displaystyle 0} as the gaussian wave packet becomes sharper, then at the moment of maximum sharpness, the phase of the wave function varies linearly with position. Then the coefficient of the quadratic term increases from 0 {\displaystyle 0} towards + ∞ {\displaystyle +\infty } , as the gaussian wave packet spreads out again.
The integral ∫ |Ψ| 2 d 3 r is also invariant, which is a statement of the conservation of probability. [ 16 ] Explicitly, P ( r ) = | Ψ | 2 = Ψ ∗ Ψ = ( a a 2 + ( ℏ t / m ) 2 ) 3 e − a r ⋅ r a 2 + ( ℏ t / m ) 2 , {\displaystyle P(r)=|\Psi |^{2}=\Psi ^{*}\Psi =\left({a \over {\sqrt {a^{2}+(\hbar t/m)^{2}}}}\right)^{3}~e^{-{a\,\mathbf {r} \cdot \mathbf {r} \over a^{2}+(\hbar t/m)^{2}}},} where r is the distance from the origin, the speed of the particle is zero, and width given by a 2 + ( ℏ t / m ) 2 a , {\displaystyle {\sqrt {a^{2}+(\hbar t/m)^{2} \over a}},} which is √ a at (arbitrarily chosen) time t = 0 while eventually growing linearly in time, as ħt /( m √ a ) , indicating wave-packet spreading . [ 17 ]
For example, if an electron wave packet is initially localized in a region of atomic dimensions (i.e., 10 −10 m) then the width of the packet doubles in about 10 −16 s. Clearly, particle wave packets spread out very rapidly indeed (in free space): [ 18 ] For instance, after 1 ms, the width will have grown to about a kilometer.
This linear growth is a reflection of the (time-invariant) momentum uncertainty: the wave packet is confined to a narrow Δ x = √ a /2 , and so has a momentum which is uncertain (according to the uncertainty principle) by the amount ħ / √ 2 a , a spread in velocity of ħ/m √ 2 a , and thus in the future position by ħt /m √ 2 a . The uncertainty relation is then a strict inequality, very far from saturation, indeed! The initial uncertainty Δ x Δ p = ħ /2 has now increased by a factor of ħt/ma (for large t ).
A gaussian 2D quantum wave function:
ψ ( x , y , t ) = ψ ( x , t ) ψ ( y , t ) {\displaystyle \psi (x,y,t)=\psi (x,t)\psi (y,t)}
ψ ( x , t ) = ( 2 a 2 π ) 1 / 4 e i ϕ ( a 4 + 4 ℏ 2 t 2 m 2 ) 1 / 4 e i k 0 x exp [ − ( x − ℏ k 0 m t ) 2 a 2 + 2 i ℏ t m ] {\displaystyle \psi (x,t)=\left({\frac {2a^{2}}{\pi }}\right)^{1/4}{\frac {e^{i\phi }}{\left(a^{4}+{\frac {4\hbar ^{2}t^{2}}{m^{2}}}\right)^{1/4}}}e^{ik_{0}x}\exp \left[-{\frac {\left(x-{\frac {\hbar k_{0}}{m}}t\right)^{2}}{a^{2}+{\frac {2i\hbar t}{m}}}}\right]}
where [ 19 ]
ϕ = − θ − ℏ k 0 2 2 m t {\displaystyle \phi =-\theta -{\frac {\hbar k_{0}^{2}}{2m}}t}
tan ( 2 θ ) = 2 ℏ t m a 2 {\displaystyle \tan(2\theta )={\frac {2\hbar t}{ma^{2}}}}
In contrast to the above Gaussian wave packet, which moves at constant group velocity, and always disperses, there exists a wave function based on Airy functions , that propagates freely without envelope dispersion, maintaining its shape, and accelerates in free space: [ 20 ] ψ = Ai [ B ℏ 2 / 3 ( x − B 3 t 2 4 m 2 ) ] e ( i B 3 t / 2 m ℏ ) [ x − ( B 3 t 2 / 6 m 2 ) ] , {\displaystyle \psi =\operatorname {Ai} \left[{\frac {B}{\hbar ^{2/3}}}\left(x-{\frac {B^{3}t^{2}}{4m^{2}}}\right)\right]e^{(iB^{3}t/2m\hbar )[x-(B^{3}t^{2}/6m^{2})]},} where, for simplicity (and nondimensionalization ), choosing ħ = 1 , m = 1/2 , and B an arbitrary constant results in ψ = Ai [ B ( x − B 3 t 2 ) ] e i B 3 t ( x − 2 3 B 3 t 2 ) . {\displaystyle \psi =\operatorname {Ai} [B(x-B^{3}t^{2})]\,e^{iB^{3}t(x-{\tfrac {2}{3}}B^{3}t^{2})}\,.}
There is no dissonance with Ehrenfest's theorem in this force-free situation, because the state is both non-normalizable and has an undefined (infinite) ⟨ x ⟩ for all times. (To the extent that it could be defined, ⟨ p ⟩ = 0 for all times, despite the apparent acceleration of the front.)
The Airy wave train is the only dispersionless wave in one dimensional free space. [ 21 ] In higher dimensions, other dispersionless waves are possible. [ 22 ]
In phase space , this is evident in the pure state Wigner quasiprobability distribution of this wavetrain, whose shape in x and p is invariant as time progresses, but whose features accelerate to the right, in accelerating parabolas. The Wigner function satisfies W ( x , p ; t ) = W ( x − B 3 t 2 , p − B 3 t ; 0 ) = 1 2 1 / 3 π B A i ( 2 2 / 3 ( B ( x − B 3 t 2 ) + ( p / B − t B 2 ) 2 ) ) = W ( x − 2 p t , p ; 0 ) . {\displaystyle {\begin{aligned}W(x,p;t)&=W(x-B^{3}t^{2},p-B^{3}t;0)\\&={\frac {1}{2^{1/3}\pi B}}\,\mathrm {Ai} \left(2^{2/3}\left(B(x-B^{3}t^{2}\right)+\left(p/B-tB^{2})^{2}\right)\right)\\&=W(x-2pt,p;0).\end{aligned}}} The three equalities demonstrate three facts:
Note the momentum distribution obtained by integrating over all x is constant. Since this is the probability density in momentum space , it is evident that the wave function itself is not normalizable.
The narrow-width limit of the Gaussian wave packet solution discussed is the free propagator kernel K . For other differential equations, this is usually called the Green's function , [ 23 ] but in quantum mechanics it is traditional to reserve the name Green's function for the time Fourier transform of K .
Returning to one dimension for simplicity, with m and ħ set equal to one, when a is the infinitesimal quantity ε , the Gaussian initial condition, rescaled so that its integral is one, ψ 0 ( x ) = 1 2 π ε e − x 2 2 ε {\displaystyle \psi _{0}(x)={1 \over {\sqrt {2\pi \varepsilon }}}e^{-{x^{2} \over 2\varepsilon }}\,} becomes a delta function , δ ( x ) , so that its time evolution, K t ( x ) = 1 2 π ( i t + ε ) e − x 2 2 ( i t + ε ) {\displaystyle K_{t}(x)={1 \over {\sqrt {2\pi (it+\varepsilon )}}}e^{-x^{2} \over 2(it+\varepsilon )}\,} yields the propagator.
Note that a very narrow initial wave packet instantly becomes infinitely wide, but with a phase which is more rapidly oscillatory at large values of x . This might seem strange—the solution goes from being localized at one point to being "everywhere" at all later times , but it is a reflection of the enormous momentum uncertainty of a localized particle, as explained above.
Further note that the norm of the wave function is infinite, which is also correct, since the square of a delta function is divergent in the same way.
The factor involving ε is an infinitesimal quantity which is there to make sure that integrals over K are well defined. In the limit that ε → 0 , K becomes purely oscillatory, and integrals of K are not absolutely convergent. In the remainder of this section, it will be set to zero, but in order for all the integrations over intermediate states to be well defined, the limit ε →0 is to be only taken after the final state is calculated.
The propagator is the amplitude for reaching point x at time t , when starting at the origin, x =0. By translation invariance, the amplitude for reaching a point x when starting at point y is the same function, only now translated, K t ( x , y ) = K t ( x − y ) = 1 2 π i t e i ( x − y ) 2 2 t . {\displaystyle K_{t}(x,y)=K_{t}(x-y)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2} \over 2t}\,.}
In the limit when t is small, the propagator goes to a delta function lim t → 0 K t ( x − y ) = δ ( x − y ) , {\displaystyle \lim _{t\to 0}K_{t}(x-y)=\delta (x-y)~,} but only in the sense of distributions : The integral of this quantity multiplied by an arbitrary differentiable test function gives the value of the test function at zero.
To see this, note that the integral over all space of K equals 1 at all times, ∫ K t ( x ) d x = 1 , {\displaystyle \int K_{t}(x)dx=1\,,} since this integral is the inner-product of K with the uniform wave function. But the phase factor in the exponent has a nonzero spatial derivative everywhere except at the origin, and so when the time is small there are fast phase cancellations at all but one point. This is rigorously true when the limit ε →0 is taken at the very end.
So the propagation kernel is the (future) time evolution of a delta function, and it is continuous, in a sense: it goes to the initial delta function at small times. If the initial wave function is an infinitely narrow spike at position y , ψ 0 ( x ) = δ ( x − y ) , {\displaystyle \psi _{0}(x)=\delta (x-y)\,,} it becomes the oscillatory wave, ψ t ( x ) = 1 2 π i t e i ( x − y ) 2 / 2 t . {\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}\,.}
Now, since every function can be written as a weighted sum of such narrow spikes, ψ 0 ( x ) = ∫ ψ 0 ( y ) δ ( x − y ) d y , {\displaystyle \psi _{0}(x)=\int \psi _{0}(y)\delta (x-y)dy\,,} the time evolution of every function ψ 0 is determined by this propagation kernel K ,
ψ t ( x ) = ∫ ψ 0 ( y ) 1 2 π i t e i ( x − y ) 2 / 2 t d y . {\displaystyle \psi _{t}(x)=\int \psi _{0}(y){1 \over {\sqrt {2\pi it}}}e^{i(x-y)^{2}/2t}dy\,.}
Thus, this is a formal way to express the fundamental solution or general solution . The interpretation of this expression is that the amplitude for a particle to be found at point x at time t is the amplitude that it started at y , times the amplitude that it went from y to x , summed over all the possible starting points . In other words, it is a convolution of the kernel K with the arbitrary initial condition ψ 0 , ψ t = K ∗ ψ 0 . {\displaystyle \psi _{t}=K*\psi _{0}\,.}
Since the amplitude to travel from x to y after a time t + t ' can be considered in two steps, the propagator obeys the composition identity, ∫ K ( x − y ; t ) K ( y − z ; t ′ ) d y = K ( x − z ; t + t ′ ) , {\displaystyle \int K(x-y;t)K(y-z;t')dy=K(x-z;t+t')~,} which can be interpreted as follows: the amplitude to travel from x to z in time t + t ' is the sum of the amplitude to travel from x to y in time t , multiplied by the amplitude to travel from y to z in time t ', summed over all possible intermediate states y . This is a property of an arbitrary quantum system, and by subdividing the time into many segments, it allows the time evolution to be expressed as a path integral . [ 24 ]
The spreading of wave packets in quantum mechanics is directly related to the spreading of probability densities in diffusion . For a particle which is randomly walking , the probability density function satisfies the diffusion equation [ 25 ] ∂ ∂ t ρ = 1 2 ∂ 2 ∂ x 2 ρ , {\displaystyle {\partial \over \partial t}\rho ={1 \over 2}{\partial ^{2} \over \partial x^{2}}\rho ,} where the factor of 2, which can be removed by rescaling either time or space, is only for convenience.
A solution of this equation is the time-varying Gaussian function ρ t ( x ) = 1 2 π t e − x 2 2 t , {\displaystyle \rho _{t}(x)={1 \over {\sqrt {2\pi t}}}e^{-x^{2} \over 2t},} which is a form of the heat kernel . Since the integral of ρ t is constant while the width is becoming narrow at small times, this function approaches a delta function at t =0, lim t → 0 ρ t ( x ) = δ ( x ) {\displaystyle \lim _{t\to 0}\rho _{t}(x)=\delta (x)} again only in the sense of distributions, so that lim t → 0 ∫ x f ( x ) ρ t ( x ) = f ( 0 ) {\displaystyle \lim _{t\to 0}\int _{x}f(x)\rho _{t}(x)=f(0)} for any test function f .
The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity, K t + t ′ = K t ∗ K t ′ , {\displaystyle K_{t+t'}=K_{t}*K_{t'}\,,} which allows diffusion to be expressed as a path integral. The propagator is the exponential of an operator H , K t ( x ) = e − t H , {\displaystyle K_{t}(x)=e^{-tH}\,,} which is the infinitesimal diffusion operator, H = − ∇ 2 2 . {\displaystyle H=-{\nabla ^{2} \over 2}\,.}
A matrix has two indices, which in continuous space makes it a function of x and x '. In this case, because of translation invariance, the matrix element K only depend on the difference of the position, and a convenient abuse of notation is to refer to the operator, the matrix elements, and the function of the difference by the same name: K t ( x , x ′ ) = K t ( x − x ′ ) . {\displaystyle K_{t}(x,x')=K_{t}(x-x')\,.}
Translation invariance means that continuous matrix multiplication, C ( x , x ″ ) = ∫ x ′ A ( x , x ′ ) B ( x ′ , x ″ ) , {\displaystyle C(x,x'')=\int _{x'}A(x,x')B(x',x'')\,,} is essentially convolution, C ( Δ ) = C ( x − x ″ ) = ∫ x ′ A ( x − x ′ ) B ( x ′ − x ″ ) = ∫ y A ( Δ − y ) B ( y ) . {\displaystyle C(\Delta )=C(x-x'')=\int _{x'}A(x-x')B(x'-x'')=\int _{y}A(\Delta -y)B(y)\,.}
The exponential can be defined over a range of t s which include complex values, so long as integrals over the propagation kernel stay convergent, K z ( x ) = e − z H . {\displaystyle K_{z}(x)=e^{-zH}\,.} As long as the real part of z is positive, for large values of x , K is exponentially decreasing, and integrals over K are indeed absolutely convergent.
The limit of this expression for z approaching the pure imaginary axis is the above Schrödinger propagator encountered, K t S c h r = K i t + ε = e − ( i t + ε ) H , {\displaystyle K_{t}^{\rm {Schr}}=K_{it+\varepsilon }=e^{-(it+\varepsilon )H}\,,} which illustrates the above time evolution of Gaussians.
From the fundamental identity of exponentiation, or path integration, K z ∗ K z ′ = K z + z ′ {\displaystyle K_{z}*K_{z'}=K_{z+z'}\,} holds for all complex z values, where the integrals are absolutely convergent so that the operators are well defined.
Thus, quantum evolution of a Gaussian, which is the complex diffusion kernel K , ψ 0 ( x ) = K a ( x ) = K a ∗ δ ( x ) {\displaystyle \psi _{0}(x)=K_{a}(x)=K_{a}*\delta (x)\,} amounts to the time-evolved state, ψ t = K i t ∗ K a = K a + i t . {\displaystyle \psi _{t}=K_{it}*K_{a}=K_{a+it}\,.}
This illustrates the above diffusive form of the complex Gaussian solutions, ψ t ( x ) = 1 2 π ( a + i t ) e − x 2 2 ( a + i t ) . {\displaystyle \psi _{t}(x)={1 \over {\sqrt {2\pi (a+it)}}}e^{-{x^{2} \over 2(a+it)}}\,.}
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https://en.wikipedia.org/wiki/Wave_packet
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Wave run-up is the height to which waves run up the slope of a revetment , bank or dike , regardless of whether the waves are breaking or not. Conversely, wave run-down is the height to which waves recede. These heights are always measured vertically (and not along the slope). The wave run-up height, denoted by R u {\displaystyle R_{u}} , R u p {\displaystyle {R_{up}}} , or z {\displaystyle z} , is a very important parameter in coastal engineering as, together with the design highest still water level, it determines the required crest height of a dike or revetment.
The first scientific measurements of wave run-up were carried out by the Lorentz Committee in preparation for the works to close off the Zuiderzee . [ 1 ] The Committee measured the wave height and wave run-up at various locations in 1920, but established that state of the art methods for measuring waves in the field during storms were inadequate. As a result, scale tests were also undertaken, but these also proved to be of very limited efficacy due to the fact that only regular waves (idealised, periodic waves with constant amplitude and a fixed time period between successive wave crests, following a sinusoidal pattern) could be modelled at the time.
The methods and technology available to the committee at the time did not permit model testing of the more realistic and complex irregular waves (consisting of varying heights, periods and directions), which provide a more accurate representation of the actual conditions faced by coastal structures and shorelines.
It was found, however, that the depth in front of the dike is very important for wave run-up and that, at least for the range of observations in the committee's measurements, the slope ratio does not play a major role. Nearly all dikes in the Netherlands at that time had a slope of 1:3.
Current knowledge indicates that during storms and on gentle coastal slopes, the significant wave height is approximately half the water depth. This relationship appears to be accurate, and the observation is more pronounced for slopes around 1:3.
This research was continued during the Zuiderzee Works , and eventually led to the (old) Delft formula for wave run-up: [ 2 ]
in which:
This formula proved to be generally applicable for smooth slopes and relatively steep (storm) waves. Subsequently, it was discovered that longer (swell) waves resulted in higher run-up. To account for this, the wave period was incorporated into the formula using the Iribarren number , ξ {\displaystyle \xi } , leading to the development of Hunt's Formula: [ 3 ]
This formula was also valid for regular waves. The Old Delft Formula and Hunt's Formula are identical for waves with a steepness of 1/64, or about 2%. For higher values of ξ {\displaystyle \xi } , Hunt's formula has a limit value: ξ > 2 . 5 {\displaystyle \xi >2{.}5} → {\displaystyle \rightarrow } R u / H = 2 . 5 {\displaystyle Ru/H=2{.}5} .
In 1988, van der Meer provided formulae for wave run-up on rubble mound breakwaters, based on tests with rock-armoured straight slopes. He also introduced a notional permeability factor P {\displaystyle P} for the structure. [ 4 ] This factor also accounts for the effect of the pore volume. Defining R u p {\displaystyle R_{up}} at the run-up level of exceedance probability p {\displaystyle p} , the formula, valid for 0.1 ≤ P ≤ 0.6 {\displaystyle 0.1\leq P\leq 0.6} and head-on waves, is:
R u p H s = { a ξ m for ξ m < 1.5 b ξ m c for ξ m > 1.5 {\displaystyle {\frac {R_{up}}{H_{s}}}={\begin{cases}a\xi _{m}&{\text{for }}\xi _{m}<1.5\\b\xi _{m}^{c}&{\text{for }}\xi _{m}>1.5\end{cases}}}
The term R u p H s {\displaystyle {\frac {R_{up}}{H_{s}}}} reaches a constant maximum value equal to d {\displaystyle d} in the case of permeable structures, i.e., P > 0.4 {\displaystyle P>0.4} . This corresponds to the region of surging waves, where there is no real wave breaking and where wave steepness and slope angle do not influence the run-up. [ 5 ] The coefficients a , b , {\displaystyle a,b,} c {\displaystyle c} and d {\displaystyle d} are presented in the table below:
The values of the coefficients highlight the considerable variability of the run-up level from one wave to another in irregular seas. [ 6 ] For run-up levels on smooth slopes, work in the Netherlands by the Technische Adviescommissie voor de Waterkeringen ( English : Technical Advisory Committee on water Defences) in 1974 discussed the reduction in run-up due to different types of surface roughness. [ 7 ]
In practical scenarios, waves are irregular, consisting of a combination of waves with varying heights, periods, and directions. These waves are typically analysed using statistical methods and spectral analysis, providing a more accurate representation of the actual conditions faced by coastal structures and shorelines. Consequently, it is not possible to define a single wave run-up value. Instead, a wave run-up with a specific probability of exceedance is used, typically set at 2%. This wave run-up represents the height exceeded by 2% of the waves in a wave field.
Research indicates that wave run-up follows a Rayleigh distribution , similar to the waves themselves. A probability of exceedance value has been chosen that is small enough to prevent overtopping waves from causing damage to the inner slope. The 2% value has been adopted internationally and was arbitrarily selected by the Dutch Waterloopkundig Laboratorium shortly before 1940. Considering the function, 1% or 5% could have also been possible. The choice of 2% was based on the duration of experimental designs, as a complete trial could be conducted in half a day.
In 1972, Jurjen Battjes , commissioned by the Dutch Technical Advisory Committee for Flood Defences, summarised the available research and provided a solid theoretical foundation. [ 8 ] [ 9 ] This work led to an improved version of Hunt's Formula, which explicitly included parameters for the angle of incidence of the waves, the effect of a berm, and the slope's roughness. However, the available experimental data on roughness and the berm were insufficient to establish a definitive formula.
Subsequent research was conducted in the following years, with an emphasis on wave overtopping as a more indicative factor for dike height than wave run-up. This research ultimately resulted in a Technical Report in 2002 by the Dutch organisation, TAW. [ 10 ] The wave run-up formula mentioned in this report remains in use, and the EurOtop manual has adopted it. The scope of validity has been further expanded in the EurOtop manual, featuring modified formulas. [ 11 ]
The EurOtop manual provides a general formula (Formula 1.4 in the manual) [ 11 ] for wave run-up:
with a maximum value around 3. The Iribarren number ξ m − 1 , 0 {\displaystyle \xi _{m-1,0}} is then used based on the period determined using the first negative moment of the wave spectrum. Additionally, γ {\displaystyle \gamma } is the reduction coefficient for the factors described below.
The following equation is valid:
in which:
A range is provided for Hillblock and Ronataille materials, as their reduction coefficient is dependent on wave height. A similar phenomenon occurs with grass. When subjected to high waves, natural grass becomes very smooth, resulting in a reduction coefficient γ f = 1 . 0 {\displaystyle \gamma _{f}=1{.}0} . However, for smaller waves — approximately 25 centimetres (9.8 in) or less — natural grass tends to be much rougher. In such cases, one may opt for a reduction coefficient of γ f {\displaystyle \gamma _{f}} below 1.0.
When dealing with short-crested waves, the highest run-up is caused by head-on waves (those with an angle of incidence, β = 0 {\displaystyle \beta =0} ) and equates to the run-up for long-crested waves. For smooth slopes, the run-up decreases slightly with β {\displaystyle \beta } . The run-down typically ranges between a third and a half of the run-up. For breakwaters and revetments constructed with rock armour , the maximum run-down level may indicate the minimum downward extension of the primary armour, and a potential upper level for introducing a berm with a smaller armour size. [ 5 ]
For wave run-down there is a similar formula: [ 12 ]
Following storm events, a layer of floating debris, known as the flood mark or flotsam, often remains on the slope. This tide mark indicates the maximum wave run-up during the preceding storm. As the flood mark is situated near the height of maximum wave run-up and water levels are generally well-documented by nearby tide stations, it is straightforward to calculate the Ru 2% of the storm by subtracting the observed storm surge level from the flood mark level.
In the past, authorities in the Netherlands systematically recorded these observations for most dikes, resulting in a collection of flood mark heights for each dike section. The statistics of flood mark heights can be utilised to determine dike height, which should comprise the design water level plus a safety height (freeboard). The freeboard at the design water level must be equal to the maximum permissible wave run-up. [ 13 ]
For a dike with an acceptable load exceedance probability per year, such as 1/500 (as with the temporary dike reinforcement in the Oosterschelde ), it is necessary to determine the 1/500 wave run-up. This can be calculated if the 1/500 wave height at the toe of the dike is known. However, this value is rarely measured and must be determined using a computational model, such as SWAN. [ 14 ] In many instances, this process can be challenging and prone to errors.
By analysing flood mark heights, which involves simply plotting the data on logarithmic paper, it is possible to directly obtain values such as the 1/500 wave run-up, and consequently the required safety height. An example of this can be observed in the accompanying photo of the run-up and flood mark lines at a dike along the Bathpolder in Zeeland . The photo shows two flood mark lines, which represent the wave run-up of two subsequent storms (on October 12, 2009, with water levels at 2.2 metres (7 ft 3 in) and 1.9 metres (6 ft 3 in) above mean sea level) in the Bathpolder. In the foreground, there is a slope with Haringman blocks, while the background features a slope of Elastocoast. The wave height during these storms was approximately 0.5 metres (1 ft 7.7 in). The wave run-up was 0.8 and 0.85 metres (2 ft 7 in and 2 ft 9 in) above the storm surge level on the Elastocoast, and 1.05 and 1.1 metres (3 ft 5 in and 3 ft 7 in) above the storm surge level on the Haringman blocks. The slope gradient here is 1:4.2. As a Haringman block measures exactly 50 centimetres (19.7 in), the run-up can be assessed in this photo. Subsequent analysis reveals that the reduction coefficient γ f for Haringman blocks here is 1.0, and for Elastocoast, it is 0.8.
To assess the safety of a dike and the durability of its grass cover, particularly on the sea or river side, a wave run-up simulator can be employed. The wave conditions for which a dike is designed are infrequent, and the strength of grass coverings varies. These dike conditions can be replicated in-situ using a wave run-up simulator, allowing the manager of the relevant flood defence system to determine if the grass cover is strong enough to withstand expected waves under extreme conditions. [ 11 ]
During these tests, the wave run-up simulator is placed on the outer slope and continuously filled with water at a constant flow rate. The flaps at the bottom of the simulator can be opened to varying extents, enabling the simulation of different wave run-up volumes. [ 15 ]
The wave run-up simulator is one method for assessing the strength of the grass cover. Another approach involves using a sod puller, which can determine the tensile strength of a sod and allows conversion of this tensile strength by an engineer into a strength under load caused by wave run-up. [ 16 ] In addition to simulating wave run-up, the simulation of wave impacts and wave overtopping can be achieved using specifically designed generators and simulators.
Wave run-up should not be confused with wave set-up (an increase in water level due to known waves) or with wind setup ( storm surge , an increase in water level due to the driving force of wind).
General reference
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https://en.wikipedia.org/wiki/Wave_run-up
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In fluid dynamics , a wind wave , or wind-generated water wave , is a surface wave that occurs on the free surface of bodies of water as a result of the wind blowing over the water's surface. The contact distance in the direction of the wind is known as the fetch . Waves in the oceans can travel thousands of kilometers before reaching land. Wind waves on Earth range in size from small ripples to waves over 30 m (100 ft) high, being limited by wind speed, duration, fetch, and water depth. [ 1 ]
When directly generated and affected by local wind, a wind wave system is called a wind sea . Wind waves will travel in a great circle route after being generated – curving slightly left in the southern hemisphere and slightly right in the northern hemisphere. After moving out of the area of fetch and no longer being affected by the local wind, wind waves are called swells and can travel thousands of kilometers. A noteworthy example of this is waves generated south of Tasmania during heavy winds that will travel across the Pacific to southern California, producing desirable surfing conditions. [ 2 ] Wind waves in the ocean are also called ocean surface waves and are mainly gravity waves , where gravity is the main equilibrium force.
Wind waves have a certain amount of randomness : subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process , in combination with the physics governing their generation, growth, propagation, and decay – as well as governing the interdependence between flow quantities such as the water surface movements, flow velocities , and water pressure . The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models .
Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves. [ 3 ] [ 4 ] [ 5 ] Waves in bodies of water may also be generated by other causes, both at the surface and underwater (such as watercraft , animals , waterfalls , landslides , earthquakes , bubbles , and impact events ).
The great majority of large breakers seen at a beach result from distant winds. Five factors influence the formation of the flow structures in wind waves: [ 6 ]
All of these factors work together to determine the size of the water waves and the structure of the flow within them.
The main dimensions associated with wave propagation are:
A fully developed sea has the maximum wave size theoretically possible for a wind of specific strength, duration, and fetch. Further exposure to that specific wind could only cause a dissipation of energy due to the breaking of wave tops and formation of "whitecaps". Waves in a given area typically have a range of heights. For weather reporting and for scientific analysis of wind wave statistics, their characteristic height over a period of time is usually expressed as significant wave height . This figure represents an average height of the highest one-third of the waves in a given time period (usually chosen somewhere in the range from 20 minutes to twelve hours), or in a specific wave or storm system. The significant wave height is also the value a "trained observer" (e.g. from a ship's crew) would estimate from visual observation of a sea state. Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm. [ 7 ]
Wave formation on an initially flat water surface by wind is started by a random distribution of normal pressure of turbulent wind flow over the water. This pressure fluctuation produces normal and tangential stresses in the surface water, which generates waves. It is usually assumed for the purpose of theoretical analysis that: [ 8 ]
The second mechanism involves wind shear forces on the water surface. John W. Miles suggested a surface wave generation mechanism that is initiated by turbulent wind shear flows based on the inviscid Orr–Sommerfeld equation in 1957. He found the energy transfer from the wind to the water surface is proportional to the curvature of the velocity profile of the wind at the point where the mean wind speed is equal to the wave speed. Since the wind speed profile is logarithmic to the water surface, the curvature has a negative sign at this point. This relation shows the wind flow transferring its kinetic energy to the water surface at their interface.
Assumptions:
Generally, these wave formation mechanisms occur together on the water surface and eventually produce fully developed waves.
For example, [ 10 ] if we assume a flat sea surface (Beaufort state 0), and a sudden wind flow blows steadily across the sea surface, the physical wave generation process follows the sequence:
Three different types of wind waves develop over time:
Ripples appear on smooth water when the wind blows, but will die quickly if the wind stops. The restoring force that allows them to propagate is surface tension . Sea waves are larger-scale, often irregular motions that form under sustained winds. These waves tend to last much longer, even after the wind has died, and the restoring force that allows them to propagate is gravity. As waves propagate away from their area of origin, they naturally separate into groups of common direction and wavelength. The sets of waves formed in this manner are known as swells. The Pacific Ocean is 19,800 km (12,300 mi) from Indonesia to the coast of Colombia and, based on an average wavelength of 76.5 m (251 ft), would have ~258,824 swells over that width.
It is sometimes alleged that out of a set of waves, the seventh wave in a set is always the largest; while this isn't the case, the waves in the middle of a given set tend to be larger than those before and after them. [ 13 ]
Individual " rogue waves " (also called "freak waves", "monster waves", "killer waves", and "king waves") much higher than the other waves in the sea state can occur. In the case of the Draupner wave , its 25 m (82 ft) height was 2.2 times the significant wave height . Such waves are distinct from tides , caused by the Moon and Sun 's gravitational pull , tsunamis that are caused by underwater earthquakes or landslides , and waves generated by underwater explosions or the fall of meteorites —all having far longer wavelengths than wind waves.
The largest ever recorded wind waves are not rogue waves, but standard waves in extreme sea states. For example, 29.1 m (95 ft) high waves were recorded on the RRS Discovery in a sea with 18.5 m (61 ft) significant wave height, so the highest wave was only 1.6 times the significant wave height. [ 14 ] The biggest recorded by a buoy (as of 2011) was 32.3 m (106 ft) high during the 2007 typhoon Krosa near Taiwan. [ 15 ]
Ocean waves can be classified based on: the disturbing force that creates them; the extent to which the disturbing force continues to influence them after formation; the extent to which the restoring force weakens or flattens them; and their wavelength or period. Seismic sea waves have a period of about 20 minutes, and speeds of 760 km/h (470 mph). Wind waves (deep-water waves) have a period up to about 20 seconds.
The speed of all ocean waves is controlled by gravity, wavelength, and water depth. Most characteristics of ocean waves depend on the relationship between their wavelength and water depth. Wavelength determines the size of the orbits of water molecules within a wave, but water depth determines the shape of the orbits. The paths of water molecules in a wind wave are circular only when the wave is traveling in deep water. A wave cannot "feel" the bottom when it moves through water deeper than half its wavelength because too little wave energy is contained in the water movement below that depth. Waves moving through water deeper than half their wavelength are known as deep-water waves. On the other hand, the orbits of water molecules in waves moving through shallow water are flattened by the proximity of the sea bottom surface. Waves in water shallower than 1/20 their original wavelength are known as shallow-water waves. Transitional waves travel through water deeper than 1/20 their original wavelength but shallower than half their original wavelength.
In general, the longer the wavelength, the faster the wave energy will move through the water. The relationship between the wavelength, period and velocity of any wave is:
where C is speed (celerity), L is the wavelength, and T is the period (in seconds). Thus the speed of the wave derives from the functional dependence L ( T ) {\displaystyle L(T)} of the wavelength on the period (the dispersion relation ).
The speed of a deep-water wave may also be approximated by:
where g is the acceleration due to gravity, 9.8 meters (32 feet) per second squared. Because g and π (3.14) are constants, the equation can be reduced to:
when C is measured in meters per second and L in meters. In both formulas the wave speed is proportional to the square root of the wavelength.
The speed of shallow-water waves is described by a different equation that may be written as:
where C is speed (in meters per second), g is the acceleration due to gravity, and d is the depth of the water (in meters). The period of a wave remains unchanged regardless of the depth of water through which it is moving. As deep-water waves enter the shallows and feel the bottom, however, their speed is reduced, and their crests "bunch up", so their wavelength shortens.
Sea state can be described by the sea wave spectrum or just wave spectrum S ( ω , Θ ) {\displaystyle S(\omega ,\Theta )} . It is composed of a wave height spectrum (WHS) S ( ω ) {\displaystyle S(\omega )} and a wave direction spectrum (WDS) f ( Θ ) {\displaystyle f(\Theta )} . Many interesting properties about the sea state can be found from the wave spectra.
WHS describes the spectral density of wave height variance ("power") versus wave frequency , with dimension { S ( ω ) } = { length 2 ⋅ time } {\displaystyle \{S(\omega )\}=\{{\text{length}}^{2}\cdot {\text{time}}\}} .
The relationship between the spectrum S ( ω j ) {\displaystyle S(\omega _{j})} and the wave amplitude A j {\displaystyle A_{j}} for a wave component j {\displaystyle j} is:
Some WHS models are listed below.
As for WDS, an example model of f ( Θ ) {\displaystyle f(\Theta )} might be:
Thus the sea state is fully determined and can be recreated by the following function where ζ {\displaystyle \zeta } is the wave elevation, ϵ j {\displaystyle \epsilon _{j}} is uniformly distributed between 0 and 2 π {\displaystyle 2\pi } , and Θ j {\displaystyle \Theta _{j}} is randomly drawn from the directional distribution function f ( Θ ) : {\displaystyle {\sqrt {f(\Theta )}}:} [ 22 ]
As waves travel from deep to shallow water, their shape changes (wave height increases, speed decreases, and length decreases as wave orbits become asymmetrical). This process is called shoaling .
Wave refraction is the process that occurs when waves interact with the sea bed to slow the velocity of propagation as a function of wavelength and period. As the waves slow down in shoaling water, the crests tend to realign at a decreasing angle to the depth contours. Varying depths along a wave crest cause the crest to travel at different phase speeds , with those parts of the wave in deeper water moving faster than those in shallow water . This process continues while the depth decreases, and reverses if it increases again, but the wave leaving the shoal area may have changed direction considerably. Rays —lines normal to wave crests between which a fixed amount of energy flux is contained—converge on local shallows and shoals. Therefore, the wave energy between rays is concentrated as they converge, with a resulting increase in wave height.
Because these effects are related to a spatial variation in the phase speed, and because the phase speed also changes with the ambient current—due to the Doppler shift —the same effects of refraction and altering wave height also occur due to current variations. In the case of meeting an adverse current the wave steepens , i.e. its wave height increases while the wavelength decreases, similar to the shoaling when the water depth decreases. [ 23 ]
Some waves undergo a phenomenon called "breaking". [ 24 ] A breaking wave is one whose base can no longer support its top, causing it to collapse. A wave breaks when it runs into shallow water , or when two wave systems oppose and combine forces. When the slope, or steepness ratio, of a wave, is too great, breaking is inevitable.
Individual waves in deep water break when the wave steepness—the ratio of the wave height H to the wavelength λ —exceeds about 0.17, so for H > 0.17 λ . In shallow water, with the water depth small compared to the wavelength, the individual waves break when their wave height H is larger than 0.8 times the water depth h , that is H > 0.8 h . [ 25 ] Waves can also break if the wind grows strong enough to blow the crest off the base of the wave.
In shallow water, the base of the wave is decelerated by drag on the seabed. As a result, the upper parts will propagate at a higher velocity than the base and the leading face of the crest will become steeper and the trailing face flatter. This may be exaggerated to the extent that the leading face forms a barrel profile, with the crest falling forward and down as it extends over the air ahead of the wave.
Three main types of breaking waves are identified by surfers or surf lifesavers . Their varying characteristics make them more or less suitable for surfing and present different dangers.
When the shoreline is near vertical, waves do not break but are reflected. Most of the energy is retained in the wave as it returns to seaward. Interference patterns are caused by superposition of the incident and reflected waves, and the superposition may cause localized instability when peaks cross, and these peaks may break due to instability. (see also clapotic waves )
Wind waves are mechanical waves that propagate along the interface between water and air ; the restoring force is provided by gravity, and so they are often referred to as surface gravity waves . As the wind blows, pressure and friction perturb the equilibrium of the water surface and transfer energy from the air to the water, forming waves. The initial formation of waves by the wind is described in the theory of Phillips from 1957, and the subsequent growth of the small waves has been modeled by Miles , also in 1957. [ 26 ] [ 27 ]
In linear plane waves of one wavelength in deep water, parcels near the surface move not plainly up and down but in circular orbits: forward above and backward below (compared to the wave propagation direction). As a result, the surface of the water forms not an exact sine wave , but more a trochoid with the sharper curves upwards—as modeled in trochoidal wave theory. Wind waves are thus a combination of transversal and longitudinal waves.
When waves propagate in shallow water , (where the depth is less than half the wavelength) the particle trajectories are compressed into ellipses . [ 29 ] [ 30 ]
In reality, for finite values of the wave amplitude (height), the particle paths do not form closed orbits; rather, after the passage of each crest, particles are displaced slightly from their previous positions, a phenomenon known as Stokes drift . [ 31 ] [ 32 ]
As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the wavelength λ, the orbital movement has decayed to less than 5% of its value at the surface. The phase speed (also called the celerity) of a surface gravity wave is—for pure periodic wave motion of small- amplitude waves—well approximated by
where
In deep water, where d ≥ 1 2 λ {\displaystyle d\geq {\frac {1}{2}}\lambda } , so 2 π d λ ≥ π {\displaystyle {\frac {2\pi d}{\lambda }}\geq \pi } and the hyperbolic tangent approaches 1 {\displaystyle 1} , the speed c {\displaystyle c} approximates
In SI units, with c deep {\displaystyle c_{\text{deep}}} in m/s, c deep ≈ 1.25 λ {\displaystyle c_{\text{deep}}\approx 1.25{\sqrt {\lambda }}} , when λ {\displaystyle \lambda } is measured in metres.
This expression tells us that waves of different wavelengths travel at different speeds. The fastest waves in a storm are the ones with the longest wavelength. As a result, after a storm, the first waves to arrive on the coast are the long-wavelength swells.
For intermediate and shallow water, the Boussinesq equations are applicable, combining frequency dispersion and nonlinear effects. And in very shallow water, the shallow water equations can be used.
If the wavelength is very long compared to the water depth, the phase speed (by taking the limit of c when the wavelength approaches infinity) can be approximated by
On the other hand, for very short wavelengths, surface tension plays an important role and the phase speed of these gravity-capillary waves can (in deep water) be approximated by
where
When several wave trains are present, as is always the case in nature, the waves form groups. In deep water, the groups travel at a group velocity which is half of the phase speed . [ 34 ] Following a single wave in a group one can see the wave appearing at the back of the group, growing, and finally disappearing at the front of the group.
As the water depth d {\displaystyle d} decreases towards the coast , this will have an effect: wave height changes due to wave shoaling and refraction . As the wave height increases, the wave may become unstable when the crest of the wave moves faster than the trough . This causes surf , a breaking of the waves.
The movement of wind waves can be captured by wave energy devices . The energy density (per unit area) of regular sinusoidal waves depends on the water density ρ {\displaystyle \rho } , gravity acceleration g {\displaystyle g} and the wave height H {\displaystyle H} (which, for regular waves, is equal to twice the amplitude , a {\displaystyle a} ):
The velocity of propagation of this energy is the group velocity .
Surfers are very interested in the wave forecasts . There are many websites that provide predictions of the surf quality for the upcoming days and weeks. Wind wave models are driven by more general weather models that predict the winds and pressures over the oceans, seas, and lakes.
Wind wave models are also an important part of examining the impact of shore protection and beach nourishment proposals. For many beach areas there is only patchy information about the wave climate, therefore estimating the effect of wind waves is important for managing littoral environments.
A wind-generated wave can be predicted based on two parameters: wind speed at 10 m above sea level and wind duration, which must blow over long periods of time to be considered fully developed. The significant wave height and peak frequency can then be predicted for a certain fetch length. [ 35 ]
Ocean water waves generate seismic waves that are globally visible on seismographs . [ 36 ] There are two principal constituents of the ocean wave-generated seismic microseism. [ 37 ] The strongest of these is the secondary microseism which is created by ocean floor pressures generated by interfering ocean waves and has a spectrum that is generally between approximately 6–12 s periods, or at approximately half of the period of the responsible interfering waves. The theory for microseism generation by standing waves was provided by Michael Longuet-Higgins in 1950 after in 1941 Pierre Bernard suggested this relation with standing waves on the basis of observations. [ 38 ] [ 39 ] The weaker primary microseism, also globally visible, is generated by dynamic seafloor pressures of propagating waves above shallower (less than several hundred meters depth) regions of the global ocean. Microseisms were first reported in about 1900, and seismic records provide long-term proxy measurements of seasonal and climate-related large-scale wave intensity in Earth's oceans [ 40 ] including those associated with anthropogenic global warming . [ 41 ] [ 42 ] [ 43 ]
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A wave tank is a laboratory setup for observing the behavior of surface waves . The typical wave tank is a box filled with liquid , usually water , leaving open or air -filled space on top. At one end of the tank, an actuator generates waves; the other end usually has a wave-absorbing surface. [ 1 ] A similar device is the ripple tank , which is flat and shallow and used for observing patterns of surface waves from above.
A wave basin is a wave tank which has a width and length of comparable magnitude, often used for testing ships, offshore structures and three-dimensional models of harbors (and their breakwaters).
A wave flume (or wave channel ) is a special sort of wave tank: the width of the flume is much less than its length. The generated waves are therefore – more or less – two-dimensional in a vertical plane (2DV), meaning that the orbital flow velocity component in the direction perpendicular to the flume side wall is much smaller than the other two components of the three-dimensional velocity vector . This makes a wave flume a well-suited facility to study near-2DV structures, like cross-sections of a breakwater . Also (3D) constructions providing little blockage to the flow may be tested, e.g. measuring wave forces on vertical cylinders with a diameter much less than the flume width. [ 3 ]
Wave flumes may be used to study the effects of water waves on coastal structures , offshore structures , sediment transport and other transport phenomena .
The waves are most often generated with a mechanical wavemaker, although there are also wind–wave flumes with (additional) wave generation by an air flow over the water – with the flume closed above by a roof above the free surface. The wavemaker frequently consists of a translating or rotating rigid wave board. Modern wavemakers are computer controlled, and can generate besides periodic waves also random waves , solitary waves , wave groups or even tsunami -like wave motion. The wavemaker is at one end of the wave flume, and at the other end is the construction being tested, or a wave absorber (a beach or special wave absorbing constructions). [ 4 ]
Often, the side walls contain glass windows, or are completely made of glass, allowing for a clear visual observation of the experiment, and the easy deployment of optical instruments (e.g. by Laser Doppler velocimetry or particle image velocimetry ).
In 2014, the first circular, combined current and wave test basin, FloWaveTT , was commissioned in The University of Edinburgh . This allows for "true" 360° waves to be generated to simulate rough storm conditions as well as scientific controlled waves in the same facility.
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A waveguide is a structure that guides waves by restricting the transmission of energy to one direction. Common types of waveguides include acoustic waveguides which direct sound , optical waveguides which direct light , and radio-frequency waveguides which direct electromagnetic waves other than light like radio waves .
Without the physical constraint of a waveguide, waves would expand into three-dimensional space and their intensities would decrease according to the inverse square law .
There are different types of waveguides for different types of waves. The original and most common meaning is a hollow conductive metal pipe used to carry high frequency radio waves , particularly microwaves . [ 1 ] Dielectric waveguides are used at higher radio frequencies, and transparent dielectric waveguides and optical fibers serve as waveguides for light. In acoustics , air ducts and horns are used as waveguides for sound in musical instruments and loudspeakers , and specially-shaped metal rods conduct ultrasonic waves in ultrasonic machining .
The geometry of a waveguide reflects its function; in addition to more common types that channel the wave in one dimension, there are two-dimensional slab waveguides which confine waves to two dimensions. The frequency of the transmitted wave also dictates the size of a waveguide: each waveguide has a cutoff wavelength determined by its size and will not conduct waves of greater wavelength; an optical fiber that guides light will not transmit microwaves which have a much larger wavelength. Some naturally occurring structures can also act as waveguides. The SOFAR channel layer in the ocean can guide the sound of whale song across enormous distances. [ 2 ] Any shape of cross section of waveguide can support EM waves. Irregular shapes are difficult to analyse. Commonly used waveguides are rectangular and circular in shape.
The uses of waveguides for transmitting signals were known even before the term was coined. The phenomenon of sound waves guided through a taut wire have been known for a long time, as well as sound through a hollow pipe such as a cave or medical stethoscope . Other uses of waveguides are in transmitting power between the components of a system such as radio, radar or optical devices. Waveguides are the fundamental principle of guided wave testing (GWT), one of the many methods of non-destructive evaluation . [ 3 ]
Specific examples:
The first structure for guiding waves was proposed by J. J. Thomson in 1893, and was first experimentally tested by Oliver Lodge in 1894. The first mathematical analysis of electromagnetic waves in a metal cylinder was performed by Lord Rayleigh in 1897. [ 6 ] : 8 For sound waves, Lord Rayleigh published a full mathematical analysis of propagation modes in his seminal work, "The Theory of Sound". [ 7 ] Jagadish Chandra Bose researched millimeter wavelengths using waveguides, and in 1897 described to the Royal Institution in London his research carried out in Kolkata. [ 8 ] [ 9 ]
The study of dielectric waveguides (such as optical fibers, see below) began as early as the 1920s, by several people, most famous of which are Rayleigh, Sommerfeld and Debye . [ 10 ] Optical fiber began to receive special attention in the 1960s due to its importance to the communications industry.
The development of radio communication initially occurred at the lower frequencies because these could be more easily propagated over large distances. The long wavelengths made these frequencies unsuitable for use in hollow metal waveguides because of the impractically large diameter tubes required. Consequently, research into hollow metal waveguides stalled and the work of Lord Rayleigh was forgotten for a time and had to be rediscovered by others. Practical investigations resumed in the 1930s by George C. Southworth at Bell Labs and Wilmer L. Barrow at MIT . Southworth at first took the theory from papers on waves in dielectric rods because the work of Lord Rayleigh was unknown to him. This misled him somewhat; some of his experiments failed because he was not aware of the phenomenon of waveguide cutoff frequency already found in Lord Rayleigh's work. Serious theoretical work was taken up by John R. Carson and Sallie P. Mead . This work led to the discovery that for the TE 01 mode in circular waveguide losses go down with frequency and at one time this was a serious contender for the format for long-distance telecommunications. [ 11 ] : 544–548
The importance of radar in World War II gave a great impetus to waveguide research, at least on the Allied side. The magnetron , developed in 1940 by John Randall and Harry Boot at the University of Birmingham in the United Kingdom, provided a good power source and made microwave radar feasible. The most important centre of US research was at the Radiation Laboratory (Rad Lab) at MIT but many others took part in the US, and in the UK such as the Telecommunications Research Establishment . The head of the Fundamental Development Group at Rad Lab was Edward Mills Purcell . His researchers included Julian Schwinger , Nathan Marcuvitz , Carol Gray Montgomery, and Robert H. Dicke . Much of the Rad Lab work concentrated on finding lumped element models of waveguide structures so that components in waveguide could be analysed with standard circuit theory. Hans Bethe was also briefly at Rad Lab, but while there he produced his small aperture theory which proved important for waveguide cavity filters , first developed at Rad Lab. The German side, on the other hand, largely ignored the potential of waveguides in radar until very late in the war. So much so that when radar parts from a downed British plane were sent to Siemens & Halske for analysis, even though they were recognised as microwave components, their purpose could not be identified.
At that time, microwave techniques were badly neglected in Germany. It was generally believed that it was of no use for electronic warfare, and those who wanted to do research work in this field were not allowed to do so.
German academics were even allowed to continue publicly publishing their research in this field because it was not felt to be important. [ 12 ] : 548–554 [ 13 ] : 1055, 1057
Immediately after World War II waveguide was the technology of choice in the microwave field. However, it has some problems; it is bulky, expensive to produce, and the cutoff frequency effect makes it difficult to produce wideband devices. Ridged waveguide can increase bandwidth beyond an octave, but a better solution is to use a technology working in TEM mode (that is, non-waveguide) such as coaxial conductors since TEM does not have a cutoff frequency. A shielded rectangular conductor can also be used and this has certain manufacturing advantages over coax and can be seen as the forerunner of the planar technologies ( stripline and microstrip ). However, planar technologies really started to take off when printed circuits were introduced. These methods are significantly cheaper than waveguide and have largely taken its place in most bands. However, waveguide is still favoured in the higher microwave bands from around Ku band upwards. [ 12 ] : 556–557 [ 14 ] : 21–27, 21–50
A propagation mode in a waveguide is one solution of the wave equations, or, in other words, the form of the wave. [ 10 ] Due to the constraints of the boundary conditions , there are only limited frequencies and forms for the wave function which can propagate in the waveguide. The lowest frequency in which a certain mode can propagate is the cutoff frequency of that mode. The mode with the lowest cutoff frequency is the fundamental mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency. [ 15 ] : 38
Propagation modes are computed by solving the Helmholtz equation alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the propagation direction (i.e. z {\displaystyle z} ). More specifically, the common approach is to first replace all unknown time-varying fields u ( x , y , z , t ) {\displaystyle u(x,y,z,t)} (assuming for simplicity to describe the fields in cartesian components) with their complex phasors representation U ( x , y , z ) {\displaystyle U(x,y,z)} , sufficient to fully describe any infinitely long single-tone signal at frequency f {\displaystyle f} , (angular frequency ω = 2 π f {\displaystyle \omega =2\pi f} ), and rewrite the Helmholtz equation and boundary conditions accordingly. Then, every unknown field is forced to have a form like U ( x , y , z ) = U ^ ( x , y ) e − γ z {\displaystyle U(x,y,z)={\hat {U}}(x,y)e^{-\gamma z}} , where the γ {\displaystyle \gamma } term represents the propagation constant (still unknown) along the direction along which the waveguide extends to infinity. The Helmholtz equation can be rewritten to accommodate such form and the resulting equality needs to be solved for γ {\displaystyle \gamma } and U ^ ( x , y ) {\displaystyle {\hat {U}}(x,y)} , yielding in the end an eigenvalue equation for γ {\displaystyle \gamma } and a corresponding eigenfunction U ^ ( x , y ) γ {\displaystyle {\hat {U}}(x,y)_{\gamma }} for each solution of the former. [ 16 ]
The propagation constant γ {\displaystyle \gamma } of the guided wave is complex, in general. For a lossless case, the propagation constant might be found to take on either real or imaginary values, depending on the chosen solution of the eigenvalue equation and on the angular frequency ω {\displaystyle \omega } . When γ {\displaystyle \gamma } is purely real, the mode is said to be "below cutoff", since the amplitude of the field phasors tends to exponentially decrease with propagation; an imaginary γ {\displaystyle \gamma } , instead, represents modes said to be "in propagation" or "above cutoff", as the complex amplitude of the phasors does not change with z {\displaystyle z} . [ 17 ]
In circuit theory , the impedance is a generalization of electrical resistance in the case of alternating current , and is measured in ohms ( Ω {\displaystyle \Omega } ). [ 10 ] A waveguide in circuit theory is described by a transmission line having a length and characteristic impedance . [ 18 ] : 2–3, 6–12 [ 19 ] : 14 [ 20 ] In other words, the impedance indicates the ratio of voltage to current of the circuit component (in this case a waveguide) during propagation of the wave. This description of the waveguide was originally intended for alternating current, but is also suitable for electromagnetic and sound waves, once the wave and material properties (such as pressure , density , dielectric constant ) are properly converted into electrical terms ( current and impedance for example). [ 21 ] : 14
Impedance matching is important when components of an electric circuit are connected (waveguide to antenna for example): The impedance ratio determines how much of the wave is transmitted forward and how much is reflected. In connecting a waveguide to an antenna a complete transmission is usually required, so an effort is made to match their impedances. [ 20 ]
The reflection coefficient can be calculated using: Γ = Z 2 − Z 1 Z 2 + Z 1 {\displaystyle \Gamma ={\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}} , where Γ {\displaystyle \Gamma } (Gamma) is the reflection coefficient (0 denotes full transmission, 1 full reflection, and 0.5 is a reflection of half the incoming voltage), Z 1 {\displaystyle Z_{1}} and Z 2 {\displaystyle Z_{2}} are the impedance of the first component (from which the wave enters) and the second component, respectively. [ 22 ]
An impedance mismatch creates a reflected wave, which added to the incoming waves creates a standing wave. An impedance mismatch can be also quantified with the standing wave ratio (SWR or VSWR for voltage), which is connected to the impedance ratio and reflection coefficient by: V S W R = | V | m a x | V | m i n = 1 + | Γ | 1 − | Γ | {\displaystyle \mathrm {VSWR} ={\frac {|V|_{\rm {max}}}{|V|_{\rm {min}}}}={\frac {1+|\Gamma |}{1-|\Gamma |}}} , where | V | m i n / m a x {\displaystyle \left|V\right|_{\rm {min/max}}} are the minimum and maximum values of the voltage absolute value , and the VSWR is the voltage standing wave ratio, which value of 1 denotes full transmission, without reflection and thus no standing wave, while very large values mean high reflection and standing wave pattern. [ 20 ]
Waveguides can be constructed to carry waves over a wide portion of the electromagnetic spectrum , but are especially useful in the microwave and optical frequency ranges. Depending on the frequency, they can be constructed from either conductive or dielectric materials. Waveguides are used for transferring both power and communication signals. [ 15 ] : 1–3 [ 23 ] : xiii–xiv
Waveguides used at optical frequencies are typically dielectric waveguides, structures in which a dielectric material with high permittivity , and thus high index of refraction , is surrounded by a material with lower permittivity. The structure guides optical waves by total internal reflection . An example of an optical waveguide is optical fiber . [ 24 ]
Other types of optical waveguide are also used, including photonic-crystal fiber , which guides waves by any of several distinct mechanisms. Guides in the form of a hollow tube with a highly reflective inner surface have also been used as light pipes for illumination applications. The inner surfaces may be polished metal, or may be covered with a multilayer film that guides light by Bragg reflection (this is a special case of a photonic-crystal fiber). One can also use small prisms around the pipe which reflect light via total internal reflection [ 25 ] —such confinement is necessarily imperfect, however, since total internal reflection can never truly guide light within a lower -index core (in the prism case, some light leaks out at the prism corners). [ 26 ]
An acoustic waveguide is a physical structure for guiding sound waves. Sound in an acoustic waveguide behaves like electromagnetic waves on a transmission line . Waves on a string, like the ones in a tin can telephone , are a simple example of an acoustic waveguide. Another example are pressure waves in the pipes of an organ . The term acoustic waveguide is also used to describe elastic waves guided in micro-scale devices, like those employed in piezoelectric delay lines and in stimulated Brillouin scattering .
Waveguides are interesting objects of study from a strictly mathematical perspective. A waveguide (or tube) is defined as type of boundary condition on the wave equation such that the wave function must be equal to zero on the boundary and that the allowed region is finite in all dimensions but one (an infinitely long cylinder is an example.) A large number of interesting results can be proven from these general conditions. It turns out that any tube with a bulge (where the width of the tube increases) admits at least one bound state that exist inside the mode gaps. The frequencies of all the bound states can be identified by using a pulse short in time. This can be shown using the variational principles. An interesting result by Jeffrey Goldstone and Robert Jaffe is that any tube of constant width with a twist, admits a bound state. [ 27 ]
Sound synthesis uses digital delay lines as computational elements to simulate wave propagation in tubes of wind instruments and the vibrating strings of string instruments . [ 28 ]
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In radio-frequency engineering and communications engineering , a waveguide is a hollow metal pipe used to carry radio waves . [ 1 ] This type of waveguide is used as a transmission line mostly at microwave frequencies, for such purposes as connecting microwave transmitters and receivers to their antennas , in equipment such as microwave ovens , radar sets, satellite communications , and microwave radio links.
The electromagnetic waves in a (metal-pipe) waveguide may be imagined as travelling down the guide in a zig-zag path, being repeatedly reflected between opposite walls of the guide. For the particular case of rectangular waveguide , it is possible to base an exact analysis on this view. Propagation in a dielectric waveguide may be viewed in the same way, with the waves confined to the dielectric by total internal reflection at its surface. Some structures, such as non-radiative dielectric waveguides and the Goubau line , use both metal walls and dielectric surfaces to confine the wave.
Depending on the frequency, waveguides can be constructed from either conductive or dielectric materials. Generally, the lower the frequency to be passed the larger the waveguide is. For example, the natural waveguide the earth forms given by the dimensions between the conductive ionosphere and the ground as well as the circumference at the median altitude of the Earth is resonant at 7.83 Hz. This is known as Schumann resonance . On the other hand, waveguides used in extremely high frequency (EHF) communications can be less than a millimeter in width.
During the 1890s theorists did the first analyses of electromagnetic waves in ducts. [ 3 ] Around 1893 J. J. Thomson derived the electromagnetic modes inside a cylindrical metal cavity. [ 3 ] In 1897 Lord Rayleigh did a definitive analysis of waveguides; he solved the boundary value problem of electromagnetic waves propagating through both conducting tubes and dielectric rods of arbitrary shape. [ 3 ] [ 4 ] [ 5 ] [ 6 ] He showed that the waves could travel without attenuation only in specific normal modes with either the electric field ( TE modes ) or magnetic field ( TM modes ), perpendicular to the direction of propagation. He also showed each mode had a cutoff frequency below which waves would not propagate. Since the cutoff wavelength for a given tube was of the same order as its width, it was clear that a hollow conducting tube could not carry radio wavelengths much larger than its diameter. In 1902 R. H. Weber observed that electromagnetic waves travel at a slower speed in tubes than in free space, and deduced the reason; that the waves travel in a "zigzag" path as they reflect from the walls. [ 3 ] [ 5 ] [ 7 ]
Prior to the 1920s, practical work on radio waves concentrated on the low frequency end of the radio spectrum, as these frequencies were better for long-range communication. [ 3 ] These were far below the frequencies that could propagate in even large waveguides, so there was little experimental work on waveguides during this period, although a few experiments were done. In a June 1, 1894 lecture, "The work of Hertz", before the Royal Society , Oliver Lodge demonstrated the transmission of 3 inch radio waves from a spark gap through a short cylindrical copper duct. [ 3 ] [ 8 ] In his pioneering 1894-1900 research on microwaves, Jagadish Chandra Bose used short lengths of pipe to conduct the waves, so some sources credit him with inventing the waveguide. [ 9 ] However, after this, the concept of radio waves being carried by a tube or duct passed out of engineering knowledge. [ 3 ]
During the 1920s the first continuous sources of high frequency radio waves were developed: the Barkhausen–Kurz tube , [ 10 ] the first oscillator which could produce power at UHF frequencies; and the split-anode magnetron which by the 1930s had generated radio waves at up to 10 GHz. [ 3 ] These made possible the first systematic research on microwaves in the 1930s. It was discovered that transmission lines used to carry lower frequency radio waves, parallel line and coaxial cable , had excessive power losses at microwave frequencies, creating a need for a new transmission method. [ 3 ] [ 10 ]
The waveguide was developed independently between 1932 and 1936 by George C. Southworth at Bell Telephone Laboratories [ 2 ] and Wilmer L. Barrow at the Massachusetts Institute of Technology , who worked without knowledge of one another. [ 3 ] [ 5 ] [ 6 ] [ 10 ] Southworth's interest was sparked during his 1920s doctoral work in which he measured the dielectric constant of water with a radio frequency Lecher line in a long tank of water. He found that if he removed the Lecher line, the tank of water still showed resonance peaks, indicating it was acting as a dielectric waveguide . [ 3 ] At Bell Labs in 1931 he resumed work in dielectric waveguides. By March 1932 he observed waves in water-filled copper pipes. Rayleigh's previous work had been forgotten, and Sergei A. Schelkunoff , a Bell Labs mathematician, did theoretical analyses of waveguides [ 3 ] [ 11 ] and rediscovered waveguide modes. In December 1933 it was realized that with a metal sheath the dielectric is superfluous and attention shifted to metal waveguides.
Barrow had become interested in high frequencies in 1930 studying under Arnold Sommerfeld in Germany. [ 3 ] At MIT beginning in 1932 he worked on high frequency antennas to generate narrow beams of radio waves to locate aircraft in fog. He invented a horn antenna and hit on the idea of using a hollow pipe as a feedline to feed radio waves to the antenna. [ 3 ] By March 1936 he had derived the propagation modes and cutoff frequency in a rectangular waveguide. [ 10 ] The source he was using had a large wavelength of 40 cm, so for his first successful waveguide experiments he used a 16-foot section of air duct, 18 inches in diameter. [ 3 ]
Barrow and Southworth became aware of each other's work a few weeks before both were scheduled to present papers on waveguides to a combined meeting of the American Physical Society and the Institute of Radio Engineers in May 1936. [ 3 ] [ 10 ] They amicably worked out credit sharing and patent division arrangements.
The development of centimeter radar during World War 2 and the first high power microwave tubes, the klystron (1938) and cavity magnetron (1940), resulted in the first widespread use of waveguide. [ 10 ] Standard waveguide "plumbing" components were manufactured, with flanges on the end which could be bolted together. After the war in the 1950s and 60s waveguides became common in commercial microwave systems, such as airport radar and microwave relay networks which were built to transmit telephone calls and television programs between cities.
In the microwave region of the electromagnetic spectrum , a waveguide normally consists of a hollow metallic conductor. These waveguides can take the form of single conductors with or without a dielectric coating, e.g. the Goubau line and helical waveguides. Hollow waveguides must be one-half wavelength or more in diameter in order to support one or more transverse wave modes.
Waveguides may be filled with pressurized gas to inhibit arcing and prevent multipaction , allowing higher power transmission. Conversely, waveguides may be required to be evacuated as part of evacuated systems (e.g. electron beam systems).
A slotted waveguide is generally used for radar and other similar applications. The waveguide serves as a feed path, and each slot is a separate radiator, thus forming an antenna. This structure has the capability of generating a radiation pattern to launch an electromagnetic wave in a specific relatively narrow and controllable direction.
A closed waveguide is an electromagnetic waveguide (a) that is tubular, usually with a circular or rectangular cross section, (b) that has electrically conducting walls, (c) that may be hollow or filled with a dielectric material, (d) that can support a large number of discrete propagating modes, though only a few may be practical, (e) in which each discrete mode defines the propagation constant for that mode, (f) in which the field at any point is describable in terms of the supported modes, (g) in which there is no radiation field, and (h) in which discontinuities and bends may cause mode conversion but not radiation. [ citation needed ]
The dimensions of a hollow metallic waveguide determine which wavelengths it can support, and in which modes. Typically the waveguide is operated so that only a single mode is present. The lowest order mode possible is generally selected. Frequencies below the guide's cutoff frequency will not propagate. It is possible to operate waveguides at higher order modes, or with multiple modes present, but this is usually impractical.
Waveguides are almost exclusively made of metal and mostly rigid structures. There are certain types of "corrugated" waveguides that have the ability to flex and bend but only used where essential since they degrade propagation properties. Due to propagation of energy in mostly air or space within the waveguide, it is one of the lowest loss transmission line types and highly preferred for high frequency applications where most other types of transmission structures introduce large losses. Due to the skin effect at high frequencies, electric current along the walls penetrates typically only a few micrometers into the metal of the inner surface. Since this is where most of the resistive loss occurs, it is important that the conductivity of interior surface be kept as high as possible. For this reason, most waveguide interior surfaces are plated with copper , silver , or gold .
Voltage standing wave ratio ( VSWR ) measurements may be taken to ensure that a waveguide is contiguous and has no leaks or sharp bends. If such bends or holes in the waveguide surface are present, this may diminish the performance of both transmitter and receiver equipment connected at either end. Poor transmission through the waveguide may also occur as a result of moisture build up which corrodes and degrades conductivity of the inner surfaces, which is crucial for low loss propagation. For this reason, waveguides are nominally fitted with microwave windows at the outer end that will not interfere with propagation but keep the elements out. Moisture can also cause fungus build up or arcing in high power systems such as radio or radar transmitters. Moisture in waveguides can typically be prevented with silica gel , a desiccant , or slight pressurization of the waveguide cavities with dry nitrogen or argon . Desiccant silica gel canisters may be attached with screw-on nibs and higher power systems will have pressurized tanks for maintaining pressure including leakage monitors. Arcing may also occur if there is a hole, tear or bump in the conducting walls, if transmitting at high power (usually 200 watts or more). Waveguide plumbing [ 12 ] is crucial for proper waveguide performance. Voltage standing waves occur when impedance mismatches in the waveguide cause energy to reflect back in the opposite direction of propagation. In addition to limiting the effective transfer of energy, these reflections can cause higher voltages in the waveguide and damage equipment.
In practice, waveguides act as the equivalent of cables for super high frequency (SHF) systems. For such applications, it is desired to operate waveguides with only one mode propagating through the waveguide. With rectangular waveguides, it is possible to design the waveguide such that the frequency band over which only one mode propagates is as high as 2:1 (i.e. the ratio of the upper band edge to lower band edge is two). The relation between the waveguide dimensions and the lowest frequency is simple: if W {\displaystyle \scriptstyle W} is the greater of its two dimensions, then the longest wavelength that will propagate is λ = 2 W {\displaystyle \lambda \;=\;2W} and the lowest frequency is thus f = c / λ = c / 2 W {\displaystyle f\;=\;c/\lambda \;=\;c/2W}
With circular waveguides, the highest possible bandwidth allowing only a single mode to propagate is only 1.3601:1. [ 13 ]
Because rectangular waveguides have a much larger bandwidth over which only a single mode can propagate, standards exist for rectangular waveguides, but not for circular waveguides. In general (but not always), standard waveguides are designed such that
The first condition is to allow for applications near band edges. The second condition limits dispersion , a phenomenon in which the velocity of propagation is a function of frequency. It also limits the loss per unit length. The third condition is to avoid evanescent-wave coupling via higher order modes. The fourth condition is that which allows a 2:1 operation bandwidth. Although it is possible to have a 2:1 operating bandwidth when the height is less than half the width, having the height exactly half the width maximizes the power that can propagate inside the waveguide before dielectric breakdown occurs.
Below is a table of standard waveguides. The waveguide name WR stands for waveguide rectangular , and the number is the inner dimension width of the waveguide in hundredths of an inch (0.01 inch = 0.254 mm) rounded to the nearest hundredth of an inch.
For the frequencies in the table above, the main advantage of waveguides over coaxial cables is that waveguides support propagation with lower loss. For lower frequencies, the waveguide dimensions become impractically large, and for higher frequencies the dimensions become impractically small (the manufacturing tolerance becomes a significant portion of the waveguide size).
Electromagnetic waveguides are analyzed by solving Maxwell's equations , or their reduced form, the electromagnetic wave equation , with boundary conditions determined by the properties of the materials and their interfaces. These equations have multiple solutions, or modes, which are eigenfunctions of the equation system. Each mode is characterized by a cutoff frequency below which the mode cannot exist in the guide. Waveguide propagation modes depend on the operating wavelength and polarization and the shape and size of the guide. The longitudinal mode of a waveguide is a particular standing wave pattern formed by waves confined in the cavity. The transverse modes are classified into different types:
Waveguides with certain symmetries may be solved using the method of separation of variables . Rectangular wave guides may be solved in rectangular coordinates. [ 16 ] : 143 Round waveguides may be solved in cylindrical coordinates. [ 16 ] : 198
In hollow, single conductor waveguides, TEM waves are not possible. This contrasts with two-conductor transmission lines used at lower frequencies; coaxial cable , parallel wire line and stripline , in which TEM mode is possible. Additionally, the propagating modes (i.e. TE and TM) inside the waveguide can be mathematically expressed as the superposition of two TEM waves. [ 17 ]
The mode with the lowest cutoff frequency is termed the dominant mode of the guide. It is common to choose the size of the guide such that only this one mode can exist in the frequency band of operation. In rectangular and circular (hollow pipe) waveguides, the dominant modes are designated the TE 1,0 mode and TE 1,1 modes respectively. [ 18 ]
A dielectric waveguide employs a solid dielectric rod rather than a hollow pipe. An optical fibre is a dielectric guide designed to work at optical frequencies. Transmission lines such as microstrip , coplanar waveguide , stripline or coaxial cable may also be considered to be waveguides.
Dielectric rod and slab waveguides are used to conduct radio waves, mostly at millimeter wave frequencies and above. [ 19 ] [ 20 ] These confine the radio waves by total internal reflection from the step in refractive index due to the change in dielectric constant at the material surface. [ 21 ] At millimeter wave frequencies and above, metal is not a good conductor, so metal waveguides can have increasing attenuation. At these wavelengths dielectric waveguides can have lower losses than metal waveguides. Optical fibre is a form of dielectric waveguide used at optical wavelengths.
One difference between dielectric and metal waveguides is that at a metal surface the electromagnetic waves are tightly confined; at high frequencies the electric and magnetic fields penetrate a very short distance into the metal. In contrast, the surface of the dielectric waveguide is an interface between two dielectrics, so the fields of the wave penetrate outside the dielectric in the form of an evanescent (non-propagating) wave. [ 21 ]
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In the physical sciences , the wavenumber (or wave number ), also known as repetency , [ 1 ] is the spatial frequency of a wave . Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of reciprocal length , expressed in SI units of cycles per metre or reciprocal metre (m −1 ). Angular wavenumber , defined as the wave phase divided by time, is a quantity with dimension of angle per length and SI units of radians per metre. [ 2 ] [ 3 ] [ 4 ] They are analogous to temporal frequency , respectively the ordinary frequency , defined as the number of wave cycles divided by time (in cycles per second or reciprocal seconds ), and the angular frequency , defined as the phase angle divided by time (in radians per second).
In multidimensional systems , the wavenumber is the magnitude of the wave vector . The space of wave vectors is called reciprocal space . Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering , such as X-ray diffraction , neutron diffraction , electron diffraction , and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck constant is the canonical momentum .
Wavenumber can be used to specify quantities other than spatial frequency. For example, in optical spectroscopy , it is often used as a unit of temporal frequency assuming a certain speed of light .
Wavenumber, as used in spectroscopy and most chemistry fields, [ 5 ] is defined as the number of wavelengths per unit distance:
where λ is the wavelength. It is sometimes called the "spectroscopic wavenumber". [ 1 ] It equals the spatial frequency . [ 6 ]
In theoretical physics, an angular wave number, defined as the number of radians per unit distance is more often used: [ 7 ]
The SI unit of spectroscopic wavenumber is the reciprocal m, written m −1 .
However, it is more common, especially in spectroscopy , to give wavenumbers in cgs units i.e., reciprocal centimeters or cm −1 , with
Occasionally in older references, the unit kayser (after Heinrich Kayser ) is used; [ 8 ] it is abbreviated as K or Ky , where 1 K = 1 cm −1 . [ 9 ]
Angular wavenumber may be expressed in the unit radian per meter (rad⋅m −1 ), or as above, since the radian is dimensionless .
The frequency of light with wavenumber ν ~ {\displaystyle {\tilde {\nu }}} is
where c {\displaystyle c} is the speed of light .
The conversion from spectroscopic wavenumber to frequency is therefore [ 10 ]
Wavenumber can also be used as unit of energy , since a photon of frequency f {\displaystyle f} has energy h f {\displaystyle hf} , where h {\displaystyle h} is Planck's constant .
The energy of a photon with wavenumber ν ~ {\displaystyle {\tilde {\nu }}} is
The conversion from spectroscopic wavenumber to energy is therefore
where energy is expressed either in J or eV .
A complex-valued wavenumber can be defined for a medium with complex-valued relative permittivity ε r {\displaystyle \varepsilon _{r}} , relative permeability μ r {\displaystyle \mu _{r}} and refraction index n as: [ 11 ]
where k 0 is the free-space wavenumber, as above. The imaginary part of the wavenumber expresses attenuation per unit distance and is useful in the study of exponentially decaying evanescent fields .
The propagation factor of a sinusoidal plane wave propagating in the positive x direction in a linear material is given by [ 12 ] : 51
where
The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x-direction.
Wavelength , phase velocity , and skin depth have simple relationships to the components of the wavenumber:
Here we assume that the wave is regular in the sense that the different quantities describing the wave such as the wavelength, frequency and thus the wavenumber are constants. See wavepacket for discussion of the case when these quantities are not constant.
In general, the angular wavenumber k (i.e. the magnitude of the wave vector ) is given by
where ν is the frequency of the wave, λ is the wavelength, ω = 2 πν is the angular frequency of the wave, and v p is the phase velocity of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a dispersion relation .
For the special case of an electromagnetic wave in a vacuum, in which the wave propagates at the speed of light, k is given by:
where E is the energy of the wave, ħ is the reduced Planck constant , and c is the speed of light in a vacuum.
For the special case of a matter wave , for example an electron wave, in the non-relativistic approximation (in the case of a free particle , that is, the particle has no potential energy):
Here p is the momentum of the particle, m is the mass of the particle, E is the kinetic energy of the particle, and ħ is the reduced Planck constant .
Wavenumber is also used to define the group velocity .
In spectroscopy , "wavenumber" ν ~ {\displaystyle {\tilde {\nu }}} (in reciprocal centimeters , cm −1 ) refers to a temporal frequency (in hertz) which has been divided by the speed of light in vacuum (usually in centimeters per second, cm⋅s −1 ):
The historical reason for using this spectroscopic wavenumber rather than frequency is that it is a convenient unit when studying atomic spectra by counting fringes per cm with an interferometer : the spectroscopic wavenumber is the reciprocal of the wavelength of light in vacuum:
which remains essentially the same in air, and so the spectroscopic wavenumber is directly related to the angles of light scattered from diffraction gratings and the distance between fringes in interferometers , when those instruments are operated in air or vacuum. Such wavenumbers were first used in the calculations of Johannes Rydberg in the 1880s. The Rydberg–Ritz combination principle of 1908 was also formulated in terms of wavenumbers. A few years later spectral lines could be understood in quantum theory as differences between energy levels, energy being proportional to wavenumber, or frequency. However, spectroscopic data kept being tabulated in terms of spectroscopic wavenumber rather than frequency or energy.
For example, the spectroscopic wavenumbers of the emission spectrum of atomic hydrogen are given by the Rydberg formula :
where R is the Rydberg constant , and n i and n f are the principal quantum numbers of the initial and final levels respectively ( n i is greater than n f for emission).
A spectroscopic wavenumber can be converted into energy per photon E by Planck's relation :
It can also be converted into wavelength of light:
where n is the refractive index of the medium . Note that the wavelength of light changes as it passes through different media, however, the spectroscopic wavenumber (i.e., frequency) remains constant.
Often spatial frequencies are stated by some authors "in wavenumbers", [ 13 ] incorrectly transferring the name of the quantity to the CGS unit cm −1 itself. [ 14 ]
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A waverider is a hypersonic aircraft design that improves its supersonic lift-to-drag ratio by using the shock waves being generated by its own flight as a lifting surface, a phenomenon known as compression lift .
The waverider remains a well-studied design for high-speed aircraft in the Mach 5 and higher hypersonic regime, although no such design has yet entered production. The Boeing X-51 scramjet demonstration aircraft was tested from 2010 to 2013. In its final test flight, it reached a speed of Mach 5.1 (5,400 km/h; 3,400 mph). [ 1 ] [ 2 ]
The waverider design concept was first developed by Terence Nonweiler of the Queen's University of Belfast , and first described in print in 1951 as a re-entry vehicle. [ 3 ] It consisted of a delta-wing platform with a low wing loading to provide considerable surface area to dump the heat of re-entry. At the time, Nonweiler was forced to use a greatly simplified 2D model of airflow around the aircraft, which he realized would not be accurate due to spanwise flow across the wing. However, he also noticed that the spanwise flow would be stopped by the shockwave being generated by the aircraft, and that if the wing was positioned to deliberately approach the shock, the spanwise flow would be trapped under wing, increasing pressure, and thus increasing lift.
In the 1950s, the British started a space program based around the Blue Streak missile , which was, at some point, to include a crewed vehicle. Armstrong-Whitworth were contracted to develop the re-entry vehicle, and unlike the U.S. space program, they decided to stick with a winged vehicle instead of a ballistic capsule . Between 1957 and 1959, they contracted Nonweiler to develop his concepts further. This work produced a pyramid -shaped design with a flat underside and short wings. Heat was conducted through the wings to the upper cool surfaces, where it was dumped into the turbulent air on the top of the wing. In 1960, work on the Blue Streak was canceled as the missile was seen as being obsolete before it could have entered service. Work then moved to the Royal Aircraft Establishment (RAE), where it continued as a research program into high-speed (Mach 4 to 7) civilian airliners . [ 4 ]
This work was discovered by engineers at North American Aviation during the early design studies of what would lead to the XB-70 bomber. They re-designed the original "classic" delta wing to incorporate drooping wing tips in order to trap the shock waves mechanically, rather than using a shock cone generated from the front of the aircraft. This mechanism also had two other beneficial effects; it reduced the amount of horizontal lifting surface at the rear of the aircraft, which helped offset a nose-down trim that occurs at high speeds, and it added more vertical surface which helped improve the directional stability, which decreased at high speed. [ citation needed ]
Nonweiler's original design used the shock wave generated by the aircraft as a way to control spanwise flow, and thereby increase the amount of air trapped under the wing in the same way as a wing fence . While working on these concepts, he noticed that it was possible to shape the wing in such a way that the shock wave generated off its leading edge would form a horizontal sheet under the craft. In this case, the airflow would not only be trapped horizontally, spanwise, but vertically as well. The only area the air above the shock wave could escape would be out the back of the sheet where the fuselage ended. Since the air was trapped between this sheet and the fuselage, a large volume of air would be trapped, much more than the more basic approach he first developed. Furthermore, since the shock surface was held at a distance from the craft, shock heating was limited to the leading edges of the wings, lowering the thermal loads on the fuselage.
In 1962 Nonweiler moved to Glasgow University to become Professor of Aerodynamics and Fluid Mechanics. That year his "Delta Wings of Shapes Amenable to Exact Shock-Wave Theory" was published by the Journal of the Royal Aeronautical Society , and earned him that society's Gold Medal . A craft generated using this model looks like a delta wing that has been broken down the center and the two sides folded downward. From the rear it looks like an upside-down V, or alternately, the " caret ", ^, and such designs are known as "caret wings". Two to three years later the concept briefly came into the public eye, due to the airliner work at the RAE that led to the prospect of reaching Australia in 90 minutes. Newspaper articles led to an appearance on Scottish Television . [ citation needed ]
Hawker Siddeley examined the caret wing waverider in the later 1960s as a part of a three-stage lunar rocket design. The first stage was built on an expanded Blue Steel , the second a waverider, and the third a nuclear-powered crewed stage. This work was generalized in 1971 to produce a two-staged reusable spacecraft. The 121-foot (37 m) long first stage was designed as a classical waverider, with air-breathing propulsion for return to the launch site. The upper stage was designed as a lifting body, and would have carried an 8000-pound (3.6 t) payload to low Earth orbit . [ citation needed ]
Nonweiler's work was based on studies of planar 2D shocks due to the difficulty understanding and predicting real-world shock patterns around 3D bodies. As the study of hypersonic flows improved, researchers were able to study waverider designs that used different shockwave shapes, the simplest being the conical shock generated by a cone. In these cases, a waverider is designed to keep the rounded shockwave attached to its wings, not a flat sheet, which increases the volume of air trapped under the surface, and thereby increases lift. [ 5 ]
Unlike the caret wing, the cone flow designs smoothly curve their wings, from near horizontal in the center, to highly drooped where they meet the shock. Like the caret wing, they have to be designed to operate at a specific speed to properly attach the shock wave to the wing's leading edge, but unlike them the entire body shape can be varied dramatically at the different design speeds, and sometimes have wingtips that curve upward to attach to the shockwave. [ citation needed ]
Further development of the conical sections, adding canopies and fuselage areas, led to the "osculating cones waverider", which develops several conical shock waves at different points on the body, blending them to produce a single shaped shock. The expansion to a wider range of compression surface flows allowed the design of waveriders with control of volume, [ 5 ] upper surface shape, engine integration and centre of pressure position. Performance improvements and off-design analysis continued until 1970. [ 6 ] [ 7 ]
During this period at least one waverider was tested at the Woomera Rocket Range , mounted on the nose of an air-launched Blue Steel missile , and a number of airframes were tested in the wind tunnel at NASA's Ames Research Center . However, during the 1970s most work in hypersonics disappeared, and the waverider along with it. [ citation needed ]
One of the many differences between supersonic and hypersonic flight concerns the interaction of the boundary layer and the shock waves generated from the nose of the aircraft. Normally the boundary layer is quite thin compared to the streamline of airflow over the wing, and can be considered separately from other aerodynamic effects. However, as the speed increases and the shock wave increasingly approaches the sides of the craft, there comes a point where the two start to interact and the flowfield becomes very complex. Long before that point, the boundary layer starts to interact with the air trapped between the shock wave and the fuselage, the air that is being used for lift on a waverider.
Calculating the effects of these interactions was beyond the abilities of aerodynamics until the introduction of useful computational fluid dynamics starting in the 1980s. In 1981, Maurice Rasmussen at the University of Oklahoma started a waverider renaissance by publishing a paper on a new 3D underside shape using these techniques. These shapes have superior lifting performance and less drag. Since then, whole families of cone -derived waveriders have been designed using more and more complex conic shocks, based on more complex software. This work eventually led to a conference in 1989, the First International Hypersonic Waverider Conference , held at the University of Maryland.
These newest shapes, the "viscous optimized waveriders", look similar to conical designs as long as the angle of the shock wave on the nose is beyond some critical angle, about 14 degrees for a Mach 6 design for instance. The angle of the shock can be controlled by widening out the nose into a curved plate of specific radius, and reducing the radius produces a smaller shock cone angle. Vehicle design starts by selecting a given angle and then developing the body shape that traps that angle, then repeating this process for different angles. For any given speed, a single shape will generate the best results.
During re-entry , hypersonic vehicles generate lift only from the underside of the fuselage . The underside, which is inclined to the flow at a high angle of attack , creates lift in reaction to the vehicle wedging the airflow downwards. The amount of lift is not particularly high, compared to a traditional wing , but more than enough to maneuver given the amount of distance the vehicle covers.
Most re-entry vehicles have been based on the blunt-nose reentry design pioneered by Theodore von Kármán . [ citation needed ] He demonstrated that a shock wave is forced to "detach" from a curved surface, forced out into a larger configuration that requires considerable energy to form. Energy expended in forming this shock wave is no longer available as heat, so this shaping can dramatically reduce the heat load on the spacecraft. Such a design has been the basis for almost every re-entry vehicle since, [ citation needed ] found on the blunt noses of the early ICBM warheads, the bottoms of the various NASA capsules, and the large nose of the Space Shuttle .
The problem with the blunt-nose system is that the resulting design creates very little lift, meaning the vehicle has problems maneuvering during re-entry. If the spacecraft is meant to be able to return to its point of launch "on command", then some sort of maneuvering will be required to counteract the fact that the Earth is turning under the spacecraft as it flies. After a single low Earth orbit , the launching point will be over 1,000 km (600 mi) to the east of the spacecraft by the time it has completed one full orbit. A considerable amount of research was dedicated to combining the blunt-nose system with wings, leading to the development of the lifting body designs in the U.S. [ citation needed ]
It was while working on one such design that Nonweiler developed the waverider. He noticed that the detachment of the shock wave over the blunt leading edges of the wings of the Armstrong-Whitworth design would allow the air on the bottom of the craft to flow spanwise and escape to the upper part of the wing through the gap between the leading edge and the detached shock wave. This loss of airflow reduced (by up to a quarter) the lift being generated by the waverider, which led to studies on how to avoid this problem and keep the flow trapped under the wing.
Nonweiler's resulting design is a delta-wing with some amount of negative dihedral — the wings are bent down from the fuselage towards the tips. When viewed from the front, the wing resembles a caret symbol ( ) in cross section , and these designs are often referred to as carets. The more modern 3D version typically looks like a rounded letter 'M'. Theoretically, a star-shaped [ clarification needed ] waverider with a frontal cross-section of a "+" or "×" could reduce drag by another 20%. The disadvantage of this design is that it has more area in contact with the shock wave and therefore has more pronounced heat dissipation problems.
Waveriders generally have sharp noses and sharp leading edges on their wings. The underside shock-surface remains attached to this. Air flowing in through the shock surface is trapped between the shock and the fuselage, and can only escape at the rear of the fuselage. With sharp edges, all the lift is retained.
Even though sharp edges get much hotter than rounded ones at the same air density, the improved lift means that waveriders can glide on re-entry at much higher altitudes where the air density is lower. A list ranking various space vehicles in order of heating applied to the airframe would have capsules at the top (re-entering quickly with very high heating loads), waveriders at the bottom (extremely long gliding profiles at high altitude), and the Space Shuttle somewhere in the middle.
Simple waveriders have substantial design problems. First, the obvious designs only work at a particular Mach number , and the amount of lift captured will change dramatically as the vehicle changes speed. Another problem is that the waverider depends on radiative cooling , possible as long as the vehicle spends most of its time at very high altitudes. However these altitudes also demand a very large wing to generate the needed lift in the thin air, and that same wing can become rather unwieldy at lower altitudes and speeds.
Because of these problems, waveriders have not found favor with practical aerodynamic designers, despite the fact that they might make long-distance hypersonic vehicles efficient enough to carry air freight .
Some researchers [ who? ] controversially [ citation needed ] claim that there are designs that overcome these problems. One candidate for a multi-speed waverider is a " caret wing ", operated at different angles of attack. A caret wing is a delta wing with longitudinal conical or triangular slots or strakes . It strongly resembles a paper airplane or rogallo wing . The correct angle of attack would become increasingly precise at higher Mach numbers, but this is a control problem that is theoretically solvable. The wing is said to perform even better if it can be constructed of tight mesh, because that reduces its drag, while maintaining lift. Such wings are said to have the unusual attribute of operating at a wide range of Mach numbers in different fluids with a wide range of Reynolds numbers .
The temperature problem can be solved with some combination of a transpiring surface, exotic materials, and possibly heat-pipes . In a transpiring surface, small amounts of a coolant such as water are pumped through small holes in the aircraft's skin (see transpiration and perspiration ). This design works for Mach 25 spacecraft re-entry shields , and therefore should work for any aircraft that can carry the weight of the coolant. Exotic materials such as carbon-carbon composite do not conduct heat but endure it, but they tend to be brittle . Heatpipes are not widely used at present. Like a conventional heat exchanger , they conduct heat better than most solid materials, but like a thermosiphon are passively pumped. The Boeing X-51A deals with external heating through the use of a tungsten nosecone and space shuttle-style heat shield tiles on its belly. Internal (engine) heating is absorbed by using the JP-7 fuel as a coolant prior to combustion. [ 8 ] Other high temperature materials, referred to as SHARP materials (typically zirconium diboride and hafnium diboride ) have been used on steering vanes for ICBM reentry vehicles since the 1970s, and are proposed for use on hypersonic vehicles. They are said to permit Mach 11 flight at 100,000 ft (30,000 m) altitudes and Mach 7 flight at sea level. These materials are more structurally rugged than the Reinforced Carbon Composite (RCC) used on the space shuttle nose and leading edges, have higher radiative and temperature tolerance properties, and do not suffer from oxidation issues that RCC needs to be protected against with coatings. [ 9 ] [ 10 ]
A surface material for waverider and hypersonic ( Mach 5 – 10) vehicles developed by scientists at the China Academy of Aerospace Aerodynamics (CAAA) in Beijing was tested during 2023. [ 11 ]
An alternative developed by RTX Corporation uses a perspiring membrane developed under work supported by the United States Air Force under Contract No. United States Air Force FA8650-20-C-7001 [ 12 ]
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Waviness is the measurement of the more widely spaced component of surface texture . It is a broader view of roughness because it is more strictly defined as "the irregularities whose spacing is greater than the roughness sampling length". It can occur from machine or work deflections , chatter, residual stress , vibrations , or heat treatment . [ 1 ] [ 2 ] Waviness should also be distinguished from flatness , both by its shorter spacing and its characteristic of being typically periodic in nature.
There are several parameters for expressing waviness height, the most common being Wa & Wt, for average waviness and total waviness , respectively. [ 3 ] In the lateral direction along the surface, the waviness spacing , Wsm, is another parameter that describes the mean spacing between periodic waviness peaks. There are numerous measurement settings which influence these resultant parameter values, which are mentioned below. One of the most important is the waviness evaluation length , which is the length in which the waviness parameters are determined. Within this length the waviness profile is determined. This is a surface texture profile that has the shorter roughness characteristics filtered out, or removed; it also does not include any profile changes due to changes in workpiece geometry that are either unintentional (flatness) or intentional (form).
Waviness is included in the ISO standards ISO 4287 [ 3 ] and ISO 16610 -21 [ 4 ] as well as the U.S. standard ASME B46.1, [ 5 ] and it is part of the surface texture symbol used in engineering drawings . [ 6 ]
The measurement of the waviness can be done with a variety of instruments, including both surface finish profilometers and roundness instruments. The nature of these instruments is continually progressing and now includes both stylus-based contact instruments as well as optical & laser-based non-contact instruments. In earlier instruments, the measurement output was inherently linked to the instrument itself, whereas there is now emerging some divergence between the instrument that collects the surface profile data and the analytical software that is able to evaluate this data.
Examples of two earlier generation instruments are the waveometer or a microtopographer . A waveometer uses a plastic tip that is connected to an electronic pickup which then measures the surface variations. The measurement is recorded as an electronic signal which is amplified and split into two signals: a high band and a low band. For measuring a ball bearing , the low band signal records variations that occur every four to seventeen times per revolution and the high band signal records variations that occur seventeen to 330 times per revolution; the low band is the waviness. These bands are transmitted to an oscilloscope for analysis. [ 7 ] [ 8 ]
Waviness measurements are not as common as roughness measurement however there are important applications. For example, waviness in bearing balls and bearing races is one of the reasons for vibrations and noise in ball bearings. Other application examples are waviness in flat milled sealing surfaces, "orange peel" on painted surfaces, and chatter on round shaft surfaces. [ 9 ] [ 10 ] [ 11 ]
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Waxes are a diverse class of organic compounds that are lipophilic , malleable solids near ambient temperatures. They include higher alkanes and lipids , typically with melting points above about 40 °C (104 °F), melting to give low viscosity liquids. Waxes are insoluble in water but soluble in nonpolar organic solvents such as hexane , benzene and chloroform . Natural waxes of different types are produced by plants and animals and occur in petroleum .
Waxes are organic compounds that characteristically consist of long aliphatic alkyl chains, although aromatic compounds may also be present. Natural waxes may contain unsaturated bonds and include various functional groups such as fatty acids , primary and secondary alcohols , ketones , aldehydes and fatty acid esters . Synthetic waxes often consist of homologous series of long-chain aliphatic hydrocarbons ( alkanes or paraffins) that lack functional groups . [ 1 ]
Waxes are synthesized by both plants and animals. Those of animal origin typically consist of wax esters derived from a variety of fatty acids and carboxylic alcohols. In waxes of plant origin, characteristic mixtures of unesterified hydrocarbons may predominate over esters. [ 2 ] The composition depends not only on species, but also on geographic location of the organism.
The best-known animal wax is beeswax , used in constructing the honeycombs of beehives, but other insects also secrete waxes. A major component of beeswax is myricyl palmitate which is an ester of triacontanol and palmitic acid . Its melting point is 62–65 °C (144–149 °F). Spermaceti occurs in large amounts in the head oil of the sperm whale . One of its main constituents is cetyl palmitate , another ester of a fatty acid and a fatty alcohol . Lanolin is a wax obtained from wool, consisting of esters of sterols . [ 1 ]
Plants secrete waxes into and on the surface of their cuticles as a way to control evaporation, wettability and hydration. [ 3 ] The epicuticular waxes of plants are mixtures of substituted long-chain aliphatic hydrocarbons, containing alkanes, alkyl esters, fatty acids, primary and secondary alcohols, diols , ketones and aldehydes. [ 2 ] From the commercial perspective, the most important plant wax is carnauba wax , a hard wax obtained from the Brazilian palm Copernicia prunifera . Containing the ester myricyl cerotate, it has many applications, such as confectionery and other food coatings, car and furniture polish, floss coating, and surfboard wax . Other more specialized vegetable waxes include jojoba oil , candelilla wax and ouricury wax .
Plant and animal based waxes or oils can undergo selective chemical modifications to produce waxes with more desirable properties than are available in the unmodified starting material. [ 4 ] This approach has relied on green chemistry approaches including olefin metathesis and enzymatic reactions and can be used to produce waxes from inexpensive starting materials like vegetable oils. [ 5 ] [ 6 ]
Although many natural waxes contain esters, paraffin waxes are hydrocarbons, mixtures of alkanes usually in a homologous series of chain lengths. These materials represent a significant fraction of petroleum. They are refined by vacuum distillation . Paraffin waxes are mixtures of saturated n- and iso- alkanes , naphthenes , and alkyl - and naphthene-substituted aromatic compounds. A typical alkane paraffin wax chemical composition comprises hydrocarbons with the general formula C n H 2 n +2 , such as hentriacontane , C 31 H 64 . The degree of branching has an important influence on the properties. Microcrystalline wax is a lesser produced petroleum based wax that contains higher percentage of isoparaffinic (branched) hydrocarbons and naphthenic hydrocarbons.
Millions of tons of paraffin waxes are produced annually. They are used in foods (such as chewing gum and cheese wrapping), in candles and cosmetics, as non-stick and waterproofing coatings and in polishes.
Montan wax is a fossilized wax extracted from coal and lignite . [ 7 ] It is very hard, reflecting the high concentration of saturated fatty acids and alcohols. Although dark brown and odorous, they can be purified and bleached to give commercially useful products.
As of 1995 [update] , about 200 million kilograms of polyethylene waxes were consumed annually. [ 3 ]
Polyethylene waxes are manufactured by one of three methods:
Each production technique generates products with slightly different properties. Key properties of low molecular weight polyethylene waxes are viscosity, density and melt point.
Polyethylene waxes produced by means of degradation or recovery from polyethylene resin streams contain very low molecular weight materials that must be removed to prevent volatilization and potential fire hazards during use. Polyethylene waxes manufactured by this method are usually stripped of low molecular weight fractions to yield a flash point >500 °F (>260 °C). Many polyethylene resin plants produce a low molecular weight stream often referred to as low polymer wax (LPW). LPW is unrefined and contains volatile oligomers, corrosive catalyst and may contain other foreign material and water. Refining of LPW to produce a polyethylene wax involves removal of oligomers and hazardous catalyst. Proper refining of LPW to produce polyethylene wax is especially important when being used in applications requiring FDA or other regulatory certification. [ citation needed ]
Waxes are mainly consumed industrially as components of complex formulations, often for coatings. The main use of polyethylene and polypropylene waxes is in the formulation of colourants for plastics . Waxes confer matting effects (i.e., to confer non-glossy finishes) and wear resistance to paints. Polyethylene waxes are incorporated into inks in the form of dispersions to decrease friction. They are employed as release agents , find use as slip agents in furniture, and confer corrosion resistance. [ 3 ]
Waxes such as paraffin wax or beeswax , and hard fats such as tallow are used to make candles , used for lighting and decoration. Another fuel type used in candle manufacturing includes soy . Soy wax is made by the hydrogenation process using soybean oil.
Waxes are used as finishes and coatings for wood products. [ 8 ] Beeswax is frequently used as a lubricant on drawer slides where wood to wood contact occurs.
Sealing wax was used to close important documents in the Middle Ages . Wax tablets were used as writing surfaces. There were different types of wax in the Middle Ages, namely four kinds of wax ( Ragusan , Montenegro , Byzantine , and Bulgarian ), "ordinary" waxes from Spain , Poland , and Riga , unrefined waxes and colored waxes (red, white, and green). [ 9 ] [ 10 ] Waxes are used to make waxed paper , impregnating and coating paper and card to waterproof it or make it resistant to staining, or to modify its surface properties. Waxes are also used in shoe polishes , wood polishes , and automotive polishes, as mold release agents in mold making , as a coating for many cheeses , and to waterproof leather and fabric. Wax has been used since antiquity as a temporary, removable model in lost-wax casting of gold , silver and other materials.
Wax with colorful pigments added has been used as a medium in encaustic painting , and is used today in the manufacture of crayons , china markers and colored pencils . Carbon paper , used for making duplicate typewritten documents was coated with carbon black suspended in wax, typically montan wax , but has largely been superseded by photocopiers and computer printers . In another context, lipstick and mascara are blends of various fats and waxes colored with pigments, and both beeswax and lanolin are used in other cosmetics . Ski wax is used in skiing and snowboarding . Also, the sports of surfing and skateboarding [ 11 ] often use wax to enhance the performance.
Some waxes are considered food-safe and are used to coat wooden cutting boards and other items that come into contact with food. Beeswax or coloured synthetic wax is used to decorate Easter eggs in Romania, Ukraine, Poland, Lithuania and the Czech Republic. Paraffin wax is used in making chocolate covered sweets.
Wax is also used in wax bullets , which are used as simulation aids, and for wax sculpturing .
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The wax argument or the sheet of wax example is a thought experiment that René Descartes created in the second of his Meditations on First Philosophy . He devised it to analyze what properties are essential for bodies, show how uncertain our knowledge of the world is compared to our knowledge of our minds, and argue for rationalism . [ 1 ] [ 2 ]
Descartes first considers all the sensible properties of a sheet of wax such as its shape, texture, size, color, and smell. He then points out that all these properties change as the wax is moved closer to a fire. The only properties that necessarily remain are extension, changeability and movability:
Let us begin by considering the commonest matters, those which we believe to be the most distinctly comprehended, to wit, the bodies which we touch and see; not indeed bodies in general, for these general ideas are usually a little more confused, but let us consider one body in particular. Let us take, for example, this piece of wax: it has been taken quite freshly from the hive, and it has not yet lost the sweetness of the honey which it contains; it still retains somewhat of the odour of the flowers from which it has been culled; its colour, its figure, its size are apparent; it is hard, cold, easily handled, and if you strike it with the finger, it will emit a sound. Finally all the things which are requisite to cause us distinctly to recognise a body, are met with in it. But notice that while I speak and approach the fire what remained of the taste is exhaled, the smell evaporates, the colour alters, the figure is destroyed, the size increases, it becomes liquid, it heats, scarcely can one handle it, and when one strikes it, no sound is emitted. Does the same wax remain after this change? We must confess that it remains; none would judge otherwise. What then did I know so distinctly in this piece of wax? It could certainly be nothing of all that the senses brought to my notice, since all these things which fall under taste, smell, sight, touch, and hearing, are found to be changed, and yet the same wax remains.
Perhaps it was what I now think, viz. that this wax was not that sweetness of honey, nor that agreeable scent of flowers, nor that particular whiteness, nor that figure, nor that sound, but simply a body which a little while before appeared to me as perceptible under these forms, and which is now perceptible under others. But what, precisely, is it that I imagine when I form such conceptions? Let us attentively consider this, and, abstracting from all that does not belong to the wax, let us see what remains. Certainly nothing remains excepting a certain extended thing which is flexible and movable.
These properties are, however, not directly perceived through the senses or imagination (the wax can be extended and moved in more ways than can be imagined). Instead, to grasp the essence of the wax, it must be done through pure reason:
We must then grant that I could not even understand through the imagination what this piece of wax is, and that it is my mind alone which perceives it.
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Wax emulsion s are stable mixtures of one or more waxes in water . Waxes and water are normally immiscible but can be brought together stably by the use of surfactants and a clever preparation process. Strictly speaking a wax emulsion should be called a wax dispersion since the wax is solid at room temperature. However, because the preparation takes place above the melting point of the wax, the actual process is called emulsification, hence the name wax emulsion. In praxis, wax dispersion is used for solvent based systems.
A wide range of emulsions based on different waxes and blends thereof are available, depending on the final application. Waxes that are found in wax emulsions can be of natural or synthetic origin. Common non-fossil natural waxes are carnaubawax, beeswax, candelilla wax or ricebran wax. Paraffin, microcrystalline and montanwax are the most used fossil natural waxes that are found in emulsions. Synthetic waxes that are used include (oxidised) LDPE and HDPE, maleic anhydride grafted polypropylene and Fischer-Tropsch waxes.
A range of different emulsifiers or surfactants are used to emulsify waxes. These can be anionic, cationic or non-ionic in nature. The most common however are fatty alcohol ethoxylates as non-ionic surfactants due to their superb stability against hard water, pH-shock and electrolytes. Some applications demand different emulsifier systems for example anionic surfactants for better hydrophobicity or cationic surfactants for better adhesion to certain materials like textile fibers.
Wax emulsions are widely used in a variety of technical applications like printing inks & lacquers, leather and textiles, paper, wood, metal, polishes, glass fiber sizing, glass bottle protection among other things. The most important properties that can be improved by the addition of wax emulsions are matting & gloss, hydrophobicity, soft touch, abrasion & rub resistance, scratch resistance, release, corrosion protection and anti-blocking. [ 1 ]
Emulsions based on natural waxes are used for coating fruits and candies and crop protection. Synthetic wax based emulsions are often used in food packaging.
Wax emulsions based on beeswax, carnauba wax and paraffin wax are used in creams and ointments.
The emergence of soybean waxes with varying properties and melt points has led to the use of vegetable wax emulsions in applications such as paper coatings, paint and ink additives, and even wet sizing for pulp and paper applications. These wax emulsions can be formulated to deliver some of the same properties that petroleum-based wax emulsions deliver, but offer advantages of being a green product and offer more consistent availability. [ 2 ]
This article about materials science is a stub . You can help Wikipedia by expanding it .
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The Waxman-Bahcall bound is a computed upper limit on the observed flux of high energy neutrinos based on the observed flux of high energy cosmic rays . Since the highest energy neutrinos are produced from interactions of utlra-high-energy cosmic rays, the observed rate of production of the latter places a limit on the former. It is named for John Bahcall and Eli Waxman. [ 1 ]
The Waxman-Bahcall limit comes from the analysis of cosmic rays at various energy levels and their respective fluxes. Cosmic rays are high energy particles, like protons or atomic nuclei that move at near the speed of light. [ 1 ] These rays can come from a variety of sources such as the Sun, the Solar System , the Milky Way galaxy, or even further beyond. [ 2 ] [ 3 ]
Upon entry into our atmosphere, these cosmic rays interact with atoms in the atmosphere, initiating cosmic-ray air showers . These showers are cascades of secondary particles, including muons and neutrinos . [ 2 ] These atmospheric neutrinos can be studied and a general plot of the energy of said neutrinos and their fluxes can be determined and created. The plot below shows the cosmic-ray energy spectrum. Caution: the energy spectrum of atmospheric neutrinos is different; also, the Waxman-Bahcall bound does not apply to atmospheric neutrinos, but to (ultra)-high-energy neutrinos from outside of our galaxy.
[ 2 ]
During Waxman's and Bahcall's research and work into neutrinos, there seemed to be a gap of very high energetic neutrinos, past the atmospheric neutrino limit, but still below the GZK limit, meaning there exists some extra-galactic high energy neutrino source yet to be detected. [ 1 ] [ 3 ]
Atmospheric neutrinos are produced in the atmosphere, about 15 km above the Earth's surface. [ 2 ] They are the result of particles, usually protons or light atomic nuclei, hitting other particles in the atmosphere and causing a shower or neutrinos into the Earths surface. [ 2 ]
Atmospheric neutrinos were successfully detected in the 1960s when experiments were able to successfully find muons that resulted from these neutrinos. [ 4 ] From that, they were able to find the energy of the neutrinos and the flux associated with them. Currently, neutrinos are able to be detected by many different experiments, such as IceCube Lab, allowing for the higher accurate measurements of their energy and fluxes .
The GZK limit exists as a limit on the highest possible cosmic-ray energy that can travel without interaction through the universe, and cosmic rays above around 5 x 10 19 eV [ 5 ] can reach Earth only from the nearby universe. The limit exists because at these higher energies, and at travel distances further than 50 Mpc, interactions of cosmic rays with the CMB photons increase. [ 5 ] With these interactions, the new cosmic-ray product particles have lower and lower energy, and cosmic rays above a few 10 20 eV do not reach Earth (except if their source would be very close). Important in this context is that the GZK interactions also produce neutrinos, called cosmogenic neutrinos. Their energy is typically one order of magnitude below the energy per nucleon of the cosmic ray particle (e.g., a 10 20 eV proton would lead to 10 19 eV neutrinos, but a 10 20 eV iron nucleus with 56 nucleons, would lead to neutrions of 56 times lower energy than for the proton case).
The Waxman - Bahcall upper bound is derived from a problem where neutrinos were discovered to have a higher energy than the atmospheric limit but still below the GZK limit discussed above. Unsure about what possible source could be the cause of these neutrinos, Waxman and Bahcall worked to cross off possible other sources, such as assist from magnetic fields , redshift correction, and sources of high energy outside the Milky Way Galaxy . [ 1 ]
The current upper bound on the intensity of muon neutrino is said to be:
I max = 1.5 ∗ 10 − 8 ∗ ξ z GeV cm − 2 s − 1 sr − 1 {\displaystyle I_{\text{max}}=1.5*10^{-8}*\xi _{z}{\text{GeV}}{\text{cm}}^{-2}{\text{s}}^{-1}{\text{sr}}^{-1}}
with the expected neutrino intensity to be 1/2 I max . [ 1 ]
Initially in the derivation for the muon neutrino intensity above, redshift factors were ignored. However, if a correction factor was included, it could also be found that the neutrinos detected above either started out at high energies and were detected at a lower energy due to redshift. [ 1 ]
It is known, however, that if red-shift is to be the prime factor in the limit, that the proton would have had to have a redshift z of less than 1. If the particle started from outside this range, as told by the GZK limit, other interactions would take place during the particles travel, and make it so that the neutrinos detected would be far below the threshold discussed. [ 1 ]
Deriving a correction factor to multiply by Imax to change the threshold of the system, it was found to be:
ξ z = ∫ 0 z max d z g ( z ) ( 1 + z ) − 7 / 2 f ( z ) ∫ 0 inf d z g ( z ) ( 1 + z ) − 5 / 2 {\displaystyle \xi _{z}={\frac {\int _{0}^{z_{\text{max}}}\mathrm {d} zg(z)(1+z)^{-7/2}f(z)}{\int _{0}^{\inf }\mathrm {d} zg(z)(1+z)^{-5/2}}}}
Working with nearby galaxies and clusters , it was found that there is no significant change on the limit from the redshift correction, and that the reason for the limit and expected values outside of the limit has to come from some other external source. [ 1 ]
Another factor to consider was the addition of the magnetic fields at the source of the neutrinos and how it might allow for increased energy of an incoming charged particle from a cosmic ray. [ 1 ] If protons can be prevented from leaving the source due to a magnetic field, then only neutrinos would be allowed to go through, meaning we would be able to see higher level neutrinos. [ 1 ] Bahcall and Waxman quickly ruled this out as a permanent option, as when there is a proto-meso interaction a charged pion is created but a proton is then also turned into a neutron . The neutron will not be affected by the field in any way, and will travel about 100 kpc when with high energies. This makes it impossible to exceed the upper bound found earlier from Waxman and Bahcall. [ 1 ]
Another theory is that the intergalactic magnetic field would be able to change the direction of the protons on their way to Earth, allowing for the neutrinos to come in relatively in a straight line. [ 6 ]
To derive this theory, Waxman and Bahcall started with the basic proton traveling with energy E , in a magnetic field B , and with correlation length λ . If the proton travels a distance of λ , the resulting angle of deflection is: [ 1 ]
angle = λ / R l {\displaystyle {\text{angle}}=\lambda /R_{l}}
Where R l is the Larmor Radius. [ 1 ]
R l = E / e B {\displaystyle R_{l}=E/eB}
If the angle is kept small, and propagating a distance l , the new deflection angle becomes: [ 1 ]
angle = l / λ ∗ λ / R l {\displaystyle {\text{angle}}={\sqrt {l/\lambda }}*\lambda /R_{l}}
Plugging in values for time, which would give us a maximum propagation distance that the particle could travel in that time, we find that the existence of a uniformly distributed inter-galactic magnetic field would have no effect on the limit. [ 6 ] [ 1 ]
When looking out into the galaxy, and starting to think about what could have caused such high energy neutrinos to appear, it was thought of that jets from Active Galactic Nuclei (AGN) were the main cause. Looking further into the details, Waxman and Bachall saw that the intensities for jets from AGN's are two times higher in magnitude than the limit discussed above. [ 7 ]
Initially, it was thought that the photons and protons were accelerated into the jets thanks to Fermi acceleration with an energy spectrum: [ 1 ]
d N d E ∝ E − 2 {\displaystyle {\frac {\mathrm {d} N}{\mathrm {d} E}}\varpropto E^{-2}}
for both protons and photons (simply plug in the values for photons or protons for either quantity). This implies the optical depth is related to E p and assuming a small optical depth allows us to have the neutrino spectrum of: [ 1 ]
d N d E ∝ E − 1 {\displaystyle {\frac {\mathrm {d} N}{\mathrm {d} E}}\varpropto E^{-1}}
Later, it was realized that the decay of neutral pions, which are created along with charged pions, cause a high energy gamma ray emission. It was then found that the large energies being seen was not a result of Compton scattering of protons and photon, but of neutral pion decay. [ 7 ] Once this emission was fixed, the intensity of the neutrinos found from AGN was under the max limit discussed above, and AGN then became a valid cause for these higher energy neutrinos if the area was optically thin and the energy burst was cause by a single interaction of a decaying neutral pion. [ 1 ]
The Gamma-Ray Bursts (GRB) fireball model has also been another candidate for the reasoning behind higher energy neutrinos. [ 8 ] [ 1 ]
The high energy neutrino model already took multiple variables into account and was a match for the limit discussed above. [ 9 ] Similar to AGN's, the GRB's are optically thin, however, unlike AGN's which needed some more assumptions to be made on how the energy was being expelled and reached to match the flux calculations, the GRB model was able to correctly match this limit. [ 9 ] [ 1 ]
The fireball model works by having the initial burst of the GRB, but then has another shock later on which goes onto explain the afterglow associated with GRB's. This second shock continues to push particles away and allows them to reach detectors on Earth within the limits discussed earlier. [ 9 ]
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Wayfinder was a wholly owned subsidiary of Vodafone [ 2 ] specializing in creating mobile navigation systems for a number of platforms such as Symbian 2nd and 3rd edition, UIQ, Windows Mobile and some other smartphones . An external Bluetooth GPS receiver is required for non GPS enabled phones. On March 12, 2010, it was announced that Vodafone was closing the company and all employees would be let go. [ 3 ]
On July 13, 2010, Wayfinder announced [ 4 ] that they would open source their software under the BSD 3-clause license . [ 5 ] [ 6 ] Source code for both the server (with import tools for map data from OpenStreetMap ) and client software for various phone operating systems (including Android , iPhone and Symbian S60 ) is available at GitHub .
Wayfinder Navigator is a mobile GPS application that provides turn-by-turn directions through a mobile phone. By downloading pre-loaded maps, Navigator can provide directions and different points of interest without Internet connection. [ 7 ]
In May 2008, Wayfinder Navigator was updated to include pre-loaded maps, social networking and map coverage of 150 countries world-wide. [ 8 ]
Wayfinder Earth is a mobile application that displays a 3D globe. The application contains over three million Points of Interest (POIs) including restaurants, train stations, bars, museums, gas stations and hospitals. Another international application, map coverage includes Europe and North America . Wayfinder Earth can set favorites, save address searches and show GPS information such as speed, position and heading as well. [ 9 ]
As the maps are downloaded from the internet on the fly so the phone requires a GPRS / UMTS connection. Maps are cached on the phone memory or memorycard. According to the site, maps can even be downloaded from the site and stored on the phone, once you have registered.
A half an hour of navigating uses approximately 200 kb of data or only 8 kb in the so-called Guide view.
Wayfinder Active was a mobile phone GPS application to track physical outdoor activities. The application calculated exercise statistics such as distance, calories burned and altitude info, so that people could set and track specific exercise goals. [ 10 ] Through an online community, activeoutdoor.com, people could track and share local routes within the community. The product was discontinued spring 2009.
Wayfinder launched the Wayfinder SpeedAlert application in December 2006. The application runs on Java and Symbian 3rd edition only. It warns the user of 15,000 speed cameras around Europe as well as when they exceed a manually set speed limit.
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Wayfinding (or way-finding ) encompasses all of the ways in which people (and animals) orient themselves in physical space and navigate from place to place. Wayfinding software is a self-service computer program that helps users to find a location, usually used indoors and installed on interactive kiosks or smartphones .
The basic process of wayfinding involves four stages:
Historically, wayfinding refers to the techniques used by travelers over land and sea to find relatively unmarked and often mislabeled routes. These include but are not limited to dead reckoning , map and compass , astronomical positioning and, more recently, global positioning . [ 2 ]
Polynesian wayfinding refers to the use of traditional wayfinding and navigation methods by the indigenous peoples of Polynesia . [ 3 ] The ancient Polynesians and Pacific Islanders mastered the methods of wayfinding to explore and settle on the islands of the Pacific, many using devices such as the Marshall Islands stick chart . With these skills, some of them were even able to navigate the ocean as well as they could navigate their own land. Despite the dangers of being out at sea for a long time, wayfinding was a way of life. [ 4 ] Today, The Polynesian Voyaging Society tries-out the traditional Polynesian ways of navigation.
Wayfinding is used in the fields of architecture , urban planning and communication design and refers to the user experience of navigating and orienting oneself within the physical environment. It has been defined as a spatial problem-solving process involving the interpretation of visual and environmental cues to navigate to a destination in a familiar or unfamiliar environment. [ 5 ]
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https://en.wikipedia.org/wiki/Wayfinding_software
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Wayne Wesolowski is a builder of miniature models. [ 1 ]
Wesolowski's models have been exhibited at the Chicago Museum of Science and Industry , [ 2 ] the Springfield, Illinois, Lincoln Home Site, [ 3 ] the West Chicago City Museum, the Batavia Depot Museum, and the National Railroad Museum . [ 4 ]
One of his more noted works is a model of Abraham Lincoln 's funeral train . This model took 4½ years to build and is 15 feet (4½ meters) long. [ 3 ] Wesolowski appeared on an episode of Tracks Ahead featuring this train and his model of Lincoln's home. [ 5 ]
Wesolowski has written scores of articles and four books on model building. [ 6 ] He has been featured in videos shown on PBS television. [ 7 ] Good Morning America selected and showed part of one tape as an example of video education. [ 8 ]
Bob Hundman of Mainline Modeler Magazine noted that "He's always leading those of us who like scratchbuilding down new roads. He's a very inventive modeler." [ 9 ]
Wesolowski holds a Ph.D. in chemistry from the University of Arizona and lectures there. [ 6 ]
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https://en.wikipedia.org/wiki/Wayne_Wesolowski
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WeNMR is a worldwide e-Infrastructure for NMR spectroscopy and structural biology . It is the largest virtual Organization in the life sciences and is supported by EGI .
WeNMR aims at bringing together complementary research teams in the structural biology and life science area into a virtual research community at a worldwide level and provide them with a platform integrating and streamlining the computational approaches necessary for NMR and SAXS data analysis and structural modelling. Access to the infrastructure is provided through a portal integrating commonly used software and GRID technology.
There are about 2 dozen computational NMR services available that can be divided into:
The three-year WeNMR project started in November 2010 as the natural successor of the eNMR project. Financial support was provided by the European Community grants 213010 (eNMR) and 261572 (WeNMR) in the 7th Framework Programme (e-Infrastructure RI-261571).
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https://en.wikipedia.org/wiki/WeNMR
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The Weak-Link Approach (WLA) is a supramolecular coordination-based assembly methodology, first introduced in 1998 by the Mirkin Group at Northwestern University . [ 1 ] This method takes advantage of hemilabile ligands -ligands that contain both strong and weak binding moieties- that can coordinate to metal centers and quantitatively assemble into a single condensed ‘closed’ structure ( Figure 1 ). Unlike other supramolecular assembly methods, the WLA allows for the synthesis of supramolecular complexes that can be modulated from rigid ‘closed’ structures to flexible ‘open’ structures through reversible binding of allosteric effectors at the structural metal centers. The approach is general and has been applied to a variety of metal centers and ligand designs including those with utility in catalysis and allosteric regulation .
There are three main components of the WLA methodology that enable the in situ control of supramolecular architecture: 1) the utilization of hemilabile ligands, 2) the choice of metal centers, and 3) the type of allosteric effector.
A key component of the WLA is the use of hemilabile ligands. [ 2 ] [ 3 ] Hemilabile ligands are polydentate chelates that contain at least two different types of bonding groups, denoted X and Y ( Figure 2 ). The first group (X) bonds strongly to the metal center, while the other group (Y) is weakly bonding and easily displaced by coordinating ligands or solvent molecules (Z). In this way, the substitutionally labile group (Y) can be displaced from the metal center yet remain available for recoordination. For WLA-generated structures, a typical ligand design consists of a phosphine -based strong binding group and a weak-binding group containing O , S , Se , or N . More recent reports have utilized N-heterocyclic carbenes (NHC) as the strong-binding moiety. By using a combination of NHC- and phosphine-based hemilabile ligands, heteroligated complexes, [ 4 ] and macrocycles [ 5 ] have been successfully synthesized, allowing access to more complex architectures with sophisticated functions.
Due to the well-developed understanding of the reactions between the hemilabile ligands and d8 metal ions, the WLA has relied extensively on this type of metal center within its methodology. Initial reports focused on the use of Rh (I), [ 1 ] but Ir (I), [ 6 ] Ni (II), [ 7 ] Pd (II), [ 8 ] and Pt (II) [ 9 ] have all been successfully employed. While d 8 metal centers dominate the WLA literature, d 6 Ru (II) [ 10 ] and d 9 Cu (I) [ 11 ] have also been utilized. Importantly, the choice of metal centers tunes the identity and selectivity of the various allosteric effectors.
The use of hemilabile ligands allows structural motifs synthesized via the WLA to be modified with small molecule effectors much like allosteric enzymes in biology . As described above, the weak Y–M bond can be easily displaced by a coordinating ligands including Cl − , CO , CH 3 CN , RCO 2 − , and a variety of nitriles /isonitriles ( Figure 2 ). Typical WLA constructs rely on the allosteric effector’s stronger affinity for the metal center versus the weakly binding Y moiety. Upon introduction of these effectors, the closed, rigid structures open to their more flexible form. The closed structures can then be reformed in situ by halide abstraction agents, such as noncoordinating silver and thallium salts, or by evacuation of the reaction chamber to remove solvent or small molecules. Recent progress has shown that the inclusion of pendent redox active transition metal groups in the WLA ligands enables control over the binding of ancillary ligands to a redox-inactive Pt(II) center via oxidation and reduction of the distal metal site ( Figure 3 ). [ 12 ] This discovery highlights that new forms of stimuli can be incorporated into the WLA for the design of novel stimuli-responsive materials.
The generality of the WLA and its ability to accommodate a multitude of functional groups has allowed the facile synthesis of both molecular and supramolecular architectures. These structures can be broadly grouped into two classes of compounds based on the coordination geometry of the “closed” complexes: 1) cis -WLA complexes and 2) trans -WLA complexes.
The majority of WLA architectures synthesized to date can be classified as cis -WLA complexes. The strong-binding moieties adopt cis -coordination geometry around the metal center in these complexes, regardless of the identities of the strong-binder. For example, the heteroligated complex shown in Figure 3 is understood to be a cis -WLA complex because both the NHC- and phosphino- groups, the strong-binding components, are cis relative to each other. Using these complexes, molecular tweezers , macrocycles, and triple-layer structures have all been successfully synthesized ( Figure 4 ). In 2017, the Mirkin group reported infinite coordination polymer particles incorporating WLA approach complexes. [ 13 ] The extended structure was successfully obtained by appending secondary terpyridine groups onto the hemilabile ligands within the WLA subunits and allowing them to selectively bind Fe(II) ions ( Figure 5 ).
The first trans -WLA complex was reported by the Mirkin group in 2017. [ 14 ] In this complex, two NHC groups adopt a trans -coordination geometry around a Pd(II) metal center due to the addition of the sterically bulky tert -butyl groups to the imidazole ring of the hemilabile ligand. Upon effector binding, a linear change of up to ~9Å was observed ( Figure 6 ). To date, only this molecular complex has been reported utilizing a trans -WLA complex.
Allosteric regulation in supramolecular structures generated via the WLA is particularly important in the context of designing and synthesizing novel, bioinspired catalytic systems, where the conformation of the complex controls the activity of the catalyst. Below are a series of different catalytic motifs that have been constructed via the WLA and a discussion of the control mechanisms that can be used to modulate catalytic activity:
The first catalytically active supramolecular structure generated via the WLA was designed to operate via a mechanism inspired by the Enzyme Linked ImmunoSorbent Assay ( ELISA ). [ 16 ] In such a supramolecular system, a target sandwiching event creates a catalyst target complex that subsequently generates chemiluminescent or fluorescent readout. For example, a homologated WLA-based Rh(I) macrocyclic structure has been developed that incorporates pyridine -bisimine Zn(II) moieties and behaves as an efficient and completely reversible allosteric modulator for the hydrolysis of 2-(hydroxypropyl)-p-nitrophenyl phosphate (HPNP), a model substrate for RNA ( Figure 7 ). [ 15 ] Significantly, the structural changes induced by small molecule regulators Cl − and CO transition this system from a catalytically inactive state to a very active one in a highly reversible fashion. Further, this system provides a highly sensitive platform for sensing chloride anions . As chloride binds to the Rh(I) centers, the complex is opened, allowing hydrolysis to occur. The hydrolysis product of the reaction (p-nitrophenolate) can be followed by UV-vis spectroscopy . As in ELISA, the WLA-generated mimic can take a small amount of target (chloride anions) and produce a large fluorescent readout that can be utilized for detection.
There are several notable conclusions that can be drawn based on the catalytic studies of this complex. The first is that the closed complex is completely inactive under hydrolysis conditions. Second, the open complex is extremely active and capable of quantitatively hydrolyzing all the HPNP substrate in less than 40 min. By simply bubbling N 2 into the solution, the reformation of the closed complex and the generation of an inactive catalyst can be achieved.
The polymerase chain reaction (PCR) is utilized in biochemistry and molecular biology for exponentially amplifying nucleic acids by making copies of a specific region of a nucleic acid target. When coupled with diagnostic probes, this technique allows one to detect a small collection of molecules under very dilute conditions. A limitation of PCR is that it only works with nucleic acid targets, and there are no known analogues of PCR for other target molecular candidates.
Using the WLA, this type of target amplification approach has been exemplified in an abiotic system. By incorporating Zn(II)- salen ligands into a supramolecular assembly , an acyl transfer reaction involving acetic anhydride and pyridylcarbinol as substrates was investigated. [ 17 ] In the absence of acetate , there is almost no catalytic activity. Once a small amount of tetrabutylammonium acetate reacts with inactive complex at its two rhodium centers that serve as structural regulatory sites, it is converted into open cavity complex, which then catalyzes the reaction ( Figure 8 ).
In the early stages of the reaction, only a minor amount of the catalyst is activated. As the reaction proceeds, more acetate is generated, which leads to the formation of more activated complex and progressively faster catalysis. This type of behavior is typical for cascade reactions including PCR. Unlike the previous example in which the catalyst produced a signal amplifier, this catalyst is a target amplifier making more copies of the target acetate. Following the reaction by gas chromatography , one observes that the generation of products follows a sigmoidal curve , indicative of a PCR-like cascade reaction system.
There was also a need to design a catalytic structure that would allow for the inclusion of mono-metallic catalyst that could be completely turned off. To this end the triple-layer motif was developed, composed of two transition metal nodes, two chemically inert blocking exterior layers, and a single catalytically active interior ligand. This complex was synthesized using the WLA and halide induced ligand rearrangement processes, and it can be reversibly activated and deactivated through small-molecule or elemental anion effector reactions that assemble and disassemble the trilayer structures. In a Al (III)-salen example, the polymerization of ε -caprolactone could be turned on and off based on the ancillary ligands and abstraction agents added to the system ( Figure 9 ). [ 18 ] Unlike with previous catalytic structures that utilized bimetallic systems, the tri-layer motif allows for the incorporation of a monometallic catalyst, opening the scope of potential catalysts that can be employed using these types of structures.
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https://en.wikipedia.org/wiki/Weak-Link_Approach
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Affinity chromatography is a method of separating a biomolecule from a mixture, based on a highly specific macromolecular binding interaction between the biomolecule and another substance. The specific type of binding interaction depends on the biomolecule of interest; antigen and antibody , enzyme and substrate , receptor and ligand , or protein and nucleic acid [ 1 ] binding interactions are frequently exploited for isolation of various biomolecules. Affinity chromatography is useful for its high selectivity and resolution of separation, [ 2 ] [ 3 ] compared to other chromatographic methods.
Affinity chromatography has the advantage of specific binding interactions between the analyte of interest (normally dissolved in the mobile phase ), and a binding partner or ligand (immobilized on the stationary phase ). In a typical affinity chromatography experiment, the ligand is attached to a solid, insoluble matrix—usually a polymer such as agarose or polyacrylamide —chemically modified to introduce reactive functional groups with which the ligand can react, forming stable covalent bonds. [ 4 ] The stationary phase is first loaded into a column to which the mobile phase is introduced. Molecules that bind to the ligand will remain associated with the stationary phase. A wash buffer is then applied to remove non-target biomolecules by disrupting their weaker interactions with the stationary phase, while the biomolecules of interest will remain bound. Target biomolecules may then be removed by applying a so-called elution buffer, which disrupts interactions between the bound target biomolecules and the ligand. The target molecule is thus recovered in the eluting solution. [ 5 ] [ page needed ]
Affinity chromatography does not require the molecular weight, charge, hydrophobicity, or other physical properties of the analyte of interest to be known, although knowledge of its binding properties is useful in the design of a separation protocol. [ 5 ] Types of binding interactions commonly exploited in affinity chromatography procedures are summarized in the table below.
Binding to the solid phase may be achieved by column chromatography whereby the solid medium is packed onto a column, the initial mixture run through the column to allow settling, a wash buffer run through the column and the elution buffer subsequently applied to the column and collected. These steps are usually done at ambient pressure. Alternatively, binding may be achieved using a batch treatment, for example, by adding the initial mixture to the solid phase in a vessel, mixing, separating the solid phase, removing the liquid phase, washing, re-centrifuging, adding the elution buffer, re-centrifuging and removing the elute.
Sometimes a hybrid method is employed such that the binding is done by the batch method, but the solid phase with the target molecule bound is packed onto a column and washing and elution are done on the column.
The ligands used in affinity chromatography are obtained from both organic and inorganic sources. Examples of biological sources are serum proteins, lectins and antibodies. Inorganic sources are moronic acid, metal chelates and triazine dyes. [ 7 ]
A third method, expanded bed absorption, which combines the advantages of the two methods mentioned above, has also been developed. The solid phase particles are placed in a column where liquid phase is pumped in from the bottom and exits at the top. The gravity of the particles ensure that the solid phase does not exit the column with the liquid phase.
Affinity columns can be eluted by changing salt concentrations, pH, pI, charge and ionic strength directly or through a gradient to resolve the particles of interest.
More recently, setups employing more than one column in series have been developed. The advantage compared to single column setups is that the resin material can be fully loaded since non-binding product is directly passed on to a consecutive column with fresh column material. These chromatographic processes are known as periodic counter-current chromatography (PCC). The resin costs per amount of produced product can thus be drastically reduced. Since one column can always be eluted and regenerated while the other column is loaded, already two columns are sufficient to make full use of the advantages. [ 8 ] Additional columns can give additional flexibility for elution and regeneration times, at the cost of additional equipment and resin costs.
Affinity chromatography can be used in a number of applications, including nucleic acid purification, protein purification [ 9 ] from cell free extracts, and purification from blood.
By using affinity chromatography, one can separate proteins that bind to a certain fragment from proteins that do not bind that specific fragment. [ 10 ] Because this technique of purification relies on the biological properties of the protein needed, it is a useful technique and proteins can be purified many folds in one step. [ 11 ] [ page needed ]
Many different affinity media exist for a variety of possible uses. [ 12 ] [ 9 ] [ 13 ] Briefly, they are (generalized) activated/functionalized that work as a functional spacer, support matrix, and eliminates handling of toxic reagents.
Amino acid media is used with a variety of serum proteins, proteins, peptides, and enzymes, as well as rRNA and dsDNA. Avidin biotin media is used in the purification process of biotin/avidin and their derivatives.
Carbohydrate bonding is most often used with glycoproteins or any other carbohydrate-containing substance; carbohydrate is used with lectins, glycoproteins, or any other carbohydrate metabolite protein. Dye ligand media is nonspecific but mimics biological substrates and proteins. Glutathione is useful for separation of GST tagged recombinant proteins. Heparin is a generalized affinity ligand, and it is most useful for separation of plasma coagulation proteins, along with nucleic acid enzymes and lipases
Hydrophobic interaction media are most commonly used to target free carboxyl groups and proteins.
Immunoaffinity media (detailed below) utilizes antigens' and antibodies' high specificity to separate; immobilized metal affinity chromatography is detailed further below and uses interactions between metal ions and proteins (usually specially tagged) to separate; nucleotide/coenzyme that works to separate dehydrogenases, kinases, and transaminases.
Nucleic acids function to trap mRNA, DNA, rRNA, and other nucleic acids/oligonucleotides. Protein A/G method is used to purify immunoglobulins.
Speciality media are designed for a specific class or type of protein/co enzyme; this type of media will only work to separate a specific protein or coenzyme.
Another use for the procedure is the affinity purification of antibodies from blood serum. If the serum is known to contain antibodies against a specific antigen (for example if the serum comes from an organism immunized against the antigen concerned) then it can be used for the affinity purification of that antigen. This is also known as Immunoaffinity Chromatography. For example, if an organism is immunised against a GST-fusion protein it will produce antibodies against the fusion-protein, and possibly antibodies against the GST tag as well. The protein can then be covalently coupled to a solid support such as agarose and used as an affinity ligand in purifications of antibody from immune serum.
For thoroughness, the GST protein and the GST-fusion protein can each be coupled separately. The serum is initially allowed to bind to the GST affinity matrix. This will remove antibodies against the GST part of the fusion protein. The serum is then separated from the solid support and allowed to bind to the GST-fusion protein matrix. This allows any antibodies that recognize the antigen to be captured on the solid support. Elution of the antibodies of interest is most often achieved using a low pH buffer such as glycine pH 2.8. The eluate is collected into a neutral tris or phosphate buffer, to neutralize the low pH elution buffer and halt any degradation of the antibody's activity. This is a nice example as affinity purification is used to purify the initial GST-fusion protein, to remove the undesirable anti-GST antibodies from the serum and to purify the target antibody.
Monoclonal antibodies can also be selected to bind proteins with great specificity, where protein is released under fairly gentle conditions. This can become of use for further research in the future. [ 14 ]
A simplified strategy is often employed to purify antibodies generated against peptide antigens . When the peptide antigens are produced synthetically, a terminal cysteine residue is added at either the N- or C-terminus of the peptide. This cysteine residue contains a sulfhydryl functional group which allows the peptide to be easily conjugated to a carrier protein (e.g. Keyhole limpet hemocyanin (KLH)). The same cysteine-containing peptide is also immobilized onto an agarose resin through the cysteine residue and is then used to purify the antibody.
Most monoclonal antibodies have been purified using affinity chromatography based on immunoglobulin -specific Protein A or Protein G , derived from bacteria. [ 15 ]
Immunoaffinity chromatography with monoclonal antibodies immobilized on monolithic column has been successfully used to capture extracellular vesicles (e.g., exosomes and exomeres) from human blood plasma by targeting tetraspanins and integrins found on the surface of the EVs. [ 16 ] [ 17 ]
Immunoaffinity chromatography is also the basis for immunochromatographic test (ICT) strips, which provide a rapid means of diagnosis in patient care. Using ICT, a technician can make a determination at a patient's bedside, without the need for a laboratory. [ 18 ] ICT detection is highly specific to the microbe causing an infection. [ 19 ]
Immobilized metal ion affinity chromatography (IMAC) is based on the specific coordinate covalent bond of amino acids, particularly histidine, to metals. This technique works by allowing proteins with an affinity for metal ions to be retained in a column containing immobilized metal ions, such as cobalt, nickel, or copper for the purification of histidine-containing proteins or peptides, iron, zinc or gallium for the purification of phosphorylated proteins or peptides. Many naturally occurring proteins do not have an affinity for metal ions, therefore recombinant DNA technology can be used to introduce such a protein tag into the relevant gene. Methods used to elute the protein of interest include changing the pH, or adding a competitive molecule, such as imidazole . [ 20 ] [ 21 ]
Possibly the most common use of affinity chromatography is for the purification of recombinant proteins. Proteins with a known affinity are protein tagged in order to aid their purification. The protein may have been genetically modified so as to allow it to be selected for affinity binding; this is known as a fusion protein. Protein tags include hexahistidine ( His ), glutathione -S-transferase (GST), maltose binding protein (MBP), and the Colicin E7 variant CL7 tag. Histidine tags have an affinity for nickel , cobalt , zinc , copper and iron ions which have been immobilized by forming coordinate covalent bonds with a chelator incorporated in the stationary phase. For elution, an excess amount of a compound able to act as a metal ion ligand, such as imidazole , is used. GST has an affinity for glutathione which is commercially available immobilized as glutathione agarose. During elution, excess glutathione is used to displace the tagged protein. CL7 has an affinity and specificity for Immunity Protein 7 (Im7) which is commercially available immobilized as Im7 agarose resin. For elution, an active and site-specific protease is applied to the Im7 resin to release the tag-free protein. [ 22 ]
Lectin affinity chromatography is a form of affinity chromatography where lectins are used to separate components within the sample. Lectins, such as concanavalin A are proteins which can bind specific alpha-D-mannose and alpha-D-glucose carbohydrate molecules. Some common carbohydrate molecules that is used in lectin affinity chromatography are Con A-Sepharose and WGA-agarose. [ 23 ] Another example of a lectin is wheat germ agglutinin which binds D-N-acetyl-glucosamine. [ 24 ] The most common application is to separate glycoproteins from non-glycosylated proteins, or one glycoform from another glycoform. [ 25 ] Although there are various ways to perform lectin affinity chromatography, the goal is extract a sugar ligand of the desired protein. [ 23 ]
Another use for affinity chromatography is the purification of specific proteins using a gel matrix that is unique to a specific protein. For example, the purification of E. coli β-galactosidase is accomplished by affinity chromatography using p-aminobenyl-1-thio-β-D-galactopyranosyl agarose as the affinity matrix. p-aminobenyl-1-thio-β-D-galactopyranosyl agarose is used as the affinity matrix because it contains a galactopyranosyl group, which serves as a good substrate analog for E. coli β-Galactosidase. This property allows the enzyme to bind to the stationary phase of the affinity matrix and β-Galactosidase is eluted by adding increasing concentrations of salt to the column. [ 26 ]
Alkaline phosphatase from E. coli can be purified using a DEAE-Cellulose matrix. A. phosphatase has a slight negative charge, allowing it to weakly bind to the positively charged amine groups in the matrix. The enzyme can then be eluted out by adding buffer with higher salt concentrations. [ 27 ]
Boronate affinity chromatography consists of using boronic acid or boronates to elute and quantify amounts of glycoproteins . Clinical adaptations have applied this type of chromatography for use in determining long term assessment of diabetic patients through analysis of their glycated hemoglobin . [ 24 ]
Affinity purification of albumin and macroglobulin contamination is helpful in removing excess albumin and α 2 -macroglobulin contamination, when performing mass spectrometry. In affinity purification of serum albumin, the stationary used for collecting or attracting serum proteins can be Cibacron Blue-Sepharose. Then the serum proteins can be eluted from the adsorbent with a buffer containing thiocyanate (SCN − ). [ 28 ]
Weak affinity chromatography [ 29 ] ( WAC ) is an affinity chromatography technique for affinity screening in drug development. [ 30 ] [ 31 ] WAC is an affinity-based liquid chromatographic technique that separates chemical compounds based on their different weak affinities to an immobilized target. The higher affinity a compound has towards the target, the longer it remains in the separation unit, and this will be expressed as a longer retention time. The affinity measure and ranking of affinity can be achieved by processing the obtained retention times of analyzed compounds. Affinity chromatography is part of a larger suite of techniques used in chemoproteomics based drug target identification.
The WAC technology is demonstrated against a number of different protein targets – proteases , kinases , chaperones and protein–protein interaction (PPI) targets. WAC has been shown to be more effective than established methods for fragment based screening. [ 31 ]
Affinity chromatography was conceived and first developed by Pedro Cuatrecasas and Meir Wilchek . [ 32 ] [ 33 ]
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https://en.wikipedia.org/wiki/Weak_affinity_chromatography
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In nuclear physics and atomic physics , weak charge , or rarely neutral weak charge , refers to the Standard Model weak interaction coupling of a particle to the Z boson . For example, for any given nuclear isotope, the total weak charge is approximately −0.99 per neutron , and +0.07 per proton . [ 1 ] It also shows an effect of parity violation during electron scattering .
This same term is sometimes also used to refer to other, different quantities, such as weak isospin [ 2 ] or weak hypercharge ; this article concerns the use of weak charge for a quantity that measures the degree of vector coupling of a fermion to the Z boson (i.e. the coupling strength of weak neutral currents ). [ 3 ]
Measurements in 2017 give the weak charge of the proton as 0.0719 ± 0.0045 . [ 4 ]
The weak charge may be summed in atomic nuclei, so that the predicted weak charge for 133 Cs (55 protons, 78 neutrons) is 55×(+0.0719) + 78×(−0.989) = −73.19, while the value determined experimentally, from measurements of parity violating electron scattering, was −72.58 . [ 5 ]
A recent study used four even-numbered isotopes of ytterbium to test the formula Q w = −0.989 N + 0.071 Z , for weak charge, with N corresponding to the number of neutrons and Z to the number of protons. The formula was found consistent to 0.1% accuracy using the 170 Yb , 172 Yb , 174 Yb , and 176 Yb isotopes of ytterbium . [ 6 ]
In the ytterbium test, atoms were excited by laser light in the presence of electric and magnetic fields, and the resulting parity violation was observed. [ 7 ] The specific transition observed was the forbidden transition from 6s 2 1 S 0 to 5d6s 3 D 1 (24489 cm −1 ). The latter state was mixed, due to weak interaction, with 6s6p 1 P 1 (25068 cm −1 ) to a degree proportional to the nuclear weak charge. [ 6 ]
This table gives the values of the electric charge (the coupling to the photon, referred to in this article as Q ϵ {\displaystyle Q_{\epsilon }} [ a ] ). Also listed are the approximate weak charge Q w {\displaystyle Q_{\mathsf {w}}} (the vector part of the Z boson coupling to fermions), weak isospin T 3 {\displaystyle T_{3}} (the coupling to the W bosons ), weak hypercharge Y w {\displaystyle Y_{\mathsf {w}}} (the coupling to the B boson) and the approximate Z boson coupling factors ( Q L {\displaystyle Q_{\boldsymbol {\mathsf {L}}}} and Q R {\displaystyle Q_{\boldsymbol {\mathsf {R}}}} in the "Theoretical" section, below).
If the variable correction terms shown for different θ w {\displaystyle \ \theta _{\mathsf {w}}\ } values are not added in, then the table's constant values for weak charge are only approximate: They happen to be exact for particles whose energies make the weak mixing angle θ w = 30 ∘ , {\displaystyle \ \theta _{\mathsf {w}}=30^{\circ }\ ,} with sin 2 θ w = 1 4 . {\displaystyle \ \sin ^{2}\theta _{\mathsf {w}}={\tfrac {1}{4}}~.} This value is very close to the typical approximately 29° angle observed in particle accelerators. The embedded formulas give the (more) exact values for when the Weinberg angle , θ w , {\displaystyle \ \theta _{\mathsf {w}}\ ,} is known.
For brevity, the table omits antiparticles. Every particle listed (except for the uncharged bosons the photon , Z boson , gluon , and Higgs boson [ c ] which are their own antiparticles) has an antiparticle with identical mass and opposite charge. All non-zero signs in the table have to be reversed for antiparticles. The paired columns labeled LEFT and RIGHT for fermions (top four rows), have to be swapped in addition to their signs being flipped.
All left-handed (regular) fermions and right-handed antifermions have T 3 = ± 1 2 , {\displaystyle \ T_{3}=\pm {\tfrac {1}{2}}\ ,} and therefore interact with the W boson . They could be referred to as "proper"-handed (that is, they have the "proper" handedness for a W ± interaction). Right-handed fermions and left-handed antifermions, on the other hand, have zero weak isospin and therefore do not interact with the W boson (except for electrical interaction); they could therefore be referred to as "wrong"-handed (i.e. they are "wrong handed" for W ± interactions). "Proper"-handed fermions are organized into isospin doublets, while "wrong"-handed fermions are represented as isospin singlets. While "wrong"-handed particles do not interact with the W boson (no charged current interactions ), all "wrong"-handed fermions known to exist do interact with the Z boson ( neutral current interactions ).
"Wrong"-handed neutrinos ( sterile neutrinos ) have never been observed, but may still exist since they would be invisible to existing detectors. [ 8 ] Sterile neutrinos play a role in speculations about the way neutrinos have masses (see Seesaw mechanism ). The above statement that the Z 0 interacts with all fermions will need an exception for sterile neutrinos inserted, if they are ever detected experimentally.
Massive fermions – except (perhaps) neutrinos [ d ] [ b ] – always exist in a superposition of left-handed and right-handed states, and never in pure chiral states. This mixing is caused by interaction with the Higgs field , which acts as an infinite source and sink of weak isospin and / or hypercharge, due to its non-zero vacuum expectation value (for further information see Higgs mechanism ).
The formula for the weak charge is derived from the Standard Model, and is given by [ 9 ] [ 10 ]
Q w = 2 T 3 − Q ϵ 4 sin 2 θ w ≈ 2 T 3 − Q ϵ , o r Q w + Q ϵ ≈ 2 T 3 = ± 1 ; {\displaystyle ~Q_{\mathsf {w}}~=~2\,T_{3}-Q_{\epsilon }\,4\,\sin ^{2}\theta _{\mathsf {w}}~\approx ~2\,T_{3}-Q_{\epsilon }\;,\qquad {\mathsf {or}}\qquad ~Q_{\mathsf {w}}+Q_{\epsilon }~\approx ~2\,T_{3}~=~\pm 1;~}
where Q w {\displaystyle ~Q_{\mathsf {w}}~} is the weak charge, [ e ] T 3 {\displaystyle T_{3}} is the weak isospin, [ f ] θ w {\displaystyle \theta _{\mathsf {w}}} is the weak mixing angle , and Q ϵ {\displaystyle \,Q_{\epsilon }\,} is the electric charge . [ a ] The approximation for the weak charge is usually valid, since the weak mixing angle typically is 29° ≈ 30° , and 4 sin 2 30 ∘ = 1 , {\displaystyle \ 4\sin ^{2}30^{\circ }=1\ ,} and 4 sin 2 29 ∘ ≈ 0.940 , {\displaystyle \;4\sin ^{2}29^{\circ }\approx 0.940\ ,} a discrepancy of only a little more than 1 in 17 .
This relation only directly applies to quarks and leptons ( fundamental particles ), since weak isospin is not clearly defined for composite particles , such as protons and neutrons, partly due to weak isospin not being conserved. One can set the weak isospin of the proton to + + 1 / 2 and of the neutron to − + 1 / 2 , [ 11 ] [ 12 ] in order to obtain approximate value for the weak charge. Equivalently, one can sum up the weak charges of the constituent quarks to get the same result.
Thus the calculated weak charge for the neutron is
Q w = 2 T 3 − 4 Q ϵ sin 2 θ w = 2 ⋅ ( − 1 2 ) = − 1 ≈ − 0.99 . {\displaystyle Q_{\mathsf {w}}=2\,T_{3}-4\,Q_{\epsilon }\,\sin ^{2}\theta _{\mathsf {w}}=2\cdot \left(-{\tfrac {1}{2}}\right)=-1~\approx ~-0.99~.}
The weak charge for the proton calculated using the above formula and a weak mixing angle of 29° is
Q w = 2 T 3 − 4 Q ϵ sin 2 θ w = 2 1 2 − 4 sin 2 29 ∘ ≈ 1 − 0.94016 = 0.05983 ≈ 0.06 ≈ 0.07 , {\displaystyle Q_{\mathsf {w}}=2\,T_{3}-4\,Q_{\epsilon }\,\sin ^{2}\theta _{\mathsf {w}}~=~2\;{\tfrac {1}{2}}-4\,\sin ^{2}29^{\circ }~\approx ~1-0.94016~=~0.05983\approx 0.06\approx 0.07~,}
a very small value, similar to the nearly zero total weak charge of charged leptons (see the table above).
Corrections arise when doing the full theoretical calculation for nucleons, however. Specifically, when evaluating Feynman diagrams beyond the tree level (i.e. diagrams containing loops), the weak mixing angle becomes dependent on the momentum scale due to the running of coupling constants , [ 10 ] and due to the fact that nucleons are composite particles.
Because weak hypercharge Y w is given by
Y w = 2 ( Q ϵ − T 3 ) {\displaystyle Y_{\mathsf {w}}=2\,(Q_{\epsilon }-T_{3})~}
the weak hypercharge Y w , weak charge Q w , and electric charge Q ≡ Q ϵ {\displaystyle \,Q\equiv Q_{\epsilon }\,} are related by
Q w + Y w = 2 Q ϵ ( 1 − 2 sin 2 θ w ) = 2 Q ϵ cos ( 2 θ w ) , {\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}=2\,Q_{\epsilon }\left(1-2\ \sin ^{2}\theta _{\mathsf {w}}\right)=2\,Q_{\epsilon }\,\cos \left(2\,\theta _{\mathsf {w}}\right)\ ,} or equivalently Q w + Y w = Q ϵ + Q ϵ ( 1 − 4 sin 2 θ w ) ≈ Q ϵ + 0 , {\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}=Q_{\epsilon }+Q_{\epsilon }\left(1-4\ \sin ^{2}\theta _{\mathsf {w}}\right)\approx Q_{\epsilon }+0\ ,}
where Y w {\displaystyle ~Y_{\mathsf {w}}~} is the weak hypercharge for left-handed fermions and right-handed antifermions, hence
Q w + Y w ≈ Q ϵ , {\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}\approx Q_{\epsilon }~,}
in the typical case, when the weak mixing angle is approximately 30°.
The Standard Model coupling of fermions to the Z boson and photon is given by: [ 13 ]
L i n t = − Ψ ¯ L [ ( Q ϵ − T 3 ) e cos θ w B μ + T 3 e sin θ w W μ 3 ] σ ¯ μ Ψ L − Ψ ¯ R [ Q ϵ e cos θ w B μ ] σ μ Ψ R , {\displaystyle {\mathcal {L}}_{\mathrm {int} }~=~-{\bar {\Psi }}_{\boldsymbol {\mathsf {L}}}\,\left[\left(Q_{\epsilon }\,-\,T_{3}\right)\,{\frac {e}{\;\cos \theta _{\mathsf {w}}}}\,B_{\mu }~+~T_{3}\,{\frac {e}{\;\sin \theta _{\mathsf {w}}\,}}W_{\mu }^{3}\;\right]\,{\bar {\sigma }}^{\mu }\,\Psi _{\boldsymbol {\mathsf {L}}}~-~{\bar {\Psi }}_{\boldsymbol {\mathsf {R}}}\,\left[\,Q_{\epsilon }{\frac {e}{\;\cos \theta _{\mathsf {w}}\;}}\,B_{\mu }\,\right]\,\sigma ^{\mu }\,\Psi _{\boldsymbol {\mathsf {R}}}~,}
where
and the expansion uses for its basis vectors the (mostly implicit) Pauli matrices from the Weyl equation : [ clarification needed ]
σ μ = ( I , σ 1 , σ 2 , σ 3 ) {\displaystyle \sigma ^{\mu }={\Bigl (}\,I\,,\;~~\sigma ^{1}\,,\;~~\sigma ^{2}\,,\;~~\sigma ^{3}\,{\Bigr )}~}
and
σ ¯ μ = ( I , − σ 1 , − σ 2 , − σ 3 ) {\displaystyle {\bar {\sigma }}^{\mu }={\Bigl (}\,I\,,\;-\sigma ^{1}\,,\;-\sigma ^{2}\,,\;-\sigma ^{3}\,{\Bigr )}~}
The fields for B and W 3 boson are related to the Z boson field Z μ , {\displaystyle Z_{\mu },} and electromagnetic field A μ {\displaystyle A_{\mu }} (photons) by
B μ = ( cos θ w ) A μ − ( sin θ w ) Z μ {\displaystyle ~B_{\mu }=\left(\,\cos \theta _{\mathsf {w}}\,\right)\,A_{\mu }-\left(\,\sin \theta _{\mathsf {w}}\,\right)Z_{\mu }~}
and
W μ 3 = ( cos θ w ) Z μ + ( sin θ w ) A μ . {\displaystyle W_{\mu }^{3}=\left(\,\cos \theta _{\mathsf {w}}\,\right)Z_{\mu }~+~\left(\,\sin \theta _{\mathsf {w}}\,\right)\,A_{\mu }~.}
By combining these relations with the above equation and separating by Z μ {\displaystyle Z_{\mu }} and A μ , {\displaystyle ~A_{\mu }~,} one obtains:
L i n t = − Ψ ¯ L [ ( Q ϵ − T 3 ) e cos θ w ( cos θ w A μ − sin θ w Z μ ) + T 3 e sin θ w ( cos θ w Z μ + sin θ w A μ ) ] σ ¯ μ Ψ L − Ψ ¯ R [ Q ϵ e cos θ w ( cos θ w A μ − sin θ w Z μ ) ] σ μ Ψ R = − e Ψ ¯ L [ Q ϵ A μ + ( T 3 − Q ϵ sin 2 θ w ) 1 cos θ w sin θ w Z μ ] σ ¯ μ Ψ L − e Ψ ¯ R [ Q ϵ A μ − Q ϵ sin 2 θ w 1 cos θ w sin θ w Z μ ] σ μ Ψ R . {\displaystyle {\begin{aligned}{\mathcal {L}}_{\mathrm {int} }~=~-{\bar {\Psi }}_{\boldsymbol {\mathsf {L}}}\left[\;\left(\,Q_{\epsilon }\,-\,T_{3}\,\right){\frac {e}{\;\cos \theta _{\mathsf {w}}\;}}\left(\;\cos \theta _{\mathsf {w}}\,A_{\mu }-\sin \theta _{\mathsf {w}}\,Z_{\mu }\;\right)\,+\,T_{3}{\frac {e}{\;\sin \theta _{\mathsf {w}}\;}}\left(\;\cos \theta _{\mathsf {w}}Z_{\mu }\,+\,\sin \theta _{\mathsf {w}}\,A_{\mu }\;\right)\right]{\bar {\sigma }}^{\mu }\Psi _{\boldsymbol {\mathsf {L}}}\\-{\bar {\Psi }}_{\boldsymbol {\mathsf {R}}}{\biggl [}Q_{\epsilon }\,{\frac {e}{\;\cos \theta _{\mathsf {w}}\;}}\left(\,\cos \theta _{\mathsf {w}}\,A_{\mu }\,-\,\sin \theta _{\mathsf {w}}\,Z_{\mu }\,\right)\;{\biggr ]}\sigma ^{\mu }\Psi _{\boldsymbol {\mathsf {R}}}\\\\~=~-~e\,{\bar {\Psi }}_{\boldsymbol {\mathsf {L}}}\left[\;Q_{\epsilon }\,A_{\mu }\,+\,\left(\;T_{3}\,-\,Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}\;\right){\frac {1}{\;\cos \theta _{\mathsf {w}}\sin \theta _{\mathsf {w}}\;}}\;Z_{\mu }\;\right]{\bar {\sigma }}^{\mu }\Psi _{\boldsymbol {\mathsf {L}}}\\~-~e\,{\bar {\Psi }}_{\boldsymbol {\mathsf {R}}}\left[\;Q_{\epsilon }\,A_{\mu }\,-\,Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}\;{\frac {1}{\;\cos \theta _{\mathsf {w}}\,\sin \theta _{\mathsf {w}}\;}}\;Z_{\mu }\;\right]\sigma ^{\mu }\Psi _{\boldsymbol {\mathsf {R}}}~.\end{aligned}}}
The Q ϵ A μ {\displaystyle Q_{\epsilon }\,A_{\mu }} term that is present for both left- and right-handed fermions represents the familiar electromagnetic interaction . The terms involving the Z boson depend on the chirality of the fermion, thus there are two different coupling strengths:
Q L = T 3 − Q ϵ sin 2 θ w {\displaystyle ~Q_{\boldsymbol {\mathsf {L}}}=T_{3}-Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}\quad } and Q R = − Q ϵ sin 2 θ w . {\displaystyle \quad Q_{\boldsymbol {\mathsf {R}}}=-Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}~.}
It is however more convenient to treat fermions as a single particle instead of treating left- and right-handed fermions separately. The Weyl basis is chosen for this derivation: [ 14 ]
Ψ ≡ ( Ψ L Ψ R ) , γ μ ≡ ( 0 σ μ σ ¯ μ 0 ) for μ = 0 , 1 , 2 , 3 ; {\displaystyle {\boldsymbol {\Psi }}\equiv {\begin{pmatrix}\Psi _{\boldsymbol {\mathsf {L}}}\\\Psi _{\boldsymbol {\mathsf {R}}}\end{pmatrix}}~,\qquad \gamma ^{\mu }\equiv {\begin{pmatrix}0&\sigma ^{\mu }\\{\bar {\sigma }}^{\mu }&0\end{pmatrix}}\quad {\text{ for }}~\mu =0,1,2,3~;} γ 5 ≡ ( − I 0 0 I ) . {\displaystyle \qquad \gamma ^{5}\equiv {\begin{pmatrix}-I&0\\~~0&I\end{pmatrix}}~.}
Thus the above expression can be written fairly compactly as:
L i n t = − e Ψ ¯ γ μ [ Q ϵ A μ + ( Q w − 2 T 3 γ 5 ) 2 sin ( 2 θ w ) Z μ ] Ψ , {\displaystyle {\mathcal {L}}_{\mathrm {int} }=-e\ {\boldsymbol {\bar {\Psi }}}\ \gamma ^{\mu }\ \left[\ Q_{\epsilon }\ A_{\mu }\;+\;{\frac {\left(\ Q_{\mathsf {w}}-2\ T_{3}\ \gamma ^{5}\ \right)}{\ 2\ \sin \left(\ 2\ \theta _{\mathsf {w}}\ \right)\ }}\;Z_{\mu }\ \right]\ {\boldsymbol {\Psi }}~,}
where
Q w ≡ 2 ( Q L + Q R ) = 2 T 3 − 4 Q ϵ sin 2 θ w . {\displaystyle Q_{\mathsf {w}}\;\equiv \;2\,\left(\,Q_{\boldsymbol {\mathsf {L}}}+Q_{\boldsymbol {\mathsf {R}}}\,\right)\;=\;2\,T_{3}-4\,Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}~.}
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The term weak continuum hypothesis can be used to refer to the hypothesis that 2 ℵ 0 < 2 ℵ 1 {\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}} , which is the negation of the second continuum hypothesis . [ 1 ] : 80 [ 2 ] : Lecture 7 [ 3 ] : 3616 It is equivalent to a weak form of ◊ on ℵ 1 {\displaystyle \aleph _{1}} . [ 4 ] : 2 [ 5 ] F. Burton Jones proved that if it is true, then every separable normal Moore space is metrizable . [ 6 ] : Theorem 5
Weak continuum hypothesis may also refer to the assertion that every uncountable set of real numbers can be placed in bijective correspondence with the set of all reals. This second assertion was Cantor 's original form of the Continuum Hypothesis (CH). Given the Axiom of Choice , it is equivalent to the usual form of CH, that 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph _{0}}=\aleph _{1}} . [ 7 ] : 155 [ 8 ] : 289
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In the Standard Model of electroweak interactions of particle physics , the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin . It is frequently denoted Y W {\displaystyle Y_{\mathsf {W}}} and corresponds to the gauge symmetry U(1) . [ 1 ] [ 2 ]
It is conserved (only terms that are overall weak-hypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with the Higgs field . Since the Higgs field vacuum expectation value is nonzero, particles interact with this field all the time even in vacuum. This changes their weak hypercharge (and weak isospin T 3 ). Only a specific combination of them, Q = T 3 + 1 2 Y W {\displaystyle \ Q=T_{3}+{\tfrac {1}{2}}\,Y_{\mathsf {W}}\ } (electric charge), is conserved.
Mathematically, weak hypercharge appears similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved in weak interactions and is zero for leptons).
In the electroweak theory SU(2) transformations commute with U(1) transformations by definition and therefore U(1) charges for the elements of the SU(2) doublet (for example lefthanded up and down quarks) have to be equal. This is why U(1) cannot be identified with U(1) em and weak hypercharge has to be introduced. [ 3 ] [ 4 ]
Weak hypercharge was first introduced by Sheldon Glashow in 1961. [ 4 ] [ 5 ] [ 6 ]
Weak hypercharge is the generator of the U(1) component of the electroweak gauge group, SU(2) × U(1) and its associated quantum field B mixes with the W 3 electroweak quantum field to produce the observed Z gauge boson and the photon of quantum electrodynamics .
The weak hypercharge satisfies the relation
Q = T 3 + 1 2 Y W , {\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\text{W}}~,}
where Q is the electric charge (in elementary charge units) and T 3 is the third component of weak isospin (the SU(2) component).
Rearranging, the weak hypercharge can be explicitly defined as:
Y W = 2 ( Q − T 3 ) {\displaystyle Y_{\rm {W}}=2(Q-T_{3})}
where "left"- and "right"-handed here are left and right chirality , respectively (distinct from helicity ).
The weak hypercharge for an anti-fermion is the opposite of that of the corresponding fermion because the electric charge and the third component of the weak isospin reverse sign under charge conjugation .
The sum of −isospin and +charge is zero for each of the gauge bosons; consequently, all the electroweak gauge bosons have
Y W = 0 . {\displaystyle \,Y_{\text{W}}=0~.}
Hypercharge assignments in the Standard Model are determined up to a twofold ambiguity by requiring cancellation of all anomalies.
For convenience, weak hypercharge is often represented at half-scale, so that
Y W = Q − T 3 , {\displaystyle \,Y_{\rm {W}}=Q-T_{3}~,}
which is equal to just the average electric charge of the particles in the isospin multiplet . [ 8 ] [ 9 ]
Weak hypercharge is related to baryon number minus lepton number via:
1 2 X + Y W = 5 2 ( B − L ) {\displaystyle {\tfrac {1}{2}}X+Y_{\rm {W}}={\tfrac {5}{2}}(B-L)\,}
where X is a conserved quantum number in GUT . Since weak hypercharge is always conserved within the Standard Model and most extensions, this implies that baryon number minus lepton number is also always conserved.
n → p + e − + ν e
Hence neutron decay conserves baryon number B and lepton number L separately, so also the difference B − L is conserved.
Proton decay is a prediction of many grand unification theories . p + ⟶ e + + π 0 ↓ 2 γ {\displaystyle {\begin{aligned}{\rm {p^{+}\longrightarrow e^{+}+}}&\ \pi ^{0}\\&\downarrow \\&2\gamma \end{aligned}}}
Hence this hypothetical proton decay would conserve B − L , even though it would individually violate conservation of both lepton number and baryon number .
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In mathematical logic , weak interpretability is a notion of translation of logical theories, introduced together with interpretability by Alfred Tarski in 1953.
Let T and S be formal theories . Slightly simplified, T is said to be weakly interpretable in S if, and only if, the language of T can be translated into the language of S in such a way that the translation of every theorem of T is consistent with S . Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas .
A generalization of weak interpretability, tolerance , was introduced by Giorgi Japaridze in 1992.
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https://en.wikipedia.org/wiki/Weak_interpretability
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Weak localization is a physical effect which occurs in disordered electronic systems at very low temperatures. The effect manifests itself as a positive correction to the resistivity of a metal or semiconductor . [ 1 ] The name emphasizes the fact that weak localization is a precursor of Anderson localization , which occurs at strong disorder.
The effect is quantum-mechanical in nature and has the following origin: In a disordered electronic system, the electron motion is diffusive rather than ballistic. That is, an electron does not move along a straight line, but experiences a series of random scatterings off impurities which results in a random walk .
The resistivity of the system is related to the probability of an electron to propagate between two given points in space. Classical physics assumes that the total probability is just the sum of the probabilities of the paths connecting the two points. However quantum mechanics tells us that to find the total probability we have to sum up the quantum-mechanical amplitudes of the paths rather than the probabilities themselves. Therefore, the correct (quantum-mechanical) formula for the probability for an electron to move from a point A to a point B includes the classical part (individual probabilities of diffusive paths) and a number of interference terms (products of the amplitudes corresponding to different paths). These interference terms effectively make it more likely that a carrier will "wander around in a circle" than it would otherwise, which leads to an increase in the net resistivity. The usual formula for the conductivity of a metal (the so-called Drude formula ) corresponds to the former classical terms, while the weak localization correction corresponds to the latter quantum interference terms averaged over disorder realizations.
The weak localization correction can be shown to come mostly from quantum interference between self-crossing paths in which an electron can propagate in the clock-wise and counter-clockwise direction around a loop. Due to the identical length of the two paths along a loop, the quantum phases cancel each other exactly and these (otherwise random in sign) quantum interference terms survive disorder averaging. Since it is much more likely to find a self-crossing trajectory in low dimensions, the weak localization effect manifests itself much more strongly in low-dimensional systems (films and wires). [ 2 ]
In a system with spin–orbit coupling , the spin of a carrier is coupled to its momentum. The spin of the carrier rotates as it goes around a self-intersecting path, and the direction of this rotation is opposite for the two directions about the loop. Because of this, the two paths along any loop interfere destructively which leads to a lower net resistivity. [ 3 ]
In two dimensions the change in conductivity from applying a magnetic field , due to either weak localization or weak anti-localization can be described by the Hikami-Larkin-Nagaoka equation: [ 3 ] [ 4 ]
Where a = 4 D e H / ℏ c {\displaystyle a=4DeH/\hbar c} , and τ , τ 1 , τ 2 , τ 3 {\displaystyle \tau ,\tau _{1},\tau _{2},\tau _{3}} are various relaxation times and σ 0 {\displaystyle \sigma _{0}} is the conductivity of the system in the absence of weak localization or weak anti-localization. This theoretically derived equation was soon restated in terms of characteristic fields, which are more directly experimentally relevant quantities: [ 5 ]
Where the characteristic fields are:
Where H 0 {\displaystyle H_{0}} is potential scattering, H i {\displaystyle H_{i}} is inelastic scattering , H S {\displaystyle H_{S}} is magnetic scattering, and H S O {\displaystyle H_{SO}} is spin-orbit scattering. For a non-magnetic sample ( H S = 0 {\displaystyle H_{S}=0} ), this can be rewritten:
ψ {\displaystyle \psi } is the digamma function . B ϕ {\displaystyle B_{\phi }} is the phase coherence characteristic field, which is roughly the magnetic field required to destroy phase coherence, B SO {\displaystyle B_{\text{SO}}} is the spin–orbit characteristic field which can be considered a measure of the strength of the spin–orbit interaction and B e {\displaystyle B_{e}} is the elastic characteristic field. The characteristic fields are better understood in terms of their corresponding characteristic lengths which are deduced from B i = ℏ / 4 e l i 2 {\displaystyle {B_{i}=\hbar /4el_{i}^{2}}} . l ϕ {\displaystyle l_{\phi }} can then be understood as the distance traveled by an electron before it loses phase coherence, l SO {\displaystyle l_{\text{SO}}} can be thought of as the distance traveled before the spin of the electron undergoes the effect of the spin–orbit interaction, and finally l e {\displaystyle l_{e}} is the mean free path .
In the limit of strong spin–orbit coupling B SO ≫ B ϕ {\displaystyle B_{\text{SO}}\gg B_{\phi }} , the equation above reduces to:
In this equation α {\displaystyle \alpha } is -1 for weak antilocalization and +1/2 for weak localization.
The strength of either weak localization or weak anti-localization falls off quickly in the presence of a magnetic field, which causes carriers to acquire an additional phase as they move around paths.
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All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.
In mathematics , especially order theory , a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set , some of whose members may be tied with each other. Weak orders are a generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders . [ 1 ]
There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic (interconvertable with no loss of information): they may be axiomatized as strict weak orderings (strictly partially ordered sets in which incomparability is a transitive relation ), as total preorders (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as ordered partitions ( partitions of the elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called a preferential arrangement based on a utility function is also possible.
Weak orderings are counted by the ordered Bell numbers . They are used in computer science as part of partition refinement algorithms, and in the C++ Standard Library . [ 2 ]
In horse racing , the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context) dead heats , so the outcome of a horse race may be modeled by a weak ordering. [ 3 ] In an example from the Maryland Hunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied for second place, with the remaining horses farther back; three horses did not finish. [ 4 ] In the weak ordering describing this outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before all the other horses that finished, and the three horses that did not finish would be placed last in the order but tied with each other.
The points of the Euclidean plane may be ordered by their distance from the origin , giving another example of a weak ordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to a common circle centered at the origin), and infinitely many points within these subsets. Although this ordering has a smallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.
Opinion polling in political elections provides an example of a type of ordering that resembles weak orderings, but is better modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another, or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they are within the margin of error of each other. However, if candidate x {\displaystyle x} is statistically tied with y , {\displaystyle y,} and y {\displaystyle y} is statistically tied with z , {\displaystyle z,} it might still be possible for x {\displaystyle x} to be clearly better than z , {\displaystyle z,} so being tied is not in this case a transitive relation . Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings. [ 5 ]
Suppose throughout that < {\displaystyle \,<\,} is a homogeneous binary relation on a set S {\displaystyle S} (that is, < {\displaystyle \,<\,} is a subset of S × S {\displaystyle S\times S} ) and as usual, write x < y {\displaystyle x<y} and say that x < y {\displaystyle x<y} holds or is true if and only if ( x , y ) ∈ < . {\displaystyle (x,y)\in \,<.\,}
Preliminaries on incomparability and transitivity of incomparability
Two elements x {\displaystyle x} and y {\displaystyle y} of S {\displaystyle S} are said to be incomparable with respect to < {\displaystyle \,<\,} if neither x < y {\displaystyle x<y} nor y < x {\displaystyle y<x} is true. [ 1 ] A strict partial order < {\displaystyle \,<\,} is a strict weak ordering if and only if incomparability with respect to < {\displaystyle \,<\,} is an equivalence relation .
Incomparability with respect to < {\displaystyle \,<\,} is always a homogeneous symmetric relation on S . {\displaystyle S.} It is reflexive if and only if < {\displaystyle \,<\,} is irreflexive (meaning that x < x {\displaystyle x<x} is always false), which will be assumed so that transitivity is the only property this "incomparability relation" needs in order to be an equivalence relation .
Define also an induced homogeneous relation ≲ {\displaystyle \,\lesssim \,} on S {\displaystyle S} by declaring that x ≲ y is true if and only if y < x is false {\displaystyle x\lesssim y{\text{ is true }}\quad {\text{ if and only if }}\quad y<x{\text{ is false}}} where importantly, this definition is not necessarily the same as: x ≲ y {\displaystyle x\lesssim y} if and only if x < y or x = y . {\displaystyle x<y{\text{ or }}x=y.} Two elements x , y ∈ S {\displaystyle x,y\in S} are incomparable with respect to < {\displaystyle \,<\,} if and only if x and y {\displaystyle x{\text{ and }}y} are equivalent with respect to ≲ {\displaystyle \,\lesssim \,} (or less verbosely, ≲ {\displaystyle \,\lesssim } -equivalent ), which by definition means that both x ≲ y and y ≲ x {\displaystyle x\lesssim y{\text{ and }}y\lesssim x} are true.
The relation "are incomparable with respect to < {\displaystyle \,<} " is thus identical to (that is, equal to) the relation "are ≲ {\displaystyle \,\lesssim } -equivalent" (so in particular, the former is transitive if and only if the latter is).
When < {\displaystyle \,<\,} is irreflexive then the property known as " transitivity of incomparability " (defined below) is exactly the condition necessary and sufficient to guarantee that the relation "are ≲ {\displaystyle \,\lesssim } -equivalent" does indeed form an equivalence relation on S . {\displaystyle S.} When this is the case, it allows any two elements x , y ∈ S {\displaystyle x,y\in S} satisfying x ≲ y and y ≲ x {\displaystyle x\lesssim y{\text{ and }}y\lesssim x} to be identified as a single object (specifically, they are identified together in their common equivalence class ).
Definition
A strict weak ordering on a set S {\displaystyle S} is a strict partial order < {\displaystyle \,<\,} on S {\displaystyle S} for which the incomparability relation induced on S {\displaystyle S} by < {\displaystyle \,<\,} is a transitive relation . [ 1 ] Explicitly, a strict weak order on S {\displaystyle S} is a homogeneous relation < {\displaystyle \,<\,} on S {\displaystyle S} that has all four of the following properties:
Properties (1), (2), and (3) are the defining properties of a strict partial order, although there is some redundancy in this list as asymmetry (3) implies irreflexivity (1), and also because irreflexivity (1) and transitivity (2) together imply asymmetry (3). [ 6 ] The incomparability relation is always symmetric and it will be reflexive if and only if < {\displaystyle \,<\,} is an irreflexive relation (which is assumed by the above definition).
Consequently, a strict partial order < {\displaystyle \,<\,} is a strict weak order if and only if its induced incomparability relation is an equivalence relation .
In this case, its equivalence classes partition S {\displaystyle S} and moreover, the set P {\displaystyle {\mathcal {P}}} of these equivalence classes can be strictly totally ordered by a binary relation , also denoted by < , {\displaystyle \,<,} that is defined for all A , B ∈ P {\displaystyle A,B\in {\mathcal {P}}} by:
Conversely, any strict total order on a partition P {\displaystyle {\mathcal {P}}} of S {\displaystyle S} gives rise to a strict weak ordering < {\displaystyle \,<\,} on S {\displaystyle S} defined by a < b {\displaystyle a<b} if and only if there exists sets A , B ∈ P {\displaystyle A,B\in {\mathcal {P}}} in this partition such that a ∈ A , b ∈ B , and A < B . {\displaystyle a\in A,b\in B,{\text{ and }}A<B.}
Not every partial order obeys the transitive law for incomparability. For instance, consider the partial order in the set { a , b , c } {\displaystyle \{a,b,c\}} defined by the relationship b < c . {\displaystyle b<c.} The pairs a , b and a , c {\displaystyle a,b{\text{ and }}a,c} are incomparable but b {\displaystyle b} and c {\displaystyle c} are related, so incomparability does not form an equivalence relation and this example is not a strict weak ordering.
For transitivity of incomparability, each of the following conditions is necessary , and for strict partial orders also sufficient :
Strict weak orders are very closely related to total preorders or (non-strict) weak orders , and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a preorder in which any two elements are comparable. [ 7 ] A total preorder ≲ {\displaystyle \,\lesssim \,} satisfies the following properties:
A total order is a total preorder which is antisymmetric, in other words, which is also a partial order . Total preorders are sometimes also called preference relations .
The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we take the converse of the complement: for a strict weak ordering < , {\displaystyle \,<,} define a total preorder ≲ {\displaystyle \,\lesssim \,} by setting x ≲ y {\displaystyle x\lesssim y} whenever it is not the case that y < x . {\displaystyle y<x.} In the other direction, to define a strict weak ordering < from a total preorder ≲ , {\displaystyle \,\lesssim ,} set x < y {\displaystyle x<y} whenever it is not the case that y ≲ x . {\displaystyle y\lesssim x.} [ 8 ]
In any preorder there is a corresponding equivalence relation where two elements x {\displaystyle x} and y {\displaystyle y} are defined as equivalent if x ≲ y and y ≲ x . {\displaystyle x\lesssim y{\text{ and }}y\lesssim x.} In the case of a total preorder the corresponding partial order on the set of equivalence classes is a total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding strict weak ordering.
A partition of a set S {\displaystyle S} is a family of non-empty disjoint subsets of S {\displaystyle S} that have S {\displaystyle S} as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition [ 9 ] and by Theodore Motzkin a list of sets . [ 10 ] An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set { a , b } {\displaystyle \{a,b\}} are { a } , { b } , {\displaystyle \{a\},\{b\},} { b } , { a } , and {\displaystyle \{b\},\{a\},\;{\text{ and }}} { a , b } . {\displaystyle \{a,b\}.}
In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition gives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in the partition, and otherwise inherit the order of the sets that contain them.
For sets of sufficiently small cardinality , a fourth axiomatization is possible, based on real-valued functions. If X {\displaystyle X} is any set then a real-valued function f : X → R {\displaystyle f:X\to \mathbb {R} } on X {\displaystyle X} induces a strict weak order on X {\displaystyle X} by setting a < b if and only if f ( a ) < f ( b ) . {\displaystyle a<b{\text{ if and only if }}f(a)<f(b).} The associated total preorder is given by setting a ≲ b if and only if f ( a ) ≤ f ( b ) {\displaystyle a{}\lesssim {}b{\text{ if and only if }}f(a)\leq f(b)} and the associated equivalence by setting a ∼ b if and only if f ( a ) = f ( b ) . {\displaystyle a{}\sim {}b{\text{ if and only if }}f(a)=f(b).}
The relations do not change when f {\displaystyle f} is replaced by g ∘ f {\displaystyle g\circ f} ( composite function ), where g {\displaystyle g} is a strictly increasing real-valued function defined on at least the range of f . {\displaystyle f.} Thus for example, a utility function defines a preference relation. In this context, weak orderings are also known as preferential arrangements . [ 11 ]
If X {\displaystyle X} is finite or countable, every weak order on X {\displaystyle X} can be represented by a function in this way. [ 12 ] However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the lexicographic order on R n . {\displaystyle \mathbb {R} ^{n}.} Thus, while in most preference relation models the relation defines a utility function up to order-preserving transformations, there is no such function for lexicographic preferences .
More generally, if X {\displaystyle X} is a set, Y {\displaystyle Y} is a set with a strict weak ordering < , {\displaystyle \,<,\,} and f : X → Y {\displaystyle f:X\to Y} is a function, then f {\displaystyle f} induces a strict weak ordering on X {\displaystyle X} by setting a < b if and only if f ( a ) < f ( b ) . {\displaystyle a<b{\text{ if and only if }}f(a)<f(b).} As before, the associated total preorder is given by setting a ≲ b if and only if f ( a ) ≲ f ( b ) , {\displaystyle a{}\lesssim {}b{\text{ if and only if }}f(a){}\lesssim {}f(b),} and the associated equivalence by setting a ∼ b if and only if f ( a ) ∼ f ( b ) . {\displaystyle a{}\sim {}b{\text{ if and only if }}f(a){}\sim {}f(b).} It is not assumed here that f {\displaystyle f} is an injective function , so a class of two equivalent elements on Y {\displaystyle Y} may induce a larger class of equivalent elements on X . {\displaystyle X.} Also, f {\displaystyle f} is not assumed to be a surjective function , so a class of equivalent elements on Y {\displaystyle Y} may induce a smaller or empty class on X . {\displaystyle X.} However, the function f {\displaystyle f} induces an injective function that maps the partition on X {\displaystyle X} to that on Y . {\displaystyle Y.} Thus, in the case of finite partitions, the number of classes in X {\displaystyle X} is less than or equal to the number of classes on Y . {\displaystyle Y.}
Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability. [ 13 ] A strict weak order that is trichotomous is called a strict total order . [ 14 ] The total preorder which is the inverse of its complement is in this case a total order .
For a strict weak order < {\displaystyle \,<\,} another associated reflexive relation is its reflexive closure , a (non-strict) partial order ≤ . {\displaystyle \,\leq .} The two associated reflexive relations differ with regard to different a {\displaystyle a} and b {\displaystyle b} for which neither a < b {\displaystyle a<b} nor b < a {\displaystyle b<a} : in the total preorder corresponding to a strict weak order we get a ≲ b {\displaystyle a\lesssim b} and b ≲ a , {\displaystyle b\lesssim a,} while in the partial order given by the reflexive closure we get neither a ≤ b {\displaystyle a\leq b} nor b ≤ a . {\displaystyle b\leq a.} For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order. [ 14 ] The reflexive closure of a strict weak ordering is a type of series-parallel partial order .
The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an n {\displaystyle n} -element set is given by the following sequence (sequence A000670 in the OEIS ):
Note that S ( n , k ) refers to Stirling numbers of the second kind .
These numbers are also called the Fubini numbers or ordered Bell numbers .
For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are three ways of partitioning the items into one singleton set and one group of two tied items, and each of these partitions gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is reversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons, which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.
Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves that add or remove a single order relation to or from a given ordering. For instance, for three elements, the ordering in which all three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict weak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weak orderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes, and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in the ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a Dedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies. For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph that has the weak orderings as its vertices, and these moves as its edges, forms a partial cube . [ 15 ]
Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron , and the dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron itself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in the corresponding weak ordering. [ 16 ] In this geometric representation the partial cube of moves on weak orderings is the graph describing the covering relation of the face lattice of the permutohedron.
For instance, for n = 3 , {\displaystyle n=3,} the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon (again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon, six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie, and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.
As mentioned above, weak orders have applications in utility theory. [ 12 ] In linear programming and other types of combinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, determined by a real-valued objective function ; the phenomenon of ties in these orderings is called "degeneracy", and several types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to prevent problems caused by degeneracy. [ 17 ]
Weak orders have also been used in computer science , in partition refinement based algorithms for lexicographic breadth-first search and lexicographic topological ordering . In these algorithms, a weak ordering on the vertices of a graph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a total order on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that is the output of the algorithm. [ 18 ]
In the Standard Library for the C++ programming language, the set and multiset data types sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering. [ 2 ]
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In chemistry , a salt or ionic compound is a chemical compound consisting of an assembly of positively charged ions ( cations ) and negatively charged ions ( anions ), [ 1 ] which results in a compound with no net electric charge (electrically neutral). The constituent ions are held together by electrostatic forces termed ionic bonds .
The component ions in a salt can be either inorganic , such as chloride (Cl − ), or organic , such as acetate ( CH 3 COO − ). Each ion can be either monatomic (termed simple ion ), such as sodium (Na + ) and chloride (Cl − ) in sodium chloride , or polyatomic , such as ammonium ( NH + 4 ) and carbonate ( CO 2− 3 ) ions in ammonium carbonate . Salts containing basic ions hydroxide (OH − ) or oxide (O 2− ) are classified as bases , such as sodium hydroxide and potassium oxide .
Individual ions within a salt usually have multiple near neighbours, so they are not considered to be part of molecules, but instead part of a continuous three-dimensional network. Salts usually form crystalline structures when solid.
Salts composed of small ions typically have high melting and boiling points , and are hard and brittle . As solids they are almost always electrically insulating , but when melted or dissolved they become highly conductive , because the ions become mobile. Some salts have large cations, large anions, or both. In terms of their properties, such species often are more similar to organic compounds.
In 1913 the structure of sodium chloride was determined by William Henry Bragg and his son William Lawrence Bragg . [ 2 ] [ 3 ] [ 4 ] This revealed that there were six equidistant nearest-neighbours for each atom, demonstrating that the constituents were not arranged in molecules or finite aggregates, but instead as a network with long-range crystalline order. [ 4 ] Many other inorganic compounds were also found to have similar structural features. [ 4 ] These compounds were soon described as being constituted of ions rather than neutral atoms , but proof of this hypothesis was not found until the mid-1920s, when X-ray reflection experiments (which detect the density of electrons), were performed. [ 4 ] [ 5 ]
Principal contributors to the development of a theoretical treatment of ionic crystal structures were Max Born , Fritz Haber , Alfred Landé , Erwin Madelung , Paul Peter Ewald , and Kazimierz Fajans . [ 6 ] Born predicted crystal energies based on the assumption of ionic constituents, which showed good correspondence to thermochemical measurements, further supporting the assumption. [ 4 ]
Many metals such as the alkali metals react directly with the electronegative halogens gases to form salts. [ 7 ] [ 8 ]
Salts form upon evaporation of their solutions . [ 9 ] Once the solution is supersaturated and the solid compound nucleates. [ 9 ] This process occurs widely in nature and is the means of formation of the evaporite minerals. [ 10 ]
Insoluble salts can be precipitated by mixing two solutions, one with the cation and one with the anion in it. Because all solutions are electrically neutral, the two solutions mixed must also contain counterions of the opposite charges. To ensure that these do not contaminate the precipitated salt, it is important to ensure they do not also precipitate. [ 11 ] If the two solutions have hydrogen ions and hydroxide ions as the counterions, they will react with one another in what is called an acid–base reaction or a neutralization reaction to form water. [ 12 ] Alternately the counterions can be chosen to ensure that even when combined into a single solution they will remain soluble as spectator ions . [ 11 ]
If the solvent is water in either the evaporation or precipitation method of formation, in many cases the ionic crystal formed also includes water of crystallization , so the product is known as a hydrate , and can have very different chemical properties compared to the anhydrous material. [ 13 ]
Molten salts will solidify on cooling to below their freezing point . [ 14 ] This is sometimes used for the solid-state synthesis of complex salts from solid reactants, which are first melted together. [ 15 ] In other cases, the solid reactants do not need to be melted, but instead can react through a solid-state reaction route . In this method, the reactants are repeatedly finely ground into a paste and then heated to a temperature where the ions in neighboring reactants can diffuse together during the time the reactant mixture remains in the oven. [ 8 ] Other synthetic routes use a solid precursor with the correct stoichiometric ratio of non-volatile ions, which is heated to drive off other species. [ 8 ]
In some reactions between highly reactive metals (usually from Group 1 or Group 2 ) and highly electronegative halogen gases, or water, the atoms can be ionized by electron transfer , [ 16 ] a process thermodynamically understood using the Born–Haber cycle . [ 17 ]
Salts are formed by salt-forming reactions
Ions in salts are primarily held together by the electrostatic forces between the charge distribution of these bodies, and in particular, the ionic bond resulting from the long-ranged Coulomb attraction between the net negative charge of the anions and net positive charge of the cations. [ 18 ] There is also a small additional attractive force from van der Waals interactions which contributes only around 1–2% of the cohesive energy for small ions. [ 19 ] When a pair of ions comes close enough for their outer electron shells (most simple ions have closed shells ) to overlap, a short-ranged repulsive force occurs, [ 20 ] due to the Pauli exclusion principle . [ 21 ] The balance between these forces leads to a potential energy well with minimum energy when the nuclei are separated by a specific equilibrium distance. [ 20 ]
If the electronic structure of the two interacting bodies is affected by the presence of one another, covalent interactions (non-ionic) also contribute to the overall energy of the compound formed. [ 22 ] Salts are rarely purely ionic, i.e. held together only by electrostatic forces. The bonds between even the most electronegative / electropositive pairs such as those in caesium fluoride exhibit a small degree of covalency . [ 23 ] [ 24 ] Conversely, covalent bonds between unlike atoms often exhibit some charge separation and can be considered to have a partial ionic character. [ 22 ] The circumstances under which a compound will have ionic or covalent character can typically be understood using Fajans' rules , which use only charges and the sizes of each ion. According to these rules, compounds with the most ionic character will have large positive ions with a low charge, bonded to a small negative ion with a high charge. [ 25 ] More generally HSAB theory can be applied, whereby the compounds with the most ionic character are those consisting of hard acids and hard bases: small, highly charged ions with a high difference in electronegativities between the anion and cation. [ 26 ] [ 27 ] This difference in electronegativities means that the charge separation, and resulting dipole moment, is maintained even when the ions are in contact (the excess electrons on the anions are not transferred or polarized to neutralize the cations). [ 28 ]
Although chemists classify idealized bond types as being ionic or covalent, the existence of additional types such as hydrogen bonds and metallic bonds , for example, has led some philosophers of science to suggest that alternative approaches to understanding bonding are required. This could be by applying quantum mechanics to calculate binding energies. [ 29 ] [ 30 ]
The lattice energy is the summation of the interaction of all sites with all other sites. For unpolarizable spherical ions, only the charges and distances are required to determine the electrostatic interaction energy. For any particular ideal crystal structure, all distances are geometrically related to the smallest internuclear distance. So for each possible crystal structure, the total electrostatic energy can be related to the electrostatic energy of unit charges at the nearest neighboring distance by a multiplicative constant called the Madelung constant [ 20 ] that can be efficiently computed using an Ewald sum . [ 31 ] When a reasonable form is assumed for the additional repulsive energy, the total lattice energy can be modelled using the Born–Landé equation , [ 32 ] the Born–Mayer equation , or in the absence of structural information, the Kapustinskii equation . [ 33 ]
Using an even simpler approximation of the ions as impenetrable hard spheres, the arrangement of anions in these systems are often related to close-packed arrangements of spheres, with the cations occupying tetrahedral or octahedral interstices . [ 34 ] [ 35 ] Depending on the stoichiometry of the salt, and the coordination (principally determined by the radius ratio ) of cations and anions, a variety of structures are commonly observed, [ 36 ] and theoretically rationalized by Pauling's rules . [ 37 ]
In some cases, the anions take on a simple cubic packing and the resulting common structures observed are:
Some ionic liquids , particularly with mixtures of anions or cations, can be cooled rapidly enough that there is not enough time for crystal nucleation to occur, so an ionic glass is formed (with no long-range order). [ 53 ]
Within any crystal, there will usually be some defects. To maintain electroneutrality of the crystals, defects that involve loss of a cation will be associated with loss of an anion, i.e. these defects come in pairs. [ 54 ] Frenkel defects consist of a cation vacancy paired with a cation interstitial and can be generated anywhere in the bulk of the crystal, [ 54 ] occurring most commonly in compounds with a low coordination number and cations that are much smaller than the anions. [ 55 ] Schottky defects consist of one vacancy of each type, and are generated at the surfaces of a crystal, [ 54 ] occurring most commonly in compounds with a high coordination number and when the anions and cations are of similar size. [ 55 ] If the cations have multiple possible oxidation states , then it is possible for cation vacancies to compensate for electron deficiencies on cation sites with higher oxidation numbers, resulting in a non-stoichiometric compound . [ 54 ] Another non-stoichiometric possibility is the formation of an F-center , a free electron occupying an anion vacancy. [ 56 ] When the compound has three or more ionic components, even more defect types are possible. [ 54 ] All of these point defects can be generated via thermal vibrations and have an equilibrium concentration. Because they are energetically costly but entropically beneficial, they occur in greater concentration at higher temperatures. Once generated, these pairs of defects can diffuse mostly independently of one another, by hopping between lattice sites. This defect mobility is the source of most transport phenomena within an ionic crystal, including diffusion and solid state ionic conductivity . [ 54 ] When vacancies collide with interstitials (Frenkel), they can recombine and annihilate one another. Similarly, vacancies are removed when they reach the surface of the crystal (Schottky). Defects in the crystal structure generally expand the lattice parameters , reducing the overall density of the crystal. [ 54 ] Defects also result in ions in distinctly different local environments, which causes them to experience a different crystal-field symmetry , especially in the case of different cations exchanging lattice sites. [ 54 ] This results in a different splitting of d-electron orbitals , so that the optical absorption (and hence colour) can change with defect concentration. [ 54 ]
Ionic compounds containing hydrogen ions (H + ) are classified as acids , and those containing electropositive cations [ 57 ] and basic anions ions hydroxide (OH − ) or oxide (O 2− ) are classified as bases . Other ionic compounds are known as salts and can be formed by acid–base reactions . [ 58 ] Salts that produce hydroxide ions when dissolved in water are called alkali salts , and salts that produce hydrogen ions when dissolved in water are called acid salts . If the compound is the result of a reaction between a strong acid and a weak base , the result is an acid salt . If it is the result of a reaction between a strong base and a weak acid , the result is a base salt . If it is the result of a reaction between a strong acid and a strong base, the result is a neutral salt. Weak acids reacted with weak bases can produce ionic compounds with both the conjugate base ion and conjugate acid ion, such as ammonium acetate .
Some ions are classed as amphoteric , being able to react with either an acid or a base. [ 59 ] This is also true of some compounds with ionic character, typically oxides or hydroxides of less-electropositive metals (so the compound also has significant covalent character), such as zinc oxide , aluminium hydroxide , aluminium oxide and lead(II) oxide . [ 60 ]
Electrostatic forces between particles are strongest when the charges are high, and the distance between the nuclei of the ions is small. In such cases, the compounds generally have very high melting and boiling points and a low vapour pressure . [ 61 ] Trends in melting points can be even better explained when the structure and ionic size ratio is taken into account. [ 62 ] Above their melting point, salts melt and become molten salts (although some salts such as aluminium chloride and iron(III) chloride show molecule-like structures in the liquid phase). [ 63 ] Inorganic compounds with simple ions typically have small ions, and thus have high melting points, so are solids at room temperature. Some substances with larger ions, however, have a melting point below or near room temperature (often defined as up to 100 °C), and are termed ionic liquids . [ 64 ] Ions in ionic liquids often have uneven charge distributions, or bulky substituents like hydrocarbon chains, which also play a role in determining the strength of the interactions and propensity to melt. [ 65 ]
Even when the local structure and bonding of an ionic solid is disrupted sufficiently to melt it, there are still strong long-range electrostatic forces of attraction holding the liquid together and preventing ions boiling to form a gas phase. [ 66 ] This means that even room temperature ionic liquids have low vapour pressures, and require substantially higher temperatures to boil. [ 66 ] Boiling points exhibit similar trends to melting points in terms of the size of ions and strength of other interactions. [ 66 ] When vapourized, the ions are still not freed of one another. For example, in the vapour phase sodium chloride exists as diatomic "molecules". [ 67 ]
Most salts are very brittle . Once they reach the limit of their strength, they cannot deform malleably , because the strict alignment of positive and negative ions must be maintained. Instead the material undergoes fracture via cleavage . [ 68 ] As the temperature is elevated (usually close to the melting point) a ductile–brittle transition occurs, and plastic flow becomes possible by the motion of dislocations . [ 68 ] [ 69 ]
The compressibility of a salt is strongly determined by its structure, and in particular the coordination number . For example, halides with the caesium chloride structure (coordination number 8) are less compressible than those with the sodium chloride structure (coordination number 6), and less again than those with a coordination number of 4. [ 70 ]
When simple salts dissolve , they dissociate into individual ions, which are solvated and dispersed throughout the resulting solution. Salts do not exist in solution. [ 71 ] In contrast, molecular compounds, which includes most organic compounds, remain intact in solution.
The solubility of salts is highest in polar solvents (such as water ) or ionic liquids , but tends to be low in nonpolar solvents (such as petrol / gasoline ). [ 72 ] This contrast is principally because the resulting ion–dipole interactions are significantly stronger than ion-induced dipole interactions, so the heat of solution is higher. When the oppositely charged ions in the solid ionic lattice are surrounded by the opposite pole of a polar molecule, the solid ions are pulled out of the lattice and into the liquid. If the solvation energy exceeds the lattice energy , the negative net enthalpy change of solution provides a thermodynamic drive to remove ions from their positions in the crystal and dissolve in the liquid. In addition, the entropy change of solution is usually positive for most solid solutes like salts, which means that their solubility increases when the temperature increases. [ 73 ] There are some unusual salts such as cerium(III) sulfate , where this entropy change is negative, due to extra order induced in the water upon solution, and the solubility decreases with temperature. [ 73 ]
The lattice energy , the cohesive forces between these ions within a solid, determines the solubility. The solubility is dependent on how well each ion interacts with the solvent, so certain patterns become apparent. For example, salts of sodium , potassium and ammonium are usually soluble in water. Notable exceptions include ammonium hexachloroplatinate and potassium cobaltinitrite . Most nitrates and many sulfates are water-soluble. Exceptions include barium sulfate , calcium sulfate (sparingly soluble), and lead(II) sulfate , where the 2+/2− pairing leads to high lattice energies. For similar reasons, most metal carbonates are not soluble in water. Some soluble carbonate salts are: sodium carbonate , potassium carbonate and ammonium carbonate .
Strong salts or strong electrolyte salts are chemical salts composed of strong electrolytes . These salts dissociate completely or almost completely in water . They are generally odorless and nonvolatile .
Strong salts start with Na__, K__, NH 4 __, or they end with __NO 3 , __ClO 4 , or __CH 3 COO. Most group 1 and 2 metals form strong salts. Strong salts are especially useful when creating conductive compounds as their constituent ions allow for greater conductivity. [ citation needed ]
Weak salts or weak electrolyte salts are composed of weak electrolytes . These salts do not dissociate well in water. They are generally more volatile than strong salts. They may be similar in odor to the acid or base they are derived from. For example, sodium acetate , CH 3 COONa, smells similar to acetic acid CH 3 COOH.
Salts are characteristically insulators . Although they contain charged atoms or clusters, these materials do not typically conduct electricity to any significant extent when the substance is solid. In order to conduct, the charged particles must be mobile rather than stationary in a crystal lattice . This is achieved to some degree at high temperatures when the defect concentration increases the ionic mobility and solid state ionic conductivity is observed. When the salts are dissolved in a liquid or are melted into a liquid , they can conduct electricity because the ions become completely mobile. For this reason, molten salts and solutions containing dissolved salts (e.g., sodium chloride in water) can be used as electrolytes . [ 75 ] This conductivity gain upon dissolving or melting is sometimes used as a defining characteristic of salts. [ 76 ]
In some unusual salts: fast-ion conductors , and ionic glasses , [ 53 ] one or more of the ionic components has a significant mobility, allowing conductivity even while the material as a whole remains solid. [ 77 ] This is often highly temperature dependent, and may be the result of either a phase change or a high defect concentration. [ 77 ] These materials are used in all solid-state supercapacitors , batteries , and fuel cells , and in various kinds of chemical sensors . [ 78 ] [ 79 ]
The colour of a salt is often different from the colour of an aqueous solution containing the constituent ions, [ 80 ] or the hydrated form of the same compound. [ 13 ]
The anions in compounds with bonds with the most ionic character tend to be colorless (with an absorption band in the ultraviolet part of the spectrum). [ 81 ] In compounds with less ionic character, their color deepens through yellow, orange, red, and black (as the absorption band shifts to longer wavelengths into the visible spectrum). [ 81 ]
The absorption band of simple cations shifts toward a shorter wavelength when they are involved in more covalent interactions. [ 81 ] This occurs during hydration of metal ions, so colorless anhydrous salts with an anion absorbing in the infrared can become colorful in solution. [ 81 ]
Salts exist in many different colors , which arise either from their constituent anions, cations or solvates . For example:
Some minerals are salts, some of which are soluble in water. [ dubious – discuss ] [ clarification needed ] Similarly, inorganic pigments tend not to be salts, because insolubility is required for fastness. Some organic dyes are salts, but they are virtually insoluble in water.
Salts can elicit all five basic tastes , e.g., salty ( sodium chloride ), sweet ( lead diacetate , which will cause lead poisoning if ingested), sour ( potassium bitartrate ), bitter ( magnesium sulfate ), and umami or savory ( monosodium glutamate ).
Salts of strong acids and strong bases (" strong salts ") are non- volatile and often odorless, whereas salts of either weak acids or weak bases (" weak salts ") may smell like the conjugate acid (e.g., acetates like acetic acid ( vinegar ) and cyanides like hydrogen cyanide ( almonds )) or the conjugate base (e.g., ammonium salts like ammonia ) of the component ions. That slow, partial decomposition is usually accelerated by the presence of water, since hydrolysis is the other half of the reversible reaction equation of formation of weak salts.
Salts have long had a wide variety of uses and applications. Many minerals are ionic. [ 82 ] Humans have processed common salt (sodium chloride) for over 8000 years, using it first as a food seasoning and preservative, and now also in manufacturing, agriculture , water conditioning, for de-icing roads, and many other uses. [ 83 ] Many salts are so widely used in society that they go by common names unrelated to their chemical identity. Examples of this include borax , calomel , milk of magnesia , muriatic acid , oil of vitriol , saltpeter , and slaked lime . [ 84 ]
Soluble salts can easily be dissolved to provide electrolyte solutions. This is a simple way to control the concentration and ionic strength . The concentration of solutes affects many colligative properties , including increasing the osmotic pressure , and causing freezing-point depression and boiling-point elevation . [ 85 ] Because the solutes are charged ions they also increase the electrical conductivity of the solution. [ 86 ] The increased ionic strength reduces the thickness of the electrical double layer around colloidal particles, and therefore the stability of emulsions and suspensions . [ 87 ]
The chemical identity of the ions added is also important in many uses. For example, fluoride containing compounds are dissolved to supply fluoride ions for water fluoridation . [ 88 ]
Solid salts have long been used as paint pigments, and are resistant to organic solvents, but are sensitive to acidity or basicity. [ 89 ] Since 1801 pyrotechnicians have described and widely used metal-containing salts as sources of colour in fireworks. [ 90 ] Under intense heat, the electrons in the metal ions or small molecules can be excited. [ 91 ] These electrons later return to lower energy states, and release light with a colour spectrum characteristic of the species present. [ 92 ] [ 93 ]
In chemical synthesis , salts are often used as precursors for high-temperature solid-state synthesis. [ 94 ]
Many metals are geologically most abundant as salts within ores . [ 95 ] To obtain the elemental materials, these ores are processed by smelting or electrolysis , in which redox reactions occur (often with a reducing agent such as carbon) such that the metal ions gain electrons to become neutral atoms. [ 96 ] [ 97 ]
According to the nomenclature recommended by IUPAC , salts are named according to their composition, not their structure. [ 98 ] In the most simple case of a binary salt with no possible ambiguity about the charges and thus the stoichiometry , the common name is written using two words. [ 99 ] The name of the cation (the unmodified element name for monatomic cations) comes first, followed by the name of the anion. [ 100 ] [ 101 ] For example, MgCl 2 is named magnesium chloride , and Na 2 SO 4 is named sodium sulfate ( SO 2− 4 , sulfate , is an example of a polyatomic ion ). To obtain the empirical formula from these names, the stoichiometry can be deduced from the charges on the ions, and the requirement of overall charge neutrality. [ 102 ]
If there are multiple different cations and/or anions, multiplicative prefixes ( di- , tri- , tetra- , ...) are often required to indicate the relative compositions, [ 103 ] and cations then anions are listed in alphabetical order. [ 104 ] For example, KMgCl 3 is named magnesium potassium trichloride to distinguish it from K 2 MgCl 4 , magnesium dipotassium tetrachloride [ 105 ] (note that in both the empirical formula and the written name, the cations appear in alphabetical order, but the order varies between them because the symbol for potassium is K). [ 106 ] When one of the ions already has a multiplicative prefix within its name, the alternate multiplicative prefixes ( bis- , tris- , tetrakis- , ...) are used. [ 107 ] For example, Ba(BrF 4 ) 2 is named barium bis(tetrafluoridobromate) . [ 108 ]
Compounds containing one or more elements which can exist in a variety of charge/ oxidation states will have a stoichiometry that depends on which oxidation states are present, to ensure overall neutrality. This can be indicated in the name by specifying either the oxidation state of the elements present, or the charge on the ions. [ 108 ] Because of the risk of ambiguity in allocating oxidation states, IUPAC prefers direct indication of the ionic charge numbers. [ 108 ] These are written as an arabic integer followed by the sign (... , 2−, 1−, 1+, 2+, ...) in parentheses directly after the name of the cation (without a space separating them). [ 108 ] For example, FeSO 4 is named iron(2+) sulfate (with the 2+ charge on the Fe 2+ ions balancing the 2− charge on the sulfate ion), whereas Fe 2 (SO 4 ) 3 is named iron(3+) sulfate (because the two iron ions in each formula unit each have a charge of 3+, to balance the 2− on each of the three sulfate ions). [ 108 ] Stock nomenclature , still in common use, writes the oxidation number in Roman numerals (... , −II, −I, 0, I, II, ...). So the examples given above would be named iron(II) sulfate and iron(III) sulfate respectively. [ 109 ] For simple ions the ionic charge and the oxidation number are identical, but for polyatomic ions they often differ. For example, the uranyl(2+) ion, UO 2+ 2 , has uranium in an oxidation state of +6, so would be called a dioxouranium(VI) ion in Stock nomenclature. [ 110 ] An even older naming system for metal cations, also still widely used, appended the suffixes -ous and -ic to the Latin root of the name, to give special names for the low and high oxidation states. [ 111 ] For example, this scheme uses "ferrous" and "ferric", for iron(II) and iron(III) respectively, [ 111 ] so the examples given above were classically named ferrous sulfate and ferric sulfate . [ citation needed ]
Common salt-forming cations include:
Common salt-forming anions (parent acids in parentheses where available) include:
Salts with varying number of hydrogen atoms replaced by cations as compared to their parent acid can be referred to as monobasic , dibasic , or tribasic , identifying that one, two, or three hydrogen atoms have been replaced; polybasic salts refer to those with more than one hydrogen atom replaced. Examples include:
Zwitterions contain an anionic and a cationic centre in the same molecule , but are not considered salts. Examples of zwitterions are amino acids , many metabolites , peptides , and proteins . [ 112 ]
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Weak selection in evolutionary biology is when individuals with different phenotypes possess similar fitness , i.e. one phenotype is weakly preferred over the other. Weak selection, therefore, is an evolutionary theory to explain the maintenance of multiple phenotypes in a stable population. [ 1 ]
Weak selection can only be used to explain the maintenance of mutations in a Moran process . [ 1 ] A Moran process is one in which birth and death are paired events, and therefore population size remains constant. If the population size was increasing, both wild type and mutant phenotypes can proliferate and the weak selection for one phenotype results in no particular selection for either. Hence weak selection requires a finite population to operate. Otherwise there would be no expectation of fixation and hence no selection .
The result of weak selection is two phenotypes with similar fixation probabilities. Weak selection works to elongate fixation time for two competing alleles . Consequently, weak selection provides a model for describing how evolution can occur in large steps in a population in which multiple alleles are maintained. [ 1 ]
There are two basic reasons that two phenotypes could have very similar fitness. One reason could be that the phenotypic differences between wild type and mutant are large but the significance of the mutation is minor. An example could be a change in pigmentation. Another reason could be that the phenotypic differences between wild type and mutant are actually small, such as tail length variation. In either case, the significance of the mutation, which is determined by the environment creating the selective pressure , is low in comparison to other mutations. Hence, almost near neutral mutations result in phenotypes that are weakly selected. [ 1 ]
Weak selection creates a situation in which the evolutionary dynamics governing the phenotype frequencies in a population are mainly driven by random fluctuations. Hence weak selection increases the impact of stochastic processes on the evolutionary dynamics of the trait being weakly selected. For example, genetic drift could cause a nearly neutral mutation to become the dominant allele in a population by wiping out the wild type. Weak selection is therefore also especially sensitive to the effects of population size. In smaller populations, a weakly selected mutation could proliferate due to stochastic processes such as genetic drift even more easily. [ 2 ]
Empirically, nonsynonymous substitutions have been reported to proliferate through weak selection in Drosophila melanogaster and Arabidopsis . These non-neutral mutations are thought to have special significance evolutionarily when they affect gene regulatory elements. This is because differential gene expression is critical development and therefore can potentially affect the morphology of an organism. Furthermore, weak selection operates in codon-usage bias resulting in differential levels of gene expression by altering the rate of transcription in mutants with non-preferred codons. Hence, even so called "silent" mutations can result in slight variations in the fitness of an organism. Additionally, gene duplication offers another way in which an apparently nonfunctional mutation can be maintained through weak selection. Differential expression of duplicate gene copies provides a mechanism through which a protein can evolve new functions. [ 3 ]
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In mathematics , a weak solution (also called a generalized solution ) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions .
Avoiding the language of distributions, one starts with a differential equation and rewrites it in such a way that no derivatives of the solution of the equation show up (the new form is called the weak formulation , and the solutions to it are called weak solutions). Somewhat surprisingly, a differential equation may have solutions that are not differentiable , and the weak formulation allows one to find such solutions.
Weak solutions are important because many differential equations encountered in modelling real-world phenomena do not admit of sufficiently smooth solutions, and the only way of solving such equations is using the weak formulation. Even in situations where an equation does have differentiable solutions, it is often convenient to first prove the existence of weak solutions and only later show that those solutions are in fact smooth enough.
Examples of equations that have weak solutions but fail to have strong solutions include the Tanaka equation and Tsirelson's stochastic differential equation .
As an illustration of the concept, consider the first-order wave equation :
where u = u ( t , x ) is a function of two real variables. To indirectly probe the properties of a possible solution u , one integrates it against an arbitrary smooth function φ {\displaystyle \varphi \,\!} of compact support , known as a test function, taking
For example, if φ {\displaystyle \varphi } is a smooth probability distribution concentrated near a point ( t , x ) = ( t ∘ , x ∘ ) {\displaystyle (t,x)=(t_{\circ },x_{\circ })} , the integral is approximately u ( t ∘ , x ∘ ) {\displaystyle u(t_{\circ },x_{\circ })} . Notice that while the integrals go from − ∞ {\displaystyle -\infty } to ∞ {\displaystyle \infty } , they are essentially over a finite box where φ {\displaystyle \varphi } is non-zero.
Thus, assume a solution u is continuously differentiable on the Euclidean space R 2 , multiply the equation ( 1 ) by a test function φ {\displaystyle \varphi } (smooth of compact support), and integrate:
Using Fubini's theorem , which allows one to interchange the order of integration, as well as integration by parts (in t for the first term and in x for the second term) this equation becomes:
(Boundary terms vanish since φ {\displaystyle \varphi } is zero outside a finite box.) We have shown that equation ( 1 ) implies equation ( 2 ) as long as u is continuously differentiable.
The key to the concept of weak solution is that there exist functions u that satisfy equation ( 2 ) for any φ {\displaystyle \varphi } , but such u may not be differentiable and so cannot satisfy equation ( 1 ). An example is u ( t , x ) = | t − x | , as one may check by splitting the integrals over regions x ≥ t and x ≤ t , where u is smooth, and reversing the above computation using integration by parts. A weak solution of equation ( 1 ) means any solution u of equation ( 2 ) over all test functions φ {\displaystyle \varphi } .
The general idea that follows from this example is that, when solving a differential equation in u , one can rewrite it using a test function φ {\displaystyle \varphi } , such that whatever derivatives in u show up in the equation, they are "transferred" via integration by parts to φ {\displaystyle \varphi } , resulting in an equation without derivatives of u . This new equation generalizes the original equation to include solutions that are not necessarily differentiable.
The approach illustrated above works in great generality. Indeed, consider a linear differential operator in an open set W in R n :
where the multi-index ( α 1 , α 2 , ..., α n ) varies over some finite set in N n and the coefficients a α 1 , α 2 , … , α n {\displaystyle a_{\alpha _{1},\alpha _{2},\dots ,\alpha _{n}}} are smooth enough functions of x in R n .
The differential equation P ( x , ∂ ) u ( x ) = 0 can, after being multiplied by a smooth test function φ {\displaystyle \varphi } with compact support in W and integrated by parts, be written as
where the differential operator Q ( x , ∂ ) is given by the formula
The number
shows up because one needs α 1 + α 2 + ⋯ + α n integrations by parts to transfer all the partial derivatives from u to φ {\displaystyle \varphi } in each term of the differential equation, and each integration by parts entails a multiplication by −1.
The differential operator Q ( x , ∂ ) is the formal adjoint of P ( x , ∂ ) (cf. adjoint of an operator ).
In summary, if the original (strong) problem was to find an | α |-times differentiable function u defined on the open set W such that
(a so-called strong solution ), then an integrable function u would be said to be a weak solution if
for every smooth function φ {\displaystyle \varphi } with compact support in W .
The notion of weak solution based on distributions is sometimes inadequate. In the case of hyperbolic systems , the notion of weak solution based on distributions does not guarantee uniqueness, and it is necessary to supplement it with entropy conditions or some other selection criterion. In fully nonlinear PDE such as the Hamilton–Jacobi equation , there is a very different definition of weak solution called viscosity solution .
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Weak stability boundary (WSB), including low-energy transfer , is a concept introduced by Edward Belbruno in 1987. The concept explained how a spacecraft could change orbits using very little fuel.
Weak stability boundary is defined for the three-body problem . This problem considers the motion of a particle P of negligible mass moving with respect to two larger bodies, P1, P2, modeled as point masses, where these bodies move in circular or elliptical orbits with respect to each other, and P2 is smaller than P1. [ 1 ]
The force between the three bodies is the classical Newtonian gravitational force . For example, P1 is the Earth, P2 is the Moon and P is a spacecraft; or P1 is the Sun, P2 is Jupiter and P is a comet, etc. This model is called the restricted three-body problem . [ 1 ] The weak stability boundary defines a region about P2 where P is temporarily captured. This region is in position-velocity space. Capture means that the Kepler energy between P and P2 is negative. This is also called weak capture. [ 1 ]
This boundary was defined for the first time by Edward Belbruno of Princeton University in 1987. [ 2 ] He described a Low-energy transfer which would allow a spacecraft to change orbits using very little fuel. It was for motion about Moon (P2) with P1 = Earth. It is defined algorithmically by monitoring cycling motion of P about the Moon and finding the region where cycling motion transitions between stable and unstable after one cycle. Stable motion means P can completely cycle about the Moon for one cycle relative to a reference section, starting in weak capture. P needs to return to the reference section with negative Kepler energy. Otherwise, the motion is called unstable , where P does not return to the reference section within one cycle or if it returns, it has non-negative Kepler energy. [ 2 ] [ 1 ]
The set of all transition points about the Moon comprises the weak stability boundary, W . The motion of P is sensitive or chaotic as it moves about the Moon within W . A mathematical proof that the motion within W is chaotic was given in 2004. [ 1 ] This is accomplished by showing that the set W about an arbitrary body P2 in the restricted three-body problem contains a hyperbolic invariant set of fractional dimension consisting of the infinitely many intersections Hyperbolic manifolds . [ 1 ]
The weak stability boundary was originally referred to as the fuzzy boundary . [ 3 ] [ 4 ] This term was used since the transition between capture and escape defined in the algorithm is not well defined and limited by the numerical accuracy. This defines a "fuzzy" location for the transition points. It is also due the inherent chaos in the motion of P near the transition points. It can be thought of as a fuzzy chaos region. As is described in an article in Discover magazine, the WSB can be roughly viewed as the fuzzy edge of a region, referred to as a gravity well , about a body (the Moon), where its force of gravity becomes small enough to be dominated by force of gravity of another body (the Earth) and the motion there is chaotic. [ 3 ]
A much more general algorithm defining W was given in 2007. [ 5 ] It defines W relative to n -cycles, where n = 1,2,3,..., yielding boundaries of order n. This gives a much more complex region consisting of the union of all the weak stability boundaries of order n. This definition was explored further in 2010. [ 6 ] The results suggested that W consists, in part, of the hyperbolic network of invariant manifolds associated to the Lyapunov orbits about the L1, L2 Lagrange points near P2. The explicit determination of the set W about P2 = Jupiter, where P1 is the Sun, is described in "Computation of Weak Stability Boundaries: Sun-Jupiter Case". [ 7 ] It turns out that a weak stability region can also be defined relative to the larger mass point, P1. A proof of the existence of the weak stability boundary about P1 was given in 2012, [ 8 ] but a different definition is used. The chaos of the motion is analytically proven in "Geometry of Weak Stability Boundaries". [ 8 ] The boundary is studied in "Applicability and Dynamical Characterization of the Associated Sets of the Algorithmic Weak Stability Boundary in the Lunar Sphere of Influence". [ 9 ]
There are a number of important applications for the weak stability boundary (WSB). Since the WSB defines a region of temporary capture, it can be used, for example, to find transfer trajectories from the Earth to the Moon that arrive at the Moon within the WSB region in weak capture, which is called ballistic capture for a spacecraft. No fuel is required for capture in this case. This was numerically demonstrated in 1987. [ 2 ] This is the first reference for ballistic capture for spacecraft and definition of the weak stability boundary. The boundary was operationally demonstrated to exist in 1991 when it was used to find a ballistic capture transfer to the Moon for Japan's Hiten spacecraft. [ 10 ] Other missions have used the same transfer type as Hiten , including Grail , Capstone , Danuri , Hakuto-R Mission 1 and SLIM . The WSB for Mars is studied in "Earth-Mars Transfers with Ballistic Capture" [ 11 ] and ballistic capture transfers to Mars are computed. The BepiColombo mission of ESA will achieve ballistic capture at the WSB of Mercury in 2025.
The WSB region can be used in the field of Astrophysics . It can be defined for stars within open star clusters . This is done in "Chaotic Exchange of Solid Material Between Planetary Systems: Implications for the Lithopanspermia Hypothesis" [ 12 ] to analyze the capture of solid material that may have arrived on the Earth early in the age of the Solar System to study the validity of the lithopanspermia hypothesis .
Numerical explorations of trajectories for P starting in the WSB region about P2 show that after the particle P escapes P2 at the end of weak capture, it moves about the primary body, P1, in a near resonant orbit, in resonance with P2 about P1. This property was used to study comets that move in orbits about the Sun in orbital resonance with Jupiter, which change resonance orbits by becoming weakly captured by Jupiter. [ 13 ] An example of such a comet is 39P/Oterma .
This property of change of resonance of orbits about P1 when P is weakly captured by the WSB of P2 has an interesting application to the field of quantum mechanics to the motion of an electron about the proton in a hydrogen atom. The transition motion of an electron about the proton between different energy states described by the Schrödinger equation is shown to be equivalent to the change of resonance of P about P1 via weak capture by P2 for a family of transitioning resonance orbits. [ 14 ] This gives a classical model using chaotic dynamics with Newtonian gravity for the motion of an electron.
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In mathematics , weak topology is an alternative term for certain initial topologies , often on topological vector spaces or spaces of linear operators , for instance on a Hilbert space . The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space ) with respect to its continuous dual . The remainder of this article will deal with this case, which is one of the concepts of functional analysis .
One may call subsets of a topological vector space weakly closed (respectively, weakly compact , etc.) if they are closed (respectively, compact , etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable , weakly analytic , etc.) if they are continuous (respectively, differentiable , analytic , etc.) with respect to the weak topology.
Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable. [ 1 ] In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence . [ 1 ] The weak topology is called topologie faible in French and schwache Topologie in German.
Let K {\displaystyle \mathbb {K} } be a topological field , namely a field with a topology such that addition, multiplication, and division are continuous . In most applications K {\displaystyle \mathbb {K} } will be either the field of complex numbers or the field of real numbers with the familiar topologies.
Both the weak topology and the weak* topology are special cases of a more general construction for pairings , which we now describe.
The benefit of this more general construction is that any definition or result proved for it applies to both the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs. This is also the reason why the weak* topology is also frequently referred to as the "weak topology"; because it is just an instance of the weak topology in the setting of this more general construction.
Suppose ( X , Y , b ) is a pairing of vector spaces over a topological field K {\displaystyle \mathbb {K} } (i.e. X and Y are vector spaces over K {\displaystyle \mathbb {K} } and b : X × Y → K {\displaystyle \mathbb {K} } is a bilinear map ).
The weak topology on Y is now automatically defined as described in the article Dual system . However, for clarity, we now repeat it.
If the field K {\displaystyle \mathbb {K} } has an absolute value | ⋅ | , then the weak topology 𝜎( X , Y , b ) on X is induced by the family of seminorms , p y : X → R {\displaystyle \mathbb {R} } , defined by
for all y ∈ Y and x ∈ X . This shows that weak topologies are locally convex .
We now consider the special case where Y is a vector subspace of the algebraic dual space of X (i.e. a vector space of linear functionals on X ).
There is a pairing, denoted by ( X , Y , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (X,Y,\langle \cdot ,\cdot \rangle )} or ( X , Y ) {\displaystyle (X,Y)} , called the canonical pairing whose bilinear map ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the canonical evaluation map , defined by ⟨ x , x ′ ⟩ = x ′ ( x ) {\displaystyle \langle x,x'\rangle =x'(x)} for all x ∈ X {\displaystyle x\in X} and x ′ ∈ Y {\displaystyle x'\in Y} . Note in particular that ⟨ ⋅ , x ′ ⟩ {\displaystyle \langle \cdot ,x'\rangle } is just another way of denoting x ′ {\displaystyle x'} i.e. ⟨ ⋅ , x ′ ⟩ = x ′ ( ⋅ ) {\displaystyle \langle \cdot ,x'\rangle =x'(\cdot )} .
In this case, the weak topology on X (resp. the weak topology on Y ), denoted by 𝜎( X , Y ) (resp. by 𝜎( Y , X ) ) is the weak topology on X (resp. on Y ) with respect to the canonical pairing ⟨ X , Y ⟩ .
The topology σ( X , Y ) is the initial topology of X with respect to Y .
If Y is a vector space of linear functionals on X , then the continuous dual of X with respect to the topology σ( X , Y ) is precisely equal to Y . [ 1 ] ( Rudin 1991 , Theorem 3.10)
Let X be a topological vector space (TVS) over K {\displaystyle \mathbb {K} } , that is, X is a K {\displaystyle \mathbb {K} } vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that X starts with the original , starting , or given topology (the reader is cautioned against using the terms " initial topology " and " strong topology " to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on X using the topological or continuous dual space X ∗ {\displaystyle X^{*}} , which consists of all linear functionals from X into the base field K {\displaystyle \mathbb {K} } that are continuous with respect to the given topology.
Recall that ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the canonical evaluation map defined by ⟨ x , x ′ ⟩ = x ′ ( x ) {\displaystyle \langle x,x'\rangle =x'(x)} for all x ∈ X {\displaystyle x\in X} and x ′ ∈ X ∗ {\displaystyle x'\in X^{*}} , where in particular, ⟨ ⋅ , x ′ ⟩ = x ′ ( ⋅ ) = x ′ {\displaystyle \langle \cdot ,x'\rangle =x'(\cdot )=x'} .
We give alternative definitions below.
Alternatively, the weak topology on a TVS X is the initial topology with respect to the family X ∗ {\displaystyle X^{*}} . In other words, it is the coarsest topology on X such that each element of X ∗ {\displaystyle X^{*}} remains a continuous function .
A subbase for the weak topology is the collection of sets of the form ϕ − 1 ( U ) {\displaystyle \phi ^{-1}(U)} where ϕ ∈ X ∗ {\displaystyle \phi \in X^{*}} and U is an open subset of the base field K {\displaystyle \mathbb {K} } . In other words, a subset of X is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form ϕ − 1 ( U ) {\displaystyle \phi ^{-1}(U)} .
From this point of view, the weak topology is the coarsest polar topology .
The weak topology is characterized by the following condition: a net ( x λ ) {\displaystyle (x_{\lambda })} in X converges in the weak topology to the element x of X if and only if ϕ ( x λ ) {\displaystyle \phi (x_{\lambda })} converges to ϕ ( x ) {\displaystyle \phi (x)} in R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } for all ϕ ∈ X ∗ {\displaystyle \phi \in X^{*}} .
In particular, if x n {\displaystyle x_{n}} is a sequence in X , then x n {\displaystyle x_{n}} converges weakly to x if
as n → ∞ for all φ ∈ X ∗ {\displaystyle \varphi \in X^{*}} . In this case, it is customary to write
or, sometimes,
If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space .
If X is a normed space, then the dual space X ∗ {\displaystyle X^{*}} is itself a normed vector space by using the norm
This norm gives rise to a topology, called the strong topology , on X ∗ {\displaystyle X^{*}} . This is the topology of uniform convergence . The uniform and strong topologies are generally different for other spaces of linear maps; see below.
The weak* topology is an important example of a polar topology .
A space X can be embedded into its double dual X** by
Thus T : X → X ∗ ∗ {\displaystyle T:X\to X^{**}} is an injective linear mapping, though not necessarily surjective (spaces for which this canonical embedding is surjective are called reflexive ). The weak-* topology on X ∗ {\displaystyle X^{*}} is the weak topology induced by the image of T : T ( X ) ⊂ X ∗ ∗ {\displaystyle T:T(X)\subset X^{**}} . In other words, it is the coarsest topology such that the maps T x , defined by T x ( ϕ ) = ϕ ( x ) {\displaystyle T_{x}(\phi )=\phi (x)} from X ∗ {\displaystyle X^{*}} to the base field R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } remain continuous.
A net ϕ λ {\displaystyle \phi _{\lambda }} in X ∗ {\displaystyle X^{*}} is convergent to ϕ {\displaystyle \phi } in the weak-* topology if it converges pointwise:
for all x ∈ X {\displaystyle x\in X} . In particular, a sequence of ϕ n ∈ X ∗ {\displaystyle \phi _{n}\in X^{*}} converges to ϕ {\displaystyle \phi } provided that
for all x ∈ X . In this case, one writes
as n → ∞ .
Weak-* convergence is sometimes called the simple convergence or the pointwise convergence . Indeed, it coincides with the pointwise convergence of linear functionals.
If X is a separable (i.e. has a countable dense subset) locally convex space and H is a norm-bounded subset of its continuous dual space, then H endowed with the weak* (subspace) topology is a metrizable topological space. [ 1 ] However, for infinite-dimensional spaces, the metric cannot be translation-invariant. [ 2 ] If X is a separable metrizable locally convex space then the weak* topology on the continuous dual space of X is separable. [ 1 ]
By definition, the weak* topology is weaker than the weak topology on X ∗ {\displaystyle X^{*}} . An important fact about the weak* topology is the Banach–Alaoglu theorem : if X is normed, then the closed unit ball in X ∗ {\displaystyle X^{*}} is weak*- compact (more generally, the polar in X ∗ {\displaystyle X^{*}} of a neighborhood of 0 in X is weak*-compact). Moreover, the closed unit ball in a normed space X is compact in the weak topology if and only if X is reflexive .
In more generality, let F be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let X be a normed topological vector space over F , compatible with the absolute value in F . Then in X ∗ {\displaystyle X^{*}} , the topological dual space X of continuous F -valued linear functionals on X , all norm-closed balls are compact in the weak* topology.
If X is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded. [ 1 ] This implies, in particular, that when X is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of X does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded). [ 1 ] Thus, even though norm-closed balls are compact, X* is not weak* locally compact .
If X is a normed space, then X is separable if and only if the weak* topology on the closed unit ball of X ∗ {\displaystyle X^{*}} is metrizable, [ 1 ] in which case the weak* topology is metrizable on norm-bounded subsets of X ∗ {\displaystyle X^{*}} . If a normed space X has a dual space that is separable (with respect to the dual-norm topology) then X is necessarily separable. [ 1 ] If X is a Banach space , the weak* topology is not metrizable on all of X ∗ {\displaystyle X^{*}} unless X is finite-dimensional. [ 3 ]
Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space L 2 ( R n {\displaystyle \mathbb {R} ^{n}} ) . Strong convergence of a sequence ψ k ∈ L 2 ( R n ) {\displaystyle \psi _{k}\in L^{2}(\mathbb {R} ^{n})} to an element ψ means that
as k → ∞ . Here the notion of convergence corresponds to the norm on L 2 .
In contrast weak convergence only demands that
for all functions f ∈ L 2 (or, more typically, all f in a dense subset of L 2 such as a space of test functions , if the sequence { ψ k } is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in C {\displaystyle \mathbb {C} } .
For example, in the Hilbert space L 2 (0,π) , the sequence of functions
form an orthonormal basis . In particular, the (strong) limit of ψ k {\displaystyle \psi _{k}} as k → ∞ does not exist. On the other hand, by the Riemann–Lebesgue lemma , the weak limit exists and is zero.
One normally obtains spaces of distributions by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on R n {\displaystyle \mathbb {R} ^{n}} ). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such as L 2 . Thus one is led to consider the idea of a rigged Hilbert space .
Suppose that X is a vector space and X # is the algebraic dual space of X (i.e. the vector space of all linear functionals on X ). If X is endowed with the weak topology induced by X # then the continuous dual space of X is X # , every bounded subset of X is contained in a finite-dimensional vector subspace of X , every vector subspace of X is closed and has a topological complement . [ 4 ]
If X and Y are topological vector spaces, the space L ( X , Y ) of continuous linear operators f : X → Y may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space Y to define operator convergence ( Yosida 1980 , IV.7 Topologies of linear maps). There are, in general, a vast array of possible operator topologies on L ( X , Y ) , whose naming is not entirely intuitive.
For example, the strong operator topology on L ( X , Y ) is the topology of pointwise convergence . For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x ∈ X :
More generally, if a family of seminorms Q defines the topology on Y , then the seminorms p q , x on L ( X , Y ) defining the strong topology are given by
indexed by q ∈ Q and x ∈ X .
In particular, see the weak operator topology and weak* operator topology .
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Predicate transformer semantics were introduced by Edsger Dijkstra in his seminal paper " Guarded commands, nondeterminacy and formal derivation of programs ". They define the semantics of an imperative programming paradigm by assigning to each statement in this language a corresponding predicate transformer : a total function between two predicates on the state space of the statement. In this sense, predicate transformer semantics are a kind of denotational semantics . Actually, in guarded commands , Dijkstra uses only one kind of predicate transformer: the well-known weakest preconditions (see below).
Moreover, predicate transformer semantics are a reformulation of Floyd–Hoare logic . Whereas Hoare logic is presented as a deductive system , predicate transformer semantics (either by weakest-preconditions or by strongest-postconditions see below) are complete strategies to build valid deductions of Hoare logic. In other words, they provide an effective algorithm to reduce the problem of verifying a Hoare triple to the problem of proving a first-order formula . Technically, predicate transformer semantics perform a kind of symbolic execution of statements into predicates: execution runs backward in the case of weakest-preconditions, or runs forward in the case of strongest-postconditions.
For a statement S and a postcondition R , a weakest precondition is a predicate Q such that for any precondition P , { P } S { R } {\displaystyle \{P\}S\{R\}} if and only if P ⇒ Q {\displaystyle P\Rightarrow Q} . In other words, it is the "loosest" or least restrictive requirement needed to guarantee that R holds after S . Uniqueness follows easily from the definition: If both Q and Q' are weakest preconditions, then by the definition { Q ′ } S { R } {\displaystyle \{Q'\}S\{R\}} so Q ′ ⇒ Q {\displaystyle Q'\Rightarrow Q} and { Q } S { R } {\displaystyle \{Q\}S\{R\}} so Q ⇒ Q ′ {\displaystyle Q\Rightarrow Q'} , and thus Q = Q ′ {\displaystyle Q=Q'} . We often use w p ( S , R ) {\displaystyle wp(S,R)} to denote the weakest precondition for statement S with respect to a postcondition R .
We use T to denote the predicate that is everywhere true and F to denote the one that is everywhere false. We shouldn't at least conceptually confuse ourselves with a Boolean expression defined by some language syntax, which might also contain true and false as Boolean scalars. For such scalars we need to do a type coercion such that we have T = predicate(true) and F = predicate(false). Such a promotion is carried out often casually, so people tend to take T as true and F as false.
We give below two equivalent weakest-preconditions for the assignment statement. In these formulas, R [ x ← E ] {\displaystyle R[x\leftarrow E]} is a copy of R where free occurrences of x are replaced by E . Hence, here, expression E is implicitly coerced into a valid term of the underlying logic: it is thus a pure expression, totally defined, terminating and without side effect.
where y is a fresh variable and not free in E and R (representing the final value of variable x )
Provided that E is well defined, we just apply the so-called one-point rule on version 1. Then
The first version avoids a potential duplication of x in R , whereas the second version is simpler when there is at most a single occurrence of x in R . The first version also reveals a deep duality between weakest-precondition and strongest-postcondition (see below).
An example of a valid calculation of wp (using version 2) for assignments with integer valued variable x is:
This means that in order for the postcondition x > 10 to be true after the assignment, the precondition x > 15 must be true before the assignment. This is also the "weakest precondition", in that it is the "weakest" restriction on the value of x which makes x > 10 true after the assignment.
For example,
As example:
Ignoring termination for a moment, we can define the rule for the weakest liberal precondition , denoted wlp , using a predicate INV , called the Loop INV ariant , typically supplied by the programmer:
To show total correctness, we also have to show that the loop terminates. For this we define a well-founded relation on the state space denoted as ( wfs , <) and define a variant function vf , such that we have:
where v is a fresh tuple of variables
Informally, in the above conjunction of three formulas:
However, the conjunction of those three is not a necessary condition. Exactly, we have
Actually, Dijkstra's Guarded Command Language (GCL) is an extension of the simple imperative language given until here with non-deterministic statements. Indeed, GCL aims to be a formal notation to define algorithms. Non-deterministic statements represent choices left to the actual implementation (in an effective programming language): properties proved on non-deterministic statements are ensured for all possible choices of implementation. In other words, weakest-preconditions of non-deterministic statements ensure
Notice that the definitions of weakest-precondition given above (in particular for while-loop ) preserve this property.
Selection is a generalization of if statement:
[ citation needed ]
Here, when two guards E i {\displaystyle E_{i}} and E j {\displaystyle E_{j}} are simultaneously true, then execution of this statement can run any of the associated statement S i {\displaystyle S_{i}} or S j {\displaystyle S_{j}} .
Repetition is a generalization of while statement in a similar way.
Refinement calculus extends GCL with the notion of specification statement . Syntactically, we prefer to write a specification statement as
which specifies a computation that starts in a state satisfying pre and is guaranteed
to end in a state satisfying post by changing only x . We call l {\displaystyle l} a logical constant employed to aid in a specification. For example, we can specify a computation that increment x by 1 as
Another example is a computation of a square root of an integer.
The specification statement appears like a primitive in the sense that it does not contain other statements. However, it is very expressive, as pre and post are arbitrary predicates. Its weakest precondition is as follows.
where s is fresh.
It combines Morgan's syntactic idea with the sharpness idea by Bijlsma, Matthews and Wiltink. [ 1 ] The very advantage of this is its capability of defining wp of goto L and other jump statements. [ 2 ]
Formalization of jump statements like goto L takes a very long bumpy process. A common belief seems to indicate the goto statement could only be argued operationally. This is probably due to a failure to recognize that goto L is actually miraculous (i.e. non-strict) and does not follow Dijkstra's coined Law of Miracle Excluded, as stood in itself. But it enjoys an extremely simple operational view from the weakest precondition perspective, which was unexpected. We define
where wpL is the weakest precondition at label L .
For goto L execution transfers control to label L at which the weakest precondition has to hold.
The way that wpL is referred to in the rule should not be taken as a big surprise. It is just w p ( L : S , Q ) {\displaystyle wp(L:S,Q)} for some Q computed to that point. This is like any wp rules, using constituent statements to give wp definitions, even though goto L appears a primitive. The rule does not require the uniqueness for locations where wpL holds within a program, so theoretically it allows the same label to appear in multiple locations as long as the weakest precondition at each location is the same wpL. The goto statement can jump to any of such locations. This actually justifies that we could place the same labels at the same location multiple times, as S ( L : L : S 1 ) {\displaystyle S(L:L:S1)} , which is the same as S ( L : S 1 ) {\displaystyle S(L:S1)} . Also, it does not imply any scoping rule, thus allowing a jump into a loop body, for example. Let us calculate wp of the following program S, which has a jump into the loop body.
Therefore,
An important variant of the weakest precondition is the weakest liberal precondition w l p ( S , R ) {\displaystyle wlp(S,R)} , which yields the weakest condition under which S either does not terminate or establishes R . It therefore differs from wp in not guaranteeing termination. Hence it corresponds to Hoare logic in partial correctness: for the statement language given above, wlp differs with wp only on while-loop , in not requiring a variant (see above).
Given S a statement and R a precondition (a predicate on the initial state), then s p ( S , R ) {\displaystyle sp(S,R)} is their strongest-postcondition : it implies any postcondition satisfied by the final state of any execution of S,
for any initial state satisfying R. In other words, a Hoare triple { P } S { Q } {\displaystyle \{P\}S\{Q\}} is provable in Hoare logic if and only if the predicate below hold:
Usually, strongest-postconditions are used in partial correctness.
Hence, we have the following relation between weakest-liberal-preconditions and strongest-postconditions:
For example, on assignment we have:
where y is fresh
Above, the logical variable y represents the initial value of variable x .
Hence,
On sequence, it appears that sp runs forward (whereas wp runs backward):
Leslie Lamport has suggested win and sin as predicate transformers for concurrent programming . [ 3 ]
This section presents some characteristic properties of predicate transformers. [ 4 ] Below, S denotes a predicate transformer (a function between two predicates on the state space) and P a predicate. For instance, S(P) may denote wp(S,P) or sp(S,P) . We keep x as the variable of the state space.
Predicate transformers of interest ( wp , wlp , and sp ) are monotonic . A predicate transformer S is monotonic if and only if:
This property is related to the consequence rule of Hoare logic .
A predicate transformer S is strict iff:
For instance, wp is artificially made strict, whereas wlp is generally not. In particular, if statement S may not terminate then w l p ( S , F ) {\displaystyle wlp(S,{\texttt {F}})} is satisfiable. We have
Indeed, T is a valid invariant of that loop.
The non-strict but monotonic or conjunctive predicate transformers are called miraculous and can also be used to define a class of programming constructs, in particular, jump statements, which Dijkstra cared less about. Those jump statements include straight goto L, break and continue in a loop and return statements in a procedure body, exception handling, etc. It turns out that all jump statements are executable miracles, [ 5 ] i.e. they can be implemented but not strict.
A predicate transformer S is terminating if:
Actually, this terminology makes sense only for strict predicate transformers: indeed, w p ( S , T ) {\displaystyle wp(S,{\texttt {T}})} is the weakest-precondition ensuring termination of S .
It seems that naming this property non-aborting would be more appropriate: in total correctness, non-termination is abortion, whereas in partial correctness, it is not.
A predicate transformer S is conjunctive iff:
This is the case for w p ( S , . ) {\displaystyle wp(S,.)} , even if statement S is non-deterministic as a selection statement or a specification statement.
A predicate transformer S is disjunctive iff:
This is generally not the case of w p ( S , . ) {\displaystyle wp(S,.)} when S is non-deterministic. Indeed, consider a non-deterministic statement S choosing an arbitrary Boolean. This statement is given here as the following selection statement :
Then, w p ( S , R ) {\displaystyle wp(S,R)} reduces to the formula R [ x ← 0 ] ∧ R [ x ← 1 ] {\displaystyle R[x\leftarrow 0]\wedge R[x\leftarrow 1]} .
Hence, w p ( S , x = 0 ∨ x = 1 ) {\displaystyle wp(S,\ x=0\vee x=1)} reduces to the tautology ( 0 = 0 ∨ 0 = 1 ) ∧ ( 1 = 0 ∨ 1 = 1 ) {\displaystyle (0=0\vee 0=1)\wedge (1=0\vee 1=1)}
Whereas, the formula w p ( S , x = 0 ) ∨ w p ( S , x = 1 ) {\displaystyle wp(S,x=0)\vee wp(S,x=1)} reduces to the wrong proposition ( 0 = 0 ∧ 1 = 0 ) ∨ ( 1 = 0 ∧ 1 = 1 ) {\displaystyle (0=0\wedge 1=0)\vee (1=0\wedge 1=1)} .
In predicate transformers semantics, expressions are restricted to terms of the logic (see above). However, this restriction seems too strong for most existing programming languages, where expressions may have side effects (call to a function having a side effect), may not terminate or abort (like division by zero ). There are many proposals to extend weakest-preconditions or strongest-postconditions for imperative expression languages and in particular for monads .
Among them, Hoare Type Theory combines Hoare logic for a Haskell -like language, separation logic and type theory . [ 9 ] This system is currently implemented as a Coq library called Ynot . [ 10 ] In this language, evaluation of expressions corresponds to computations of strongest-postconditions .
Probabilistic Predicate Transformers are an extension of predicate transformers for probabilistic programs .
Indeed, such programs have many applications in cryptography (hiding of information using some randomized noise), distributed systems (symmetry breaking). [ 11 ]
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A weakless universe is a hypothetical universe that contains no weak interactions , but is otherwise very similar to our own universe.
In particular, a weakless universe is constructed to have atomic physics and chemistry identical to standard atomic physics and chemistry. The dynamics of a weakless universe includes a period of Big Bang nucleosynthesis , star formation , stars with sufficient fuel to burn for billions of years, stellar nuclear synthesis of heavy elements and also supernovae that distribute the heavy elements into the interstellar medium.
The strength of the weak interaction is an outstanding problem in modern particle physics . A theory should ideally explain why the weak interaction is 32 orders of magnitude stronger than gravity ; this is known as the hierarchy problem . There are various models that address the hierarchy problem in a dynamical and natural way, for example, supersymmetry , technicolor , warped extra dimensions , and so on.
An alternative approach to explaining the hierarchy problem is to invoke the anthropic principle : One assumes that there are many other patches of the universe (or multiverse ) in which physics is very different. In particular one can assume that the “ landscape ” of possible universes contains ones where the weak force has a different strength compared to our own. In such a scenario observers would presumably evolve wherever they can. If the observed strength of the weak force is then vital for the emergence of observers, this would explain why the weak force is indeed observed with this strength. Barr and others argued [ citation needed ] that if one only allows the electroweak symmetry breaking scale to vary between universes, keeping all other parameters fixed, atomic physics would change in ways that would not allow life as we know it.
Anthropic arguments have recently been boosted by the realization that string theory has many possible solutions, or vacua, called the “ string landscape ”, and by Steven Weinberg 's prediction of the cosmological constant by anthropic reasoning. [ citation needed ]
The hypothetical universe without the weak interaction is meant to serve as a counter-example to the anthropic approach to the hierarchy problem. For this “ weakless universe ”, other parameters are varied as the electroweak breaking scale is changed. Indeed, string theory implies that the landscape is very big and diverse. The ostensible habitability of the weakless universe implies that anthropic reasoning alone cannot explain the hierarchy problem, unless the available vacua in the landscape are severely restricted for some other reason.
One of the biggest obstacles for a habitable weakless universe is the necessary existence of stars. Main sequence stars work through fusing two protons to deuterium as a first step, which proceeds through weak interactions. In the weakless universe of Harnik, Kribs , and Perez [ 1 ] this is overcome by ensuring a high primordial deuterium to hydrogen ratio during Big Bang Nucleosynthesis (BBN). This permits long-lived stars fueled by direct deuterium-proton burning to helium, which proceeds through strong interactions. The high initial deuterium/hydrogen ratio (~1:3 by mass) is arranged by simply reducing the overall baryon to photon ratio, which allows the BBN deuterium to be produced at a lower temperature where the Coulomb barrier protects deuterium from immediate fusion into 4 He .
Another potential problem for a weakless universe is that supernova explosions are necessarily neutrinoless. The resulting efficiency of production and dispersion of heavy elements (in particular, oxygen) into the interstellar medium for subsequent incorporation into habitable planets has been questioned by Clavelli and White. [ 2 ]
Baryogenesis and leptogenesis within the Standard Model rely on the weak interaction: For matter not to be wiped off by anti-matter at the very early universe, the universe must either have to start with a different amount of each (i.e. initial non-zero baryon number), or admit Sakharov's conditions to baryogenesis. In the latter case, there are two options:
Harnik, Kribs, and Perez argue that the Standard Model does not explain the observed size of the baryon asymmetry either, and that their weakless universe model only focuses on the time where the asymmetry already exists. [ 1 ]
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In mathematics , a topological space is said to be weakly contractible if all of its homotopy groups are trivial.
It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible .
Define S ∞ {\displaystyle S^{\infty }} to be the inductive limit of the spheres S n , n ≥ 1 {\displaystyle S^{n},n\geq 1} . Then this space is weakly contractible. Since S ∞ {\displaystyle S^{\infty }} is moreover a CW-complex, it is also contractible. See Contractibility of unit sphere in Hilbert space for more.
The Long Line is an example of a space which is weakly contractible, but not contractible. This does not contradict Whitehead theorem since the Long Line does not have the homotopy type of a CW-complex.
Another prominent example for this phenomenon is the Warsaw circle .
This topology-related article is a stub . You can help Wikipedia by expanding it .
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In model theory , a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets .
A linearly ordered structure, M , with language L including an ordering relation <, is called weakly o-minimal if every parametrically definable subset of M is a finite union of convex (definable) subsets. A theory is weakly o-minimal if all its models are weakly o-minimal.
Note that, in contrast to o-minimality , it is possible for a theory to have models that are weakly o-minimal and to have other models that are not weakly o-minimal. [ 1 ]
In an o-minimal structure ( M , < , . . . ) {\displaystyle (M,<,...)} the definable sets in M {\displaystyle M} are finite unions of points and intervals, where interval stands for a sets of the form I = { r ∈ M : a < r < b } {\displaystyle I=\{r\in M\,:\,a<r<b\}} , for some a and b in M ∪ { ± ∞ } {\displaystyle M\cup \{\pm \infty \}} . For weakly o-minimal structures ( M , < , . . . ) {\displaystyle (M,<,...)} this is relaxed so that the definable sets in M are finite unions of convex definable sets. A set C {\displaystyle C} is convex if whenever a and b are in C {\displaystyle C} , a < b and c ∈ M {\displaystyle M} satisfies that a < c < b , then c is in C . Points and intervals are of course convex sets, but there are convex sets that are not either points or intervals, as explained below.
If we have a weakly o-minimal structure expanding ( R ,<), the real ordered field, then the structure will be o-minimal. The two notions are different in other settings though. For example, let R be the ordered field of real algebraic numbers with the usual ordering < inherited from R . Take a transcendental number, say π , and add a unary relation S to the structure given by the subset (− π , π ) ∩ R . Now consider the subset A of R defined by the formula
so that the set consists of all strictly positive real algebraic numbers that are less than π . The set is clearly convex, but cannot be written as a finite union of points and intervals whose endpoints are in R . To write it as an interval one would either have to include the endpoint π , which isn't in R , or one would require infinitely many intervals, such as the union
Since we have a definable set that isn't a finite union of points and intervals, this structure is not o-minimal. However, it is known that the structure is weakly o-minimal, and in fact the theory of this structure is weakly o-minimal. [ 2 ]
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In mathematics , a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space , due to Élie Cartan . Geometrically the spaces are defined as complete Riemannian manifolds such that any two points can be exchanged by an isometry , the symmetric case being when the isometry is required to have period two. The classification of weakly symmetric spaces relies on that of periodic automorphisms of complex semisimple Lie algebras . They provide examples of Gelfand pairs , although the corresponding theory of spherical functions in harmonic analysis , known for symmetric spaces, has not yet been developed.
This differential geometry -related article is a stub . You can help Wikipedia by expanding it .
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The Wealden iron industry was located in the Weald of south-eastern England . It was formerly an important industry, producing a large proportion of the bar iron made in England in the 16th century and most British cannon until about 1770. Ironmaking in the Weald used ironstone from various clay beds, and was fuelled by charcoal made from trees in the heavily wooded landscape. The industry in the Weald declined when ironmaking began to be fuelled by coke made from coal , which does not occur accessibly in the area.
Iron ore in the form of siderite , commonly known as iron stone or historically as mine, occurs in patches or bands in the Cretaceous clays of the Weald. Differing qualities of ore were extracted and mixed by experienced smelters to give the best results. Sites of opencast quarries survive from the pre-Roman and Roman eras, but medieval ore extraction was mainly done by digging a series of minepits about five metres in diameter and up to twelve metres deep with material being winched up in baskets suspended from a wooden tripod. This was less destructive of the land as spoil from one pit was used to backfill the previous pit allowing continued land use.
The fuel for smelting was charcoal , which needed to be produced as close as possible to the smelting sites because it would crumble to dust if transported far by cart over rough tracks. Wood was also needed for pre-roasting the ore on open fires, a process which broke down the lumps or nodules and converted the carbonate into oxide. Large areas of woodland were available in the Weald and coppicing woodlands could provide a sustainable source of wood. Sustainable charcoal production for a post-medieval blast furnace required the timber production from a 3 miles (4.8 km) radius of a furnace in a landscape that was a quarter to a third wooded. Forging and finishing of the iron from bloomeries and blast furnaces also required large quantities of charcoal and was usually carried out at a separate site.
Water power became important with the introduction of blast furnaces and finery forges in the late medieval period. Blast furnaces needed to operate continuously for as long as possible and a series of ponds were often created in a valley to give a sustainable flow for the waterwheel . A campaign, as the production run was known, usually ran from October through to late spring when streams began to dry up, although Lamberhurst Furnace driven by the River Teise ran continuously for more than three years in the 1740s. Finery forges with three or four waterwheels to drive bellows and hammers needed more water than a furnace at times, although continuity was not as important. They tended to be sited downstream from a furnace if they were in the same valley. Ponds were created by building a dam known as a pond bay, which often served as a road, across one of the many valleys in the undulating Wealden landscape. [ 1 ] In 1754 one furnace was so drought-stricken that its manager considered hiring workmen to turn the wheel as a treadmill . [ 2 ] This need for continuous water power was an incentive in the development of the water-returning engine , a waterwheel driven by water raised by a steam engine pump.
So far only about two dozen sites have been identified where iron was made before the Roman invasion , mostly scattered across East Sussex and the Vale of Kent . A large site at Broadfield, Crawley is the westernmost place where smelting has been ascertained, although there is a possible site associated with an Iron Age enclosure at Piper's Copse near Northchapel in the western Weald. Continuity of pottery styles from the Iron Age into the early Roman period makes precise dating of many sites to before or after the Roman conquest difficult. Carbon dating has identified a site at Cullinghurst Wood, Hartfield to between 350 and 750 BC. [ 3 ]
During his invasions of Britain in 55 and 54 BC Julius Caesar noted iron production near the coast, possibly at known sites at Sedlescombe and Crowhurst Park near Hastings .
View of the 13th fairway of Beauport Park golf course, beneath which are remains of the Roman ironworks.
Beauport Park , where evidence has been found of probably the third largest iron works in the whole Roman Empire . [ 4 ] The Romans made full use of the brown- and ochre-coloured stone in the Weald , and many of their roads there are the means of transport for the ore, and were extensively metalled with slag from iron smelting. [ 5 ] The sites of about 113 bloomeries have been identified as Roman, mainly in East Sussex . [ 6 ] The Weald was in this period one of the most important iron-producing regions in Roman Britain . Excavations at a few sites have produced tiles of the Classis Britannica , suggesting that they were actually run by, or were supplying iron to this Roman fleet. Total iron production has been estimated at 750 tons per year, but under 200 tons per year after 250 AD. [ 7 ]
The invasion and settlement of the Weald by Saxons seems to have brought a complete end to the Romano-British iron industry. No evidence of iron smelting has been found after the end of Roman rule until the ninth century when a primitive bloomery was built at Millbrook on Ashdown Forest , with a small hearth for reheating the blooms nearby. The date of this site has been established by radiocarbon and archaeomagnetic methods. The technology used there was similar to a slightly earlier furnace excavated in the eastern Netherlands, indicating that knowledge of Romano-British methods had been completely lost and replaced by the Saxons' own method. Evidence of forging of iron blooms in settlements close to the South Downs does indicate that smelting may have been going on at other undiscovered sites. It was usual for settlements concentrated along the Downs to have outlying parcels of land in the Weald for summer grazing. It is likely that smelting was carried out during the summer and the iron blooms taken back to the main settlement to work on in the winter. [ 8 ]
In all some 30 unpowered medieval bloomery sites are known in the Weald, but most of these remain undated. Accounts survive of the operation of just one, at Tudeley near Tonbridge in the mid-14th century. [ 9 ]
From about the 14th century, water-power began to be applied to bloomeries , but fewer than ten such sites are suspected.
A new ironmaking process was devised in the Namur region of what is now Belgium in the 15th century. This spread to the pays de Bray on the eastern boundary of Normandy and then to the Weald. The new smelting process involved a blast furnace and finery forge . It was introduced in about 1490 at Queenstock in Buxted parish. [ 10 ] The number of ironworks increased greatly from about 1540.
Nearly 180 sites in all were used for this process, having a furnace, a forge or both between the 15th century and 18th century. Waterpower was the means of operating the bellows in the blast furnaces and for operating bellows and helve hammers in finery forges . Scattered through the Weald are ponds still to be found called ’Furnace Pond’ or ’Hammer Pond’. The iron was used for making household utensils, nails and hinges ; and for casting cannon . The first blast furnace was recorded at Buxted in 1490.
The industry was at its peak towards the end of Queen Elizabeth I 's reign. Most works were small, but at Brenchley one ironmaster employed 200 men . Most of them would have been engaged in mining ore and cutting wood (for charcoal ), as the actual ironworks only required a small workforce. The wars fought during the reign of Henry VIII increased the need for armaments , and the Weald became the centre of an armaments industry . Cast-iron cannon were made in the Weald from 1543 when Buxted 's Ralf Hogge cast the first iron cannon for his unlikely employer: a Sussex vicar who was gunstonemaker to the king.
In the 16th century and the early 17th century, the Weald was a major source of iron for manufacture in London , peaking at over 9000 tons per year in the 1590s. [ 11 ] However, after 1650, Wealden production became increasingly focused on the production of cannon ; and bar iron was only produced for local consumption. This decline may have begun as early as the 1610s, when Midland ironware began to be sold in London . Certainly after Swedish iron began to be imported in large quantities after the Restoration , Wealden bar iron seems to have been unable to compete in the London market.
Cannon production was a major activity in the Weald until the end of the Seven Years' War , but a cut in the price paid by the Board of Ordnance drove several Wealden ironmasters into bankruptcy. They were unable to match the much lower price that was acceptable to the Scottish Carron Company , whose fuel was coke . A few ironworks continued operating on a very small scale. With no local source of mineral coal, the Wealden iron industry was unable to compete with the new coke-fired ironworks of the Industrial Revolution . The last to close was the forge at Ashburnham . Little survives of the furnace and forge buildings, although there are still scores of the industry's hammer and furnace ponds scattered throughout the Weald. [ 12 ]
Steel production was never widespread in the Weald, with most high quality steel being imported from Spain, the Middle East, or Germany. A steel forge was built upstream from Newbridge Furnace on Ashdown Forest around 1505 but had ceased production by 1539. The Sydney family, with mills at Robertsbridge forge and at Sandhurst in Kent, produced steel using skilled German workers, but faced strong competition from German suppliers. In the 17th century a steel forge existed at Warbleton in Sussex. [ 13 ]
The Lamberhurst Foundry is believed to have been the maker in 1710–14 of some of the earliest cast-iron railings produced in England, which they made for St Paul's Cathedral , despite the objections of Christopher Wren , who did not want a fence around the Cathedral at all, and said that if there had to be one it should be of wrought rather than cast iron. [ 14 ] The railings surrounded the cathedral, including seven gates. It weighed two hundred tons and cost six pence a pound. [ 14 ] The total cost was £11,202. [ 14 ] No further railings are known to have been cast in the Weald. [ 15 ] Other early uses of cast iron railings were at Cambridge Senate House and at St Martin-in-the-Fields , London. [ 14 ]
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Wear is the damaging, gradual removal or deformation of material at solid surfaces . Causes of wear can be mechanical (e.g., erosion ) or chemical (e.g., corrosion ). The study of wear and related processes is referred to as tribology .
Wear in machine elements , together with other processes such as fatigue and creep , causes functional surfaces to degrade, eventually leading to material failure or loss of functionality. Thus, wear has large economic relevance as first outlined in the Jost Report . [ 1 ] Abrasive wear alone has been estimated to cost 1–4% of the gross national product of industrialized nations. [ 2 ]
Wear of metals occurs by plastic displacement of surface and near-surface material and by detachment of particles that form wear debris . The particle size may vary from millimeters to nanometers . [ 3 ] This process may occur by contact with other metals, nonmetallic solids, flowing liquids, solid particles or liquid droplets entrained in flowing gasses. [ 4 ]
The wear rate is affected by factors such as type of loading (e.g., impact, static, dynamic), type of motion (e.g., sliding , rolling ), temperature , and lubrication , in particular by the process of deposition and wearing out of the boundary lubrication layer. [ 5 ] Depending on the tribosystem , different wear types and wear mechanisms can be observed.
Types of wear are identified by relative motion , the nature of disturbance at the worn surface or "mechanism", and whether it effects a self regenerative or base layer. [ 6 ]
Wear mechanisms are the physical disturbance. For example, the mechanism of adhesive wear is adhesion . Wear mechanisms and/or sub-mechanisms frequently overlap and occur in a synergistic manner, producing a greater rate of wear than the sum of the individual wear mechanisms. [ 7 ]
Adhesive wear can be found between surfaces during frictional contact and generally refers to unwanted displacement and attachment of wear debris and material compounds from one surface to another. [ 8 ] Two adhesive wear types can be distinguished: [ citation needed ]
Generally, adhesive wear occurs when two bodies slide over or are pressed into each other, which promote material transfer. This can be described as plastic deformation of very small fragments within the surface layers. [ citation needed ] The asperities or microscopic high points ( surface roughness ) found on each surface affect the severity of how fragments of oxides are pulled off and added to the other surface, partly due to strong adhesive forces between atoms, [ 9 ] but also due to accumulation of energy in the plastic zone between the asperities during relative motion.
The type of mechanism and the amplitude of surface attraction varies between different materials but are amplified by an increase in the density of "surface energy". Most solids will adhere on contact to some extent. However, oxidation films, lubricants and contaminants naturally occurring generally suppress adhesion, [ 10 ] and spontaneous exothermic chemical reactions between surfaces generally produce a substance with low energy status in the absorbed species. [ 11 ]
Adhesive wear can lead to an increase in roughness and the creation of protrusions (i.e., lumps) above the original surface. In industrial manufacturing, this is referred to as galling , which eventually breaches the oxidized surface layer and connects to the underlying bulk material, enhancing the possibility for a stronger adhesion [ 11 ] and plastic flow around the lump.
A simple model for the wear volume for adhesive wear, V {\displaystyle V} , can be described by: [ 12 ] [ 13 ]
V = K W L H v {\displaystyle V=K{\frac {WL}{H_{v}}}}
where W {\displaystyle W} is the load, K {\displaystyle K} is the wear coefficient, L {\displaystyle L} is the sliding distance, and H v {\displaystyle H_{v}} is the hardness.
Abrasive wear occurs when a hard rough surface slides across a softer surface. [ 9 ] ASTM International defines it as the loss of material due to hard particles or hard protuberances that are forced against and move along a solid surface. [ 14 ]
Abrasive wear is commonly classified according to the type of contact and the contact environment. [ 15 ] The type of contact determines the mode of abrasive wear. The two modes of abrasive wear are known as two-body and three-body abrasive wear. Two-body wear occurs when the grits or hard particles remove material from the opposite surface. The common analogy is that of material being removed or displaced by a cutting or plowing operation. Three-body wear occurs when the particles are not constrained, and are free to roll and slide down a surface. The contact environment determines whether the wear is classified as open or closed. An open contact environment occurs when the surfaces are sufficiently displaced to be independent of one another
There are a number of factors which influence abrasive wear and hence the manner of material removal. Several different mechanisms have been proposed to describe the manner in which the material is removed. Three commonly identified mechanisms of abrasive wear are: [ citation needed ]
Plowing occurs when material is displaced to the side, away from the wear particles, resulting in the formation of grooves that do not involve direct material removal. The displaced material forms ridges adjacent to grooves, which may be removed by subsequent passage of abrasive particles.
Cutting occurs when material is separated from the surface in the form of primary debris, or microchips, with little or no material displaced to the sides of the grooves. This mechanism closely resembles conventional machining.
Fragmentation occurs when material is separated from a surface by a cutting process and the indenting abrasive causes localized fracture of the wear material. These cracks then freely propagate locally around the wear groove, resulting in additional material removal by spalling . [ 15 ]
Abrasive wear can be measured as loss of mass by the Taber Abrasion Test according to ISO 9352 or ASTM D 4060.
The wear volume for single-abrasive wear, V {\displaystyle V} , can be described by: [ 13 ]
V = α β W L H v = K W L H v {\displaystyle V=\alpha \beta {\frac {WL}{H_{v}}}=K{\frac {WL}{H_{v}}}}
where W {\displaystyle W} is the load, α {\displaystyle \alpha } is the shape factor of an asperity (typically ~ 0.1), β {\displaystyle \beta } is the degrees of wear by an asperity (typically 0.1 to 1.0), K {\displaystyle K} is the wear coefficient, L {\displaystyle L} is the sliding distance, and H v {\displaystyle H_{v}} is the hardness.
Surface fatigue is a process in which the surface of a material is weakened by cyclic loading, which is one type of general material fatigue. Fatigue wear is produced when the wear particles are detached by cyclic crack growth of microcracks on the surface. These microcracks are either superficial cracks or subsurface cracks.
Fretting wear is the repeated cyclical rubbing between two surfaces. Over a period of time fretting which will remove material from one or both surfaces in contact. It occurs typically in bearings, although most bearings have their surfaces hardened to resist the problem. Another problem occurs when cracks in either surface are created, known as fretting fatigue. It is the more serious of the two phenomena because it can lead to catastrophic failure of the bearing. An associated problem occurs when the small particles removed by wear are oxidized in air. The oxides are usually harder than the underlying metal, so wear accelerates as the harder particles abrade the metal surfaces further. Fretting corrosion acts in the same way, especially when water is present. Unprotected bearings on large structures like bridges can suffer serious degradation in behaviour, especially when salt is used during the winter to deice the highways carried by the bridges. The problem of fretting corrosion was involved in the Silver Bridge tragedy and the Mianus River Bridge accident.
Erosive wear can be defined as an extremely short sliding motion and is executed within a short time interval. Erosive wear is caused by the impact of particles of solid or liquid against the surface of an object. [ 10 ] [ 16 ] The impacting particles gradually remove material from the surface through repeated deformations and cutting actions. [ 17 ] It is a widely encountered mechanism in industry. Due to the nature of the conveying process, piping systems are prone to wear when abrasive particles have to be transported. [ 18 ]
The rate of erosive wear is dependent upon a number of factors. The material characteristics of the particles, such as their shape, hardness, impact velocity and impingement angle are primary factors along with the properties of the surface being eroded. The impingement angle is one of the most important factors and is widely recognized in literature. [ 19 ] For ductile materials, the maximum wear rate is found when the impingement angle is approximately 30°, whilst for non-ductile materials the maximum wear rate occurs when the impingement angle is normal to the surface. [ 19 ] A detailed theoretical analysis of dependency of the erosive wear on the inclination angle and material properties is provided in. [ 20 ]
For a given particle morphology, the erosion rate, E {\displaystyle E} , can be fit with a power law dependence on velocity: [ 16 ]
E = k v n {\displaystyle E=kv^{n}}
where k {\displaystyle k} is a constant, v {\displaystyle v} is velocity, and n {\displaystyle n} is a velocity exponent. n {\displaystyle n} is typically between 2 - 2.5 for metals and 2.5 - 3 for ceramics.
Corrosion and oxidation wear occurs both in lubricated and dry contacts. The fundamental cause are chemical reactions between the worn material and the corroding medium. [ 21 ] Wear caused by a synergistic action of tribological stresses and corrosion is also called tribocorrosion .
Impact wear is caused by contact between two bodies. Unlike erosive wear, impact wear always occurs at the same, well-defined place. If the impact is repeated, then usually with constant kinetic energy at the moment of impact. The frequency of impacts can vary. Wear can occur on both bodies, but usually, one body has significantly higher hardness and toughness and its wear is neglected.
Other, less common types of wear are cavitation and diffusive wear. [ 6 ]
Under nominal operation conditions, the wear rate normally changes in three different stages: [ citation needed ]
The wear rate is strongly influenced by the operating conditions and the formation of tribofilms . The secondary stage is shortened with increasing severity of environmental conditions, such as high temperatures, strain rates and stresses.
So-called wear maps, demonstrating wear rate under different operation condition, are used to determine stable operation points for tribological contacts. Wear maps also show dominating wear modes under different loading conditions. [ citation needed ]
In explicit wear tests simulating industrial conditions between metallic surfaces, there are no clear chronological distinction between different wear-stages due to big overlaps and symbiotic relations between various friction mechanisms. Surface engineering and treatments are used to minimize wear and extend the components working life. [ 1 ] [ 22 ]
Several standard test methods exist for different types of wear to determine the amount of material removal during a specified time period under well-defined conditions. ASTM International Committee G-2 standardizes wear testing for specific applications, which are periodically updated. The Society for Tribology and Lubrication Engineers (STLE) has documented a large number of frictional, wear and lubrication tests. Standardized wear tests are used to create comparative material rankings for a specific set of test parameter as stipulated in the test description. To obtain more accurate predictions of wear in industrial applications it is necessary to conduct wear testing under conditions simulating the exact wear process.
An attrition test is a test that is carried out to measure the resistance of a granular material to wear.
The Reye–Archard–Khrushchov wear law is the classic wear prediction model. [ 23 ]
The wear coefficient is a physical coefficient used to measure, characterize and correlate the wear of materials.
Lubricant analysis is an alternative, indirect way of measuring wear. Here, wear is detected by the presence of wear particles in a liquid lubricant. To gain further insights into the nature of the particles, chemical (such as XRF, ICP-OES), structural (such as ferrography ) or optical analysis (such as light microscopy ) can be performed. [ 24 ]
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Wear OS [ a ] , formerly Android Wear , [ 9 ] is a closed-source Android distribution designed for smartwatches and other wearable computers, [ 10 ] [ 11 ] [ 12 ] developed by Google . [ 2 ] [ 3 ] Wear OS is designed to pair with mobile phones running Android ( version 6.0 "Marshmallow" or newer) or iOS (version 10.0 or newer), [ 13 ] providing mobile notifications into a smartwatch form factor and integration with the Google Assistant technology. [ 14 ]
Wear OS supports Bluetooth , NFC , Wi-Fi , [ 15 ] 3G , and LTE connectivity, as well as a range of features and applications provided through Google Play . Watch face styles include round, square and rectangular. Hardware manufacturing partners include Asus , Broadcom , Fossil , HTC , Intel , LG , MediaTek , Imagination Technologies , Motorola , New Balance , Xiaomi , Qualcomm , Samsung , Huawei , Skagen , Polar , TAG Heuer , Suunto , and Mobvoi . [ 16 ]
The operating system was first released in 2014 as Android Wear , and took its current name in 2018. [ 17 ] Analysts estimate that over 720,000 Android Wear smartwatches were shipped in 2014, the year of its launch. [ 18 ] By mid-October 2022, the Wear OS app had more than 50 million downloads. [ 19 ] Wear OS was estimated to account for 17.3% of the smartwatch market in Q3 2021, behind Apple's 21.8%. Samsung accounts for the majority of Wear OS devices sold, [ 20 ] due to its switch back from Tizen to Wear OS in 2021. [ 21 ]
The platform was announced on March 18, 2014, along with the release of a developer preview. At the same time, companies such as Motorola , Samsung , LG , HTC and Asus were announced as partners. [ 22 ] On June 25, 2014, at Google I/O , the Samsung Gear Live and LG G Watch were launched, along with further details about Android Wear. The LG G Watch is the first Android Wear smartwatch to be released and shipped.
Motorola's Moto 360 was released on September 5, 2014.
On December 10, 2014, an update started to roll out, adding new features including a watch face API and changed the software to be based on Android 5.0 "Lollipop" . [ 23 ]
The LG G Watch [ 24 ] and Samsung Gear Live [ 25 ] started shipping in July 2014, while the Motorola Moto 360 [ 26 ] began shipping in September 2014. The next batch of Android Wear devices, which arrived at the end of 2014, included the Asus ZenWatch , [ 27 ] the Sony SmartWatch 3 , [ 28 ] and the LG G Watch R . [ 29 ] As of March 2015 [update] , the latest Wear OS devices are the LG Watch Urbane , [ 30 ] and the Huawei Watch . [ 31 ] [ needs update ]
On August 31, 2015, Google launched a Wear OS app for iOS version 8.2 or newer, allowing limited support for receiving iOS notifications on smartwatches running Wear OS. [ 32 ] As of September 2015 [update] , only the LG Watch Urbane and Huawei Watch are supported, but Google announced support for more smartwatch models. [ 13 ] [ 33 ]
In March 2018, Android Wear was rebranded as Wear OS. It was stated that the renaming "better reflects our technology, vision, and most important of all — the people who wear our watches." [ 34 ] In September 2018, Google announced Wear OS 2.0, which made the personalized Google feed (replacing Google Now ) and new fitness tracking platform Google Fit accessible from the watch face, and redesigned the notification area to use a scrolling pane rather than pages, and support automatically generated smart replies (as on Android Pie ). [ 35 ] [ 36 ] In November 2018, the underlying platform of Wear OS was upgraded to a version of Android Pie. [ 37 ]
In January 2021, Google completed its acquisition of wearables manufacturer Fitbit ; upon its announcement of the purchase in November 2019, Google's head of hardware Rick Osterloh stated that it would be "an opportunity to invest even more in Wear OS as well as introduce Made by Google wearable devices into the market." [ 38 ] [ 39 ]
In May 2021 at Google I/O, Google announced a major update to the platform, internally known as Wear OS 3.0. It incorporates a new visual design inspired by Android 12 , and Fitbit exercise tracking features. Google also announced a partnership with Samsung Electronics , who is collaborating with Google to unify its Tizen -based smartwatch platform with Wear OS, and has committed to using Wear OS on its future smartwatch products. The underlying codebase was also upgraded to Android 11 . [ 40 ] [ 41 ] Wear OS 3.0 will be available to Wear OS devices running Qualcomm Snapdragon Wear 4100 system on chip , and will be an opt-in upgrade requiring a factory reset to install. [ 42 ]
Wear OS can synchronize notifications from a paired device, and supports voice control with the "OK Google" hotword along with gesture-based input. [ 43 ] Wear OS integrates with Google services such as the Google Assistant and Google Mobile Services (including Gmail , Google Maps , and Google Wallet ), as well as third-party watch apps from Play Store . [ 44 ] [ 45 ] From the watch face, the user can swipe up to access their notifications, down to access a quick settings panel, from the left to view their personalized Google feed, and the right to view Google Fit . [ 43 ] Via Google Fit and similar applications, Wear OS supports ride and run tracking, and devices containing heart rate sensors can perform a reading on-demand, or at intervals throughout the day. [ 46 ] The watch can control media being played on streamed on paired devices. [ 45 ] [ 44 ]
[ 57 ]
Brings Android 8.0 Oreo features to smartwatches
Wear OS App version: 2.10 [ 65 ]
Health features to Google Fit:
Hardware improvements:
Social integration:
Removed:
[ 73 ]
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https://en.wikipedia.org/wiki/Wear_OS
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The wear coefficient is a physical coefficient used to measure, characterize and correlate the wear of materials.
Traditionally, the wear of materials has been characterized by weight loss and wear rate. However, studies have found that wear coefficient is more suitable. The reason being that it takes the wear rate, the applied load, and the hardness of the wear pin into account. Although, measurement variations by an order of 10-1 have been observed, the variations can be minimized if suitable precautions are taken. [ 1 ] [ 2 ]
A wear volume versus distance curve can be divided into at least two regimes, the transient wear regime and the steady-state wear regime. The volume or weight loss is initially curvilinear . The wear rate per unit sliding distance in the transient wear regime decreases until it has reached a constant value in the steady-state wear regime. Hence the standard wear coefficient value obtained from a volume loss versus distance curve is a function of the sliding distance. [ 3 ]
The steady-state wear equation was proposed as: [ 2 ]
V = K P L 3 H {\displaystyle V=K{\frac {PL}{3H}}}
where H {\displaystyle H} is the Brinell hardness expressed as Pascals, V {\displaystyle V} is the volumetric loss, P {\displaystyle P} is the normal load, and L {\displaystyle L} is the sliding distance. K {\displaystyle K} is the dimensionless standard wear coefficient.
(For example , for two mild steel surfaces , Brinell hardness number HB is 120 ,
and dimension of Brinell hardness number is [kgf/mm2] ,
so S.I. value of Brinell hardness , expressing it in Pascal (is N/mm2), is 9.8E6 times as large , so it is 1.176E9 .
If these surfaces slide over eachother over a length of 1 meter ,
and top is loaded with 1 kg (yielding a force of 9.8 Newton),
and using K value 7E-3 from table on the right of this text ,
then volume of steel removed is : V = 7E-3 * 9.8 * 1 /3 /1.176E9 = 1.94E-11 m3 = 0.019 mm3 ).
Therefore, the wear coefficient K {\displaystyle K} in the abrasive model is defined as: [ 2 ]
K = 3 H V P L {\displaystyle K={\frac {3HV}{PL}}}
As V {\displaystyle V} can be estimated from weight loss W {\displaystyle W} and the density ρ {\displaystyle \rho } , the wear coefficient can also be expressed as: [ 2 ]
K = 3 H W P L ρ {\displaystyle K={\frac {3HW}{PL\rho }}}
As the standard method uses the total volume loss and the total sliding distance, there is a need to define the net steady-state wear coefficient:
K N = 3 H V s P L s {\displaystyle K_{N}={\frac {3HV_{s}}{PL_{s}}}}
where L s {\displaystyle L_{s}} is the steady-state sliding distance, and V s {\displaystyle V_{s}} is the steady-state wear volume.
With regard to the sliding wear model K can be expressed as: [ 4 ]
K = V A p L {\displaystyle K={\frac {V}{A_{p}L}}}
where A p {\displaystyle A_{p}} is the plastically deformed zone.
If the coefficient of friction is defined as: [ 4 ]
μ = F t P {\displaystyle \mu ={\frac {F_{t}}{P}}}
where F t {\displaystyle F_{t}} is the tangential force.
Then K can be defined for abrasive wear as work done to create abrasive wear particles by cutting V u {\displaystyle Vu} to external work done F L {\displaystyle FL} : [ 4 ]
K = 3 μ H V μ P L = 3 μ V u F L ≈ V u F L {\displaystyle K={\frac {3\mu HV}{\mu PL}}=3\mu {\frac {Vu}{FL}}\approx {\frac {Vu}{FL}}}
In an experimental situation the hardness of the uppermost layer of material in the contact may not be known with any certainty, consequently, the ratio K H {\displaystyle {\frac {K}{H}}} is more useful; this is known as the dimensional wear coefficient or the specific wear rate . This is usually quoted in units of mm 3 N −1 m −1 . [ 5 ]
As metal matrix composite (MMC) materials have become to be used more often due to their better physical, mechanical and tribological properties compared to matrix materials it is necessary to adjust the equation.
The proposed equation is: [ 2 ]
K = 3 g 1 d ( 1 − f v ) g 3 f v L [ 1 − e x p ( − g 3 f v L d ( 1 − f v ) ) ] {\displaystyle K={\frac {3g_{1}d(1-f_{v})}{g_{3}f_{v}L}}\left[1-exp\left({\frac {-g_{3}f_{v}L}{d(1-f_{v})}}\right)\right]}
where g 3 {\displaystyle g_{3}} is a function of the average particle diameter d {\displaystyle d} , f v {\displaystyle f_{v}} is the volume fraction of particles. g 1 {\displaystyle g_{1}} is a function of the applied load P {\displaystyle P} , the pin hardness H {\displaystyle H} and the gradient m A {\displaystyle m_{A}} of the V c {\displaystyle V_{c}} curve at L = 0 {\displaystyle L=0} .
g 1 = H m A P {\displaystyle g_{1}={\frac {Hm_{A}}{P}}}
Therefore, the effects of load and pin hardness can be shown: [ 2 ]
K = 3 H m A d ( 1 − f v ) P L g 3 f v L [ 1 − e x p ( − g 3 f v L d ( 1 − f v ) ) ] {\displaystyle K={\frac {3Hm_{A}d(1-f_{v})}{PLg_{3}f_{v}L}}\left[1-exp\left({\frac {-g_{3}f_{v}L}{d(1-f_{v})}}\right)\right]}
As wear testing is a time-consuming process, it was shown to be possible to save time by using a predictable method. [ 3 ]
Materials science
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https://en.wikipedia.org/wiki/Wear_coefficient
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Weather pains , weather-related pain , or meteoropathy is a phenomenon that occurs when people with conditions such as arthritis or limb injuries claim to feel pain, particularly with changes in barometric pressure , humidity or other weather phenomena. Scientific evidence , however, does not support a connection between weather and pain, and concludes that it is largely or entirely due to perceptual errors such as confirmation bias , [ 1 ] [ 2 ] with the possible exception being headaches and migraines. [ 3 ] [ 4 ]
A hypothetical relationship between changes in weather and pain has been documented since the classical Roman age, with Hippocrates in about 400 B.C. perhaps being the first to claim a connection. Anecdotal evidence provided by people such as Monica Seles and widely used expressions such as "aches and pain, coming rains", "feeling under the weather", and "ill health due to evil winds" reinforce the popular opinion that this effect is real, [ 5 ] despite the lack of scientific evidence supporting this contention.
The first publication to document a change in pain perception associated with the weather was the American Journal of the Medical Sciences in 1887. This involved a single case report describing a person with phantom limb pain, and it concluded that "approaching storms, dropping barometric pressure and rain were associated with increased pain complaint." [ 6 ]
Most investigations examining the relationship between weather and pain have studied people diagnosed with arthritis . After reviewing many case reports, Rentshler reported in the Journal of the American Medical Association in 1929 that there was strong evidence that "warm weather is beneficial and barometric pressure changes are detrimental to patients with arthritis." [ 6 ]
Countering the 1929 barometric pressure claim, in a 2016 article entitled "Do Your Aches, Pains Predict Rain?" professor of atmospheric sciences Dennis Driscoll is reported as stating: "People need to realize that the pressure changes associated with storms are rather small." Driscoll observes that the changes associated with a storm are about equivalent to what a person experiences in going up an elevator in a tall building. So far, there have not been many reports of people with arthritis hobbled by elevator rides in the medical literature. [ 2 ]
A study published in the British Medical Journal in 2017 examined reports of joint or back pain from millions of doctor visits between 2008 and 2012 as recorded by Medicare , the U.S. health system for the elderly. It compared these to rain data as recorded by the National Oceanic and Atmospheric Administration , but found no correlation at all. [ 7 ] The study concluded that:
Data on millions of outpatient visits of older Americans linked to data on daily rainfall showed no relation between rainfall and outpatient visits for joint or back pain... This was the case both among the older overall population and among patients with rheumatoid arthritis in particular. [ 8 ]
According to the Mayo clinic , migraines may be triggered by certain changes in the weather. [ 3 ] The NHS says "...weather changes are thought to trigger chemical and electrical changes in the brain. This irritates nerves, leading to a headache." [ 4 ] A 2023 study published in the journal of the American Headache Society found that "low barometric pressure, barometric pressure changes, higher humidity, and rainfall were associated with an increased number of headache occurrences". [ 9 ]
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https://en.wikipedia.org/wiki/Weather_pains
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Accelerated photo-ageing of polymers in SEPAP units is the controlled polymer degradation and polymer coating degradation under lab or natural conditions.
The prediction of the ageing of plastic materials is a subject that concerns both users and manufacturers. It covers plastic materials (polymers, fillers and various additives) or intermediates that are the transformers that use their thermoplastic property for the manufacture of objects by processes such as extrusion, injection molding, etc.
The reliability of the materials is one of the many guarantees that are increasingly required for all the manufactured objects. It can be integrated into the "sustainable development" approach. However, predicting the behavior of a material or an industrial part over time is a delicate process because many parameters must be taken into account.
The resistance to "natural" ageing itself is variable. It depends on temperature, sunshine (climate, latitude, humidity, ...) and on many other factors (physical constraints, level of pollution, ...) that are difficult to assess accurately. The simulation of this ageing by the use of artificial light sources and other physical constraints (temperature, sprinkling of water simulating rain, ...) has been the subject of developments that are the basis of several standards, ISO, ASTM, etc.
After all, accelerating this ageing to offer, for example, ten-year guarantees or validate stabilizing agents is a complex approach that must be based on solid science. Other applications, such as those of materials that must degrade quickly in the environment, are also concerned by this approach.
It has long been known that most ageing of these materials is based on a chemical reaction called "radical oxidation". Under the influence of external stresses that generate primary radicals attacking chemical bonds (especially the most abundant ones, between carbon and hydrogen), reactions occur with atmospheric oxygen. This led to the formation of many chemical entities, among which hydroperoxides and peroxides were the key products; they are both stable enough to be detected and reactive enough to break down into many by-products such as ketones, alcohols, acids, ... which are easily detectable by spectroscopic methods. Another important element, the decomposition of one of these peroxidized groups (like hydrogen peroxide, H 2 O 2 ) generates two new radicals, which leads to a self-acceleration of ageing.
These elementary chemical reactions lead more or less quickly to a deterioration of the physical properties of polymer materials and their precise analysis using infrared spectroscopy methods makes it possible both to understand the degradation mechanism and to make predictions about the long-term behavior of polymers.
Polypropylene, a common material in our everyday environment, is a very significant example of this approach. Its chemical structure where many tertiary carbons are present (bound to three carbon atoms and only one hydrogen) makes it a particularly sensitive material to ageing. Its use in the absence of stabilizing agents, in the form of film for example, is completely impossible without finding degradation (in a few days it quickly becomes opaque and brittle).
Sunlight (whose wavelengths on earth are greater than 295 nm) is among the main factors affecting the natural ageing of plastics along with temperature and atmospheric oxygen. However, if the influence of temperature can be analyzed separately (ageing in the dark), it is not the same for photo-ageing which is always associated with a temperature effect, it is also often rightly qualified as "photo-thermal".
The simulation of photothermal ageing is generally done by exposing samples in centers approved for their geographical location (Arizona, Florida, South of France) and their ability to know precisely the exposure conditions (duration and intensity of sunshine, temperature, humidity level, etc.). Sometimes mirror systems make it possible to intensify the radiation. The simulation can also be carried out in the laboratory, we generally use xenon lamps whose spectrum, after eliminating short wavelengths, is very similar to that of the sun. Most instruments allow control of light intensity, temperature of the surrounding environment, humidity level and water sprinklers can be programmed to simulate the effect of rain.
The use of xenon lamps is based on a similarity with the solar spectrum but that the principles of photochemistry (in particular the existence of vibrational relaxations of excited states) do not exclude the use of other light sources to simulate or accelerate photothermal ageing. Mercury-vapor lamps, properly filtered, have a discontinuous spectrum with discrete radiations (unlike the spectra of xenon and the sun which are continuous). This UV emission of Hg lamps also makes it possible to predict the durability of polymer materials formulated under use.
As early as 1978, the principles mentioned above led to the design of specific units by the Laboratory of Molecular and Macromolecular Photochemistry, now integrated into the Institute of Chemistry of Clermont-Ferrand ( https://iccf.uca.fr ). One of these units, referenced SEPAP 12–24, was long built and marketed by ATLAS MTT (picture 1) until the release of a new SEPAP MHE model in 2014 (picture 2) ( https://www.atlas-mts.com ).
In the SEPAP 12-24 unit, light excitation is provided by four 400 Watts medium-pressure mercury vapor lamps placed at the four corners of a parallelepiped. These lamps, whose shortest wavelengths are eliminated by a borosilicate glass envelope, have lifetime of 5000 hours. The temperature of the exposed surfaces (and not of the surrounding environment) is maintained and controlled by a thermoprobe in contact with a reference film of the same composition as the samples to be exposed. This temperature can vary from 45 °C to 80 °C and a good compromise between photochemical excitation and thermal excitation is always ensured at the level of the samples. 24 samples of about 1X5 cm are positioned on a metallic sample holder rotating at a constant speed in the center of the unit to ensure homogeneous illumination of all samples. The sample size is suitable for monitoring chemical evolution, with a low conversion rate, by infrared spectroscopy. SEPAP 12-24 enclosures must be calibrated using polyethylene calibration films. The detailed analysis of the mechanism of chemical evolution that controls degradation could be proposed for a large number of polymers [3,4] and it could be verified that this mechanism was identical to that which intervened in natural ageing on approved site or during real outdoor use. Today, a dozen French and European standards refer to these enclosures (agricultural films, cables) and about twenty companies have included SEPAP tests in their specifications for their subcontractors.
The new SEPAP MHE (Medium and High Energy) unit is equipped with a single medium-pressure mercury source with variable power allowing a first level of acceleration corresponding to that of the SEPAP 12-24 unit and a second level allowing an acceleration about 3 times higher (Ultra-Acceleration). It was developed by CNEP, Renault, PSA, PolyOne and Atlas-Ametek. The source has a central position and the samples are fixed on a sample holder animated by a uniform rotational movement around the source.
The analysis of the chemical evolution under the accelerated conditions of a SEPAP 12-24 or MHE units and the analysis of the chemical evolution in an early phase of exposure in outdoor use in the field (1 year or more) make it possible to define an acceleration factor if we know how to discern in the mechanism the formation of a "critical product" representative of the reaction pattern. This acceleration factor cannot be universal for all families of formulated materials that evolve according to very different reaction mechanisms, but it can be determined for each family of polymers. For example, it is close to 12 (1 month = 1 year in the field in the South of France) for the reference polyethylene. These acceleration factors have indeed been determined in very specific cases of polymers of well-defined formulations and exposed in forms that allow to take into account the diffusion of oxygen (avoid any oxygen starvation) and the migrations of stabilizers ("reservoir" effect).
The SEPAP MHE unit allows, for example, to simulate a year of exposure of a polypropylene in the south of France in 300 hours (on average acceleration) or 100 hours (in ultra-acceleration mode).
Can photo-ageing be further accelerated? There are many ways to achieve this, but there is a great risk of no longer being representative of natural ageing. From the photochemical point of view, multi-photonic effects are for example to be feared, just as the oxygen starvation may occur very quickly and strongly disrupt the degradation mechanisms. The ultra-accelerated approach developed in the SEPAP MHE unit makes it possible to solve in particular the problem of very long-term stability required for certain applications (cable-stayed bridges, photovoltaic panels, wind turbines, ...) or the need to be able to homologate a new material very quickly (automotive industry, ...).
It is first of all its physical role (leaching) that has been highlighted in particular in polyolefins (polyethylene, polypropylene). Polar degradation products and low molecular weights can be removed from the surface of the material and thus mask the ageing phenomenon. It is possible to operate the SEPAP MHE with periodic sprinklers of water by avoiding too abundant sprinkling that can lead to an underestimation of ageing. Too frequent water sprinkling can also lead to premature extraction of low molecular weight stabilizers and wrongly disqualify polymeric materials.
To examine the combined role of water with other physico-chemical constraints (Ultraviolet – heat – oxygen), a prototype SEPAP 12-24 H unit was developed. In this unit the sample holder is immersed in temperature-controlled liquid water that is re-oxygenated in outdoor circulation.
In 1986, the work of the Laboratory of Molecular and Macromolecular Photochemistry led to the creation of a transfer center CNEP to put its skills in the photo-ageing of polymer materials at the service of manufacturers, either to analyze failures of their materials or to conduct studies of collective interest.
Studies to predict the behavior of polymeric materials subjected to different environmental constraints (sunlight, heat with or without moisture) or failure analyses of polymer parts can be carried out in collaboration with the R&D departments of manufacturers. The CNEP can also be a partner in collaborative projects led by industrialists on an innovative research theme.
The Centre National d'Evaluation de Photoprotection is now associated with about sixty companies and annually carries out more than 450 studies covering all areas of application of polymers including works of art. It is also approved at the French national level as a "Technological Resources Center".
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https://en.wikipedia.org/wiki/Weather_testing_of_polymers
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Weatherization ( American English ) or weatherproofing ( British English ) is the practice of protecting a building and its interior from the elements, particularly from sunlight , precipitation , and wind , and of modifying a building to reduce energy consumption and optimize energy efficiency .
Weatherization is distinct from building insulation , although building insulation requires weatherization for proper functioning. Many types of insulation can be thought of as weatherization, because they block drafts or protect from cold winds. Whereas insulation primarily reduces conductive heat flow, weatherization primarily reduces convective heat flow.
In the United States, buildings use one third of all energy consumed and two thirds of all electricity. [ 1 ] Due to the high energy usage, they are a major source of the pollution that causes urban air quality problems and pollutants that contribute to climate change. Building energy usage accounts for 49 percent of sulfur dioxide emissions, 25 percent of nitrous oxide emissions, and 10 percent of particulate emissions. [ 2 ]
Typical weatherization procedures include:
The phrase "whole-house weatherization" extends the traditional definition of weatherization to include installation of modern, energy-saving heating and cooling equipment, or repair of old, inefficient equipment (furnaces, boilers, water heaters, programmable thermostats, air conditioners, and so on). The "Whole-House" approach also looks at how the house performs as a system. [ 5 ]
Weatherization generally does not cause indoor air quality problems by adding new pollutants to the air. (There are a few exceptions, such as caulking, that can sometimes emit pollutants.) However, measures such as installing storm windows, weather stripping, caulking, and blown-in wall insulation can reduce the amount of outdoor air infiltrating into a home. Consequently, after weatherization, concentrations of indoor air pollutants from sources inside the home can increase. [ 6 ]
Weatherization may have a negative impact on indoor air quality, if done improperly, exacerbating respiratory conditions especially among occupants with pre-existing respiratory illnesses. [ 6 ] This may occur because of a drastic decrease in air exchange rate in the home, introduction of new chemicals, and poor management of indoor moisture due to a poorly performed weatherization work. Low air exchange rates may lead to higher concentrations of pollutants in the air when ventilation is not sufficiently addressed during weatherization work. However, the situation may be different in case of a house situated in an area with high outdoor air pollution levels such as in close proximity (<200 m) from a busy major road. In such a scenario, a more airtight building envelope can actually offer protection against infiltration of outdoor air pollution. [ 7 ] The same is true for the protection offered by tighter building envelopes during wildfire events that cause elevated levels of outdoor air pollution. [ 8 ]
Weatherization is a set of measures and practices aimed at improving the energy efficiency of a building or home, primarily to reduce energy consumption and lower utility bills. The main goal of weatherization [ 9 ] is to make a structure more comfortable and cost-effective to live in, especially during extreme weather conditions. It involves making various improvements to a building's insulation, air sealing, and overall energy systems.
The American Council for an Energy-Efficient Economy estimates that up to February 2018 [update] over 7 million homes have been weatherized, giving yearly savings of 2.6 TWh of electricity, 7.9 TWh (27 × 10 ^ 12 Btu ) of fossil gas and 3.2 million metric tons (3.5 million short tons) of reduced carbon dioxide emissions. [ 10 ] The US Department of Energy estimates weatherization returns $2.69 for each dollar spent on the program, realized in energy and non-energy benefits. Families whose homes are weatherized are expected to save $358 on their first year's utility bills. [ citation needed ]
Low Income Home Energy Assistance Programs in many states work side by side with WAP to provide both immediate and long-term solutions to energy poverty . [ 11 ]
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https://en.wikipedia.org/wiki/Weatherization
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Weatherstripping is the process of sealing openings such as doors, windows, and trunks from the waters above. The term can also refer to the materials used to carry out such sealing processes. The goal of weatherstripping is to prevent rain and water from entering entirely or partially and accomplishes this by either returning or rerouting water. A secondary goal of weatherstripping is to keep interior air in, thus saving energy on heating and air conditioning.
Automotive weatherstripping is used extensively aboard automobiles, and can be found anywhere the interior compartment must be sealed from the environment. It must be functional and cohesive with the body design of the vehicle. In addition to factors standard to weatherstripping, additional factors must be considered for vehicles, specifically in the engineering of the parts. For example, the weatherstripping must function the same while the vehicle is parked and at full speed; be flexible to accommodate motion vibrations; endure extreme temperatures of hot and cold; withstand long periods of sun exposure; and resist automotive liquids such as oil, gasoline, and windshield washer fluid (methanol). Weatherstripping also plays a part in maintaining satisfactory ride quality in the vehicle, being partially responsible for sealing noise out from the passenger compartment.
Prevents water leaks: If the gaps are too large, rain and moisture can easily seep into the car, causing water damage to the interior. Prevents Increased noise: Larger gaps allow more external noise to enter the cabin, making the ride less comfortable. Prevent Drafts and temperature fluctuations: Poor seals can lead to drafts entering the car, making it difficult to maintain a desired temperature. Impact Resistance & Vibration Dampening – Reduces the effects of vibrations and absorbs shocks to prevent small amount of damage during door impact
Automobile flex when going over bumps, and vibrations cause relative motions between the relatively fixed body and movable parts like doors, windows, and sunroofs. This movement could allow water in the vehicle so the weatherstrip must compensate by filling the gap. Furthermore, this relative movement can cause noises such as squeaks, rattles, and creaks to be heard within the vehicle.
Considering a standard four-door vehicle, the doors require 20 feet (6.1 meters) or more of material per door, windows require upwards of 10 feet (3.0 meters), and trunks require large amounts.
Automotive weatherstripping can fail because of age or use. [ 1 ] Poorly performing weatherstripping should be reported to the car dealership if the vehicle is under warranty, as fixes may be known.
Weatherstripping around openings – especially doors and windows – is used in buildings to keep out weather, increase interior comfort, lower utility bills, [ 2 ] and reduce noise. Builder weatherstripping can be made from felt; [ 3 ] vinyl, rubber, or poly foam; [ 2 ] [ 3 ] EPDM cellular rubber and vinyl tubing; [ 4 ] and metals such as brass and aluminum. [ 5 ]
Weatherstripping can be used on windows to seal them on all sides. Metal caps on the window top [ 5 ] and on sashes [ 2 ] redirect rain to drip off instead of infiltrating. Foam or gasket weatherstripping can be applied to the sides and sashes. [ 6 ]
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https://en.wikipedia.org/wiki/Weatherstripping
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Web Compatibility Test for Mobile Browsers, often called the Mobile Acid test, [ 1 ] despite not being a true Acid test , [ 2 ] is a test page published and promoted by the World Wide Web Consortium (W3C) to expose web page rendering flaws in mobile web browsers and other applications that render HTML . [ 3 ] It was developed in the spirit of the Acid test by the Web Standards Project to test the relevant parts that a mobile browser needs to support. The browser has to accomplish 16 different subtests indicated by a 4 x 4 image of green or red squares.
A second version of the Web Compatibility Test for Mobile Browsers was released in January 2010, this time testing HTML5 elements. [ 4 ] The second test does not have an official explanation page, only a direct link to the test is available.
The mobile Acid test tests a variety of web standards published by the World Wide Web Consortium and the Internet Engineering Task Force .
Specifically, the mobile Acid test tests: [ 5 ]
The second version of the test tests the following elements: [ 6 ]
A green square indicates that the browser fully supports its assigned feature. A square colored red or a different color indicates that the feature is not fully supported. [ 7 ] The second test shows a percentage bar indicating the percent of elements supported. [ 6 ]
Due to the wide variety of web engines used at the time for mobile browsers, results varied between browsers used. [ 8 ] Safari on iOS 3 received a 15/16 score on the first test and the Palm Pre web browser scored a 13/16 in revision 1.47 of the first test. [ 9 ] In 2010, Firefox Mobile for Android scored a 75% while Safari scored a 67%. [ 4 ] By 2012, versions of Chrome , Safari, and Firefox had scores of 80% or over on the second test. The most common failure on the second test was <input type='date'>, with a 61.45% failure rate. [ 10 ]
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https://en.wikipedia.org/wiki/Web_Compatibility_Test_for_Mobile_Browsers
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Web Services for Devices or Web Services on Devices ( WSD ) is a Microsoft API to enable programming connections to web service enabled devices, such as printers, scanners and file shares. [ 1 ] Such devices conform to the Devices Profile for Web Services (DPWS). It is an extensible framework that serves as a replacement for older Windows networking functions and a common framework for allowing access to new device APIs.
The Microsoft Web Services for Devices API (WSDAPI) uses WS-Discovery for device discovery.
Devices that connect to the WSDAPI must implement the DPWS. [ 2 ]
This computing article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Web_Services_for_Devices
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Web badges , buttons or stickers are small images on web pages , typically part of the footer . They can be used for promotion, stating compliance with web standards or to comply with an application's terms of service . [ 1 ] They are sometimes referred to as 88x31 or 80x15 , common image resolutions for web buttons.
These were first popularized as "Best viewed in..." buttons by Netscape and Microsoft during the browser wars of the late 1990s.
This computing article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Web_badge
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A web hosting control panel is a web-based interface provided by a web hosting service that allows users to manage their servers and hosted services. Examples include cPanel , Plesk , ispmanager, My20i, CloudPanel, OpenPanel, and Enhance. For more examples, see comparison of web hosting control panels .
Web hosting control panels can include the following modules:
This computing article is a stub . You can help Wikipedia by expanding it .
This World Wide Web –related article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Web_hosting_control_panel
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Web operations ( WebOps ) is a domain of expertise within IT systems management that involves the deployment, operation, maintenance, tuning, and repair of web-based applications and systems. [ 1 ] WebOps is also increasingly acknowledged as crucial to the success of digital marketing teams, and shows up as part of the MarTech (marketing technology) ecosystem. [ 2 ]
Historically, operations were seen as a late phase of the Waterfall model development process. After engineering had built a software product, and QA had verified it as correct, it would be handed to a support staff to operate the working software. Such a view assumed that software was mostly immutable in production and that usage would be mostly stable. Increasingly, "a web application involves many specialists, but it takes people in web ops to ensure that everything works together throughout an application's lifetime". [ 3 ] The role is gaining respect as a distinct specialty among developers and managers, and is considered by many to be a subset of the larger DevOps movement.
With the rise of web technologies since mid-1995, specialists have emerged that understand the complexities of running a web application. Earlier examples of IT operations teams exist, such as the Network Operations Center (NOC) and the Database Administration (DBA) function.
Web applications are unique in many ways, presenting challenges that other software types do not have to deal with:
In this sense, WebOps simply refers to DevOps for web applications .
Web operations teams are tasked with a variety of responsibilities, including:
Typically, web operations personnel are familiar with the TCP/IP stack, the http protocol, HTML page markup, and Rich Internet applications (RIAs) such as AJAX and the like.
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https://en.wikipedia.org/wiki/Web_operations
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Webcomics (also known as online comics or Internet comics ) are comics published on the internet, such as on a website or a mobile app . While many webcomics are published exclusively online, others are also published in magazines , newspapers , or comic books .
Webcomics can be compared to self-published print comics in that anyone with an Internet connection can publish their own webcomic. Readership levels vary widely; many are read only by the creator's immediate friends and family, while some of the most widely read have audiences of well over one million readers. [ 1 ] [ 2 ] [ 3 ] Webcomics range from traditional comic strips and graphic novels to avant garde comics, and cover many genres , styles , and subjects. [ 4 ] They sometimes take on the role of a comic blog . [ 5 ] The term web cartoonist is sometimes used to refer to someone who creates webcomics.
There are several differences between webcomics and print comics. With webcomics the restrictions of traditional books, newspapers or magazines can be lifted, allowing artists and writers to take advantage of the web's unique capabilities.
The creative freedom webcomics provide allows artists to work in nontraditional styles. Clip art or photo comics (also known as fumetti ) are two types of webcomics that do not use traditional artwork. A Softer World , for example, is made by overlaying photographs with strips of typewriter-style text. [ 6 ] As in the constrained comics tradition, a few webcomics, such as Dinosaur Comics by Ryan North , are created with most strips having art copied exactly from one (or a handful of) template comics and only the text changing. [ 7 ] Pixel art , such as that created by Richard Stevens of Diesel Sweeties , is similar to that of sprite comics but instead uses low-resolution images created by the artist themself. [ 8 ] However, it is also common for some artists to use traditional styles, similar to those typically published in newspapers or comic books.
Webcomics that are independently published are not subject to the content restrictions of book publishers or newspaper syndicates , enjoying an artistic freedom similar to underground and alternative comics . Some webcomics stretch the boundaries of taste, taking advantage of the fact that Internet censorship is virtually nonexistent in countries like the United States. [ 4 ] The content of webcomics can still cause problems, such as Leisure Town artist Tristan Farnon 's legal trouble after creating a profane Dilbert parody, [ 9 ] or the Catholic League 's protest of artist Eric Millikin 's "blasphemous treatment of Jesus." [ 10 ]
Webcomic artists use many formats throughout the world. Comic strips , generally consisting of three or four panels , have been a common format for many artists. Other webcomic artists use the format of traditional printed comic books and graphic novels , sometimes with the plan of later publishing books.
Scott McCloud , an early advocate of webcomics since 1998, [ 11 ] pioneered the idea of the " infinite canvas " where, rather than being confined to normal print dimensions, artists are free to spread out in any direction indefinitely with their comics. [ 12 ] [ 13 ] Such a format proved highly successful in South-Korean webcomics when JunKoo Kim implemented an infinite scrolling mechanism in the platform Webtoon in 2004. [ 14 ] In 2009, French web cartoonist Balak described Turbomedia , a format for webcomics where a reader only views one panel at a time, in which the reader decides their own reading rhythm by going forward one panel at a time. [ 15 ] Some web cartoonists, such as political cartoonist Mark Fiore or Charley Parker with Argon Zark! , incorporate animations or interactive elements into their webcomics. [ 16 ]
The first comics to be shared through the Internet were Eric Millikin 's Witches and Stitches , which he started uploading on CompuServe in 1985. [ 17 ] [ 18 ] Services such as CompuServe and Usenet were used before the World Wide Web started to rise in popularity in 1993. Early webcomics were often derivatives from strips in college newspapers , [ citation needed ] but when the Web became widely popular in the mid-1990s, more people started creating comics exclusively for this medium. By 2000, various webcomic creators were financially successful and webcomics became more artistically recognized. Unique genres and styles became popular during this period.
The 2010s also saw the rise of webtoons in South Korea , where the form has become very prominent. This decade had also seen an increasingly larger number of successful webcomics being adapted into animated series in China and Japan.
In March 1995, artist Bebe Williams launched one of the first webcomics collectives, Art Comics Daily . [ 19 ] Newspaper comic strip syndicates also launched websites in the mid-1990s.
Other webcomics collectives followed, with many launching in the next decade. In March 2000, Chris Crosby , Crosby's mother Teri, and other artists founded Keenspot . [ 20 ] [ 21 ] In July 2000, Austin Osueke launched eigoMANGA , publishing original online manga , referred to as "webmanga".
In 2001, the subscription webcomics site Cool Beans World was launched. Contributors included UK-based comic book creators Pat Mills , Simon Bisley , John Bolton , and Kevin O'Neill , and the author Clive Barker . [ 22 ] Serialised content included Scarlet Traces and Marshal Law .
In March 2001, Shannon Denton and Patrick Coyle launched Komikwerks .com serving free strips from comics and animation professionals. The site launched with 9 titles including Steve Conley's Astounding Space Thrills , Jason Kruse's The World of Quest , and Bernie Wrightson 's The Nightmare Expeditions .
On March 2, 2002, Joey Manley founded Modern Tales , offering subscription-based webcomics. [ 23 ] The Modern Tales spin-off serializer followed in October 2002, then came girlamatic and Graphic Smash in March and September 2003 respectively.
By 2005, webcomics hosting had become a business in its own right, with sites such as Webcomics Nation . [ 24 ]
Traditional comic book publishers, such as Marvel Comics and Slave Labour Graphics , did not begin making serious digital efforts until 2006 and 2007. [ 25 ] DC Comics launched its web comic imprint, Zuda Comics in October 2007. [ 26 ] The site featured user submitted comics in a competition for a professional contract to produce web comics. In July 2010, it was announced that DC was closing down Zuda. [ 27 ]
Some creators of webcomics are able to do so professionally through various revenue channels. Webcomic artists may sell merchandise based on their work, such as T-shirts and toys, or they may sell print versions or compilations of their webcomic. [ 28 ] Webcomic creators can also sell online advertisements on their websites . [ 29 ] In the second half of the 2000s, webcomics became less financially sustainable due to the rise of social media and consumers' disinterest in certain kinds of merchandise. Crowdfunding through Kickstarter and Patreon have also become sources of income for web cartoonists. [ 30 ]
Webcomics have been used by some cartoonists as a path towards syndication in newspapers . [ 31 ] Since the mid-1990s, Scott McCloud advocated for micropayments systems as a source of income for web cartoonists, but micropayment systems have not been popular with artists or readers. [ 32 ]
Many webcomics artists have received honors for their work. In 2006, Gene Luen Yang 's graphic novel American Born Chinese , originally published as a webcomic on Modern Tales , was the first graphic novel to be nominated for a National Book Award . [ 33 ] Don Hertzfeldt 's animated film based on his webcomics, Everything Will Be OK , won the 2007 Sundance Film Festival Jury Award in Short Filmmaking, a prize rarely bestowed on an animated film. [ 34 ]
Many traditionally print-comics focused organizations have added award categories for comics published on the web. The Eagle Awards established a Favorite Web-based Comic category in 2000, and the Ignatz Awards followed the next year by introducing an Outstanding Online Comic category in 2001. After having nominated webcomics in several of their traditional print-comics categories, the Eisner Awards began awarding comics in the Best Digital Comic category in 2005. In 2006 the Harvey Awards established a Best Online Comics Work category, and in 2007 the Shuster Awards began an Outstanding Canadian Web Comic Creator Award. In 2012 the National Cartoonists Society gave their first Reuben Award for "On-line comic strips." [ 35 ]
Other awards focus exclusively on webcomics. The Web Cartoonists' Choice Awards [ 36 ] [ 37 ] consist of a number of awards that were handed out annually from 2001 to 2008. The Dutch Clickburg Webcomic Awards (also known as the Clickies) has been handed out four times between 2005 and 2010. The awards require the recipient to be active in the Benelux countries, with the exception of one international award. [ 38 ]
Though webcomics are typically published primarily on the World Wide Web, often webcomic creators decide to also print self-published books of their work. In some cases, web cartoonists may get publishing deals in which comic books are created of their work. Sometimes, these books are published by mainstream comics publishers who are traditionally aimed at the direct market of comic books stores. [ 39 ] Some web cartoonists may pursue print syndication in established newspapers or magazines .
The traditional audience base for webcomics and print comics are vastly different, and webcomic readers do not necessarily go to bookstores. For some web cartoonists, a print release may be considered the "goal" of a webcomic series, while for others, comic books are "just another way to get the content out." [ 40 ] Webcomics have been seen by some artists as a potential new path towards syndication in newspapers . According to Jeph Jacques ( Questionable Content ), "there's no real money" in syndication for webcomic artists. Some artists are not able to syndicate their work in newspapers because their comics are targeted to a specific niche audience and would not be popular with a broader readership. [ 41 ]
Many webcomics are published primarily in English , this being a major language in Australia, Canada, India, the United States, and the United Kingdom. Cultures surrounding non-anglophone webcomics have thrived in countries such as China, France, India, Japan, and South Korea. [ citation needed ]
Webcomics have been a popular medium in India since the early 2000s. Indian webcomics are successful as they reach a large audience for free [ 43 ] and they are frequently used by the country's younger generation to spread social awareness on topics such as politics and feminism . These webcomics achieve a large amount of exposure by being spread through social media . [ 44 ]
In China, Chinese webcomics have become a popular way to criticize the communist government and politicians in the country. Many webcomics by popular artists get shared around the country thanks to social networks such as Sina Weibo and WeChat . Many titles will often be censored or taken down by the government. [ citation needed ]
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The Weber number ( We ) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces. [ 1 ] It is named after Moritz Weber (1871–1951). [ 2 ] It can be thought of as a measure of the relative importance of the fluid's inertia compared to its surface tension . The quantity is useful in analyzing thin film flows and the formation of droplets and bubbles.
The Weber number may be written as:
where
The above is the force perspective to define the Weber number. We can also define it using energy perspective as the ratio of the kinetic energy on impact to the surface energy,
where
and
The Weber number appears in the incompressible Navier-Stokes equations through a free surface boundary condition. [ 3 ]
For a fluid of constant density ρ {\displaystyle \rho } and dynamic viscosity μ {\displaystyle \mu } , at the free surface interface there is a balance between the normal stress and the curvature force associated with the surface tension:
Where n ^ {\displaystyle {\widehat {\bf {n}}}} is the unit normal vector to the surface, T {\displaystyle \mathbb {T} } is the Cauchy stress tensor , and ∇ ⋅ {\displaystyle \nabla \cdot } is the divergence operator . The Cauchy stress tensor for an incompressible fluid takes the form:
Introducing the dynamic pressure p d = p − ρ g ⋅ x {\displaystyle p_{d}=p-\rho {\bf {g}}\cdot {\bf {x}}} and, assuming high Reynolds number flow, it is possible to nondimensionalize the variables with the scalings:
The free surface boundary condition in nondimensionalized variables is then:
Where Fr {\displaystyle {\text{Fr}}} is the Froude number , Re {\displaystyle {\text{Re}}} is the Reynolds number, and We {\displaystyle {\text{We}}} is the Weber number. The influence of the Weber number can then be quantified relative to gravitational and viscous forces.
One application of the Weber number is the study of heat pipes. When the momentum flux in the vapor core of the heat pipe is high, there is a possibility that the shear stress exerted on the liquid in the wick can be large enough to entrain droplets into the vapor flow. The Weber number is the dimensionless parameter that determines the onset of this phenomenon called the entrainment limit (Weber number greater than or equal to 1). In this case the Weber number is defined as the ratio of the momentum in the vapor layer divided by the surface tension force restraining the liquid, where the characteristic length is the surface pore size.
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In geometry , the Weber problem , named after Alfred Weber , is one of the most famous problems in location theory . It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to n destination points, where different destination points are associated with different costs per unit distance.
The Weber problem generalizes the geometric median , which assumes transportation costs per unit distance are the same for all destination points, and the problem of computing the Fermat point , the geometric median of three points. For this reason it is sometimes called the Fermat–Weber problem, although the same name has also been used for the unweighted geometric median problem. The Weber problem is in turn generalized by the attraction–repulsion problem , which allows some of the costs to be negative, so that greater distance from some points is better.
In the triangle case, the Fermat problem consists in locating a point D with respect to three points A, B, C in such a way that the sum of the distances between D and each of the three other points is minimized. It was formulated by the famous French mathematician Pierre de Fermat before 1640, and it can be seen as the true beginning of both location theory, and space-economy. Torricelli found a geometrical solution to this problem around 1645, but it still had no direct numerical solution more than 325 years later. E. Weiszfeld published a paper in 1937 with an algorithm for the Fermat-Weber problem. As the paper was published in Tohoku Mathematical journal, and Weiszfeld immigrated to USA and changed his name to Vaszoni, his work was not widely known. [ 1 ] Kuhn and Kuenne [ 2 ] independently found a similar iterative solution for the general Fermat problem in 1962, and, in 1972, Tellier [ 3 ] found a direct numerical solution to the Fermat triangle problem, which is trigonometric. Kuhn and Kuenne's solution applies to the case of polygons having more than three sides, which is not the case with Tellier's solution for reasons explained further on.
The Weber problem consists, in the triangle case, in locating a point D with respect to three points A, B, C in such a way that the sum of the transportation costs between D and each of the three other points is minimized. The Weber problem is a generalization of the Fermat problem since it involves both equal and unequal attractive forces (see below), while the Fermat problem only deals with equal attractive forces. It was first formulated, and solved geometrically in the triangle case, by Thomas Simpson in 1750. [ 4 ] It was later popularized by Alfred Weber in 1909. [ 5 ] Kuhn and Kuenne's iterative solution found in 1962, and Tellier's solution found in 1972 apply to the Weber triangle problem as well as to the Fermat one. Kuhn and Kuenne's solution applies also to the case of polygons having more than three sides.
In its simplest version, the attraction-repulsion problem consists in locating a point D with respect to three points A 1 , A 2 and R in such a way that the attractive forces exerted by points A 1 , A 2 , and the repulsive force exerted by point R cancel each other out as it must do at the optimum. It constitutes a generalization of both the Fermat and Weber problems. It was first formulated and solved, in the triangle case, in 1985 by Luc-Normand Tellier . [ 6 ] In 1992, Chen, Hansen, Jaumard and Tuy found a solution to the Tellier problem for the case of polygons having more than three sides.
Evangelista Torricelli ’s geometrical solution of the Fermat triangle problem stems from two observations:
It can be proved that the first observation implies that, at the optimum, the angles between the AD, BD, CD straight lines must be equal to 360° / 3 = 120°. Torricelli deduced from that conclusion that:
Simpson's geometrical solution of the so-called "Weber triangle problem" (which was first formulated by Thomas Simpson in 1750) directly derives from Torricelli's solution. Simpson and Weber stressed the fact that, in a total transportation minimization problem, the advantage to get closer to each attraction point A, B or C depends on what is carried and on its transportation cost. Consequently, the advantage of getting one kilometer closer to A, B or C varies, and the ∠ ADB , ∠ ADC , ∠ BDC angles no more need to be equal to 120°.
Simpson demonstrated that, in the same way as, in the Fermat triangle problem case, the constructed triangles △ ABE , △ ACF , △ BCG were equilateral because the three attractive forces were equal, in the Weber triangle problem case, the constructed triangles △ ABE , △ ACF , △ BCG , where E, F, G are located outside the △ ABC triangle, must be proportional to the attractive forces of the location system.
The solution is such that:
A third triangle of forces △ ACF , where F is located outside the △ ABC triangle, can be drawn based on the AC side, and a third circumference can be traced round that triangle. That third circumference crosses the two previous ones at the same point D .
A geometrical solution exists for the attraction-repulsion triangle problem. Its discovery is rather recent. [ 7 ] That geometrical solution differs from the two previous ones since, in this case, the two constructed force triangles overlap the △ A 1 A 2 R location triangle (where A 1 and A 2 are attraction points, and R , a repulsion one), while, in the preceding cases, they never did.
This solution is such that:
This solution is useless if one of the forces is greater than the sum of the two other ones or if the angles are not compatible. In some cases, no force is larger than the two other ones, and the angles are not compatible; then, the optimal location lies at the point that exerts the greater attractive force.
More than 332 years separate the first formulation of the Fermat triangle problem and the discovery of its non-iterative numerical solution, while a geometrical solution existed for almost all that period of time. Is there an explanation for that? That explanation lies in the possibility of the origins of the three vectors oriented towards the three attraction points not coinciding. If those origins do coincide and lie at the optimum location P , the vectors oriented towards A, B, C , and the sides of the △ ABC location triangle form the six angles ∠1, ∠2, ∠3, ∠4, ∠5, ∠6 , and the three vectors form the ∠ α A , ∠ α B , ∠ α C angles. It is easy to write the following six equations linking six unknowns (the angles ∠1, ∠2, ∠3, ∠4, ∠5, ∠6 ) with six known values (angles ∠ A , ∠ B , ∠ C , whose values are given, and angles ∠ α A , ∠ α B , ∠ α C , whose values depend only on the relative magnitude of the three attractive forces pointing towards the A, B, C attraction points):
∠ 1 + ∠ 2 = ∠ C ; ∠ 3 + ∠ 4 = ∠ A ; ∠ 5 + ∠ 6 = ∠ B ; ∠ 1 + ∠ 6 + ∠ α A = 180 ∘ ; ∠ 2 + ∠ 3 + ∠ α B = 180 ∘ ; ∠ 4 + ∠ 5 + ∠ α C = 180 ∘ . {\displaystyle {\begin{aligned}\angle 1+\angle 2&=\angle C;\\\angle 3+\angle 4&=\angle A;\\\angle 5+\angle 6&=\angle B;\\[4pt]\angle 1+\angle 6+\angle \alpha _{A}&=180^{\circ };\\\angle 2+\angle 3+\angle \alpha _{B}&=180^{\circ };\\\angle 4+\angle 5+\angle \alpha _{C}&=180^{\circ }.\end{aligned}}}
Unfortunately, this system of six simultaneous equations with six unknowns is undetermined, and the possibility of the origins of the three vectors oriented towards the three attraction points not coinciding explains why. In the case of non-coincidence, we observe that all the six equations are still valid. However, the optimal location P has disappeared because of the triangular hole that exists inside the triangle. In fact, as Tellier (1972) [ 8 ] has shown, that triangular hole had exactly the same proportions as the "forces triangles" we drew in Simpson's geometrical solution.
In order to solve the problem, we must add to the six simultaneous equations a seventh requirement, which states that there should be no triangular hole in the middle of the location triangle. In other words, the origins of the three vectors must coincide.
Tellier's solution of the Fermat and Weber triangle problems involves three steps:
Tellier (1985) [ 9 ] extended the Fermat–Weber problem to the case of repulsive forces. Let us examine the triangle case where there are two attractive forces w A 1 , w A 2 , and one repulsive force w R . Here as in the previous case, the possibility exists for the origins of the three vectors not to coincide. So the solution must require their coinciding. Tellier's trigonometric solution of this problem is the following:
When the number of forces is larger than three, it is no longer possible to determine the angles separating the various forces without taking into account the geometry of the location polygon. Geometric and trigonometric methods are then powerless. Iterative optimizing methods are used in such cases. Kuhn and Kuenne (1962) [ 10 ] suggested an algorithm based on iteratively reweighted least squares generalizing Weiszfeld's algorithm for the unweighted problem . Their method is valid for the Fermat and Weber problems involving many forces, but not for the attraction–repulsion problem. In this method, to find an approximation to the point y minimizing the weighted sum of distances ∑ i = 1 n w i ‖ x i − y ‖ , {\displaystyle \sum _{i=1}^{n}w_{i}\,\|x_{i}-y\|,} an initial approximation to the solution y 0 is found, and then at each stage of the algorithm is moved closer to the optimal solution by setting y j + 1 to be the point minimizing the sum of weighted squared distances ∑ i = 1 n w i ‖ x i − y j ‖ ‖ x i − y ‖ 2 {\displaystyle \sum _{i=1}^{n}{\frac {w_{i}}{\|x_{i}-y_{j}\|}}\|x_{i}-y\|^{2}} where the initial weights w i of the input points are divided by the distances from each point to the approximation from the previous stage.
As the unique optimal solution to a weighted least squares problem, each successive approximation may be found as a weighted average: y j + 1 = ∑ i = 1 n w i x i | x i − y j | ∑ i = 1 n w i | x i − y j | {\displaystyle y_{j+1}={\frac {\displaystyle \sum _{i=1}^{n}{\frac {w_{i}x_{i}}{|x_{i}-y_{j}|}}}{\displaystyle \sum _{i=1}^{n}{\frac {w_{i}}{|x_{i}-y_{j}|}}}}}
The Varignon frame provides an experimental solution of the Weber problem.
For the attraction–repulsion problem one has instead to resort to the algorithm proposed by Chen, Hansen, Jaumard and Tuy (1992). [ 11 ]
In the world of spatial economics , repulsive forces are omnipresent. Land values are the main illustration of them. In fact a substantial portion of land value theory , both rural and urban, can be summed up in the following way.
In the case where everybody is attracted by a single attraction point (the rural market or the urban central business district), competition between the various bidders who all want to locate at the center will generate land values that will transform the unique attraction point of the system into a repulsion point from the land value point of view, and, at the equilibrium, each inhabitant and activity will be located at the point where the attractive and the repulsive forces exerted by the center on them will cancel out.
The Tellier problem preceded the emergence of the New Economic Geography . It is seen by Ottaviano and Thisse (2005) [ 12 ] as a prelude to the New Economic Geography (NEG) that developed in the 1990s, and earned Paul Krugman a Nobel Memorial Prize in Economic Sciences in 2008. The concept of attractive force is akin to the NEG concept of agglomeration or centripetal force, and the concept of repulsive force is akin to the NEG concept of dispersal or centrifugal force.
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In web development , a webhook is a method of augmenting or altering the behavior of a web page or web application with custom callbacks . These callbacks may be maintained, modified, and managed by third-party users who need not be affiliated with the originating website or application. In 2007, Jeff Lindsay coined the term webhook from the computer programming term hook . [ 1 ]
Webhooks are "user-defined HTTP callbacks". [ 2 ] They are usually triggered by some event, such as pushing code to a repository, [ 3 ] a purchase, a comment being posted to a blog [ 4 ] and many more use cases. [ 5 ] When that event occurs, the source site makes an HTTP request to the URL configured for the webhook. Users can configure them to cause events on one site to invoke behavior on another.
Common uses are to trigger builds with continuous integration systems [ 6 ] or to notify bug tracking systems . [ 7 ] Because webhooks use HTTP, they can be integrated into web services without adding new infrastructure. [ 8 ]
When the client (the originating website or application) makes a webhook call to the third-party user's server, the incoming POST request should be authenticated to avoid a spoofing attack and its timestamp verified to avoid a replay attack . [ 9 ] Different techniques to authenticate the client are used:
The sender may choose to keep a constant list of IP addresses from which requests will be sent. This is not a sufficient security measure on its own, but it is useful for when the receiving endpoint is behind a firewall or NAT .
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webinos (Secure WebOS Application Environment) is a computing platform for the development of software components that are independent of the utilized computer hardware or operating system . At the same time, webinos is the name of the EU-funded project aiming to deliver this platform.
The webinos platform is based on open-source software . Its objective is to enable web applications and services to be used and shared consistently and securely over a broad spectrum of converged and connected devices (cross-platform and cross-domain), including mobile, PC, home media (TV) and in-car units. [ 1 ] More than 5,400 developers have already downloaded the webinos operating system. [ 2 ]
The webinos technology has been built on HTML5, widget and device API standards. Thus common web browsers such as Mozilla Firefox or Google Chrome can run webinos-enabled apps. To handle the cross device challenges, three central concepts have been followed: First, a method of binding devices to individuals, and for the individuals to declare their identities is the concept of “Personal Zones”. It is built up from internet agents (Personal Zone Hub) and device agents (Personal Zone Proxy) that communicate and identify each other and ensure that transmitted data is safeguarded. Furthermore, the “Remoting and Discovery” approach enables devices to broadcast their services as well as applications to discover these services and a protocol for invoking these services. Finally, a virtualized network overlays the physical networks, allowing devices to communicate optimally. This “Overlay network” runs over the internet or over local Bluetooth. [ 3 ]
The core webinos architecture is based on widget and web runtimes, which consist of rendering components, policy and permission frameworks, packaging components and extended APIs. To realize the cross device communication, webinos has split the packaging, policy and API extensions from the renderer. By loosely coupling these hitherto monolithic components, it is far easier to expose the application centric components to other devices.
The addressed challenges comprise: how to provision and adapt security across a range of devices, services, networks as well as how individuals can gain control over the privacy aspects of their web presence regardless of the service that is being used.
The project receives 10 million Euros co-funding, under the EU FP7 ICT Programme , No 257103,and has a total budget of 14 million Euros. [ 1 ] Webinos has been initiated by a research consortium with the Fraunhofer Institute for Open Communication Systems , Fraunhofer FOKUS, at the helm and will run for three years starting in September 2010. [ 4 ]
More than 30 partners are represented within the consortium:
webinos Fraunhofer FOKUS
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Read the Wiktionary entry "webmaster"
You can also:
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A webnovela or foronovela is a phenomenon in Latin American and European countries like Argentina , Brazil , Chile , Colombia , Czech Republic , Dominican Republic , Estonia , Mexico , Peru , Poland , Puerto Rico , Romania , Russia , Serbia , Slovakia , Spain , Uruguay and Venezuela among others. The term "webnovela", translated to English means "web novel". It is like a television soap opera with actors and soundtrack among other traits. Its base principally is fan fiction , where fans of a certain serial or soap opera (in Spanish telenovela) or certain actor or actress create a story based on them.
Usually, a webnovela contains one or more of the following traits:
A variation of the webnovela is the cyberserie . The cyberserie are more like North American series. They have seasons, and are composed usually of Anglo-Saxon actors.
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A Website content writer or web content writer is a person who specializes in providing content for websites . Every website has a specific target audience and requires the most relevant content to attract business. Content should contain keywords (specific business-related terms, which internet users might use in order to search for services or products) aimed towards improving a website's SEO . A website content writer who also has knowledge of the SEO process is referred to as an SEO Content Writer.
Most story pieces are centered on marketing products or services , though this is not always the case. Some websites are informational only and do not sell a product or service. These websites are often news sites or blogs. Informational sites educate the reader with complex information that is easy to understand and retain.
There is a growing demand for skilled web content writing on the Internet . Quality content often translates into higher revenues for online businesses .
Website owners and managers depend on content writers to perform several major tasks:
Website content writing aims for relevance and search-ability. Relevance means that the website text should be useful and beneficial to readers. Search-ability indicates the usage of keywords to help search engines direct users to websites that meet their search criteria.
There are various ways through which websites come up with article writing, and one of them is outsourcing content writing. However, it is riskier than other options, as not all writers can write content specific to the web.
Content can be written for various purposes in various forms. The most popular forms of content writing are:
The content in website differs based on the product or service it is used for.
Writing online is different from composing and constructing content for printed materials. Web users tend to scan text instead of reading it closely, skipping what they perceive to be unnecessary information and hunting for what they regard as most relevant. It is estimated that seventy-nine percent of users scan web content. [ 2 ] It is also reported that it takes twenty-five percent more time to scan content online compared to print content. [ 3 ] Web content writers must have the skills to insert paragraphs and headlines containing keywords for search engine optimization, as well as to make sure their composition is clear, to reach their target market . They need to be skilled writers and good at engaging an audience as well as understanding the needs of web users.
Website content writing is frequently outsourced to external providers, such as individual web copywriters or for larger or more complex projects, a specialized digital marketing agency. It shall be said that most of the content writers also spend time learning about digital marketing with more focus on Search Engine Optimization , Pay Per Click , Social Media Optimization etc. so that they can develop right content which can help clients with marketing business easily.
Digital marketing agencies combine copy-writing services with a range of editorial and associated services, that may include brand positioning, message consulting, social media , SEO consulting, developmental and copy editing, proofreading , fact checking , layout , content syndication, and design.
Outsourcing allows businesses to focus on core competencies and to benefit from the specialized knowledge of professional copywriters and editors.
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The Webster's test is a qualitative urine test used to detect the presence of trinitrotoluene and its metabolites . [ 1 ] The test was developed in 1917 by T. A. Webster [ 2 ] [ 3 ] in London as a way to test for trinitrotoluene poisoning. A positive test results in a purple color for the acidified urine samples.
This medical diagnostic article is a stub . You can help Wikipedia by expanding it .
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Webtag [ 1 ] is an on-line bioinformatics tool providing oligonucleotide sequences (usually called tags or anchors) that are absent from a specified genome . These tags can be appended to gene specific primers for reverse transcriptase polymerase chain reaction ( RT-PCR ) experiments, circumventing genomic DNA contamination.
RT-PCR is a technique used for the detection of even very low copy mRNA transcripts . [ 2 ] The sensitivity of the technique also makes it susceptible to DNA contamination. Since PCR is unable to distinguish between cDNA targets and genomic DNA contamination, false positives and/or erroneous quantitative results are possible. [ 3 ] [ 4 ]
In order to overcome genomic DNA contamination in transcriptional studies, reverse template-specific polymerase chain reaction, a modification of RT-PCR is used. The possibility of using tags whose sequences are not found in the genome further improves reverse specific polymerase chain reaction experiments. The use of anchors, or tags, in the 5' region of a gene specific primer or poly-T tail allows for RNA-specific amplification, and constitutes a viable strategy. Techniques such as RS-PCR [ 5 ] and (EXACT) RT-PCR are based on the integration of such tags (unique sequences not present in genomic DNA) in the 5' end of the first strand cDNA, permitting RNA-specific amplification without loss of sensitivity.
This web based service builds on the Tagenerator [ 6 ] tool, but is very fast because all tags are pre-generated and stored in a database. It is also a significant improvement since Webtag takes into account the interactions of the tag with the primers to be used in the experiment. Having it as a web based service also means that the molecular biologist doesn't have to download and install software with all the dependencies on their own computer.
Webtag generates tags that combine genome absence with good priming properties for RT-PCR based experiments. The use of such tags will deliberately not result in PCR amplification of genomic DNA, permitting the exclusive amplification of cDNA, therefore circumventing the effects of genomic DNA contamination in an RNA sample.
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Webwereld was a Dutch online newspaper about IT by the International Data Group . It was the oldest Dutch technology website until it was discontinued in 2020. [ 1 ]
Webwereld was founded in 1995 by Oscar Kneppers, who got the idea after visiting Silicon Valley in the summer of that year. [ 1 ] Another Dutch tech website Tweakers.net was founded in 1998 after Femme Taken concluded that the moderation on Webwereld was too strict. [ 2 ]
In August 2011, Webwereld published about court documents in the Apple Inc. v. Samsung Electronics Co. lawsuit. [ 3 ] In October 2011, Webwereld started Lektober ( portmanteau of leak and October in Dutch) where they publicized about a security bug every day of the month in a website of a well-known Dutch organization. [ 4 ] In 2011, Trans Link Systems considered suing Webwereld because they sold RFID writers that could be used for free traveling in Dutch public transport. [ 5 ]
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A wedge is a triangular shaped tool , a portable inclined plane , and one of the six simple machines . It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a force applied to its blunt end into forces perpendicular ( normal ) to its inclined surfaces. The mechanical advantage of a wedge is given by the ratio of the length of its slope to its width. [ 1 ] [ 2 ] Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle.
The force is applied on a flat, broad surface. This energy is transported to the pointy, sharp end of the wedge, hence the force is transported.
The wedge simply transports energy in the form of friction and collects it to the pointy end, consequently breaking the item.
Wedges have existed for thousands of years. They were first made of simple stone. Perhaps the first example of a wedge is the hand axe (see also Olorgesailie ), which is made by chipping stone, generally flint , to form a bifacial edge, or wedge. A wedge is a simple machine that transforms lateral force and movement of the tool into a transverse splitting force and movement of the workpiece. The available power is limited by the effort of the person using the tool, but because power is the product of force and movement, the wedge amplifies the force by reducing the movement. This amplification, or mechanical advantage is the ratio of the input speed to output speed. For a wedge, this is given by 1/tanα, where α is the tip angle. The faces of a wedge are modeled as straight lines to form a sliding or prismatic joint .
The origin of the wedge is unknown. In ancient Egyptian quarries , bronze wedges were used to break away blocks of stone used in construction. Wooden wedges that swelled after being saturated with water were also used. Some indigenous peoples of the Americas used antler wedges for splitting and working wood to make canoes , dwellings and other objects.
Wedges are used to lift heavy objects, separating them from the surface upon which they rest. [ 3 ]
Consider a block that is to be lifted by a wedge. As the wedge slides under the block, the block slides up the sloped side of a wedge. This lifts the weight F B of the block. The horizontal force F A needed to lift the block is obtained by considering the velocity of the wedge v A and the velocity of the block v B . If we assume the wedge does not dissipate or store energy, then the power into the wedge equals the power out.
Or
The velocity of the block is related to the velocity of the wedge by the slope of the side of the wedge. If the angle of the wedge is α then
which means that the mechanical advantage
M A = F B F A = 1 tan α . {\displaystyle MA={\frac {F_{\mathrm {B} }}{F_{\mathrm {A} }}}={\frac {1}{\tan {\alpha }}}.}
Thus, the smaller the angle α the greater the ratio of the lifting force to the applied force on the wedge. This is the mechanical advantage of the wedge. This formula for mechanical advantage applies to cutting edges and splitting operations, as well as to lifting.
They can also be used to separate objects, such as blocks of cut stone. Splitting mauls and splitting wedges are used to split wood along the grain.
A narrow wedge with a relatively long taper , used to finely adjust the distance between objects is called a gib, and is commonly used in machine tool adjustment.
The tips of forks and nails are also wedges, as they split and separate the material into which they are pushed or driven; the shafts may then hold fast due to friction.
The blade is a compound inclined plane, consisting of two inclined planes placed so that the planes meet at one edge. When the edge where the two planes meet is pushed into a solid or fluid substance, it overcomes the resistance of materials to separate by transferring the force exerted against the material into two opposing forces normal to the faces of the blade.
The blade's first known use by humans was the sharp edge of a flint stone that was used to cleave or split animal tissue, e.g. cutting meat. The use of iron or other metals led to the development of knives for those kinds of tasks. The blade of the knife allowed humans to cut meat, fibers, and other plant and animal materials with much less force than it would take to tear them apart by simply pulling with their hands. Other examples are plows , which separate soil particles, scissors which separate fabric, axes which separate wood fibers, and chisels and planes which separate wood.
Wedges, saws and chisels can separate thick and hard materials, such as wood, solid stone and hard metals and they do so with much less force, waste of material, and with more precision, than crushing , which is the application of the same force over a wider area of the material to be separated.
Other examples of wedges are found in drill bits , which produce circular holes in solids. The two edges of a drill bit are sharpened, at opposing angles, into a point and that edge is wound around the shaft of the drill bit. When the drill bit spins on its axis of rotation, the wedges are forced into the material to be separated. The resulting cut in the material is in the direction of rotation of the drill bit, while the helical shape of a bit allows the removal of the cut material.
Wedges can also be used to hold objects in place, such as engine parts ( poppet valves ), bicycle parts ( stems and eccentric bottom brackets ), and doors . A wedge-type door stop (door wedge) functions largely because of the friction generated between the bottom of the door and the wedge, and the wedge and the floor (or other surface).
The mechanical advantage or MA of a wedge can be calculated by dividing the height of the wedge by the wedge's width: [ 1 ]
The more acute , or narrow, the angle of a wedge, the greater the ratio of the length of its slope to its width, and thus the more mechanical advantage it will yield. [ 2 ]
A wedge will bind when the wedge included angle is less than the arctangent of the coefficient of friction between the wedge and the material. Therefore, in an elastic material such as wood, friction may bind a narrow wedge more easily than a wide one. This is why the head of a splitting maul has a much wider angle than that of an axe.
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In topology , the wedge sum is a "one-point union" of a family of topological spaces . Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints x 0 {\displaystyle x_{0}} and y 0 {\displaystyle y_{0}} ) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification x 0 ∼ y 0 : {\displaystyle x_{0}\sim y_{0}:} X ∨ Y = ( X ⨿ Y ) / ∼ , {\displaystyle X\vee Y=(X\amalg Y)\;/{\sim },}
where ∼ {\displaystyle \,\sim \,} is the equivalence closure of the relation { ( x 0 , y 0 ) } . {\displaystyle \left\{\left(x_{0},y_{0}\right)\right\}.} More generally, suppose ( X i ) i ∈ I {\displaystyle \left(X_{i}\right)_{i\in I}} is an indexed family of pointed spaces with basepoints ( p i ) i ∈ I . {\displaystyle \left(p_{i}\right)_{i\in I}.} The wedge sum of the family is given by: ⋁ i ∈ I X i = ∐ i ∈ I X i / ∼ , {\displaystyle \bigvee _{i\in I}X_{i}=\coprod _{i\in I}X_{i}\;/{\sim },} where ∼ {\displaystyle \,\sim \,} is the equivalence closure of the relation { ( p i , p j ) : i , j ∈ I } . {\displaystyle \left\{\left(p_{i},p_{j}\right):i,j\in I\right\}.} In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints ( p i ) i ∈ I , {\displaystyle \left(p_{i}\right)_{i\in I},} unless the spaces ( X i ) i ∈ I {\displaystyle \left(X_{i}\right)_{i\in I}} are homogeneous .
The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to homeomorphism).
Sometimes the wedge sum is called the wedge product , but this is not the same concept as the exterior product , which is also often called the wedge product.
The wedge sum of two circles is homeomorphic to a figure-eight space . The wedge sum of n {\displaystyle n} circles is often called a bouquet of circles , while a wedge product of arbitrary spheres is often called a bouquet of spheres .
A common construction in homotopy is to identify all of the points along the equator of an n {\displaystyle n} -sphere S n {\displaystyle S^{n}} . Doing so results in two copies of the sphere, joined at the point that was the equator: S n / ∼ = S n ∨ S n . {\displaystyle S^{n}/{\sim }=S^{n}\vee S^{n}.}
Let Ψ {\displaystyle \Psi } be the map Ψ : S n → S n ∨ S n , {\displaystyle \Psi :S^{n}\to S^{n}\vee S^{n},} that is, of identifying the equator down to a single point. Then addition of two elements f , g ∈ π n ( X , x 0 ) {\displaystyle f,g\in \pi _{n}(X,x_{0})} of the n {\displaystyle n} -dimensional homotopy group π n ( X , x 0 ) {\displaystyle \pi _{n}(X,x_{0})} of a space X {\displaystyle X} at the distinguished point x 0 ∈ X {\displaystyle x_{0}\in X} can be understood as the composition of f {\displaystyle f} and g {\displaystyle g} with Ψ {\displaystyle \Psi } : f + g = ( f ∨ g ) ∘ Ψ . {\displaystyle f+g=(f\vee g)\circ \Psi .}
Here, f , g : S n → X {\displaystyle f,g:S^{n}\to X} are maps which take a distinguished point s 0 ∈ S n {\displaystyle s_{0}\in S^{n}} to the point x 0 ∈ X . {\displaystyle x_{0}\in X.} Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at s 0 , {\displaystyle s_{0},} the point common to the wedge sum of the underlying spaces.
The wedge sum can be understood as the coproduct in the category of pointed spaces . Alternatively, the wedge sum can be seen as the pushout of the diagram X ← { ∙ } → Y {\displaystyle X\leftarrow \{\bullet \}\to Y} in the category of topological spaces (where { ∙ } {\displaystyle \{\bullet \}} is any one-point space).
Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes ) under which the fundamental group of the wedge sum of two spaces X {\displaystyle X} and Y {\displaystyle Y} is the free product of the fundamental groups of X {\displaystyle X} and Y . {\displaystyle Y.}
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Weed science is a scientific discipline concerned with plants that may be considered weeds , their effects on human activities, and their management [ 1 ] "a branch of applied ecology that attempts to modify the environment against natural evolutionary trends.". [ 2 ]
Weeds have existed since humans began settled agriculture have existed since the advent of settled agriculture around 10,000 years ago it has been suggested that the most common characteristic of the ancestors of our presently dominant crop plants is their willingness—their tendency to be successful, to thrive, in disturbed habitats, mostly those around human dwellings. Farmers have likely always been aware of weeds in their crops, although the evidence for their awareness and concern is nearly all anecdotal.
Unlike other agricultural sciences like entomology or plant pathology , the emergence of weed science is comparatively recent, occurring largely within the 20th century and coinciding with the development of herbicides .
Weeds are controlled in much of the world by hand ( roguing ) or with crude hoes. The size of a farmer's holding and yield per unit area are limited by several things and paramount among them is the rapidity with which a family can weed its crops. More human labor may be expended to weed crops than on any other single human enterprise, and most of that labor is expended by women. Weed control in the Western world and other developed areas of the world is done by sophisticated machines and by substituting chemical energy (herbicides) for mechanical and human energy. There is a relationship between the way farmers control weeds and the ability of a nation to feed its people. Successful weed management is one of the essential ingredients to maintain and increase food production.
In 1923, Clark and Fletcher suggested that the "annual losses due to the occurrence of pernicious weeds on farm land in Canada, although acknowledged in a general way, are far greater than is realized." [ 3 ] They thought this was because "farmers gave little critical attention to the weeds growing among their crops." They did not deny that farmers were aware of the weeds only that they could do little about them. Many of the same weeds described by Clark and Fletcher are shown in most current weed identification books. In spite of continued research to mitigate weeds annually many of the same species continue to be problematic.
The first U.S. Congressional appropriation for weed control was made in 1901 for work on control of johnsongrass , 23 years after Congress had appropriated funds for work on cotton insect pests. Petroleum-based herbicides were first used on California crop land in 1924 and soon became widely accepted in Southwestern states. In 1942, oils were used extensively for weed control in carrots and subsequently were used in forest nurseries. French scientists sprayed apple trees with dinitro dyes to control mosses, algae, and lichens. Some noticed that grasses that were wet from the spray did not die and that observation, or, more likely, a series of observations, led to the use of dinitros as herbicides for selective control of broadleaved weeds in cereals and flax. Sinox ( sodium dinitro cresylate ) was developed by Pastac in 1933. [ 4 ] It was the first selective organic herbicide introduced in the US. From the early 1930s until about 1945, it was used extensively in grains, clover hay, grass seed crops, peas, cane berries, onions, and lawns.
Timmons, writing in 1970 reported that "available literature indicates that relatively few agricultural leaders and farmers became interested in weeds as a problem before 1200 A.D. or even before 1500 A.D." [ 5 ] The “critical attention” Clark and Fletcher thought was absent increased slowly, primarily because the general attitude seemed to be that “weeds were a curse which must be endured, and about which little could be done except by methods which were incidental to crop production, and by laborious supplemental hand methods."
Jethro Tull, in 1731, appealed for greater attention to weeds:
It is needless to go about to compute the value of the damage weeds do, since all experienced husbandmen know it to be
very great, and would unanimously agree to extirpate their whole race as entirely as in England they have done the wolves, though much more innocent and less rapacious than weeds.
Farmers however were bound by their inability to do much about weeds except by laborious hand methods.
Insects cause both human health and crop problems. Weeds, with a few exceptions, do not cause direct harm to humans. Those that do such as poison ivy and poison oak can be avoided. Poisonous weeds have never been widespread as a weed of crops nor of great concern to the majority of people. Many weeds aggravated human allergies but many did not and other common plants are also allergenic. Insects and insecticides were respectively causes of and solutions to human disease problems. Weeds and herbicides were not and less attention was paid to them. Weeds and herbicides were agricultural problems. They were not of general societal concern. There were a few scientists interested in the study of weeds and in developing techniques to reduce the crop losses caused by weeds. There were only three full-time weed experts in 1934 and only a few part-time ones.
By late 1951, 46 state agricultural experiment stations had active weed research programs and most of them were working on weed control with herbicides. Now all colleges of agriculture in land-grant universities have weed scientists and a well-developed weed management program.
Weed science has been strongly influenced by herbicides and mechanical technology developed by supporting industries, by research by weed scientists, and, ultimately, used by farmers. Herbicides greatly expanded the opportunities and range of methods for vegetation management and weed control. The definition accepted by the Weed Science Society of America (WSSA) is “A chemical substance or cultured biological organism used to kill or suppress the growth of plants.”
Weed scientists have tended to focus on results and progress. Modern agriculture in the world's developed nations has addressed but not eliminated most weed problems through extensive use of herbicides and the more recent development of herbicide resistant crops through genetic modification. These methods while undeniably successful for their intended purpose also have created environmental, non-target species, and human health problems. Farmers in the world's developing nations use some herbicides but newer herbicides and the necessary application technology are often unavailable or too expensive. Weeds are always present in these farmer's fields but often the most available, affordable control methods are mechanical weeding, usually with animal power, or by hand, and most of the labor is provided by women. Neither the hypothesis that more energy is expended for the weeding man's crops than for any other single human task nor the corollary hypothesis that women do most of the world's weeding has been verified, but they are widely accepted.
Weed science no longer focuses exclusively on agriculture, with applications in industrial activities like maintaining railroad rights-of-way, controlling invasive species (including aquatic weeds) in natural areas and sports/park/home lawn care
While any plant can be a weed, approximately 250 plant species are sufficiently troublesome, cosmopolitan and economically injurious to warrant targeted research into their biology to assist in their management and control. [ 6 ] Examples of some of these troublesome weed species in North America are Palmer amaranth ( Amaranthus palmeri S. Watson) , common lambsquarter ( Chenopodium album L. ), horseweed ( Erigeron canadensis L.) , morningglory ( Ipomoea spp.) , waterhemp ( Amaranthus tuberculatus (Moq.) J. D. Sauer) and common ragweed ( Ambrosia artemisiifolia L. ). [ 7 ] Some weed science researchers provide extension resources for farmers and land managers by trialing a variety of weed control tools and tactics on a specific weed, publishing the results and providing recommendations for their future management. Other researchers may study the biology of weed seeds in order to determine how long weed seeds can remain viable in a soil. Much of this research is conducted at public land-grant universities.
Another aspect of weed science research is concerned with generating knowledge about the active ingredients of herbicides. Specifically, evaluating the response of weeds and crops to different combinations of herbicides at varying rates, droplet sizes, and environmental conditions within different cropping systems containing different weed communities. Some herbicides become more effective when mixed together (syngerism), while other herbicide combinations reduce the overall control ( antagonism ). [ 6 ] Some weed science researchers trial a variety of herbicides applications and combinations on weeds to evaluate their impact on the weeds and crops. Most of this research is conducted by private companies.
Another aspect of weed science is concerned with how herbicides move in the environment after they are applied. Some herbicides degrade very quickly in sunlight and can be made inactive before entering the plant while others can persist for years in the soil after they are applied, causing problems for future crops. Registering herbicides can cost hundreds of millions of dollars in order to demonstrate how the chemical moves and degrades within the environment it is applied. [ 6 ] Most of this research is conducted by private companies in order to be able to register their products for sale.
Many professional societies exist for weed scientists to publish research findings and support future research. The Weed Science Society of America (WSSA) is a non-profit professional society that hosts annual conferences and regional conferences in the United States as well as provides guidance on the use of herbicides. For example, the WSSA created a categorization system of herbicides by mode of action that is adopted by all herbicide manufacturers to clearly communicate the way in which each herbicide product impacts plant physiology. By encouraging herbicide applicators to use different modes of action, this reduces the likelihood of herbicide resistant populations of weeds from occurring, thereby stewarding the long-term ability of these weed control products.
Many other professional societies exist for weed science that support the current research challenges of a given country or region. For example, the Indian Society of Weed Science , the Weed Science Society of Japan , and more.
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Weerman degradation , also named Weerman reaction , is a name reaction in organic chemistry . It is named after Rudolf Adrian Weerman , who discovered it in 1910. [ 1 ] In general, it is an organic reaction in carbohydrate chemistry in which amides are degraded by sodium hypochlorite , forming an aldehyde with one less carbon. [ 2 ] Some have regarded it as an extension of the Hofmann rearrangement . [ 3 ]
The Weermann degradation could be executed with α-hydroxy-substituted carbonic acid amides. For example, sugar .
During the degradation of α-hydroxy-substituted carbonic acid amides, the carbon chain shortens by one carbon-atom. [ 2 ]
The reaction proceeds very slowly at room temperature, therefore the reaction mixture is heated up to 60-65 °C .
The reaction mechanism is that of the related Hofmann degradation . [ 2 ]
At first the carbonic acid amide (1) reacts with the sodium hypochlorite. After the separation of water and chloride an amine with a free bond is built 2 . The intermediate (3) is generated by rearrangement. In the next step a hydrolysis takes place. Water is added at the carbon-atom with the number ' 1' . A hydroxylic group is generated. The last step is that an acidic amide is separated and the aldehyde (4) is generated.
Additionally the Weerman degradation could be executed with α,β-unsaturated carbonic acid amides. For example, acrylamide .
During the degradation of α,β-unsaturated amides, the carbon chain shortens about one carbon-atom, too. [ 2 ]
The reaction is very slow at room temperature, therefore the reaction mixture is heated up to 60–65 °C .
The reaction mechanism is that of the related Hofmann degradation . [ 2 ]
At first the carbonic acid amide (1) reacts with the sodium hypochlorite. After separate water and chloride an amine with a free bond is built 2 . The intermediate (3) is generated by rearrangement. At this point two different mechanisms are possible. In the mechanism above two methanol molecules reacts with the intermediate. So is the compound (4) generated. After this carbon dioxide, water, ammonium and methanol are separated in different steps. At least it is protonated into an aldehyde (5) .
Until the intermediate (3) the mechanism is the same like above. Then only one methanol-atom is added 4 . With a protonation water, methanol and carbon dioxide are separated. An ammonium ion (5) is generated. During the hydrolysis a hydroxylic group is built 6 . An aldehyde (7) is generated by separating an ammonium ion.
One study demonstrated the direct oxidation of glucose to arabinose by the same sodium hypochlorite, skipping the aldonic acid and aldoamide steps. [ 4 ] [ 5 ] For example, the general degradation of D -gluconamide into D -arabinose:
On top of that, the Weerman test could be used to show whether a hydroxylic group is beside the amido group. This reaction is only important in a historical sense because it is slow yielding and thus rarely used.
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Wehrlite is an ultramafic and ultrabasic rock that is a mixture of olivine and clinopyroxene . It is a subdivision of the peridotites .
The nomenclature allows up to a few percent of orthopyroxene . Accessory minerals include ilmenite , chromite , magnetite and an aluminium-bearing mineral ( plagioclase , spinel or garnet ). [ 1 ]
Wehrlites occur as mantle xenoliths and in ophiolites . Another occurrence is as cumulate in gabbro and norite layered intrusions . [ 1 ] Some meteorites can also be classified as wehrlites (e.g. NWA 4797 ). [ 2 ]
Wehrlite is named after Alois Wehrle . [ 3 ] He was born 1791 in Kroměříž , Czech Republic (then Kremsier in Mähren) and was a professor at the "Ungarische Bergakademie" (Hungarian Mining School) in Banská Štiavnica , Slovakia (then Schemnitz, Kingdom of Hungary ). [ 4 ]
This igneous rock -related article is a stub . You can help Wikipedia by expanding it .
This geochemistry article is a stub . You can help Wikipedia by expanding it .
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Wei-Shou Hu (born November 5, 1951) is a Taiwanese-American chemical engineer. He is currently the Distinguished McKnight University Professor of Chemical Engineering and Material Science at the University of Minnesota .
Hu earned his B.S. in agricultural chemistry from National Taiwan University in 1974 and his Ph.D. in biochemical engineering from the Massachusetts Institute of Technology under the guidance of Daniel I.C. Wang in 1983. He has been a professor with the University of Minnesota since 1983. Hu has long impacted the field of cell culture bioprocessing since its infancy by steadfastly introducing quantitative and systematic analysis into this field. His work, which covers areas such as modeling and controlling cell metabolism, modulating glycosylation, and process data mining, has helped shape the advances of biopharmaceutical process
technology. He recently led an industrial consortium to embark on genomic research on Chinese hamster ovary cells, the main workhorse of biomanufacturing, and to promote post-genomic research in cell bioprocessing. [ 1 ]
Hu's research focuses on the field of cell culture bioprocessing, particularly metabolic control of the physiological state of the cell. [ 2 ] [ 3 ] In addition to his work with Chinese hamster ovary cells , his work has enabled the use of process engineering for cell therapy, especially with liver cells. [ 4 ] [ 5 ] Hu has written four different biotechnology books. [ 6 ] [ 7 ] [ 8 ] [ 9 ] One of his articles is cited by 63. [ 10 ]
He is the 2005 recipient of the Marvin Johnson Award from the American Chemical Society , the distinguished service award of Society of Biological Engineers, a special award from Asia Pacific Biochemical Engineering Conference (2009), and the Amgen Award from Engineering Conferences International, as well as both the distinguished service award and the Division award from the Food, Pharmaceuticals and Bioengineering Division of the American Institute of Chemical Engineers. [ 1 ] He has authored the books Bioseparations , Cell Culture Technology for Pharmaceutical and Cell-Based Therapies and Cell Culture Bioprocess Engineering
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Wei-Ta Fang ( Chinese : 方偉達 ; pinyin : Fāng Wěidá ) is a Taiwanese wetland scientist and environmental educator, a distinguished professor, vice dean of the College of Science, and the director of the Graduate Institute of Sustainability Management and Environmental Education, National Taiwan Normal University . [ 1 ] President of the Society of Wetland Scientists (SWS) Asia Chapter. [ 2 ]
Fang was born in Kaohsiung , Taiwan on 14 February 1966. He earned a Bachelor of Arts in land economics and administration from National Taipei University in 1989. He completed a master's degree in environmental planning (MEP) from Arizona State University , in 1994, followed by a second master's degree in landscape architecture in design studies (MDes.S.) from the Harvard Graduate School of Design in 2001. He obtained a Ph.D. from the Department of Ecosystem Science and Management, Texas A&M University in 2005. [ 3 ]
He served as a specialist in the Taipei Land Management Bureau in 1991 and 1992 and a senior specialist in charge of environmental education and environmental impact assessments (EIAs) at Taiwan's Environmental Protection Administration (EPA) Headquarters from 1994 to 2006. He was also co-principal investigator (co-PI) for the National Environmental Literacy Survey in Taiwan during 2012 and 2020. He is currently serving as distinguished professor, vice dean of the College of Science, and as director of the Graduate Institute of Sustainability Management and Environmental Education, National Taiwan Normal University, and is president of the Society of Wetland Scientists (SWS) Asia Chapter. [ 4 ] and the president of Taiwan Wetland Society. He was awarded the designation of visiting research fellow from the director, Dr. Xingyuan He at the Northeast Institute of Geography and Agroecology, Chinese Academy of Sciences in Changchun, China in March 2016 [ 4 ] His currently studies focus on the psychological, social, and physical environmental characteristics to predict pro-environmental behavioral changes, such as smartphone usage children during COVID-19 periods.
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The Weibel instability is a plasma instability present in homogeneous or nearly homogeneous electromagnetic plasmas which possess an anisotropy in momentum (velocity) space. This anisotropy is most generally understood as two temperatures in different directions. Burton Fried showed that this instability can be understood more simply as the superposition of many counter-streaming beams. In this sense, it is like the two-stream instability except that the perturbations are electromagnetic and result in filamentation as opposed to electrostatic perturbations which would result in charge bunching. In the linear limit the instability causes exponential growth of electromagnetic fields in the plasma which help restore momentum space isotropy. In very extreme cases, the Weibel instability is related to one- or two-dimensional stream instabilities .
Consider an electron-ion plasma in which the ions are fixed and the electrons are hotter in the y-direction than in x or z-direction.
To see how magnetic field perturbation would grow, suppose a field B = B cos kx spontaneously arises from noise. The Lorentz force then bends the electron trajectories with the result that upward-moving-ev x B electrons congregate at B and downward-moving ones at A [ clarification needed ] . The resulting current j = − e n v e {\displaystyle j=-env_{e}} sheets generate magnetic field that enhances the original field and thus
perturbation grows.
Weibel instability is also common in astrophysical plasmas, such as collisionless shock formation in supernova remnants and γ {\displaystyle \gamma } -ray bursts.
As a simple example of Weibel instability, consider an electron beam with density n b 0 {\displaystyle n_{b0}} and initial velocity v 0 z {\displaystyle v_{0}\mathbf {z} } propagating in a plasma of density n p 0 = n b 0 {\displaystyle n_{p0}=n_{b0}} with velocity − v 0 z {\displaystyle -v_{0}\mathbf {z} } . The analysis below will show how an electromagnetic perturbation in the form of a plane wave gives rise to a Weibel instability in this simple anisotropic plasma system. We assume a non-relativistic plasma for simplicity.
We assume there is no background electric or magnetic field i.e. B 0 = E 0 = 0 {\displaystyle \mathbf {B_{0}} =\mathbf {E_{0}} =0} . The perturbation will be taken as an electromagnetic wave propagating along x ^ {\displaystyle \mathbf {\hat {x}} } i.e. k = k x ^ {\displaystyle \mathbf {k} =k\mathbf {\hat {x}} } . Assume the electric field has the form
With the assumed spatial and time dependence, we may use ∂ ∂ t → − i ω {\displaystyle \textstyle {\frac {\partial }{\partial t}}\rightarrow -i\omega } and ∇ → i k x ^ {\displaystyle \nabla \rightarrow ik\mathbf {\hat {x}} } . From Faraday's Law, we may obtain the perturbation magnetic field
Consider the electron beam. We assume small perturbations, and so linearize the velocity v b = v b 0 + v b 1 {\displaystyle \mathbf {v_{b}} =\mathbf {v_{b0}} +\mathbf {v_{b1}} } and density n b = n b 0 + n b 1 {\displaystyle n_{b}=n_{b0}+n_{b1}} . The goal is to find the perturbation electron beam current density
where second-order terms have been neglected. To do that, we start with the fluid momentum equation for the electron beam
which can be simplified by noting that ∂ v b 0 ∂ t = ∇ ⋅ v b 0 = 0 {\displaystyle \textstyle {\frac {\partial \mathbf {v_{b0}} }{\partial t}}=\nabla \cdot \mathbf {v_{b0}} =0} and neglecting second-order terms. With the plane wave assumption for the derivatives, the momentum equation becomes
We can decompose the above equations in components, paying attention to the cross product at the far right, and obtain the non-zero components of the beam velocity perturbation:
To find the perturbation density n b 1 {\displaystyle n_{b1}} , we use the fluid continuity equation for the electron beam
which can again be simplified by noting that ∂ n b 0 ∂ t = ∇ n b 0 = 0 {\displaystyle \textstyle {\frac {\partial n_{b0}}{\partial t}}=\nabla n_{b0}=0} and neglecting second-order terms. The result is
Using these results, we may use the equation for the beam perturbation current density given above to find
Analogous expressions can be written for the perturbation current density of the left-moving plasma. By noting that the x-component of the perturbation current density is proportional to v 0 {\displaystyle v_{0}} , we see that with our assumptions for the beam and plasma unperturbed densities and velocities the x-component of the net current density will vanish, whereas the z-components, which are proportional to v 0 2 {\displaystyle v_{0}^{2}} , will add. The net current density perturbation is therefore
The dispersion relation can now be found from Maxwell's Equations:
where c = 1 ϵ 0 μ 0 {\displaystyle \textstyle c={\frac {1}{\sqrt {\epsilon _{0}\mu _{0}}}}} is the speed of light in free space. By defining the effective plasma frequency ω p 2 = 2 n b 0 e 2 ϵ 0 m {\displaystyle \textstyle \omega _{p}^{2}={\frac {2n_{b0}e^{2}}{\epsilon _{0}m}}} , the equation above results in
This bi-quadratic equation may be easily solved to give the dispersion relation
In the search for instabilities, we look for ℑ ( ω ) ≠ 0 {\displaystyle \Im (\omega )\neq 0} ( k {\displaystyle k} is assumed real). Therefore, we must take the dispersion relation/mode corresponding to the minus sign in the equation above.
To gain further insight on the instability, it is useful to harness our non-relativistic assumption v 0 ≪ c {\displaystyle v_{0}\ll c} to simplify the square root term, by noting that
The resulting dispersion relation is then much simpler
ω {\displaystyle \omega } is purely imaginary. Writing ω = i γ {\displaystyle \omega =i\gamma }
we see that ℑ ( ω ) > 0 {\displaystyle \Im (\omega )>0} , indeed corresponding to an instability.
The electromagnetic fields then have the form
Therefore, the electric and magnetic fields are 90 ∘ {\displaystyle 90^{\circ }} out of phase, and by noting that
so we see this is a primarily magnetic perturbation although there is a non-zero electric perturbation. The magnetic field growth results in the characteristic filamentation structure of Weibel instability. Saturation will happen when the growth rate γ {\displaystyle \gamma } is on the order of the electron cyclotron frequency
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The Weibull modulus is a dimensionless parameter of the Weibull distribution . It represents the width of a probability density function (PDF) in which a higher modulus is a characteristic of a narrower distribution of values. Use case examples include biological and brittle material failure analysis , where modulus is used to describe the variability of failure strength for materials.
The Weibull distribution, represented as a cumulative distribution function (CDF), is defined by:
F ( x ) = 1 − exp ( − ( x − x u x 0 ) m ) {\displaystyle F(x)=1-\exp \left(-\left({\frac {x-x_{u}}{x_{0}}}\right)^{m}\right)}
in which m is the Weibull modulus. [ 1 ] x 0 {\displaystyle x_{0}} is a parameter found during the fit of data to the Weibull distribution and represents an input value for which ~67% of the data is encompassed. As m increases, the CDF distribution more closely resembles a step function at x 0 {\displaystyle x_{0}} , which correlates with a sharper peak in the probability density function (PDF) defined by:
f ( x ) = ( m x 0 ) ( x − x u x 0 ) m − 1 exp ( − ( x − x u x 0 ) m ) {\displaystyle f(x)=\left({\frac {m}{x_{0}}}\right)\left({\frac {x-x_{u}}{x_{0}}}\right)^{m-1}\exp \left(-\left({\frac {x-x_{u}}{x_{0}}}\right)^{m}\right)}
Failure analysis often uses this distribution, [ 2 ] as a CDF of the probability of failure F of a sample, as a function of applied stress σ, in the form:
F ( σ ) = 1 − exp [ − ( σ σ 0 ) m ] {\displaystyle F(\sigma )=1-\exp \left[-\left({\frac {\sigma }{\sigma _{0}}}\right)^{m}\right]}
Failure stress of the sample, σ, is substituted for the x {\displaystyle x} property in the above equation. The initial property x u {\displaystyle x_{u}} is assumed to be 0, an unstressed, equilibrium state of the material.
In the plotted figure of the Weibull CDF, it is worth noting that the plotted functions all intersect at a stress value of 50 MPa, the characteristic strength for the distributions, even though the value of the Weibull moduli vary. It is also worth noting in the plotted figure of the Weibull PDF that a higher Weibull modulus results in a steeper slope within the plot.
The Weibull distribution can also be multi-modal, in which there would be multiple reported x 0 {\displaystyle x_{0}} values and multiple reported moduli, m. The CDF for a bimodal Weibull distribution has the following form, [ 3 ] when applied to materials failure analysis:
F ( σ ) = 1 − ϕ exp [ − ( σ σ 01 ) m 1 ] − ( 1 − ϕ ) exp [ − ( σ σ 02 ) m 2 ] {\displaystyle F(\sigma )=1-\phi \exp \left[-\left({\frac {\sigma }{\sigma _{01}}}\right)^{m_{1}}\right]-(1-\phi )\exp \left[-\left({\frac {\sigma }{\sigma _{02}}}\right)^{m_{2}}\right]}
This represents a material which fails by two different modes. In this equation m 1 is the modulus for the first mode, and m 2 is the modulus for the second mode. Φ is the fraction of the sample set which fail by the first mode. The corresponding PDF is defined by: f ( σ ) = ϕ ( m 1 σ 01 ) ( σ σ 01 ) m 1 − 1 exp [ − ( σ σ 01 ) m 1 ] + ( 1 − ϕ ) ( m 2 σ 02 ) ( σ σ 02 ) m 2 − 1 exp [ − ( σ σ 02 ) m 2 ] {\displaystyle f(\sigma )=\phi \left({\frac {m_{1}}{\sigma _{01}}}\right)\left({\frac {\sigma }{\sigma _{01}}}\right)^{m_{1}-1}\exp \left[-\left({\frac {\sigma }{\sigma _{01}}}\right)^{m_{1}}\right]+(1-\phi )\left({\frac {m_{2}}{\sigma _{02}}}\right)\left({\frac {\sigma }{\sigma _{02}}}\right)^{m_{2}-1}\exp \left[-\left({\frac {\sigma }{\sigma _{02}}}\right)^{m_{2}}\right]}
Examples of a bimodal Weibull PDF and CDF are plotted in the figures of this article with values of the characteristic strength being 40 and 120 MPa, the Weibull moduli being 4 and 10, and the value of Φ is 0.5, corresponding to 50% of the specimens failing by each failure mode.
The complement of the cumulative Weibull distribution function can be expressed as:
P = 1 − F {\displaystyle P=1-F}
Where P corresponds to the probability of survival of a specimen for a given stress value. Thus, it follows that:
P ( σ ) = 1 − [ 1 − exp [ − ( σ σ 0 ) m ] ] = exp [ − ( σ σ 0 ) m ] {\displaystyle P(\sigma )=1-\left[1-\exp \left[-({\frac {\sigma }{\sigma _{0}}})^{m}\right]\right]=\exp \left[-({\frac {\sigma }{\sigma _{0}}})^{m}\right]}
where m is the Weibull modulus. If the probability is plotted vs the stress, we find that the graph is sigmoidal, as shown in the figure above. Taking advantage of the fact that the exponential is the base of the natural logarithm, the above equation can be rearranged to:
ln [ ln ( 1 1 − F ) ] = ln [ ( σ σ 0 ) m ] {\displaystyle \ln \left[\ln \left({\frac {1}{1-F}}\right)\right]=\ln \left[\left({\frac {\sigma }{\sigma _{0}}}\right)^{m}\right]}
Which, using the properties of logarithms, can also be expressed as:
ln [ ln ( 1 1 − F ) ] = m ln ( σ ) − m ln ( σ 0 ) {\displaystyle \ln \left[\ln \left({\frac {1}{1-F}}\right)\right]=m\ln(\sigma )-m\ln(\sigma _{0})}
When the left side of this equation is plotted as a function of the natural logarithm of stress, a linear plot can be created which has a slope of the Weibull modulus, m, and an x-intercept of ln ( σ 0 ) {\displaystyle \ln(\sigma _{0})} .
Looking at the plotted linearization of the CDFs from above it can be seen that all of the lines intersect the x-axis at the same point because all of the functions have the same value of the characteristic strength. The slopes vary because of the differing values of the Weibull moduli.
Standards organizations have created multiple standards for measuring and reporting values of Weibull parameters, along with other statistical analyses of strength data:
When applying a Weibull distribution to a set of data the data points must first be put in ranked order. For the use case of failure analysis specimens' failure strengths are ranked in ascending order, i.e. from lowest to greatest strength. A probability of failure is then assigned to each failure strength measured, ASTM C1239-13 [ 4 ] uses the following formula:
F ( σ ) = i − 0.5 N {\displaystyle F(\sigma )={\frac {i-0.5}{N}}}
where i {\displaystyle i} is the specimen number as ranked and N {\displaystyle N} is the total number of specimens in the sample. From there F {\displaystyle F} can be plotted against failure strength to obtain a Weibull CDF. The Weibull parameters, modulus and characteristic strength, can be obtained from fitting or using the linearization method detailed above.
Weibull statistics are often used for ceramics and other brittle materials. [ 8 ] [ 9 ] They have also been applied to other fields as well such as meteorology where wind speeds are often described using Weibull statistics. [ 10 ] [ 11 ] [ 12 ]
For ceramics and other brittle materials, the maximum stress that a sample can be measured to withstand before failure may vary from specimen to specimen, even under identical testing conditions. This is related to the distribution of physical flaws present in the surface or body of the brittle specimen, since brittle failure processes originate at these weak points. Much work has been done to describe brittle failure with the field of linear elastic fracture mechanics and specifically with the development of the ideas of the stress intensity factor and Griffith Criterion . When flaws are consistent and evenly distributed, samples will behave more uniformly than when flaws are clustered inconsistently. This must be taken into account when describing the strength of the material, so strength is best represented as a distribution of values rather than as one specific value.
Consider strength measurements made on many small samples of a brittle ceramic material. If the measurements show little variation from sample to sample, the calculated Weibull modulus will be high, and a single strength value would serve as a good description of the sample-to-sample performance. It may be concluded that its physical flaws, whether inherent to the material itself or resulting from the manufacturing process, are distributed uniformly throughout the material. If the measurements show high variation, the calculated Weibull modulus will be low; this reveals that flaws are clustered inconsistently, and the measured strength will be generally weak and variable. Products made from components of low Weibull modulus will exhibit low reliability and their strengths will be broadly distributed. With careful manufacturing processes Weibull moduli of up to 98 have been seen for glass fibers tested in tension. [ 13 ]
A table is provided with the Weibull moduli for several common materials. However, it is important to note that the Weibull modulus is a fitting parameter from strength data, and therefore the reported value may vary from source to source. It also is specific to the sample preparation and testing method, and subject to change if the analysis or manufacturing process changes.
Studies examining organic brittle materials highlight the consistency and variability of the Weibull modulus within naturally occurring ceramics such as human dentin and abalone nacre. Research on human dentin [ 14 ] samples indicates that the Weibull modulus remains stable across different depths or locations within the tooth, with an average value of approximately 4.5 and a range between 3 and 6. Variations in the modulus suggest differences in flaw populations between individual teeth, thought to be caused by random defects introduced during specimen preparation. Speculation exists regarding a potential decrease in the Weibull modulus with age due to changes in flaw distribution and stress sensitivity. Failure in dentin typically initiates at these flaws, which can be intrinsic or extrinsic in origin, arising from factors such as cavity preparation, wear, damage, or cyclic loading.
Studies on the abalone shell illustrate its unique structural adaptations, sacrificing tensile strength perpendicular to its structure to enhance strength parallel to the tile arrangement. The Weibull modulus of abalone nacre samples [ 15 ] is determined to be 1.8, indicating a moderate degree of variability in strength among specimens.
The Weibull modulus of quasi-brittle materials correlates with the decline in the slope of the energy barrier spectrum, as established in fracture mechanics models. This relationship allows for the determination of both the fracture energy barrier spectrum decline slope and the Weibull modulus, while keeping factors like crack interaction and defect-induced degradation in consideration. Temperature dependence and variations due to crack interactions or stress field interactions are observed in the Weibull modulus of quasi-brittle materials. Damage accumulation leads to a rapid decrease in the Weibull modulus, resulting in a right-shifted distribution with a smaller Weibull modulus as damage increases. [ 16 ]
Weibull analysis is also used in quality control and "life analysis" [ 17 ] for products. A higher Weibull modulus allows for companies to more confidently predict the life of their product for use in determining warranty periods.
A further method to determine the strength of brittle materials has been described by the Wikibook contribution Weakest link determination by use of three parameter Weibull statistics .
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In mathematics , the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely . It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician Karl Weierstrass (1815–1897).
Weierstrass M-test. Suppose that ( f n ) is a sequence of real- or complex-valued functions defined on a set A , and that there is a sequence of non-negative numbers ( M n ) satisfying the conditions
Then the series
converges absolutely and uniformly on A .
A series satisfying the hypothesis is called normally convergent . The result is often used in combination with the uniform limit theorem . Together they say that if, in addition to the above conditions, the set A is a topological space and the functions f n are continuous on A , then the series converges to a continuous function.
Consider the sequence of functions
Since the series ∑ n = 1 ∞ M n {\displaystyle \sum _{n=1}^{\infty }M_{n}} converges and M n ≥ 0 for every n , then by the Cauchy criterion ,
For the chosen N ,
(Inequality (1) follows from the triangle inequality .)
The sequence S n ( x ) is thus a Cauchy sequence in R or C , and by completeness , it converges to some number S ( x ) that depends on x . For n > N we can write
Since N does not depend on x , this means that the sequence S n of partial sums converges uniformly to the function S . Hence, by definition, the series ∑ k = 1 ∞ f k ( x ) {\displaystyle \sum _{k=1}^{\infty }f_{k}(x)} converges uniformly.
Analogously, one can prove that ∑ k = 1 ∞ | f k ( x ) | {\displaystyle \sum _{k=1}^{\infty }|f_{k}(x)|} converges uniformly.
A more general version of the Weierstrass M-test holds if the common codomain of the functions ( f n ) is a Banach space , in which case the premise
is to be replaced by
where ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fréchet derivative .
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In mathematics, the Weierstrass Nullstellensatz is a version of the intermediate value theorem over a real closed field . It says: [ 1 ] [ 2 ]
Since F is real-closed, F ( i ) is algebraically closed, hence f ( x ) can be written as u ∏ i ( x − α i ) {\displaystyle u\prod _{i}(x-\alpha _{i})} , where u ∈ F {\displaystyle u\in F} is the leading coefficient and α j ∈ F ( i ) {\displaystyle \alpha _{j}\in F(i)} are the roots of f . Since each nonreal root α j = a j + i b j {\displaystyle \alpha _{j}=a_{j}+ib_{j}} can be paired with its conjugate α ¯ j = a j − i b j {\displaystyle {\overline {\alpha }}_{j}=a_{j}-ib_{j}} (which is also a root of f ), we see that f can be factored in F [ x ] as a product of linear polynomials and polynomials of the form ( x − α j ) ( x − α ¯ j ) = ( x − a j ) 2 + b j 2 {\displaystyle (x-\alpha _{j})(x-{\overline {\alpha }}_{j})=(x-a_{j})^{2}+b_{j}^{2}} , b j ≠ 0 {\displaystyle b_{j}\neq 0} .
If f changes sign between a and b , one of these factors must change sign. But ( x − a j ) 2 + b j 2 {\displaystyle (x-a_{j})^{2}+b_{j}^{2}} is strictly positive for all x in any formally real field, hence one of the linear factors x − α j {\displaystyle x-\alpha _{j}} , α j ∈ F {\displaystyle \alpha _{j}\in F} , must change sign between a and b ; i.e., the root α j {\displaystyle \alpha _{j}} of f satisfies a < α j < b {\displaystyle a<\alpha _{j}<b} .
This algebra -related article is a stub . You can help Wikipedia by expanding it .
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In mathematics , the Weierstrass product inequality states that for any real numbers 0 ≤ x 1 , ..., x n ≤ 1 we have
and similarly, for 0 ≤ x 1 , ..., x n ,
where S n = x 1 + x 2 + x 3 + x 4 + . . . . + x n . {\displaystyle S_{n}=x_{1}+x_{2}+x_{3}+x_{4}+....+x_{n}.}
The inequality is named after the German mathematician Karl Weierstrass .
The inequality with the subtractions can be proven easily via mathematical induction . The one with the additions is proven identically. We can choose n = 1 {\displaystyle n=1} as the base case and see that for this value of n {\displaystyle n} we get
which is indeed true. Assuming now that the inequality holds for all natural numbers up to n > 1 {\displaystyle n>1} , for n + 1 {\displaystyle n+1} we have:
which concludes the proof.
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The Weigert-Meyer Rule is a concept in urology and radiology that describes the anatomic relationship of two ureters in a duplicated renal collecting system, as well as the resulting patterns of hydronephrosis , obstruction, and reflux. [ 1 ]
In most cases, the upper renal moiety drains to the more medial ureteral orifice and the lower renal moiety tends to drain to the more lateral orifice. [ 2 ] The lower moiety often has a shorter muscular tunnel through the bladder wall due to the more lateral insertion, and is therefore more likely to reflux. [ 2 ] The upper moiety has higher risk of obstruction due to greater likelihood of ureterocele or ectopic insertion. [ 2 ] [ 3 ] [ 4 ]
This pattern was first described by German pathologist, Dr. Karl Weigert in 1877. This was further investigated by German pathologist, Dr. Robert Meyer , who described it as a "rule" in 1946. [ 2 ] [ 3 ]
This medical article is a stub . You can help Wikipedia by expanding it .
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In mathematics , a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is a matrix W {\displaystyle W} with entries from the set { 0 , 1 , − 1 } {\displaystyle \{0,1,-1\}} such that:
Where W T {\displaystyle W^{\mathsf {T}}} is the transpose of W {\displaystyle W} and I n {\displaystyle I_{n}} is the identity matrix of order n {\displaystyle n} . The weight w {\displaystyle w} is also called the degree of the matrix. For convenience, a weighing matrix of order n {\displaystyle n} and weight w {\displaystyle w} is often denoted by W ( n , w ) {\displaystyle W(n,w)} . [ 3 ]
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale , the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement. [ 1 ] [ 2 ]
Some properties are immediate from the definition. If W {\displaystyle W} is a W ( n , w ) {\displaystyle W(n,w)} , then:
A weighing matrix is a generalization of a Hadamard matrix , which does not allow zero entries. [ 3 ] As two special cases, a W ( n , n ) {\displaystyle W(n,n)} is a Hadamard matrix [ 3 ] and a W ( n , n − 1 ) {\displaystyle W(n,n-1)} is equivalent to a conference matrix .
Weighing matrices take their name from the problem of measuring the weight of multiple objects. If a measuring device has a statistical variance of σ 2 {\displaystyle \sigma ^{2}} , then measuring the weights of N {\displaystyle N} objects and subtracting the (equally imprecise) tare weight will result in a final measurement with a variance of 2 σ 2 {\displaystyle 2\sigma ^{2}} . [ 4 ] It is possible to increase the accuracy of the estimated weights by measuring different subsets of the objects, especially when using a balance scale where objects can be put on the opposite measuring pan where they subtract their weight from the measurement.
An order n {\displaystyle n} matrix W {\displaystyle W} can be used to represent the placement of n {\displaystyle n} objects—including the tare weight—in n {\displaystyle n} trials. Suppose the left pan of the balance scale adds to the measurement and the right pan subtracts from the measurement. Each element of this matrix w i j {\displaystyle w_{ij}} will have:
Let x {\displaystyle \mathbf {x} } be a column vector of the measurements of each of the n {\displaystyle n} trials, let e {\displaystyle \mathbf {e} } be the errors to these measurements each independent and identically distributed with variance σ 2 {\displaystyle \sigma ^{2}} , and let y {\displaystyle \mathbf {y} } be a column vector of the true weights of each of the n {\displaystyle n} objects. Then we have:
Assuming that W {\displaystyle W} is non-singular , we can use the method of least-squares to calculate an estimate of the true weights:
The variance of the estimated y {\displaystyle \mathbf {y} } vector cannot be lower than σ 2 / n {\displaystyle \sigma ^{2}/n} , and will be minimum if and only if W {\displaystyle W} is a weighing matrix. [ 4 ] [ 5 ]
Weighing matrices appear in the engineering of spectrometers , image scanners, [ 6 ] and optical multiplexing systems. [ 5 ] The design of these instruments involve an optical mask and two detectors that measure the intensity of light. The mask can either transmit light to the first detector, absorb it, or reflect it toward the second detector. The measurement of the second detector is subtracted from the first, and so these three cases correspond to weighing matrix elements of 1, 0, and −1 respectively. As this is essentially the same measurement problem as in the previous section, the usefulness of weighing matrices also applies. [ 6 ]
An orthogonal design of order n {\displaystyle n} and type ( s 1 , … , s u ) {\displaystyle (s_{1},\dots ,s_{u})} where s i {\displaystyle s_{i}} are positive integers, is an n × n {\displaystyle n\times n} matrix whose entries are in the set { 0 , ± x 1 , … , ± x u } {\displaystyle \{0,\pm x_{1},\dots ,\pm x_{u}\}} , where x i {\displaystyle x_{i}} are commuting variables. Additionally, an orthogonal design must satisfy:
This constraint is also equivalent to the rows of X {\displaystyle X} being orthogonal and each row having exactly s i {\displaystyle s_{i}} occurrences of x i {\displaystyle x_{i}} . [ 7 ] An orthogonal design can be denoted as O D ( n ; s 1 , … , s u ) {\displaystyle \mathrm {OD} (n;s_{1},\dots ,s_{u})} . [ 8 ] An orthogonal design of one variable is a weighing matrix, and so the two fields of study are connected. [ 7 ] Because of this connection, new orthogonal designs can be discovered by way of weighing matrices. [ 9 ]
Note that when weighing matrices are displayed, the symbol − {\displaystyle -} is used to represent −1. Here are some examples:
This is a W ( 2 , 2 ) {\displaystyle W(2,2)} :
This is a W ( 4 , 3 ) {\displaystyle W(4,3)} :
This is a W ( 7 , 4 ) {\displaystyle W(7,4)} :
Another W ( 7 , 4 ) {\displaystyle W(7,4)} :
Which is circulant , i.e. each row is a cyclic shift of the previous row. Such a matrix is called a C W ( n , k ) {\displaystyle CW(n,k)} and is determined by its first row.
Circulant weighing matrices are of special interest since their algebraic structure makes them easier for classification. Indeed, we know that a circulant weighing matrix of order n {\displaystyle n} and weight k {\displaystyle k} must be of square weight. So, weights 1 , 4 , 9 , 16 , . . . {\displaystyle 1,4,9,16,...} are permissible and weights k ≤ 25 {\displaystyle k\leq 25} have been completely classified. [ 10 ] Two special (and actually, extreme) cases of circulant weighing matrices are (A) circulant Hadamard matrices which are conjectured not to exist unless their order is less than 5. This conjecture, the circulant Hadamard conjecture first raised by Ryser, is known to be true for many orders but is still open . (B) C W ( n , k ) {\displaystyle CW(n,k)} of weight k = s 2 {\displaystyle k=s^{2}} and minimal order n {\displaystyle n} exist if s {\displaystyle s} is a prime power and such a circulant weighing matrix can be obtained by signing the complement of a finite projective plane .
Since all C W ( n , k ) {\displaystyle CW(n,k)} for k ≤ 25 {\displaystyle k\leq 25} have been classified, the first open case is C W ( 105 , 36 ) {\displaystyle CW(105,36)} .
The first open case for a general weighing matrix (certainly not a circulant) is W ( 35 , 25 ) {\displaystyle W(35,25)} .
Two weighing matrices are considered to be equivalent if one can be obtained from the other by a series of permutations and negations of the rows and columns of the matrix. The classification of weighing matrices is complete for cases where w ≤ 5 {\displaystyle w\leq 5} as well as all cases where n ≤ 15 {\displaystyle n\leq 15} . [ 11 ] However, very little has been done beyond this with exception to classifying circulant weighing matrices. [ 12 ] [ 13 ]
One major open question about weighing matrices is their existence: for which values of n {\displaystyle n} and w {\displaystyle w} does there exist a W ( n , w ) {\displaystyle W(n,w)} ? The following conjectures have been proposed about the existence of W ( n , w ) {\displaystyle W(n,w)} : [ 7 ]
Although the last three conjectures are statements on orthogonal designs, it has been shown that the existence of an orthogonal design O D ( n ; s 1 , … , s u ) {\displaystyle \mathrm {OD} (n;s_{1},\dots ,s_{u})} is equivalent to the existence of X 1 , … , X u {\displaystyle X_{1},\dots ,X_{u}} weighing matrices of order n {\displaystyle n} where X i {\displaystyle X_{i}} has weight s i {\displaystyle s_{i}} . [ 7 ]
An equally important but often overlooked question about weighing matrices is their enumeration: for a given n {\displaystyle n} and w {\displaystyle w} , how many W ( n , w ) {\displaystyle W(n,w)} 's are there?
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Weight distribution is the apportioning of weight within a vehicle , especially cars , airplanes , and trains . Typically, it is written in the form x / y , where x is the percentage of weight in the front, and y is the percentage in the back.
In a vehicle which relies on gravity in some way, weight distribution directly affects a variety of vehicle characteristics, including handling , acceleration , traction , and component life. For this reason weight distribution varies with the vehicle's intended usage. For example, a drag car maximizes traction at the rear axle while countering the reactionary pitch-up torque. It generates this counter-torque by placing a small amount of counterweight at a great distance forward of the rear axle.
In the airline industry , load balancing is used to evenly distribute the weight of passengers, cargo, and fuel throughout an aircraft, so as to keep the aircraft's center of gravity close to its center of pressure to avoid losing pitch control. In military transport aircraft, it is common to have a loadmaster as a part of the crew; their responsibilities include calculating accurate load information for center of gravity calculations, and ensuring cargo is properly secured to prevent its shifting.
In large aircraft and ships, multiple fuel tanks and pumps are often used, so that as fuel is consumed, the remaining fuel can be positioned to keep the vehicle balanced, and to reduce stability problems associated with the free surface effect .
In the trucking industry , individual axle weight limits require balancing the cargo when the gross vehicle weight nears the legal limit.
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A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average . Weight functions occur frequently in statistics and analysis , and are closely related to the concept of a measure . Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" [ 1 ] and "meta-calculus". [ 2 ]
In the discrete setting, a weight function w : A → R + {\displaystyle w\colon A\to \mathbb {R} ^{+}} is a positive function defined on a discrete set A {\displaystyle A} , which is typically finite or countable . The weight function w ( a ) := 1 {\displaystyle w(a):=1} corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts.
If the function f : A → R {\displaystyle f\colon A\to \mathbb {R} } is a real -valued function , then the unweighted sum of f {\displaystyle f} on A {\displaystyle A} is defined as
but given a weight function w : A → R + {\displaystyle w\colon A\to \mathbb {R} ^{+}} , the weighted sum or conical combination is defined as
One common application of weighted sums arises in numerical integration .
If B is a finite subset of A , one can replace the unweighted cardinality | B | of B by the weighted cardinality
If A is a finite non-empty set, one can replace the unweighted mean or average
by the weighted mean or weighted average
In this case only the relative weights are relevant.
Weighted means are commonly used in statistics to compensate for the presence of bias . For a quantity f {\displaystyle f} measured multiple independent times f i {\displaystyle f_{i}} with variance σ i 2 {\displaystyle \sigma _{i}^{2}} , the best estimate of the signal is obtained by averaging all the measurements with weight w i = 1 / σ i 2 {\textstyle w_{i}=1/{\sigma _{i}^{2}}} , and the resulting variance is smaller than each of the independent measurements σ 2 = 1 / ∑ i w i {\textstyle \sigma ^{2}=1/\sum _{i}w_{i}} . The maximum likelihood method weights the difference between fit and data using the same weights w i {\displaystyle w_{i}} .
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities . More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.
In regressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable , a distributed lag function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
The terminology weight function arises from mechanics : if one has a collection of n {\displaystyle n} objects on a lever , with weights w 1 , … , w n {\displaystyle w_{1},\ldots ,w_{n}} (where weight is now interpreted in the physical sense) and locations x 1 , … , x n {\displaystyle {\boldsymbol {x}}_{1},\dotsc ,{\boldsymbol {x}}_{n}} , then the lever will be in balance if the fulcrum of the lever is at the center of mass
which is also the weighted average of the positions x i {\displaystyle {\boldsymbol {x}}_{i}} .
In the continuous setting, a weight is a positive measure such as w ( x ) d x {\displaystyle w(x)\,dx} on some domain Ω {\displaystyle \Omega } , which is typically a subset of a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , for instance Ω {\displaystyle \Omega } could be an interval [ a , b ] {\displaystyle [a,b]} . Here d x {\displaystyle dx} is Lebesgue measure and w : Ω → R + {\displaystyle w\colon \Omega \to \mathbb {R} ^{+}} is a non-negative measurable function . In this context, the weight function w ( x ) {\displaystyle w(x)} is sometimes referred to as a density .
If f : Ω → R {\displaystyle f\colon \Omega \to \mathbb {R} } is a real -valued function , then the unweighted integral
can be generalized to the weighted integral
Note that one may need to require f {\displaystyle f} to be absolutely integrable with respect to the weight w ( x ) d x {\displaystyle w(x)\,dx} in order for this integral to be finite.
If E is a subset of Ω {\displaystyle \Omega } , then the volume vol( E ) of E can be generalized to the weighted volume
If Ω {\displaystyle \Omega } has finite non-zero weighted volume, then we can replace the unweighted average
by the weighted average
If f : Ω → R {\displaystyle f\colon \Omega \to {\mathbb {R} }} and g : Ω → R {\displaystyle g\colon \Omega \to {\mathbb {R} }} are two functions, one can generalize the unweighted bilinear form
to a weighted bilinear form
See the entry on orthogonal polynomials for examples of weighted orthogonal functions .
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A weight machine is an exercise machine used for weight training that uses gravity as the primary source of resistance and a combination of simple machines to convey that resistance to the person using the machine. Each of the simple machines (pulley, lever, wheel, incline) changes the mechanical advantage of the overall machine relative to the weight.
A stack machine —also called a stack or rack —has a set of rectangular plates that are pierced by a vertical bar which has holes drilled in it to accept a pin. [ 1 ] Each of the plates has a channel on its underside (or a hole through the middle, as visible in the picture) that aligns with one of the holes. When the pin is inserted through the channel into the hole, all of the plates above the pin rest upon it, and are lifted when the bar rises. The plates below do not rise. This allows the same machine to provide several levels of resistance over the same range of motion with an adjustment that requires very little force to accomplish in itself.
The means of lifting the bar varies. Some machines have a roller at the top of the bar that sits on a lever. When the lever is raised the bar can go up and the roller moves along the lever, allowing the bar to stay vertical. On some machines the bar is attached to a hinge on the lever, which causes swaying in the bar and the plates as the lever goes up and down. On other machines the bar is attached to a cable or belt, which runs through pulleys or over a wheel. The other end of the cable will either be a handle or strap that the user holds or wraps around some body part, or will be attached to a lever, adding further simple machines to the mechanical chain.
Usually, each plate is marked with a number. On some machines these numbers give the actual weight of the plate and those above it. On some, the number gives the force at the user's actuation point with the machine. And on some machines the number is simply an index counting the number of plates being lifted.
The early Nautilus machines were a combination of lever and cable machines. They also had optional, fixed elements such as a chinning bar. Universal Gym Equipment pioneered the multi-station style of machines. [ 2 ]
Plate-loaded machines (such as the Smith machine or sled-type leg press ) use standard barbell plates instead of captive stacks of plates. They combine a bar-end on which to hang the plates with several simple machines to convey the force to the user.
The plate-loaded machines will often have a very high mechanical advantage, due to the need to make room for large plates over a large range of motion following a path that causes them to converge at one end or the other. Also, the motion will generally not be vertical, and the net resistance is equal to the cosine of the angle at which it is moving relative to vertical.
For example, consider an incline press machine that is a single-lever machine that has the plates halfway up the lever from the handles to the fulcrum , and begins moving the plates at a 45- degree angle from vertical. The lever will provide a leverage advantage of 2:1, and the incline will have an advantage of 1:√2/2, for a net mechanical advantage of (4/√2):1 ≈ 2.8:1 . Thus 50 kg ( ~491 N ) of plates will apply to the user only an equaling weight of 18 kg or a force of ~174 N at the beginning of the motion.
On the other end of the spectrum may be a bent-over-row machine that is designed with the user's grip between the plates and the fulcrum. This amplifies the force needed by the user relative to the weight of the plates.
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https://en.wikipedia.org/wiki/Weight_machine
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Weight transfer and load transfer are two expressions used somewhat confusingly to describe two distinct effects: [ 1 ]
In the automobile industry, weight transfer customarily refers to the change in load borne by different wheels during acceleration. [ 2 ] This would be more properly referred to as load transfer , [ 1 ] [ 3 ] and that is the expression used in the motorcycle industry, [ 4 ] [ 5 ] while weight transfer on motorcycles, to a lesser extent on automobiles, and cargo movement on either is due to a change in the CoM location relative to the wheels. This article uses this latter pair of definitions.
In wheeled vehicles , load transfer is the measurable change of load borne by different wheels during acceleration (both longitudinal and lateral). [ 3 ] This includes braking , and deceleration (which is an acceleration at a negative rate). [ 6 ] No motion of the center of mass relative to the wheels is necessary, and so load transfer may be experienced by vehicles with no suspension at all. Load transfer is a crucial concept in understanding vehicle dynamics . The same is true in bikes, though only longitudinally. [ 4 ]
The major forces that accelerate a vehicle occur at the tires ' contact patches . Since these forces are not directed through the vehicle's CoM, one or more moments are generated whose forces are the tires' traction forces at pavement level, the other one (equal but opposed) is the mass inertia located at the CoM and the moment arm is the distance from pavement surface to CoM. It is these moments that cause variation in the load distributed between the tires. Often this is interpreted by the casual observer as a pitching or rolling motion of the vehicles body. A perfectly rigid vehicle, without suspension that would not exhibit pitching or rolling of the body, still undergoes load transfer. However, the pitching and rolling of the body of a non-rigid vehicle adds some (small) weight transfer due to the (small) CoM horizontal displacement with respect to the wheel's axis suspension vertical travel and also due to deformation of the tires i.e. contact patch displacement relative to wheel.
Lowering the CoM towards the ground is one method of reducing load transfer. As a result load transfer is reduced in both the longitudinal and lateral directions. Another method of reducing load transfer is by increasing the wheel spacings. Increasing the vehicle's wheelbase (length) reduces longitudinal load transfer while increasing the vehicle's track (width) reduces lateral load transfer. Most high performance automobiles are designed to sit as low as possible and usually have an extended wheelbase and track.
One way to calculate the effect of load transfer, keeping in mind that this article uses "load transfer" to mean the phenomenon commonly referred to as "weight transfer" in the automotive world, is with the so-called "weight transfer equation":
where Δ W e i g h t f r o n t {\displaystyle \Delta \mathrm {Weight} _{\mathrm {front} }} is the change in load borne by the front wheels, a {\displaystyle a} is the longitudinal acceleration, g {\displaystyle g} is the acceleration of gravity , h {\displaystyle h} is the center of mass height, b {\displaystyle b} is the wheelbase, m {\displaystyle m} is the total vehicle mass, and w {\displaystyle w} is the total vehicle weight. [ 7 ] [ 8 ]
Weight transfer involves the actual (relatively small) movement of the vehicle CoM relative to the wheel axes due to displacement of the chassis as the suspension complies, or of cargo or liquids within the vehicle, which results in a redistribution of the total vehicle load between the individual tires.
Weight transfer occurs as the vehicle's CoM shifts during automotive maneuvers. Acceleration causes the sprung mass to rotate about a geometric axis resulting in relocation of the CoM. Front-back weight transfer is proportional to the change in the longitudinal location of the CoM to the vehicle's wheelbase, and side-to-side weight transfer (summed over front and rear) is proportional to the ratio of the change in the CoM's lateral location to the vehicle's track.
Liquids, such as fuel, readily flow within their containers, causing changes in the vehicle's CoM. As fuel is consumed, not only does the position of the CoM change, but the total weight of the vehicle is also reduced.
By way of example, when a vehicle accelerates, a weight transfer toward the rear wheels can occur. An outside observer might witness this as the vehicle visibly leans to the back, or squats . Conversely, under braking, weight transfer toward the front of the car can occur. Under hard braking it might be clearly visible even from inside the vehicle as the nose dives toward the ground (most of this will be due to load transfer). Similarly, during changes in direction (lateral acceleration), weight transfer to the outside of the direction of the turn can occur.
Weight transfer is generally of far less practical importance than load transfer , for cars and SUVs at least. For instance in a 0.9g turn, a car with a track of 1650 mm and a CoM height of 550 mm will see a load transfer of 30% of the vehicle weight, that is the outer wheels will see 60% more load than before, and the inners 60% less. Total available grip will drop by around 6% as a result of this load transfer. At the same time, the CoM of the vehicle will typically move laterally and vertically, relative to the contact patch by no more than 30 mm, leading to a weight transfer of less than 2%, and a corresponding reduction in grip of 0.01%.
Load transfer causes the available traction at all four wheels to vary as the car brakes, accelerates, or turns. This bias to one pair of tires doing more "work" than the other pair results in a net loss of total available traction. The net loss can be attributed to the phenomenon known as tire load sensitivity .
An exception is during positive acceleration when the engine power is driving two or fewer wheels. In this situation where all the tires are not being utilized load transfer can be advantageous. As such, the most powerful cars are almost never front wheel drive , as the acceleration itself causes the front wheels' traction to decrease. This is why sports cars usually have either rear wheel drive or all wheel drive (and in the all wheel drive case, the power tends to be biased toward the rear wheels under normal conditions).
If (lateral) load transfer reaches the tire loading on one end of a vehicle, the inside wheel on that end will lift, causing a change in handling characteristic. If it reaches half the weight of the vehicle it will start to roll over. Some large trucks will roll over before skidding, while passenger vehicles and small trucks usually roll over only when they leave the road. Fitting racing tires to a tall or narrow vehicle and then driving it hard may lead to rollover.
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https://en.wikipedia.org/wiki/Weight_transfer
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