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Tachycardia: an elevated resting heart rate. In general an electrocardiogram (ECG) is required to identify the type of tachycardia.
Pulsatile This description of the pulse implies the intrinsic physiology of systole and diastole. Scientifically, systole and diastole are forces that expand and contract the pulmonary and systemic circulations.
A collapsing pulse is a sign of hyperdynamic circulation, which can be seen in AR or PDA. | Pulse | Wikipedia | 102 | 41600 | https://en.wikipedia.org/wiki/Pulse | Biology and health sciences | Medical procedures | null |
Common palpable sites
Sites can be divided into peripheral pulses and central pulses. Central pulses include the carotid, femoral, and brachial pulses.
Upper limb
Axillary pulse: located inferiorly of the lateral wall of the axilla
Brachial pulse: located on the inside of the upper arm near the elbow, frequently used in place of carotid pulse in infants (brachial artery)
Radial pulse: located on the lateral of the wrist (radial artery). It can also be found in the anatomical snuff box. Commonly, the radial pulse is measured with three fingers. The finger closest to the heart is used to occlude the pulse pressure, the middle finger is used get a crude estimate of the blood pressure, and the finger most distal to the heart (usually the ring finger) is used to nullify the effect of the ulnar pulse as the two arteries are connected via the palmar arches (superficial and deep).
Ulnar pulse: located on the medial of the wrist (ulnar artery).
Lower limb
Femoral pulse: located in the inner thigh, at the mid-inguinal point, halfway between the pubic symphysis and anterior superior iliac spine (femoral artery).
Popliteal pulse: Above the knee in the popliteal fossa, found by holding the bent knee. The patient bends the knee at approximately 124°, and the health care provider holds it in both hands to find the popliteal artery in the pit behind the knee (popliteal artery).
Dorsalis pedis pulse: located on top of the foot, immediately lateral to the extensor of hallucis longus (dorsalis pedis artery).
Tibialis posterior pulse: located on the medial side of the ankle, 2 cm inferior and 2 cm posterior to the medial malleolus (posterior tibial artery). It is easily palpable over Pimenta's Point.
Head and neck | Pulse | Wikipedia | 406 | 41600 | https://en.wikipedia.org/wiki/Pulse | Biology and health sciences | Medical procedures | null |
Carotid pulse: located in the neck (carotid artery). The carotid artery should be palpated gently and while the patient is sitting or lying down. Stimulating its baroreceptors with low palpitation can provoke severe bradycardia or even stop the heart in some sensitive persons. Also, a person's two carotid arteries should not be palpated at the same time. Doing so may limit the flow of blood to the head, possibly leading to fainting or brain ischemia. It can be felt between the anterior border of the sternocleidomastoid muscle, above the hyoid bone and lateral to the thyroid cartilage.
Facial pulse: located on the mandible (lower jawbone) on a line with the corners of the mouth (facial artery).
Temporal pulse: located on the temple directly in front of the ear (superficial temporal artery).
Although the pulse can be felt in multiple places in the head, people should not normally hear their heartbeats within the head. This is called pulsatile tinnitus, and it can indicate several medical disorders.
Torso
Apical pulse: located in the 5th left intercostal space, 1.25 cm lateral to the mid-clavicular line. In contrast with other pulse sites, the apical pulse site is unilateral, and measured not under an artery, but below the heart itself (more specifically, the apex of the heart). | Pulse | Wikipedia | 302 | 41600 | https://en.wikipedia.org/wiki/Pulse | Biology and health sciences | Medical procedures | null |
Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye. The fundamental difference between radiometry and photometry is that radiometry gives the entire optical radiation spectrum, while photometry is limited to the visible spectrum. Radiometry is distinct from quantum techniques such as photon counting.
The use of radiometers to determine the temperature of objects and gasses by measuring radiation flux is called pyrometry. Handheld pyrometer devices are often marketed as infrared thermometers.
Radiometry is important in astronomy, especially radio astronomy, and plays a significant role in Earth remote sensing. The measurement techniques categorized as radiometry in optics are called photometry in some astronomical applications, contrary to the optics usage of the term.
Spectroradiometry is the measurement of absolute radiometric quantities in narrow bands of wavelength.
Radiometric quantities
Integral and spectral radiometric quantities
Integral quantities (like radiant flux) describe the total effect of radiation of all wavelengths or frequencies, while spectral quantities (like spectral power) describe the effect of radiation of a single wavelength or frequency . To each integral quantity there are corresponding spectral quantities, defined as the quotient of the integrated quantity by the range of frequency or wavelength considered. For example, the radiant flux Φe corresponds to the spectral power Φe, and Φe,.
Getting an integral quantity's spectral counterpart requires a limit transition. This comes from the idea that the precisely requested wavelength photon existence probability is zero. Let us show the relation between them using the radiant flux as an example:
Integral flux, whose unit is W:
Spectral flux by wavelength, whose unit is :
where is the radiant flux of the radiation in a small wavelength interval .
The area under a plot with wavelength horizontal axis equals to the total radiant flux.
Spectral flux by frequency, whose unit is :
where is the radiant flux of the radiation in a small frequency interval .
The area under a plot with frequency horizontal axis equals to the total radiant flux.
The spectral quantities by wavelength and frequency are related to each other, since the product of the two variables is the speed of light ():
or or
The integral quantity can be obtained by the spectral quantity's integration: | Radiometry | Wikipedia | 477 | 41625 | https://en.wikipedia.org/wiki/Radiometry | Physical sciences | Electromagnetic radiation | Physics |
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.
The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid.
History
The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians, while Sarah Hart sees it named as such "because the properties of this curve are so beautiful".
Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery speculated that such a simple curve must have been known to the ancients, citing similar work by Carpus of Antioch described by Iamblichus. English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa, but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost. Galileo Galilei's name was put forward at the end of the 19th century and at least one author reports credit being given to Marin Mersenne. Beginning with the work of Moritz Cantor and Siegmund Günther, scholars now assign priority to French mathematician Charles de Bovelles based on his description of the cycloid in his Introductio in geometriam, published in 1503. In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel. | Cycloid | Wikipedia | 429 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
Galileo originated the term cycloid and was the first to make a serious study of the curve. According to his student Evangelista Torricelli, in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discovered the ratio was roughly 3:1, which is the true value, but he incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible. Around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem. However, this work was not published until 1693 (in his Traité des Indivisibles).
Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviani, who were able to produce a quadrature. This result and others were published by Torricelli in 1644, which is also the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647. | Cycloid | Wikipedia | 306 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (by John Wallis and Antoine de Lalouvère) was judged to be adequate. While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's Tractatus Duo, giving Wren priority for the first published proof.
Fifteen years later, Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of which is a cycloid.
Equations
The cycloid through the origin, generated by a circle of radius rolling over the -axis on the positive side (), consists of the points , with
where is a real parameter corresponding to the angle through which the rolling circle has rotated. For given , the circle's centre lies at .
The Cartesian equation is obtained by solving the -equation for and substituting into the -equation:or, eliminating the multiple-valued inverse cosine:When is viewed as a function of , the cycloid is differentiable everywhere except at the cusps on the -axis, with the derivative tending toward or near a cusp (where ). The map from to is differentiable, in fact of class ∞, with derivative 0 at the cusps.
The slope of the tangent to the cycloid at the point is given by . | Cycloid | Wikipedia | 488 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
A cycloid segment from one cusp to the next is called an arch of the cycloid, for example the points with and .
Considering the cycloid as the graph of a function , it satisfies the differential equation:
If we define as the height difference from the cycloid's vertex (the point with a horizontal tangent and ), then we have:
Involute
The involute of the cycloid has exactly the same shape as the cycloid it originates from. This can be visualized as the path traced by the tip of a wire initially lying on a half arch of the cycloid: as it unrolls while remaining tangent to the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length).
Demonstration | Cycloid | Wikipedia | 169 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
This demonstration uses the rolling-wheel definition of cycloid, as well as the instantaneous velocity vector of a moving point, tangent to its trajectory. In the adjacent picture, and are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, and coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, and traverse two cycloid curves. Considering the red line connecting and at a given time, one proves the line is always tangent to the lower arc at and orthogonal to the upper arc at . Let be the point in common between the upper and lower circles at the given time. Then:
are colinear: indeed the equal rolling speed gives equal angles , and thus . The point lies on the line therefore and analogously . From the equality of and one has that also . It follows .
If is the meeting point between the perpendicular from to the line segment and the tangent to the circle at , then the triangle is isosceles, as is easily seen from the construction: and . For the previous noted equality between and then and is isosceles.
Drawing from the orthogonal segment to , from the straight line tangent to the upper circle, and calling the meeting point, one sees that is a rhombus using the theorems on angles between parallel lines
Now consider the velocity of . It can be seen as the sum of two components, the rolling velocity and the drifting velocity , which are equal in modulus because the circles roll without skidding. is parallel to , while is tangent to the lower circle at and therefore is parallel to . The rhombus constituted from the components and is therefore similar (same angles) to the rhombus because they have parallel sides. Then , the total velocity of , is parallel to because both are diagonals of two rhombuses with parallel sides and has in common with the contact point . Thus the velocity vector lies on the prolongation of . Because is tangent to the cycloid at , it follows that also coincides with the tangent to the lower cycloid at .
Analogously, it can be easily demonstrated that is orthogonal to (the other diagonal of the rhombus). | Cycloid | Wikipedia | 453 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
This proves that the tip of a wire initially stretched on a half arch of the lower cycloid and fixed to the upper circle at will follow the point along its path without changing its length because the speed of the tip is at each moment orthogonal to the wire (no stretching or compression). The wire will be at the same time tangent at to the lower arc because of the tension and the facts demonstrated above. (If it were not tangent there would be a discontinuity at and consequently unbalanced tension forces.) | Cycloid | Wikipedia | 107 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
Area
Using the above parameterization , the area under one arch, is given by:
This is three times the area of the rolling circle.
Arc length
The arc length of one arch is given by
Another geometric way to calculate the length of the cycloid is to notice that when a wire describing an involute has been completely unwrapped from half an arch, it extends itself along two diameters, a length of . This is thus equal to half the length of arch, and that of a complete arch is .
From the cycloid's vertex (the point with a horizontal tangent and ) to any point within the same arch, the arc length squared is , which is proportional to the height difference ; this property is the basis for the cycloid's isochronism. In fact, the arc length squared is equal to the height difference multiplied by the full arch length .
Cycloidal pendulum
If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the string is constrained to be tangent to one of its arches, and the pendulum's length L is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle, L = 4r), the bob of the pendulum also traces a cycloid path. Such a pendulum is isochronous, with equal-time swings regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by:
where is the angle that the straight part of the string makes with the vertical axis, and is given by
where is the "amplitude", is the radian frequency of the pendulum and g the gravitational acceleration.
The 17th-century Dutch mathematician Christiaan Huygens discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be used in navigation. | Cycloid | Wikipedia | 390 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
Related curves
Several curves are related to the cycloid.
Trochoid: generalization of a cycloid in which the point tracing the curve may be inside the rolling circle (curtate) or outside (prolate).
Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line.
Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line.
Hypotrochoid: generalization of a hypocycloid where the generating point may not be on the edge of the rolling circle.
Epitrochoid: generalization of an epicycloid where the generating point may not be on the edge of the rolling circle.
All these curves are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the similitude ratio of curve to evolute is 1 + 2q.
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Other uses
The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used by Wallace K. Harrison in the design of the Hopkins Center at Dartmouth College in Hanover, New Hampshire.
Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves. Later work indicates that curtate cycloids do not serve as general models for these curves, which vary considerably. | Cycloid | Wikipedia | 384 | 41638 | https://en.wikipedia.org/wiki/Cycloid | Mathematics | Two-dimensional space | null |
In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wave to the incident wave, with each expressed as phasors. For example, it is used in optics to calculate the amount of light that is reflected from a surface with a different index of refraction, such as a glass surface, or in an electrical transmission line to calculate how much of the electromagnetic wave is reflected by an impedance discontinuity. The reflection coefficient is closely related to the transmission coefficient. The reflectance of a system is also sometimes called a reflection coefficient.
Different specialties have different applications for the term.
Transmission lines
In telecommunications and transmission line theory, the reflection coefficient is the ratio of the complex amplitude of the reflected wave to that of the incident wave. The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance Z0. The reference impedance used is typically the characteristic impedance of a transmission line that's involved, but one can speak of reflection coefficient without any actual transmission line being present. In terms of the forward and reflected waves determined by the voltage and current, the reflection coefficient is defined as the complex ratio of the voltage of the reflected wave () to that of the incident wave (). This is typically represented with a (capital gamma) and can be written as:
It can as well be defined using the currents associated with the reflected and forward waves, but introducing a minus sign to account for the opposite orientations of the two currents:
The reflection coefficient may also be established using other field or circuit pairs of quantities whose product defines power resolvable into a forward and reverse wave. For instance, with electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the forward wave (or magnetic fields, again with a minus sign); the ratio of each wave's electric field E to its magnetic field H is again an impedance Z0 (equal to the impedance of free space in a vacuum). Similarly in acoustics one uses the acoustic pressure and velocity respectively. | Reflection coefficient | Wikipedia | 455 | 41641 | https://en.wikipedia.org/wiki/Reflection%20coefficient | Physical sciences | Optics | Physics |
In the accompanying figure, a signal source with internal impedance possibly followed by a transmission line of characteristic impedance is represented by its Thévenin equivalent, driving the load . For a real (resistive) source impedance , if we define using the reference impedance then the source's maximum power is delivered to a load , in which case implying no reflected power. More generally, the squared-magnitude of the reflection coefficient denotes the proportion of that power that is reflected back to the source, with the power actually delivered toward the load being .
Anywhere along an intervening (lossless) transmission line of characteristic impedance , the magnitude of the reflection coefficient will remain the same (the powers of the forward and reflected waves stay the same) but with a different phase. In the case of a short circuited load (), one finds at the load. This implies the reflected wave having a 180° phase shift (phase reversal) with the voltages of the two waves being opposite at that point and adding to zero (as a short circuit demands).
Relation to load impedance
The reflection coefficient is determined by the load impedance at the end of the transmission line, as well as the characteristic impedance of the line. A load impedance of terminating a line with a characteristic impedance of will have a reflection coefficient of
This is the coefficient at the load. The reflection coefficient can also be measured at other points on the line. The magnitude of the reflection coefficient in a lossless transmission line is constant along the line (as are the powers in the forward and reflected waves). However its phase will be shifted by an amount dependent on the electrical distance from the load. If the coefficient is measured at a point meters from the load, so the electrical distance from the load is radians, the coefficient at that point will be
Note that the phase of the reflection coefficient is changed by twice the phase length of the attached transmission line. That is to take into account not only the phase delay of the reflected wave, but the phase shift that had first been applied to the forward wave, with the reflection coefficient being the quotient of these. The reflection coefficient so measured, , corresponds to an impedance which is generally dissimilar to present at the far side of the transmission line. | Reflection coefficient | Wikipedia | 461 | 41641 | https://en.wikipedia.org/wiki/Reflection%20coefficient | Physical sciences | Optics | Physics |
The complex reflection coefficient (in the region , corresponding to passive loads) may be displayed graphically using a Smith chart. The Smith chart is a polar plot of , therefore the magnitude of is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to ). Its evolution along a transmission line is likewise described by a rotation of around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to ) can directly be read. Before the advent of modern electronic computers, the Smith chart was of particular use as a sort of analog computer for this purpose.
Standing wave ratio
The standing wave ratio (SWR) is determined solely by the magnitude of the reflection coefficient:
Along a lossless transmission line of characteristic impedance Z0, the SWR signifies the ratio of the voltage (or current) maxima to minima (or what it would be if the transmission line were long enough to produce them). The above calculation assumes that has been calculated using Z0 as the reference impedance. Since it uses only the magnitude of , the SWR intentionally ignores the specific value of the load impedance ZL responsible for it, but only the magnitude of the resulting impedance mismatch. That SWR remains the same wherever measured along a transmission line (looking towards the load) since the addition of a transmission line length to a load only changes the phase, not magnitude of . While having a one-to-one correspondence with reflection coefficient, SWR is the most commonly used figure of merit in describing the mismatch affecting a radio antenna or antenna system. It is most often measured at the transmitter side of a transmission line, but having, as explained, the same value as would be measured at the antenna (load) itself.
Seismology
Reflection coefficient is used in feeder testing for reliability of medium.
Optics and microwaves
In optics and electromagnetics in general, reflection coefficient can refer to either the amplitude reflection coefficient described here, or the reflectance, depending on context. Typically, the reflectance is represented by a capital R, while the amplitude reflection coefficient is represented by a lower-case r. These related concepts are covered by Fresnel equations in classical optics.
Acoustics
Acousticians use reflection coefficients to understand the effect of different materials on their acoustic environments. | Reflection coefficient | Wikipedia | 474 | 41641 | https://en.wikipedia.org/wiki/Reflection%20coefficient | Physical sciences | Optics | Physics |
Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration that matches its natural frequency. When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger amplitude. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.
All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; this frequency is known as a resonant frequency or resonance frequency. When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude (with more force) than when the same force is applied at other, non-resonant frequencies.
The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a relative maximum within the system. Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude oscillations in the system due to the storage of vibrational energy.
Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, orbital resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters).
The term resonance (from Latin resonantia, 'echo', from resonare, 'resound') originated from the field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck.
Overview | Resonance | Wikipedia | 430 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple and distinct resonant frequencies.
Examples
A familiar example is a playground swing, which acts as a pendulum. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations.
Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. For example, when hard objects like metal, glass, or wood are struck, there are brief resonant vibrations in the object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale, such as electrons in atoms. Other examples of resonance include:
Timekeeping mechanisms of modern clocks and watches, e.g., the balance wheel in a mechanical watch and the quartz crystal in a quartz watch
Tidal resonance of the Bay of Fundy
Acoustic resonances of musical instruments and the human vocal tract
Shattering of a crystal wineglass when exposed to a musical tone of the right pitch (its resonant frequency)
Friction idiophones, such as making a glass object (glass, bottle, vase) vibrate by rubbing around its rim with a fingertip
Electrical resonance of tuned circuits in radios and TVs that allow radio frequencies to be selectively received
Creation of coherent light by optical resonance in a laser cavity
Orbital resonance as exemplified by some moons of the Solar System's giant planets and resonant groups such as the plutinos
Material resonances in atomic scale are the basis of several spectroscopic techniques that are used in condensed matter physics
Electron spin resonance
Mössbauer effect
Nuclear magnetic resonance | Resonance | Wikipedia | 459 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Linear systems
Resonance manifests itself in many linear and nonlinear systems as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.
Since many linear and nonlinear systems that oscillate are modeled as harmonic oscillators near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An RLC circuit is used to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, these connections for higher-order linear systems with multiple inputs and outputs are generalized.
The driven, damped harmonic oscillator
Consider a damped mass on a spring driven by a sinusoidal, externally applied force. Newton's second law takes the form
where m is the mass, x is the displacement of the mass from the equilibrium point, F0 is the driving amplitude, ω is the driving angular frequency, k is the spring constant, and c is the viscous damping coefficient. This can be rewritten in the form
where
is called the undamped angular frequency of the oscillator or the natural frequency,
is called the damping ratio.
Many sources also refer to ω0 as the resonant frequency. However, as shown below, when analyzing oscillations of the displacement x(t), the resonant frequency is close to but not the same as ω0. In general the resonant frequency is close to but not necessarily the same as the natural frequency. The RLC circuit example in the next section gives examples of different resonant frequencies for the same system. | Resonance | Wikipedia | 444 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
The general solution of Equation () is the sum of a transient solution that depends on initial conditions and a steady state solution that is independent of initial conditions and depends only on the driving amplitude F0, driving frequency ω, undamped angular frequency ω0, and the damping ratio ζ. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.
It is possible to write the steady-state solution for x(t) as a function proportional to the driving force with an induced phase change φ,
where
The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument.
Resonance occurs when, at certain driving frequencies, the steady-state amplitude of x(t) is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies. Looking at the amplitude of x(t) as a function of the driving frequency ω, the amplitude is maximal at the driving frequency
ωr is the resonant frequency for this system. Again, the resonant frequency does not equal the undamped angular frequency ω0 of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ω0, but the maximum response is at the resonant frequency.
Also, ωr is only real and non-zero if , so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large.
The pendulum
For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains
but the definitions of ω0 and ζ change based on the physics of the system. For a pendulum of length ℓ and small displacement angle θ, Equation () becomes
and therefore
RLC series circuits | Resonance | Wikipedia | 486 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Consider a circuit consisting of a resistor with resistance R, an inductor with inductance L, and a capacitor with capacitance C connected in series with current i(t) and driven by a voltage source with voltage vin(t). The voltage drop around the circuit is
Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze the frequency response of this circuit. Taking the Laplace transform of Equation (),
where I(s) and Vin(s) are the Laplace transform of the current and input voltage, respectively, and s is a complex frequency parameter in the Laplace domain. Rearranging terms,
Voltage across the capacitor
An RLC circuit in series presents several options for where to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is
or
Define for this circuit a natural frequency and a damping ratio,
The ratio of the output voltage to the input voltage becomes
H(s) is the transfer function between the input voltage and the output voltage. This transfer function has two poles–roots of the polynomial in the transfer function's denominator–at
and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for , the magnitude of these poles is the natural frequency ω0 and that for , our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis.
Evaluating H(s) along the imaginary axis , the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of Equation () instead of the Laplace transform. The transfer function, which is also complex, can be written as a gain and phase,
A sinusoidal input voltage at frequency ω results in an output voltage at the same frequency that has been scaled by G(ω) and has a phase shift Φ(ω). The gain and phase can be plotted versus frequency on a Bode plot. For the RLC circuit's capacitor voltage, the gain of the transfer function H(iω) is
Note the similarity between the gain here and the amplitude in Equation (). Once again, the gain is maximized at the resonant frequency | Resonance | Wikipedia | 493 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Here, the resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.
Voltage across the inductor
The resonant frequency need not always take the form given in the examples above. For the RLC circuit, suppose instead that the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain the voltage across the inductor is
using the same definitions for ω0 and ζ as in the previous example. The transfer function between Vin(s) and this new Vout(s) across the inductor is
This transfer function has the same poles as the transfer function in the previous example, but it also has two zeroes in the numerator at . Evaluating H(s) along the imaginary axis, its gain becomes
Compared to the gain in Equation () using the capacitor voltage as the output, this gain has a factor of ω2 in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency is
So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now larger than the natural frequency, though it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different choices of output is not contradictory. As shown in Equation (), the voltage drop across the circuit is divided among the three circuit elements, and each element has different dynamics. The capacitor's voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, whereas the inductor's voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency where it tends to oscillate, the different dynamics of each circuit element make each element resonate at a slightly different frequency.
Voltage across the resistor
Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor is
and using the same natural frequency and damping ratio as in the capacitor example the transfer function is
This transfer function also has the same poles as the previous RLC circuit examples, but it only has one zero in the numerator at s = 0. For this transfer function, its gain is | Resonance | Wikipedia | 505 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
The resonant frequency that maximizes this gain is
and the gain is one at this frequency, so the voltage across the resistor resonates at the circuit's natural frequency and at this frequency the amplitude of the voltage across the resistor equals the input voltage's amplitude.
Antiresonance
Some systems exhibit antiresonance that can be analyzed in the same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately small rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined.
Suppose that the output voltage of interest in the RLC circuit is the voltage across the inductor and the capacitor combined in series. Equation () showed that the sum of the voltages across the three circuit elements sums to the input voltage, so measuring the output voltage as the sum of the inductor and capacitor voltages combined is the same as vin minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of the voltage drop across the resistor equals the amplitude of vin, and therefore the voltage across the inductor and capacitor combined has zero amplitude. We can show this with the transfer function.
The sum of the inductor and capacitor voltages is
Using the same natural frequency and damping ratios as the previous examples, the transfer function is
This transfer has the same poles as the previous examples but has zeroes at
Evaluating the transfer function along the imaginary axis, its gain is
Rather than look for resonance, i.e., peaks of the gain, notice that the gain goes to zero at ω = ω0, which complements our analysis of the resistor's voltage. This is called antiresonance, which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to the zeroes of the transfer function, which were shown in Equation () and were on the imaginary axis. | Resonance | Wikipedia | 465 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Relationships between resonance and frequency response in the RLC series circuit example
These RLC circuit examples illustrate how resonance is related to the frequency response of the system. Specifically, these examples illustrate:
How resonant frequencies can be found by looking for peaks in the gain of the transfer function between the input and output of the system, for example in a Bode magnitude plot
How the resonant frequency for a single system can be different for different choices of system output
The connection between the system's natural frequency, the system's damping ratio, and the system's resonant frequency
The connection between the system's natural frequency and the magnitude of the transfer function's poles, pointed out in Equation (), and therefore a connection between the poles and the resonant frequency
A connection between the transfer function's zeroes and the shape of the gain as a function of frequency, and therefore a connection between the zeroes and the resonant frequency that maximizes gain
A connection between the transfer function's zeroes and antiresonance
The next section extends these concepts to resonance in a general linear system.
Generalizing resonance and antiresonance for linear systems
Next consider an arbitrary linear system with multiple inputs and outputs. For example, in state-space representation a third order linear time-invariant system with three inputs and two outputs might be written as
where ui(t) are the inputs, xi(t) are the state variables, yi(t) are the outputs, and A, B, C, and D are matrices describing the dynamics between the variables.
This system has a transfer function matrix whose elements are the transfer functions between the various inputs and outputs. For example,
Each Hij(s) is a scalar transfer function linking one of the inputs to one of the outputs. The RLC circuit examples above had one input voltage and showed four possible output voltages–across the capacitor, across the inductor, across the resistor, and across the capacitor and inductor combined in series–each with its own transfer function. If the RLC circuit were set up to measure all four of these output voltages, that system would have a 4×1 transfer function matrix linking the single input to each of the four outputs.
Evaluated along the imaginary axis, each Hij(iω) can be written as a gain and phase shift, | Resonance | Wikipedia | 486 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Peaks in the gain at certain frequencies correspond to resonances between that transfer function's input and output, assuming the system is stable.
Each transfer function Hij(s) can also be written as a fraction whose numerator and denominator are polynomials of s.
The complex roots of the numerator are called zeroes, and the complex roots of the denominator are called poles. For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonate and at which frequencies. In particular, any stable or marginally stable, complex conjugate pair of poles with imaginary components can be written in terms of a natural frequency and a damping ratio as
as in Equation (). The natural frequency ω0 of that pole is the magnitude of the position of the pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general,
Complex conjugate pairs of poles near the imaginary axis correspond to a peak or resonance in the frequency response in the vicinity of the pole's natural frequency. If the pair of poles is on the imaginary axis, the gain is infinite at that frequency.
Complex conjugate pairs of zeroes near the imaginary axis correspond to a notch or antiresonance in the frequency response in the vicinity of the zero's frequency, i.e., the frequency equal to the magnitude of the zero. If the pair of zeroes is on the imaginary axis, the gain is zero at that frequency.
In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation (). The second generalization relating zeroes to antiresonance is observed in Equation (). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition ζ < 1/.
Standing waves | Resonance | Wikipedia | 414 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
A physical system can have as many natural frequencies as it has degrees of freedom and can resonate near each of those natural frequencies. A mass on a spring, which has one degree of freedom, has one natural frequency. A double pendulum, which has two degrees of freedom, can have two natural frequencies. As the number of coupled harmonic oscillators increases, the time it takes to transfer energy from one to the next becomes significant. Systems with very large numbers of degrees of freedom can be thought of as continuous rather than as having discrete oscillators.
Energy transfers from one oscillator to the next in the form of waves. For example, the string of a guitar or the surface of water in a bowl can be modeled as a continuum of small coupled oscillators and waves can travel along them. In many cases these systems have the potential to resonate at certain frequencies, forming standing waves with large-amplitude oscillations at fixed positions. Resonance in the form of standing waves underlies many familiar phenomena, such as the sound produced by musical instruments, electromagnetic cavities used in lasers and microwave ovens, and energy levels of atoms.
Standing waves on a string
When a string of fixed length is driven at a particular frequency, a wave propagates along the string at the same frequency. The waves reflect off the ends of the string, and eventually a steady state is reached with waves traveling in both directions. The waveform is the superposition of the waves.
At certain frequencies, the steady state waveform does not appear to travel along the string. At fixed positions called nodes, the string is never displaced. Between the nodes the string oscillates and exactly halfway between the nodes–at positions called anti-nodes–the oscillations have their largest amplitude.
For a string of length with fixed ends, the displacement of the string perpendicular to the -axis at time is
where
is the amplitude of the left- and right-traveling waves interfering to form the standing wave,
is the wave number,
is the frequency.
The frequencies that resonate and form standing waves relate to the length of the string as
where is the speed of the wave and the integer denotes different modes or harmonics. The standing wave with oscillates at the fundamental frequency and has a wavelength that is twice the length of the string. The possible modes of oscillation form a harmonic series.
Resonance in complex networks | Resonance | Wikipedia | 491 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
A generalization to complex networks of coupled harmonic oscillators shows that such systems have a finite number of natural resonant frequencies, related to the topological structure of the network itself. In particular, such frequencies result related to the eigenvalues of the network's Laplacian matrix. Let be the adjacency matrix describing the topological structure of the network and the corresponding Laplacian matrix, where is the diagonal matrix of the degrees of the network's nodes. Then, for a network of classical and identical harmonic oscillators, when a sinusoidal driving force is applied to a specific node, the global resonant frequencies of the network are given by where are the eigenvalues of the Laplacian .
Types
Mechanical
Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, engineers must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.
Avoiding resonance disasters is a major concern in every building, tower, and bridge construction project. As a countermeasure, shock mounts can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The Taipei 101 building relies on a —a tuned mass damper—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that does not typically occur. Buildings in seismic zones are often constructed to take into account the oscillating frequencies of expected ground motion. In addition, engineers designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.
Clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal.
The cadence of runners has been hypothesized to be energetically favorable due to resonance between the elastic energy stored in the lower limb and the mass of the runner.
International Space Station | Resonance | Wikipedia | 455 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
The rocket engines for the International Space Station (ISS) are controlled by an autopilot. Ordinarily, uploaded parameters for controlling the engine control system for the Zvezda module make the rocket engines boost the International Space Station to a higher orbit. The rocket engines are hinge-mounted, and ordinarily the crew does not notice the operation. On January 14, 2009, however, the uploaded parameters made the autopilot swing the rocket engines in larger and larger oscillations, at a frequency of 0.5 Hz. These oscillations were captured on video, and lasted for 142 seconds.
Acoustic
Acoustic resonance is a branch of mechanical resonance that is concerned with the mechanical vibrations across the frequency range of human hearing, in other words sound. For humans, hearing is normally limited to frequencies between about 20 Hz and 20,000 Hz (20 kHz), Many objects and materials act as resonators with resonant frequencies within this range, and when struck vibrate mechanically, pushing on the surrounding air to create sound waves. This is the source of many percussive sounds we hear.
Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of, and tension on, a drum membrane.
Like mechanical resonance, acoustic resonance can result in catastrophic failure of the object at resonance. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass, although this is difficult in practice.
Electrical
Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedance of the circuit is at a minimum in a series circuit or at maximum in a parallel circuit (usually when the transfer function peaks in absolute value). Resonance in circuits are used for both transmitting and receiving wireless communications such as television, cell phones and radio.
Optical | Resonance | Wikipedia | 394 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
An optical cavity, also called an optical resonator, is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers. Light confined in the cavity reflects multiple times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called "modes". Longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross-section of the beam. Ring resonators and whispering galleries are examples of optical resonators that do not form standing waves.
Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them; flat mirrors are not often used because of the difficulty of aligning them precisely. The geometry (resonator type) must be chosen so the beam remains stable, i.e., the beam size does not continue to grow with each reflection. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity.
Optical cavities are designed to have a very large Q factor. A beam reflects a large number of times with little attenuation—therefore the frequency line width of the beam is small compared to the frequency of the laser.
Additional optical resonances are guided-mode resonances and surface plasmon resonance, which result in anomalous reflection and high evanescent fields at resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are usually excited by a subwavelength grating.
Orbital | Resonance | Wikipedia | 380 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet.
Atomic, particle, and molecular
Nuclear magnetic resonance (NMR) is the name given to a physical resonance phenomenon involving the observation of specific quantum mechanical magnetic properties of an atomic nucleus in the presence of an applied, external magnetic field. Many scientific techniques exploit NMR phenomena to study molecular physics, crystals, and non-crystalline materials through NMR spectroscopy. NMR is also routinely used in advanced medical imaging techniques, such as in magnetic resonance imaging (MRI).
All nuclei containing odd numbers of nucleons have an intrinsic magnetic moment and angular momentum. A key feature of NMR is that the resonant frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonant frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency.
Electron paramagnetic resonance, otherwise known as electron spin resonance (ESR), is a spectroscopic technique similar to NMR, but uses unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be paramagnetic.
The Mössbauer effect is the resonant and recoil-free emission and absorption of gamma ray photons by atoms bound in a solid form. | Resonance | Wikipedia | 508 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
Resonance in particle physics appears in similar circumstances to classical physics at the level of quantum mechanics and quantum field theory. Resonances can also be thought of as unstable particles, with the formula in the Universal resonance curve section of this article applying if Γ is the particle's decay rate and Ω is the particle's mass M. In that case, the formula comes from the particle's propagator, with its mass replaced by the complex number M + iΓ. The formula is further related to the particle's decay rate by the optical theorem.
Disadvantages
A column of soldiers marching in regular step on a narrow and structurally flexible bridge can set it into dangerously large amplitude oscillations. On April 12, 1831, the Broughton Suspension Bridge near Salford, England collapsed while a group of British soldiers were marching across. Since then, the British Army has had a standing order for soldiers to break stride when marching across bridges, to avoid resonance from their regular marching pattern affecting the bridge.
Vibrations of a motor or engine can induce resonant vibration in its supporting structures if their natural frequency is close to that of the vibrations of the engine. A common example is the rattling sound of a bus body when the engine is left idling.
Structural resonance of a suspension bridge induced by winds can lead to its catastrophic collapse. Several early suspension bridges in Europe and United States were destroyed by structural resonance induced by modest winds. The collapse of the Tacoma Narrows Bridge on 7 November 1940 is characterized in physics as a classic example of resonance. It has been argued by Robert H. Scanlan and others that the destruction was instead caused by aeroelastic flutter, a complicated interaction between the bridge and the winds passing through it—an example of a self oscillation, or a kind of "self-sustaining vibration" as referred to in the nonlinear theory of vibrations.
Q factor
The Q factor or quality factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, and characterizes the bandwidth of a resonator relative to its center frequency.
A high value for Q indicates a lower rate of energy loss relative to the stored energy, i.e., the system is lightly damped. The parameter is defined by the equation:
. | Resonance | Wikipedia | 457 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
The higher the Q factor, the greater the amplitude at the resonant frequency, and the smaller the bandwidth, or range of frequencies around resonance occurs. In electrical resonance, a high-Q circuit in a radio receiver is more difficult to tune, but has greater selectivity, and so would be better at filtering out signals from other stations. High Q oscillators are more stable.
Examples that normally have a low Q factor include door closers (Q=0.5). Systems with high Q factors include tuning forks (Q=1000), atomic clocks and lasers (Q≈1011).
Universal resonance curve
The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system, and is usually not exactly symmetric about the resonant frequency, as illustrated for the simple harmonic oscillator above.
For a lightly damped linear oscillator with a resonance frequency , the intensity of oscillations when the system is driven with a driving frequency is typically approximated by the following formula that is symmetric about the resonance frequency:
Where the susceptibility links the amplitude of the oscillator to the driving force in frequency space:
The intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, or Cauchy distribution, and this response is found in many physical situations involving resonant systems. is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonant frequency. The linewidth is inversely proportional to the Q factor, which is a measure of the sharpness of the resonance.
In radio engineering and electronics engineering, this approximate symmetric response is known as the universal resonance curve, a concept introduced by Frederick E. Terman in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and Q values. | Resonance | Wikipedia | 430 | 41660 | https://en.wikipedia.org/wiki/Resonance | Physical sciences | Waves | null |
A shield is a piece of personal armour held in the hand, which may or may not be strapped to the wrist or forearm. Shields are used to intercept specific attacks, whether from close-ranged weaponry like spears or long ranged projectiles such as arrows. They function as means of active blocks, as well as to provide passive protection by closing one or more lines of engagement during combat.
Shields vary greatly in size and shape, ranging from large panels that protect the user's whole body to small models (such as the buckler) that were intended for hand-to-hand-combat use. Shields also vary a great deal in thickness; whereas some shields were made of relatively deep, absorbent, wooden planking to protect soldiers from the impact of spears and crossbow bolts, others were thinner and lighter and designed mainly for deflecting blade strikes (like the roromaraugi or qauata). Finally, shields vary greatly in shape, ranging in roundness to angularity, proportional length and width, symmetry and edge pattern; different shapes provide more optimal protection for infantry or cavalry, enhance portability, provide secondary uses such as ship protection or as a weapon and so on.
In prehistory and during the era of the earliest civilisations, shields were made of wood, animal hide, woven reeds or wicker. In classical antiquity, the Barbarian Invasions and the Middle Ages, they were normally constructed of poplar tree, lime or another split-resistant timber, covered in some instances with a material such as leather or rawhide and often reinforced with a metal boss, rim or banding. They were carried by foot soldiers, knights and cavalry.
Depending on time and place, shields could be round, oval, square, rectangular, triangular, bilabial or scalloped. Sometimes they took on the form of kites or flatirons, or had rounded tops on a rectangular base with perhaps an eye-hole, to look through when used with combat. The shield was held by a central grip or by straps with some going over or around the user's arm and one or more being held by the hand. | Shield | Wikipedia | 433 | 41698 | https://en.wikipedia.org/wiki/Shield | Technology | Armour | null |
Often shields were decorated with a painted pattern or an animal representation to show their army or clan. It was common for Aristocratic officials such and knights, barons, dukes, and kings to have their shields painted with customary designs known as a coat of arms. These designs developed into systematized heraldic devices during the High Middle Ages for purposes of battlefield identification. Even after the introduction of gunpowder and firearms to the battlefield, shields continued to be used by certain groups. In the 18th century, for example, Scottish Highland fighters liked to wield small shields known as targes, and as late as the 19th century, some non-industrialized peoples (such as Zulu warriors) employed them when waging wars.
In the 20th and 21st century, shields have been used by military and police units that specialize in anti-terrorist actions, hostage rescue, riot control and siege-breaking.
History
Prehistory
The first prototype of the shield was believed to be created in the Late Neolithic Age. However the oldest surviving shields date to sometime in the Bronze Age. The oldest form of shield was a protection device designed to block attacks by hand weapons, such as swords, axes and maces, or ranged weapons like sling-stones and arrows. Shields have varied greatly in construction over time and place. Sometimes shields were made of metal, but wood or animal hide construction was much more common; wicker and even turtle shells have been used. Many surviving examples of metal shields are generally felt to be ceremonial rather than practical, for example the Yetholm-type shields of the Bronze Age, or the Iron Age Battersea shield.
Ancient | Shield | Wikipedia | 322 | 41698 | https://en.wikipedia.org/wiki/Shield | Technology | Armour | null |
Size and weight varied greatly. Lightly armored warriors relying on speed and surprise would generally carry light shields (pelte) that were either small or thin. Heavy troops might be equipped with robust shields that could cover most of the body. Many had a strap called a guige that allowed them to be slung over the user's back when not in use or on horseback. During the 14th–13th century BC, the Sards or Shardana, working as mercenaries for the Egyptian pharaoh Ramses II, utilized either large or small round shields against the Hittites. The Mycenaean Greeks used two types of shields: the "figure-of-eight" shield and a rectangular "tower" shield. These shields were made primarily from a wicker frame and then reinforced with leather. Covering the body from head to foot, the figure-of-eight and tower shield offered most of the warrior's body a good deal of protection in hand-to-hand combat. The Ancient Greek hoplites used a round, bowl-shaped wooden shield that was reinforced with bronze and called an aspis. The aspis was also the longest-lasting and most famous and influential of all of the ancient Greek shields. The Spartans used the aspis to create the Greek phalanx formation. Their shields offered protection not only for themselves but for their comrades to their left.
Examples of Germanic wooden shields circa 350 BC – 500 AD survive from weapons sacrifices in Danish bogs.
The heavily armored Roman legionaries carried large shields (scuta) that could provide far more protection, but made swift movement a little more difficult. The scutum originally had an oval shape, but gradually the curved tops and sides were cut to produce the familiar rectangular shape most commonly seen in the early Imperial legions. Famously, the Romans used their shields to create a tortoise-like formation called a testudo in which entire groups of soldiers would be enclosed in an armoured box to provide protection against missiles. Many ancient shield designs featured incuts of one sort or another. This was done to accommodate the shaft of a spear, thus facilitating tactics requiring the soldiers to stand close together forming a wall of shields.
Post-classical | Shield | Wikipedia | 443 | 41698 | https://en.wikipedia.org/wiki/Shield | Technology | Armour | null |
Typical in the early European Middle Ages were round shields with light, non-splitting wood like linden, fir, alder, or poplar, usually reinforced with leather cover on one or both sides and occasionally metal rims, encircling a metal shield boss. These light shields suited a fighting style where each incoming blow is intercepted with the boss in order to deflect it. The Normans introduced the kite shield around the 10th century, which was rounded at the top and tapered at the bottom. This gave some protection to the user's legs, without adding too much to the total weight of the shield. The kite shield predominantly features enarmes, leather straps used to grip the shield tight to the arm. Used by foot and mounted troops alike, it gradually came to replace the round shield as the common choice until the end of the 12th century, when more efficient limb armour allowed the shields to grow shorter, and be entirely replaced by the 14th century.
As body armour improved, knight's shields became smaller, leading to the familiar heater shield style. Both kite and heater style shields were made of several layers of laminated wood, with a gentle curve in cross section. The heater style inspired the shape of the symbolic heraldic shield that is still used today. Eventually, specialised shapes were developed such as the bouche, which had a lance rest cut into the upper corner of the lance side, to help guide it in combat or tournament. Free standing shields called pavises, which were propped up on stands, were used by medieval crossbowmen who needed protection while reloading.
In time, some armoured foot knights gave up shields entirely in favour of mobility and two-handed weapons. Other knights and common soldiers adopted the buckler, giving rise to the term "swashbuckler". The buckler is a small round shield, typically between 8 and 16 inches (20–40 cm) in diameter. The buckler was one of very few types of shield that were usually made of metal. Small and light, the buckler was easily carried by being hung from a belt; it gave little protection from missiles and was reserved for hand-to-hand combat where it served both for protection and offence. The buckler's use began in the Middle Ages and continued well into the 16th century. | Shield | Wikipedia | 470 | 41698 | https://en.wikipedia.org/wiki/Shield | Technology | Armour | null |
In Italy, the targa, parma, and rotella were used by common people, fencers and even knights. The development of plate armour made shields less and less common as it eliminated the need for a shield. Lightly armoured troops continued to use shields after men-at-arms and knights ceased to use them. Shields continued in use even after gunpowder powered weapons made them essentially obsolete on the battlefield. In the 18th century, the Scottish clans used a small, round targe that was partially effective against the firearms of the time, although it was arguably more often used against British infantry bayonets and cavalry swords in close-in fighting.
During the 19th century, non-industrial cultures with little access to guns were still using war shields. Zulu warriors carried large lightweight shields called Ishlangu made from a single ox hide supported by a wooden spine. This was used in combination with a short spear (iklwa) and/or club. Other African shields include Glagwa from Cameroon or Nguba from Congo.
Modern
Law enforcement shields
Shields for protection from armed attack are still used by many police forces around the world. These modern shields are usually intended for two broadly distinct purposes. The first type, riot shields, are used for riot control and can be made from metal or polymers such as polycarbonate Lexan or Makrolon or boPET Mylar. These typically offer protection from relatively large and low velocity projectiles, such as rocks and bottles, as well as blows from fists or clubs. Synthetic riot shields are normally transparent, allowing full use of the shield without obstructing vision. Similarly, metal riot shields often have a small window at eye level for this purpose. These riot shields are most commonly used to block and push back crowds when the users stand in a "wall" to block protesters, and to protect against shrapnel, projectiles like stones and bricks, molotov cocktails, and during hand-to-hand combat.
The second type of modern police shield is the bullet-resistant ballistic shield, also called tactical shield. These shields are typically manufactured from advanced synthetics such as Kevlar and are designed to be bulletproof, or at least bullet resistant. Two types of shields are available:
Light level IIIA shields are designed to stop pistol cartridges.
Heavy level III and IV shields are designed to stop rifle cartridges. | Shield | Wikipedia | 479 | 41698 | https://en.wikipedia.org/wiki/Shield | Technology | Armour | null |
Tactical shields often have a firing port so that the officer holding the shield can fire a weapon while being protected by the shield, and they often have a bulletproof glass viewing port. They are typically employed by specialist police, such as SWAT teams in high risk entry and siege scenarios, such as hostage rescue and breaching gang compounds, as well as in antiterrorism operations.
Law enforcement shields often have a large signs stating "POLICE" (or the name of a force, such as "US MARSHALS") to indicate that the user is a law enforcement officer.
Gallery
List
Aspis
Ballistic shield
Battersea Shield
Buckler
Escutcheon (heraldic shield)
Glagwa
Heater shield
Kite shield
Nguni shield
Pavise
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Scutum (shield)
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Yetholm-type shields
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Enarmes
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Testudo formation | Shield | Wikipedia | 193 | 41698 | https://en.wikipedia.org/wiki/Shield | Technology | Armour | null |
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in decibels. A ratio higher than 1:1 (greater than 0 dB) indicates more signal than noise.
SNR is an important parameter that affects the performance and quality of systems that process or transmit signals, such as communication systems, audio systems, radar systems, imaging systems, and data acquisition systems. A high SNR means that the signal is clear and easy to detect or interpret, while a low SNR means that the signal is corrupted or obscured by noise and may be difficult to distinguish or recover. SNR can be improved by various methods, such as increasing the signal strength, reducing the noise level, filtering out unwanted noise, or using error correction techniques.
SNR also determines the maximum possible amount of data that can be transmitted reliably over a given channel, which depends on its bandwidth and SNR. This relationship is described by the Shannon–Hartley theorem, which is a fundamental law of information theory.
SNR can be calculated using different formulas depending on how the signal and noise are measured and defined. The most common way to express SNR is in decibels, which is a logarithmic scale that makes it easier to compare large or small values. Other definitions of SNR may use different factors or bases for the logarithm, depending on the context and application.
Definition
One definition of signal-to-noise ratio is the ratio of the power of a signal (meaningful input) to the power of background noise (meaningless or unwanted input):
where is average power. Both signal and noise power must be measured at the same or equivalent points in a system, and within the same system bandwidth.
The signal-to-noise ratio of a random variable () to random noise is:
where E refers to the expected value, which in this case is the mean square of .
If the signal is simply a constant value of , this equation simplifies to:
If the noise has expected value of zero, as is common, the denominator is its variance, the square of its standard deviation .
The signal and the noise must be measured the same way, for example as voltages across the same impedance. Their root mean squares can alternatively be used according to: | Signal-to-noise ratio | Wikipedia | 499 | 41706 | https://en.wikipedia.org/wiki/Signal-to-noise%20ratio | Technology | Basics_4 | null |
where is root mean square (RMS) amplitude (for example, RMS voltage).
Decibels
Because many signals have a very wide dynamic range, signals are often expressed using the logarithmic decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as
and
In a similar manner, SNR may be expressed in decibels as
Using the definition of SNR
Using the quotient rule for logarithms
Substituting the definitions of SNR, signal, and noise in decibels into the above equation results in an important formula for calculating the signal to noise ratio in decibels, when the signal and noise are also in decibels:
In the above formula, P is measured in units of power, such as watts (W) or milliwatts (mW), and the signal-to-noise ratio is a pure number.
However, when the signal and noise are measured in volts (V) or amperes (A), which are measures of amplitude, they must first be squared to obtain a quantity proportional to power, as shown below:
Dynamic range
The concepts of signal-to-noise ratio and dynamic range are closely related. Dynamic range measures the ratio between the strongest un-distorted signal on a channel and the minimum discernible signal, which for most purposes is the noise level. SNR measures the ratio between an arbitrary signal level (not necessarily the most powerful signal possible) and noise. Measuring signal-to-noise ratios requires the selection of a representative or reference signal. In audio engineering, the reference signal is usually a sine wave at a standardized nominal or alignment level, such as 1 kHz at +4 dBu (1.228 VRMS).
SNR is usually taken to indicate an average signal-to-noise ratio, as it is possible that instantaneous signal-to-noise ratios will be considerably different. The concept can be understood as normalizing the noise level to 1 (0 dB) and measuring how far the signal 'stands out'.
Difference from conventional power
In physics, the average power of an AC signal is defined as the average value of voltage times current; for resistive (non-reactive) circuits, where voltage and current are in phase, this is equivalent to the product of the rms voltage and current: | Signal-to-noise ratio | Wikipedia | 481 | 41706 | https://en.wikipedia.org/wiki/Signal-to-noise%20ratio | Technology | Basics_4 | null |
But in signal processing and communication, one usually assumes that so that factor is usually not included while measuring power or energy of a signal. This may cause some confusion among readers, but the resistance factor is not significant for typical operations performed in signal processing, or for computing power ratios. For most cases, the power of a signal would be considered to be simply
Alternative definition
An alternative definition of SNR is as the reciprocal of the coefficient of variation, i.e., the ratio of mean to standard deviation of a signal or measurement:
where is the signal mean or expected value and is the standard deviation of the noise, or an estimate thereof. Notice that such an alternative definition is only useful for variables that are always non-negative (such as photon counts and luminance), and it is only an approximation since . It is commonly used in image processing, where the SNR of an image is usually calculated as the ratio of the mean pixel value to the standard deviation of the pixel values over a given neighborhood.
Sometimes SNR is defined as the square of the alternative definition above, in which case it is equivalent to the more common definition:
This definition is closely related to the sensitivity index or d, when assuming that the signal has two states separated by signal amplitude , and the noise standard deviation does not change between the two states.
The Rose criterion (named after Albert Rose) states that an SNR of at least 5 is needed to be able to distinguish image features with certainty. An SNR less than 5 means less than 100% certainty in identifying image details.
Yet another alternative, very specific, and distinct definition of SNR is employed to characterize sensitivity of imaging systems; see Signal-to-noise ratio (imaging).
Related measures are the "contrast ratio" and the "contrast-to-noise ratio".
Modulation system measurements
Amplitude modulation
Channel signal-to-noise ratio is given by
where W is the bandwidth and is modulation index
Output signal-to-noise ratio (of AM receiver) is given by
Frequency modulation
Channel signal-to-noise ratio is given by
Output signal-to-noise ratio is given by
Noise reduction
All real measurements are disturbed by noise. This includes electronic noise, but can also include external events that affect the measured phenomenon — wind, vibrations, the gravitational attraction of the moon, variations of temperature, variations of humidity, etc., depending on what is measured and of the sensitivity of the device. It is often possible to reduce the noise by controlling the environment. | Signal-to-noise ratio | Wikipedia | 504 | 41706 | https://en.wikipedia.org/wiki/Signal-to-noise%20ratio | Technology | Basics_4 | null |
Internal electronic noise of measurement systems can be reduced through the use of low-noise amplifiers.
When the characteristics of the noise are known and are different from the signal, it is possible to use a filter to reduce the noise. For example, a lock-in amplifier can extract a narrow bandwidth signal from broadband noise a million times stronger.
When the signal is constant or periodic and the noise is random, it is possible to enhance the SNR by averaging the measurements. In this case the noise goes down as the square root of the number of averaged samples.
Digital signals
When a measurement is digitized, the number of bits used to represent the measurement determines the maximum possible signal-to-noise ratio. This is because the minimum possible noise level is the error caused by the quantization of the signal, sometimes called quantization noise. This noise level is non-linear and signal-dependent; different calculations exist for different signal models. Quantization noise is modeled as an analog error signal summed with the signal before quantization ("additive noise").
This theoretical maximum SNR assumes a perfect input signal. If the input signal is already noisy (as is usually the case), the signal's noise may be larger than the quantization noise. Real analog-to-digital converters also have other sources of noise that further decrease the SNR compared to the theoretical maximum from the idealized quantization noise, including the intentional addition of dither.
Although noise levels in a digital system can be expressed using SNR, it is more common to use Eb/No, the energy per bit per noise power spectral density.
The modulation error ratio (MER) is a measure of the SNR in a digitally modulated signal.
Fixed point
For n-bit integers with equal distance between quantization levels (uniform quantization) the dynamic range (DR) is also determined.
Assuming a uniform distribution of input signal values, the quantization noise is a uniformly distributed random signal with a peak-to-peak amplitude of one quantization level, making the amplitude ratio 2n/1. The formula is then:
This relationship is the origin of statements like "16-bit audio has a dynamic range of 96 dB". Each extra quantization bit increases the dynamic range by roughly 6 dB. | Signal-to-noise ratio | Wikipedia | 471 | 41706 | https://en.wikipedia.org/wiki/Signal-to-noise%20ratio | Technology | Basics_4 | null |
Assuming a full-scale sine wave signal (that is, the quantizer is designed such that it has the same minimum and maximum values as the input signal), the quantization noise approximates a sawtooth wave with peak-to-peak amplitude of one quantization level and uniform distribution. In this case, the SNR is approximately
Floating point
Floating-point numbers provide a way to trade off signal-to-noise ratio for an increase in dynamic range. For n-bit floating-point numbers, with n-m bits in the mantissa and m bits in the exponent:
The dynamic range is much larger than fixed-point but at a cost of a worse signal-to-noise ratio. This makes floating-point preferable in situations where the dynamic range is large or unpredictable. Fixed-point's simpler implementations can be used with no signal quality disadvantage in systems where dynamic range is less than 6.02m. The very large dynamic range of floating-point can be a disadvantage, since it requires more forethought in designing algorithms.
Optical signals
Optical signals have a carrier frequency (about and more) that is much higher than the modulation frequency. This way the noise covers a bandwidth that is much wider than the signal itself. The resulting signal influence relies mainly on the filtering of the noise. To describe the signal quality without taking the receiver into account, the optical SNR (OSNR) is used. The OSNR is the ratio between the signal power and the noise power in a given bandwidth. Most commonly a reference bandwidth of 0.1 nm is used. This bandwidth is independent of the modulation format, the frequency and the receiver. For instance an OSNR of 20 dB/0.1 nm could be given, even the signal of 40 GBit DPSK would not fit in this bandwidth. OSNR is measured with an optical spectrum analyzer.
Types and abbreviations
Signal to noise ratio may be abbreviated as SNR and less commonly as S/N. PSNR stands for peak signal-to-noise ratio. GSNR stands for geometric signal-to-noise ratio. SINR is the signal-to-interference-plus-noise ratio. | Signal-to-noise ratio | Wikipedia | 447 | 41706 | https://en.wikipedia.org/wiki/Signal-to-noise%20ratio | Technology | Basics_4 | null |
Other uses
While SNR is commonly quoted for electrical signals, it can be applied to any form of signal, for example isotope levels in an ice core, biochemical signaling between cells, or financial trading signals. The term is sometimes used metaphorically to refer to the ratio of useful information to false or irrelevant data in a conversation or exchange. For example, in online discussion forums and other online communities, off-topic posts and spam are regarded as that interferes with the of appropriate discussion.
SNR can also be applied in marketing and how business professionals manage information overload. Managing a healthy signal to noise ratio can help business executives improve their KPIs (Key Performance Indicators).
Similar concepts
The signal-to-noise ratio is similar to Cohen's d given by the difference of estimated means divided by the standard deviation of the data and is related to the test statistic in the t-test. | Signal-to-noise ratio | Wikipedia | 182 | 41706 | https://en.wikipedia.org/wiki/Signal-to-noise%20ratio | Technology | Basics_4 | null |
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect to time, and the oscillations at different points throughout the wave are in phase. The locations at which the absolute value of the amplitude is minimum are called nodes, and the locations where the absolute value of the amplitude is maximum are called antinodes.
Standing waves were first described scientifically by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container. Franz Melde coined the term "standing wave" (German: stehende Welle or Stehwelle) around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings.
This phenomenon can occur because the medium is moving in the direction opposite to the movement of the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. The most common cause of standing waves is the phenomenon of resonance, in which standing waves occur inside a resonator due to interference between waves reflected back and forth at the resonator's resonant frequency.
For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy.
Moving medium
As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges. Such waves are often exploited by glider pilots.
Standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom. A requirement for this in river currents is a flowing water with shallow depth in which the inertia of the water overcomes its gravity due to the supercritical flow speed (Froude number: 1.7 – 4.5, surpassing 4.5 results in direct standing wave) and is therefore neither significantly slowed down by the obstacle nor pushed to the side. Many standing river waves are popular river surfing breaks.
Opposing waves | Standing wave | Wikipedia | 428 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
As an example of the second type, a standing wave in a transmission line is a wave in which the distribution of current, voltage, or field strength is formed by the superposition of two waves of the same frequency propagating in opposite directions. The effect is a series of nodes (zero displacement) and anti-nodes (maximum displacement) at fixed points along the transmission line. Such a standing wave may be formed when a wave is transmitted into one end of a transmission line and is reflected from the other end by an impedance mismatch, i.e., discontinuity, such as an open circuit or a short. The failure of the line to transfer power at the standing wave frequency will usually result in attenuation distortion.
In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved. The result is a partial standing wave, which is a superposition of a standing wave and a traveling wave. The degree to which the wave resembles either a pure standing wave or a pure traveling wave is measured by the standing wave ratio (SWR).
Another example is standing waves in the open ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, and are the source of microbaroms and microseisms.
Mathematical description
This section considers representative one- and two-dimensional cases of standing waves. First, an example of an infinite length string shows how identical waves traveling in opposite directions interfere to produce standing waves. Next, two finite length string examples with different boundary conditions demonstrate how the boundary conditions restrict the frequencies that can form standing waves. Next, the example of sound waves in a pipe demonstrates how the same principles can be applied to longitudinal waves with analogous boundary conditions.
Standing waves can also occur in two- or three-dimensional resonators. With standing waves on two-dimensional membranes such as drumheads, illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase. These nodal line patterns are called Chladni figures. In three-dimensional resonators, such as musical instrument sound boxes and microwave cavity resonators, there are nodal surfaces. This section includes a two-dimensional standing wave example with a rectangular boundary to illustrate how to extend the concept to higher dimensions.
Standing wave on an infinite length string | Standing wave | Wikipedia | 504 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
To begin, consider a string of infinite length along the x-axis that is free to be stretched transversely in the y direction.
For a harmonic wave traveling to the right along the string, the string's displacement in the y direction as a function of position x and time t is
The displacement in the y-direction for an identical harmonic wave traveling to the left is
where
ymax is the amplitude of the displacement of the string for each wave,
ω is the angular frequency or equivalently 2π times the frequency f,
λ is the wavelength of the wave.
For identical right- and left-traveling waves on the same string, the total displacement of the string is the sum of yR and yL,
Using the trigonometric sum-to-product identity ,
Equation () does not describe a traveling wave. At any position x, y(x,t) simply oscillates in time with an amplitude that varies in the x-direction as . The animation at the beginning of this article depicts what is happening. As the left-traveling blue wave and right-traveling green wave interfere, they form the standing red wave that does not travel and instead oscillates in place.
Because the string is of infinite length, it has no boundary condition for its displacement at any point along the x-axis. As a result, a standing wave can form at any frequency.
At locations on the x-axis that are even multiples of a quarter wavelength,
the amplitude is always zero. These locations are called nodes. At locations on the x-axis that are odd multiples of a quarter wavelength
the amplitude is maximal, with a value of twice the amplitude of the right- and left-traveling waves that interfere to produce this standing wave pattern. These locations are called anti-nodes. The distance between two consecutive nodes or anti-nodes is half the wavelength, λ/2.
Standing wave on a string with two fixed ends | Standing wave | Wikipedia | 392 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
Next, consider a string with fixed ends at and . The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. Suppose that at the fixed end a sinusoidal force is applied that drives the string up and down in the y-direction with a small amplitude at some frequency f. In this situation, the driving force produces a right-traveling wave. That wave reflects off the right fixed end and travels back to the left, reflects again off the left fixed end and travels back to the right, and so on. Eventually, a steady state is reached where the string has identical right- and left-traveling waves as in the infinite-length case and the power dissipated by damping in the string equals the power supplied by the driving force so the waves have constant amplitude.
Equation () still describes the standing wave pattern that can form on this string, but now Equation () is subject to boundary conditions where at and because the string is fixed at and because we assume the driving force at the fixed end has small amplitude. Checking the values of y at the two ends,
This boundary condition is in the form of the Sturm–Liouville formulation. The latter boundary condition is satisfied when . L is given, so the boundary condition restricts the wavelength of the standing waves to
Waves can only form standing waves on this string if they have a wavelength that satisfies this relationship with L. If waves travel with speed v along the string, then equivalently the frequency of the standing waves is restricted to
The standing wave with oscillates at the fundamental frequency and has a wavelength that is twice the length of the string. Higher integer values of n correspond to modes of oscillation called harmonics or overtones. Any standing wave on the string will have n + 1 nodes including the fixed ends and n anti-nodes.
To compare this example's nodes to the description of nodes for standing waves in the infinite length string, Equation () can be rewritten as
In this variation of the expression for the wavelength, n must be even. Cross multiplying we see that because L is a node, it is an even multiple of a quarter wavelength,
This example demonstrates a type of resonance and the frequencies that produce standing waves can be referred to as resonant frequencies.
Standing wave on a string with one fixed end | Standing wave | Wikipedia | 480 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
Next, consider the same string of length L, but this time it is only fixed at . At , the string is free to move in the y direction. For example, the string might be tied at to a ring that can slide freely up and down a pole. The string again has small damping and is driven by a small driving force at .
In this case, Equation () still describes the standing wave pattern that can form on the string, and the string has the same boundary condition of at . However, at where the string can move freely there should be an anti-node with maximal amplitude of y. Equivalently, this boundary condition of the "free end" can be stated as at , which is in the form of the Sturm–Liouville formulation. The intuition for this boundary condition at is that the motion of the "free end" will follow that of the point to its left.
Reviewing Equation (), for the largest amplitude of y occurs when , or
This leads to a different set of wavelengths than in the two-fixed-ends example. Here, the wavelength of the standing waves is restricted to
Equivalently, the frequency is restricted to
In this example n only takes odd values. Because L is an anti-node, it is an odd multiple of a quarter wavelength. Thus the fundamental mode in this example only has one quarter of a complete sine cycle–zero at and the first peak at –the first harmonic has three quarters of a complete sine cycle, and so on.
This example also demonstrates a type of resonance and the frequencies that produce standing waves are called resonant frequencies.
Standing wave in a pipe
Consider a standing wave in a pipe of length L. The air inside the pipe serves as the medium for longitudinal sound waves traveling to the right or left through the pipe. While the transverse waves on the string from the previous examples vary in their displacement perpendicular to the direction of wave motion, the waves traveling through the air in the pipe vary in terms of their pressure and longitudinal displacement along the direction of wave motion. The wave propagates by alternately compressing and expanding air in segments of the pipe, which displaces the air slightly from its rest position and transfers energy to neighboring segments through the forces exerted by the alternating high and low air pressures. Equations resembling those for the wave on a string can be written for the change in pressure Δp due to a right- or left-traveling wave in the pipe. | Standing wave | Wikipedia | 498 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
where
pmax is the pressure amplitude or the maximum increase or decrease in air pressure due to each wave,
ω is the angular frequency or equivalently 2π times the frequency f,
λ is the wavelength of the wave.
If identical right- and left-traveling waves travel through the pipe, the resulting superposition is described by the sum
This formula for the pressure is of the same form as Equation (), so a stationary pressure wave forms that is fixed in space and oscillates in time.
If the end of a pipe is closed, the pressure is maximal since the closed end of the pipe exerts a force that restricts the movement of air. This corresponds to a pressure anti-node (which is a node for molecular motions, because the molecules near the closed end cannot move). If the end of the pipe is open, the pressure variations are very small, corresponding to a pressure node (which is an anti-node for molecular motions, because the molecules near the open end can move freely). The exact location of the pressure node at an open end is actually slightly beyond the open end of the pipe, so the effective length of the pipe for the purpose of determining resonant frequencies is slightly longer than its physical length. This difference in length is ignored in this example. In terms of reflections, open ends partially reflect waves back into the pipe, allowing some energy to be released into the outside air. Ideally, closed ends reflect the entire wave back in the other direction.
First consider a pipe that is open at both ends, for example an open organ pipe or a recorder. Given that the pressure must be zero at both open ends, the boundary conditions are analogous to the string with two fixed ends,
which only occurs when the wavelength of standing waves is
or equivalently when the frequency is
where v is the speed of sound.
Next, consider a pipe that is open at (and therefore has a pressure node) and closed at (and therefore has a pressure anti-node). The closed "free end" boundary condition for the pressure at can be stated as , which is in the form of the Sturm–Liouville formulation. The intuition for this boundary condition at is that the pressure of the closed end will follow that of the point to its left. Examples of this setup include a bottle and a clarinet. This pipe has boundary conditions analogous to the string with only one fixed end. Its standing waves have wavelengths restricted to
or equivalently the frequency of standing waves is restricted to | Standing wave | Wikipedia | 505 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
For the case where one end is closed, n only takes odd values just like in the case of the string fixed at only one end.
So far, the wave has been written in terms of its pressure as a function of position x and time. Alternatively, the wave can be written in terms of its longitudinal displacement of air, where air in a segment of the pipe moves back and forth slightly in the x-direction as the pressure varies and waves travel in either or both directions. The change in pressure Δp and longitudinal displacement s are related as
where ρ is the density of the air. In terms of longitudinal displacement, closed ends of pipes correspond to nodes since air movement is restricted and open ends correspond to anti-nodes since the air is free to move. A similar, easier to visualize phenomenon occurs in longitudinal waves propagating along a spring.
We can also consider a pipe that is closed at both ends. In this case, both ends will be pressure anti-nodes or equivalently both ends will be displacement nodes. This example is analogous to the case where both ends are open, except the standing wave pattern has a phase shift along the x-direction to shift the location of the nodes and anti-nodes. For example, the longest wavelength that resonates–the fundamental mode–is again twice the length of the pipe, except that the ends of the pipe have pressure anti-nodes instead of pressure nodes. Between the ends there is one pressure node. In the case of two closed ends, the wavelength is again restricted to
and the frequency is again restricted to
A Rubens tube provides a way to visualize the pressure variations of the standing waves in a tube with two closed ends.
2D standing wave with a rectangular boundary
Next, consider transverse waves that can move along a two dimensional surface within a rectangular boundary of length Lx in the x-direction and length Ly in the y-direction. Examples of this type of wave are water waves in a pool or waves on a rectangular sheet that has been pulled taut. The waves displace the surface in the z-direction, with defined as the height of the surface when it is still.
In two dimensions and Cartesian coordinates, the wave equation is
where
z(x,y,t) is the displacement of the surface,
c is the speed of the wave.
To solve this differential equation, let's first solve for its Fourier transform, with
Taking the Fourier transform of the wave equation, | Standing wave | Wikipedia | 497 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
This is an eigenvalue problem where the frequencies correspond to eigenvalues that then correspond to frequency-specific modes or eigenfunctions. Specifically, this is a form of the Helmholtz equation and it can be solved using separation of variables. Assume
Dividing the Helmholtz equation by Z,
This leads to two coupled ordinary differential equations. The x term equals a constant with respect to x that we can define as
Solving for X(x),
This x-dependence is sinusoidal–recalling Euler's formula–with constants Akx and Bkx determined by the boundary conditions. Likewise, the y term equals a constant with respect to y that we can define as
and the dispersion relation for this wave is therefore
Solving the differential equation for the y term,
Multiplying these functions together and applying the inverse Fourier transform, z(x,y,t) is a superposition of modes where each mode is the product of sinusoidal functions for x, y, and t,
The constants that determine the exact sinusoidal functions depend on the boundary conditions and initial conditions. To see how the boundary conditions apply, consider an example like the sheet that has been pulled taut where z(x,y,t) must be zero all around the rectangular boundary. For the x dependence, z(x,y,t) must vary in a way that it can be zero at both and for all values of y and t. As in the one dimensional example of the string fixed at both ends, the sinusoidal function that satisfies this boundary condition is
with kx restricted to
Likewise, the y dependence of z(x,y,t) must be zero at both and , which is satisfied by
Restricting the wave numbers to these values also restricts the frequencies that resonate to
If the initial conditions for z(x,y,0) and its time derivative ż(x,y,0) are chosen so the t-dependence is a cosine function, then standing waves for this system take the form | Standing wave | Wikipedia | 420 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
So, standing waves inside this fixed rectangular boundary oscillate in time at certain resonant frequencies parameterized by the integers n and m. As they oscillate in time, they do not travel and their spatial variation is sinusoidal in both the x- and y-directions such that they satisfy the boundary conditions. The fundamental mode, and , has a single antinode in the middle of the rectangle. Varying n and m gives complicated but predictable two-dimensional patterns of nodes and antinodes inside the rectangle.
From the dispersion relation, in certain situations different modes–meaning different combinations of n and m–may resonate at the same frequency even though they have different shapes for their x- and y-dependence. For example, if the boundary is square, , the modes and , and , and and all resonate at
Recalling that ω determines the eigenvalue in the Helmholtz equation above, the number of modes corresponding to each frequency relates to the frequency's multiplicity as an eigenvalue.
Standing wave ratio, phase, and energy transfer
If the two oppositely moving traveling waves are not of the same amplitude, they will not cancel completely at the nodes, the points where the waves are 180° out of phase, so the amplitude of the standing wave will not be zero at the nodes, but merely a minimum. Standing wave ratio (SWR) is the ratio of the amplitude at the antinode (maximum) to the amplitude at the node (minimum). A pure standing wave will have an infinite SWR. It will also have a constant phase at any point in space (but it may undergo a 180° inversion every half cycle). A finite, non-zero SWR indicates a wave that is partially stationary and partially travelling. Such waves can be decomposed into a superposition of two waves: a travelling wave component and a stationary wave component. An SWR of one indicates that the wave does not have a stationary component – it is purely a travelling wave, since the ratio of amplitudes is equal to 1. | Standing wave | Wikipedia | 427 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
A pure standing wave does not transfer energy from the source to the destination. However, the wave is still subject to losses in the medium. Such losses will manifest as a finite SWR, indicating a travelling wave component leaving the source to supply the losses. Even though the SWR is now finite, it may still be the case that no energy reaches the destination because the travelling component is purely supplying the losses. However, in a lossless medium, a finite SWR implies a definite transfer of energy to the destination.
Examples
One easy example to understand standing waves is two people shaking either end of a jump rope. If they shake in sync the rope can form a regular pattern of waves oscillating up and down, with stationary points along the rope where the rope is almost still (nodes) and points where the arc of the rope is maximum (antinodes).
Acoustic resonance
Standing waves are also observed in physical media such as strings and columns of air. Any waves traveling along the medium will reflect back when they reach the end. This effect is most noticeable in musical instruments where, at various multiples of a vibrating string or air column's natural frequency, a standing wave is created, allowing harmonics to be identified. Nodes occur at fixed ends and anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are available. At the open end of a pipe the anti-node will not be exactly at the end as it is altered by its contact with the air and so end correction is used to place it exactly. The density of a string will affect the frequency at which harmonics will be produced; the greater the density the lower the frequency needs to be to produce a standing wave of the same harmonic.
Visible light
Standing waves are also observed in optical media such as optical waveguides and optical cavities. Lasers use optical cavities in the form of a pair of facing mirrors, which constitute a Fabry–Pérot interferometer. The gain medium in the cavity (such as a crystal) emits light coherently, exciting standing waves of light in the cavity. The wavelength of light is very short (in the range of nanometers, 10−9 m) so the standing waves are microscopic in size. One use for standing light waves is to measure small distances, using optical flats. | Standing wave | Wikipedia | 477 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
X-rays
Interference between X-ray beams can form an X-ray standing wave (XSW) field. Because of the short wavelength of X-rays (less than 1 nanometer), this phenomenon can be exploited for measuring atomic-scale events at material surfaces. The XSW is generated in the region where an X-ray beam interferes with a diffracted beam from a nearly perfect single crystal surface or a reflection from an X-ray mirror. By tuning the crystal geometry or X-ray wavelength, the XSW can be translated in space, causing a shift in the X-ray fluorescence or photoelectron yield from the atoms near the surface. This shift can be analyzed to pinpoint the location of a particular atomic species relative to the underlying crystal structure or mirror surface. The XSW method has been used to clarify the atomic-scale details of dopants in semiconductors, atomic and molecular adsorption on surfaces, and chemical transformations involved in catalysis.
Mechanical waves
Standing waves can be mechanically induced into a solid medium using resonance. One easy to understand example is two people shaking either end of a jump rope. If they shake in sync, the rope will form a regular pattern with nodes and antinodes and appear to be stationary, hence the name standing wave. Similarly a cantilever beam can have a standing wave imposed on it by applying a base excitation. In this case the free end moves the greatest distance laterally compared to any location along the beam. Such a device can be used as a sensor to track changes in frequency or phase of the resonance of the fiber. One application is as a measurement device for dimensional metrology.
Seismic waves
Standing surface waves on the Earth are observed as free oscillations of the Earth.
Faraday waves
The Faraday wave is a non-linear standing wave at the air-liquid interface induced by hydrodynamic instability. It can be used as a liquid-based template to assemble microscale materials. | Standing wave | Wikipedia | 406 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
Seiches
A seiche is an example of a standing wave in an enclosed body of water. It is characterised by the oscillatory behaviour of the water level at either end of the body and typically has a nodal point near the middle of the body where very little change in water level is observed. It should be distinguished from a simple storm surge where no oscillation is present. In sizeable lakes, the period of such oscillations may be between minutes and hours, for example Lake Geneva's longitudinal period is 73 minutes and its transversal seiche has a period of around 10 minutes, while Lake Huron can be seen to have resonances with periods between 1 and 2 hours. See Lake seiches. | Standing wave | Wikipedia | 150 | 41741 | https://en.wikipedia.org/wiki/Standing%20wave | Physical sciences | Waves | Physics |
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Lord Kelvin in terms of a macroscopic relation between thermodynamic work and heat transfer as defined in thermodynamics, but the kelvin was redefined by international agreement in 2019 in terms of phenomena that are now understood as manifestations of the kinetic energy of free motion of microscopic particles such as atoms, molecules, and electrons. From the thermodynamic viewpoint, for historical reasons, because of how it is defined and measured, this microscopic kinetic definition is regarded as an "empirical" temperature. It was adopted because in practice it can generally be measured more precisely than can Kelvin's thermodynamic temperature.
A thermodynamic temperature of zero is of particular importance for the third law of thermodynamics. By convention, it is reported on the Kelvin scale of temperature in which the unit of measurement is the kelvin (unit symbol: K). For comparison, a temperature of 295 K corresponds to 21.85 °C and 71.33 °F.
Overview
Thermodynamic temperature, as distinct from SI temperature, is defined in terms of a macroscopic Carnot cycle. Thermodynamic temperature is of importance in thermodynamics because it is defined in purely thermodynamic terms. SI temperature is conceptually far different from thermodynamic temperature. Thermodynamic temperature was rigorously defined historically long before there was a fair knowledge of microscopic particles such as atoms, molecules, and electrons.
The International System of Units (SI) specifies the international absolute scale for measuring temperature, and the unit of measure kelvin (unit symbol: K) for specific values along the scale. The kelvin is also used for denoting temperature intervals (a span or difference between two temperatures) as per the following example usage: "A 60/40 tin/lead solder is non-eutectic and is plastic through a range of 5 kelvins as it solidifies." A temperature interval of one degree Celsius is the same magnitude as one kelvin.
The magnitude of the kelvin was redefined in 2019 in relation to the physical property underlying thermodynamic temperature: the kinetic energy of atomic free particle motion. The revision fixed the Boltzmann constant at exactly (J/K). | Thermodynamic temperature | Wikipedia | 496 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The microscopic property that imbues material substances with a temperature can be readily understood by examining the ideal gas law, which relates, per the Boltzmann constant, how heat energy causes precisely defined changes in the pressure and temperature of certain gases. This is because monatomic gases like helium and argon behave kinetically like freely moving perfectly elastic and spherical billiard balls that move only in a specific subset of the possible motions that can occur in matter: that comprising the three translational degrees of freedom. The translational degrees of freedom are the familiar billiard ball-like movements along the X, Y, and Z axes of 3D space (see Fig. 1, below). This is why the noble gases all have the same specific heat capacity per atom and why that value is lowest of all the gases.
Molecules (two or more chemically bound atoms), however, have internal structure and therefore have additional internal degrees of freedom (see Fig. 3, below), which makes molecules absorb more heat energy for any given amount of temperature rise than do the monatomic gases. Heat energy is born in all available degrees of freedom; this is in accordance with the equipartition theorem, so all available internal degrees of freedom have the same temperature as their three external degrees of freedom. However, the property that gives all gases their pressure, which is the net force per unit area on a container arising from gas particles recoiling off it, is a function of the kinetic energy borne in the freely moving atoms' and molecules' three translational degrees of freedom.
Fixing the Boltzmann constant at a specific value, along with other rule making, had the effect of precisely establishing the magnitude of the unit interval of SI temperature, the kelvin, in terms of the average kinetic behavior of the noble gases. Moreover, the starting point of the thermodynamic temperature scale, absolute zero, was reaffirmed as the point at which zero average kinetic energy remains in a sample; the only remaining particle motion being that comprising random vibrations due to zero-point energy.
Absolute zero of temperature | Thermodynamic temperature | Wikipedia | 421 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Temperature scales are numerical. The numerical zero of a temperature scale is not bound to the absolute zero of temperature. Nevertheless, some temperature scales have their numerical zero coincident with the absolute zero of temperature. Examples are the International SI temperature scale, the Rankine temperature scale, and the thermodynamic temperature scale. Other temperature scales have their numerical zero far from the absolute zero of temperature. Examples are the Fahrenheit scale and the Celsius scale.
At the zero point of thermodynamic temperature, absolute zero, the particle constituents of matter have minimal motion and can become no colder. Absolute zero, which is a temperature of zero kelvins (0 K), precisely corresponds to −273.15 °C and −459.67 °F. Matter at absolute zero has no remaining transferable average kinetic energy and the only remaining particle motion is due to an ever-pervasive quantum mechanical phenomenon called ZPE (zero-point energy). Though the atoms in, for instance, a container of liquid helium that was precisely at absolute zero would still jostle slightly due to zero-point energy, a theoretically perfect heat engine with such helium as one of its working fluids could never transfer any net kinetic energy (heat energy) to the other working fluid and no thermodynamic work could occur.
Temperature is generally expressed in absolute terms when scientifically examining temperature's interrelationships with certain other physical properties of matter such as its volume or pressure (see Gay-Lussac's law), or the wavelength of its emitted black-body radiation. Absolute temperature is also useful when calculating chemical reaction rates (see Arrhenius equation). Furthermore, absolute temperature is typically used in cryogenics and related phenomena like superconductivity, as per the following example usage:
"Conveniently, tantalum's transition temperature (T) of 4.4924 kelvin is slightly above the 4.2221 K boiling point of helium." | Thermodynamic temperature | Wikipedia | 404 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Boltzmann constant
The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE (zero-point energy) is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of chaos, i.e., unpredictability, to rebound kinetics; it is as likely that there will be less ZPE-induced particle motion after a given collision as more. This random nature of ZPE is why it has no net effect upon either the pressure or volume of any bulk quantity (a statistically significant quantity of particles) of gases. However, in temperature condensed matter; e.g., solids and liquids, ZPE causes inter-atomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the real-world effects that ZPE has on substances can vary as one alters a thermodynamic system (for example, due to ZPE, helium won't freeze unless under a pressure of at least 2.5 MPa (25 bar)), ZPE is very much a form of thermal energy and may properly be included when tallying a substance's internal energy.
Rankine scale
Though there have been many other temperature scales throughout history, there have been only two scales for measuring thermodynamic temperature which have absolute zero as their null point (0): The Kelvin scale and the Rankine scale.
Throughout the scientific world where modern measurements are nearly always made using the International System of Units, thermodynamic temperature is measured using the Kelvin scale. The Rankine scale is part of English engineering units and finds use in certain engineering fields, particularly in legacy reference works. The Rankine scale uses the degree Rankine (symbol: °R) as its unit, which is the same magnitude as the degree Fahrenheit (symbol: °F).
A unit increment of one kelvin is exactly 1.8 times one degree Rankine; thus, to convert a specific temperature on the Kelvin scale to the Rankine scale, , and to convert from a temperature on the Rankine scale to the Kelvin scale, . Consequently, absolute zero is "0" for both scales, but the melting point of water ice (0 °C and 273.15 K) is 491.67 °R. | Thermodynamic temperature | Wikipedia | 488 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
To convert temperature intervals (a span or difference between two temperatures), the formulas from the preceding paragraph are applicable; for instance, an interval of 5 kelvin is precisely equal to an interval of 9 degrees Rankine.
Modern redefinition of the kelvin
For 65 years, between 1954 and the 2019 revision of the SI, a temperature interval of one kelvin was defined as the difference between the triple point of water and absolute zero. The 1954 resolution by the International Bureau of Weights and Measures (known by the French-language acronym BIPM), plus later resolutions and publications, defined the triple point of water as precisely 273.16 K and acknowledged that it was "common practice" to accept that due to previous conventions (namely, that 0 °C had long been defined as the melting point of water and that the triple point of water had long been experimentally determined to be indistinguishably close to 0.01 °C), the difference between the Celsius scale and Kelvin scale is accepted as 273.15 kelvins; which is to say, 0 °C corresponds to 273.15 kelvins. The net effect of this as well as later resolutions was twofold: 1) they defined absolute zero as precisely 0 K, and 2) they defined that the triple point of special isotopically controlled water called Vienna Standard Mean Ocean Water occurred at precisely 273.16 K and 0.01 °C. One effect of the aforementioned resolutions was that the melting point of water, while very close to 273.15 K and 0 °C, was not a defining value and was subject to refinement with more precise measurements.
The 1954 BIPM standard did a good job of establishing—within the uncertainties due to isotopic variations between water samples—temperatures around the freezing and triple points of water, but required that intermediate values between the triple point and absolute zero, as well as extrapolated values from room temperature and beyond, to be experimentally determined via apparatus and procedures in individual labs. This shortcoming was addressed by the International Temperature Scale of 1990, or ITS90, which defined 13 additional points, from 13.8033 K, to 1,357.77 K. While definitional, ITS90 had—and still has—some challenges, partly because eight of its extrapolated values depend upon the melting or freezing points of metal samples, which must remain exceedingly pure lest their melting or freezing points be affected—usually depressed. | Thermodynamic temperature | Wikipedia | 492 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The 2019 revision of the SI was primarily for the purpose of decoupling much of the SI system's definitional underpinnings from the kilogram, which was the last physical artifact defining an SI base unit (a platinum/iridium cylinder stored under three nested bell jars in a safe located in France) and which had highly questionable stability. The solution required that four physical constants, including the Boltzmann constant, be definitionally fixed.
Assigning the Boltzmann constant a precisely defined value had no practical effect on modern thermometry except for the most exquisitely precise measurements. Before the revision, the triple point of water was exactly 273.16 K and 0.01 °C and the Boltzmann constant was experimentally determined to be , where the "(51)" denotes the uncertainty in the two least significant digits (the 03) and equals a relative standard uncertainty of 0.37 ppm. Afterwards, by defining the Boltzmann constant as exactly , the 0.37 ppm uncertainty was transferred to the triple point of water, which became an experimentally determined value of (). That the triple point of water ended up being exceedingly close to 273.16 K after the SI revision was no accident; the final value of the Boltzmann constant was determined, in part, through clever experiments with argon and helium that used the triple point of water for their key reference temperature.
Notwithstanding the 2019 revision, water triple-point cells continue to serve in modern thermometry as exceedingly precise calibration references at 273.16 K and 0.01 °C. Moreover, the triple point of water remains one of the 14 calibration points comprising ITS90, which spans from the triple point of hydrogen (13.8033 K) to the freezing point of copper (1,357.77 K), which is a nearly hundredfold range of thermodynamic temperature.
Relationship of temperature, motions, conduction, and thermal energy
Nature of kinetic energy, translational motion, and temperature | Thermodynamic temperature | Wikipedia | 415 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The thermodynamic temperature of any bulk quantity of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as translational motion. These simple movements in the three X, Y, and Z–axis dimensions of space means the particles move in the three spatial degrees of freedom. This particular form of kinetic energy is sometimes referred to as kinetic temperature. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the ideal gas law's formula and is embodied in the gas laws.
Though the kinetic energy borne exclusively in the three translational degrees of freedom comprise the thermodynamic temperature of a substance, molecules, as can be seen in Fig. 3, can have other degrees of freedom, all of which fall under three categories: bond length, bond angle, and rotational. All three additional categories are not necessarily available to all molecules, and even for molecules that can experience all three, some can be "frozen out" below a certain temperature. Nonetheless, all those degrees of freedom that are available to the molecules under a particular set of conditions contribute to the specific heat capacity of a substance; which is to say, they increase the amount of heat (kinetic energy) required to raise a given amount of the substance by one kelvin or one degree Celsius.
The relationship of kinetic energy, mass, and velocity is given by the formula . Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy, and precisely the same temperature, as those with four times the mass but half the velocity.
The extent to which the kinetic energy of translational motion in a statistically significant collection of atoms or molecules in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the Boltzmann constant (symbol: ). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an individual particles' translational motion as follows:
where:
is the mean kinetic energy for an individual particle
is the thermodynamic temperature of the bulk quantity of the substance | Thermodynamic temperature | Wikipedia | 479 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
While the Boltzmann constant is useful for finding the mean kinetic energy in a sample of particles, it is important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in Fig. 1 above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in Fig. 2 shows the speed distribution of 5500 K helium atoms. They have a most probable speed of 4.780 km/s (0.2092 s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the x–axis to the right). This graph uses inverse speed for its x-axis so the shape of the curve can easily be compared to the curves in Fig. 5 below. In both graphs, zero on the x-axis represents infinite temperature. Additionally, the x- and y-axes on both graphs are scaled proportionally.
High speeds of translational motion
Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are very fast and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool cesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature. Formulas for calculating the velocity and speed of translational motion are given in the following footnote. | Thermodynamic temperature | Wikipedia | 424 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
It is neither difficult to imagine atomic motions due to kinetic temperature, nor distinguish between such motions and those due to zero-point energy. Consider the following hypothetical thought experiment, as illustrated in Fig. 2.5 at left, with an atom that is exceedingly close to absolute zero. Imagine peering through a common optical microscope set to 400 power, which is about the maximum practical magnification for optical microscopes. Such microscopes generally provide fields of view a bit over 0.4 mm in diameter. At the center of the field of view is a single levitated argon atom (argon comprises about 0.93% of air) that is illuminated and glowing against a dark backdrop. If this argon atom was at a beyond-record-setting one-trillionth of a kelvin above absolute zero, and was moving perpendicular to the field of view towards the right, it would require 13.9 seconds to move from the center of the image to the 200-micron tick mark; this travel distance is about the same as the width of the period at the end of this sentence on modern computer monitors. As the argon atom slowly moved, the positional jitter due to zero-point energy would be much less than the 200-nanometer (0.0002 mm) resolution of an optical microscope. Importantly, the atom's translational velocity of 14.43 microns per second constitutes all its retained kinetic energy due to not being precisely at absolute zero. Were the atom precisely at absolute zero, imperceptible jostling due to zero-point energy would cause it to very slightly wander, but the atom would perpetually be located, on average, at the same spot within the field of view. This is analogous to a boat that has had its motor turned off and is now bobbing slightly in relatively calm and windless ocean waters; even though the boat randomly drifts to and fro, it stays in the same spot in the long term and makes no headway through the water. Accordingly, an atom that was precisely at absolute zero would not be "motionless", and yet, a statistically significant collection of such atoms would have zero net kinetic energy available to transfer to any other collection of atoms. This is because regardless of the kinetic temperature of the second collection of atoms, they too experience the effects of zero-point energy. Such are the consequences of statistical mechanics and the nature of thermodynamics.
Internal motions of molecules and internal energy | Thermodynamic temperature | Wikipedia | 503 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
As mentioned above, there are other ways molecules can jiggle besides the three translational degrees of freedom that imbue substances with their kinetic temperature. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain their internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements; these are all types of internal degrees of freedom. This makes molecules distinct from monatomic substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom (the X, Y, and Z axis). Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an internal temperature. Even though these motions are called "internal", the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as internal energy is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-local thermodynamic equilibrium (LTE) phenomena such as combustion, the sublimation of solids, and the diffusion of hot gases in a partial vacuum.
The kinetic energy stored internally in molecules causes substances to contain more heat energy at any given temperature and to absorb additional internal energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions, is not contributing to the molecules' translational motions at that same instant. This extra kinetic energy simply increases the amount of internal energy that substance absorbs for a given temperature rise. This property is known as a substance's specific heat capacity. | Thermodynamic temperature | Wikipedia | 444 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Different molecules absorb different amounts of internal energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances' molecules possess more internal degrees of freedom than others do. For instance, room-temperature nitrogen, which is a diatomic molecule, has five active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has five-thirds the specific heat capacity per mole (a specific number of molecules) as do the monatomic gases. Another example is gasoline (see table showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.
Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons
Heat conduction is the diffusion of thermal energy from hot parts of a system to cold parts. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term bulk in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever thermal energy diffuses within an isolated system, temperature differences within the system decrease (and entropy increases).
One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in Fig. 1. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can move forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; convection speeds this process even more. | Thermodynamic temperature | Wikipedia | 404 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Translational motion in solids, however, takes the form of phonons (see Fig. 4 at right). Phonons are constrained, quantized wave packets that travel at the speed of sound of a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is usually inefficient and such solids are considered thermal insulators (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.
Metals however, are not restricted to only phonon-based heat conduction. Thermal energy conducts through metals extraordinarily quickly because instead of direct molecule-to-molecule collisions, the vast majority of thermal energy is mediated via very light, mobile conduction electrons. This is why there is a near-perfect correlation between metals' thermal conductivity and their electrical conductivity. Conduction electrons imbue metals with their extraordinary conductivity because they are delocalized (i.e., not tied to a specific atom) and behave rather like a sort of quantum gas due to the effects of zero-point energy (for more on ZPE, see Note 1 below). Furthermore, electrons are relatively light with a rest mass only that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Isaac Newton wrote with his third law of motion,
However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, thermal energy is readily borne by mobile conduction electrons. Additionally, because they are delocalized and very fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.
Diffusion of thermal energy: black-body radiation | Thermodynamic temperature | Wikipedia | 445 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Thermal radiation is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal photons (known as black-body radiation). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even individual molecules with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. In any bulk quantity of a substance at equilibrium, black-body photons are emitted across a range of wavelengths in a spectrum that has a bell curve-like shape called a Planck curve (see graph in Fig. 5 at right). The top of a Planck curve (the peak emittance wavelength) is located in a particular part of the electromagnetic spectrum depending on the temperature of the black-body. Substances at extreme cryogenic temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short gamma rays (see ).
Black-body radiation diffuses thermal energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.
As established by the Stefan–Boltzmann law, the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black-body at 824 K (just short of glowing dull red) emits 60 times the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, black-body radiation can be the principal mechanism by which thermal energy escapes a system.
Table of thermodynamic temperatures
The table below shows various points on the thermodynamic scale, in order of increasing temperature.
Heat of phase changes | Thermodynamic temperature | Wikipedia | 394 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The kinetic energy of particle motion is just one contributor to the total thermal energy in a substance; another is phase transitions, which are the potential energy of molecular bonds that can form in a substance as it cools (such as during condensing and freezing). The thermal energy required for a phase transition is called latent heat. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to break chemical bonds (such as during evaporation and melting). Almost everyone is familiar with the effects of phase transitions; for instance, steam at 100 °C can cause severe burns much faster than the 100 °C air from a hair dryer. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.
Even though thermal energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic alloys exhibit no temperature change whatsoever while they undergo them (see Fig. 7, below right). Consider one particular type of phase transition: melting. When a solid is melting, crystal lattice chemical bonds are being broken apart; the substance is transitioning from what is known as a more ordered state to a less ordered state. In Fig. 7, the melting of ice is shown within the lower left box heading from blue to green.
At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are, on average, at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are all-or-nothing forces: they either hold fast, or break; there is no in-between state. Consequently, when a substance is at its melting point, every joule of added thermal energy only breaks the bonds of a specific quantity of its atoms or molecules, converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional thermal energy cannot make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), thermal energy must be removed from a substance. | Thermodynamic temperature | Wikipedia | 472 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
As stated above, the thermal energy required for a phase transition is called latent heat. In the specific cases of melting and freezing, it is called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements. If the substance is one of the monatomic gases (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole. Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the thermal energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals' ratios are even greater, typically in the range of 400 to 1200 times. The phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a one-degree increase.
Water's sizable enthalpy of vaporization is why one's skin can be burned so quickly as steam condenses on it (heading from red to green in Fig. 7 above); water vapors (gas phase) are liquefied on the skin with releasing a large amount of energy (enthalpy) to the environment including the skin, resulting in skin damage. In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient wet-bulb temperature that is dependent on relative humidity); the water evaporation on the skin takes a large amount of energy from the environment including the skin, reducing the skin temperature. Water's highly energetic enthalpy of vaporization is also an important factor underlying why solar pool covers (floating, insulated blankets that cover swimming pools when the pools are not in use) are so effective at reducing heating costs: they prevent evaporation. (In other words, taking energy from water when it is evaporated is limited.) For instance, the evaporation of just 20 mm of water from a 1.29-meter-deep pool chills its water .
Internal energy | Thermodynamic temperature | Wikipedia | 498 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The total energy of all translational and internal particle motions, including that of conduction electrons, plus the potential energy of phase changes, plus zero-point energy of a substance comprise the internal energy of it.
Internal energy at absolute zero
As a substance cools, different forms of internal energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions is liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic energy or temperature decreases); the internal motions of molecules diminish (their internal energy or temperature decreases); conduction electrons (if the substance is an electrical conductor) travel somewhat slower; and black-body radiation's peak emittance wavelength increases (the photons' energy decreases). When particles of a substance are as close as possible to complete rest and retain only ZPE (zero-point energy)-induced quantum mechanical motion, the substance is at the temperature of absolute zero ( = 0).
Whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero internal energy; one must be very precise with what one means by internal energy. Often, all the phase changes that can occur in a substance, will have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, = 0 helium remains liquid at room pressure (Fig. 9 at right) and must be under a pressure of at least to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21 joules per mole) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures.
A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars, or hundreds of gigapascals). These are known as solid–solid phase transitions wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one. | Thermodynamic temperature | Wikipedia | 455 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The above complexities make for rather cumbersome blanket statements regarding the internal energy in = 0 substances. Regardless of pressure though, what can be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a closest-packed arrangement (see Fig. 8, above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy. One can also say that for a given substance at constant pressure, absolute zero is the point of lowest enthalpy (a measure of work potential that takes internal energy, pressure, and volume into consideration). Lastly, all = 0 substances contain zero kinetic thermal energy.
Practical applications for thermodynamic temperature
Thermodynamic temperature is useful not only for scientists, it can also be useful for lay-people in many disciplines involving gases. By expressing variables in absolute terms and applying Gay-Lussac's law of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a cold pressure of 200 kPa, then its absolute pressure is 300 kPa. Room temperature ("cold" in tire terms) is 296 K. If the tire temperature is 20 °C hotter (20 kelvins), the solution is calculated as = 6.8% greater thermodynamic temperature and absolute pressure; that is, an absolute pressure of 320 kPa, which is a of 220 kPa.
Relationship to ideal gas law
The thermodynamic temperature is closely linked to the ideal gas law and its consequences. It can be linked also to the second law of thermodynamics. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratio of two temperatures and is the same in all absolute scales.
Strictly speaking, the temperature of a system is well-defined only if it is at thermal equilibrium. From a microscopic viewpoint, a material is at thermal equilibrium if the quantity of heat between its individual particles cancel out. There are many possible scales of temperature, derived from a variety of observations of physical phenomena. | Thermodynamic temperature | Wikipedia | 470 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Loosely stated, temperature differences dictate the direction of heat between two systems such that their combined energy is maximally distributed among their lowest possible states. We call this distribution "entropy". To better understand the relationship between temperature and entropy, consider the relationship between heat, work and temperature illustrated in the Carnot heat engine. The engine converts heat into work by directing a temperature gradient between a higher temperature heat source, , and a lower temperature heat sink, , through a gas filled piston. The work done per cycle is equal in magnitude to net heat taken up, which is sum of the heat taken up by the engine from the high-temperature source, plus the waste heat given off by the engine, < 0. The efficiency of the engine is the work divided by the heat put into the system or
where is the work done per cycle. Thus the efficiency depends only on .
Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures and must have the same efficiency, that is to say, the efficiency is the function of only temperatures
In addition, a reversible heat engine operating between a pair of thermal reservoirs at temperatures and must have the same efficiency as one consisting of two cycles, one between and another (intermediate) temperature , and the second between and . If this were not the case, then energy (in the form of ) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles as an engine design choice, and any reversible engine between the same reservoir at and must be equally efficient regardless of the engine design.
If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as below.
Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at . We also have used the fact that the heat passes through the intermediate thermal reservoir at without losing its energy. (I.e., is not lost during its passage through the reservoir at .) This fact can be proved by the following.
In order to have the consistency in the last equation, the heat flown from the engine 2 to the intermediate reservoir must be equal to the heat flown out from the reservoir to the engine 3. | Thermodynamic temperature | Wikipedia | 503 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
With this understanding of , and , mathematically,
But since the first function is not a function of , the product of the final two functions must result in the removal of as a variable. The only way is therefore to define the function as follows:
and
so that
I.e. the ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our ; it is a matter of convenience and convention that we choose . Choosing then one fixed reference temperature (i.e. triple point of water), we establish the thermodynamic temperature scale.
Such a definition coincides with that of the ideal gas derivation; also it is this definition of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of and , and hence derive that the (complete) Carnot cycle is isentropic:
Substituting this back into our first formula for efficiency yields a relationship in terms of temperature:
Note that for the efficiency is 100% and that efficiency becomes greater than 100% for , which is unrealistic. Subtracting 1 from the right hand side of the Equation (4) and the middle portion gives and thus
The generalization of this equation is the Clausius theorem, which proposes the existence of a state function (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by
where the subscript rev indicates heat transfer in a reversible process. The function is the entropy of the system, mentioned previously, and the change of around any cycle is zero (as is necessary for any state function). The Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid a logic loop, we should first define entropy through statistical mechanics):
For a constant-volume system (so no mechanical work ) in which the entropy is a function of its internal energy , and the thermodynamic temperature is therefore given by
so that the reciprocal of the thermodynamic temperature is the rate of change of entropy with respect to the internal energy at the constant volume.
History | Thermodynamic temperature | Wikipedia | 451 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Guillaume Amontons (1663–1705) published two papers in 1702 and 1703 that may be used to credit him as being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. He made the discovery while endeavoring to improve upon the air thermometers in use at the time. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation led him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240 °C—only 33.15 degrees short of the true value of −273.15 °C. Amonton's discovery of a one-to-one relationship between absolute temperature and absolute pressure was rediscovered a century later and popularized within the scientific community by Joseph Louis Gay-Lussac. Today, this principle of thermodynamics is commonly known as Gay-Lussac's law but is also known as Amonton's law.
In 1742, Anders Celsius (1701–1744) created a "backwards" version of the modern Celsius temperature scale. In Celsius's original scale, zero represented the boiling point of water and 100 represented the melting point of ice. In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that ice's melting point was effectively unaffected by pressure. He also determined with remarkable precision how water's boiling point varied as a function of atmospheric pressure. He proposed that zero on his temperature scale (water's boiling point) would be calibrated at the mean barometric pressure at mean sea level. | Thermodynamic temperature | Wikipedia | 451 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Coincident with the death of Anders Celsius in 1744, the botanist Carl Linnaeus (1707–1778) effectively reversed Celsius's scale upon receipt of his first thermometer featuring a scale where zero represented the melting point of ice and 100 represented water's boiling point. The custom-made Linnaeus-thermometer, for use in his greenhouses, was made by Daniel Ekström, Sweden's leading maker of scientific instruments at the time. For the next 204 years, the scientific and thermometry communities worldwide referred to this scale as the centigrade scale. Temperatures on the centigrade scale were often reported simply as degrees or, when greater specificity was desired, degrees centigrade. The symbol for temperature values on this scale was °C (in several formats over the years). Because the term centigrade was also the French-language name for a unit of angular measurement (one-hundredth of a right angle) and had a similar connotation in other languages, the term "centesimal degree" was used when very precise, unambiguous language was required by international standards bodies such as the International Bureau of Weights and Measures (BIPM). The 9th CGPM (General Conference on Weights and Measures and the CIPM (International Committee for Weights and Measures formally adopted degree Celsius (symbol: °C) in 1948.
In his book Pyrometrie (1777) completed four months before his death, Johann Heinrich Lambert (1728–1777), sometimes incorrectly referred to as Joseph Lambert, proposed an absolute temperature scale based on the pressure/temperature relationship of a fixed volume of gas. This is distinct from the volume/temperature relationship of gas under constant pressure that Guillaume Amontons discovered 75 years earlier. Lambert stated that absolute zero was the point where a simple straight-line extrapolation reached zero gas pressure and was equal to −270 °C.
Notwithstanding the work of Guillaume Amontons 85 years earlier, Jacques Alexandre César Charles (1746–1823) is often credited with discovering (circa 1787), but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was . | Thermodynamic temperature | Wikipedia | 448 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Joseph Louis Gay-Lussac (1778–1850) published work in 1802 (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called Charles's law and is one of the gas laws. His are the first known formulas to use the number 273 for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C).
William Thomson (1824–1907), also known as Lord Kelvin, wrote in his 1848 paper "On an Absolute Thermometric Scale" of the need for a scale whereby infinite cold (absolute zero) was the scale's zero point, and which used the degree Celsius for its unit increment. Like Gay-Lussac, Thomson calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the kelvin thermodynamic temperature scale. Thomson's value of −273 was derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of −0.00366 expressed to five significant digits is −273.22 °C which is remarkably close to the true value of −273.15 °C.
In the paper he proposed to define temperature using idealized heat engines. In detail, he proposed that, given three heat reservoirs at temperatures , if two reversible heat engines (Carnot engine), one working between and another between , can produce the same amount of mechanical work by letting the same amount of heat pass through, then define .
Note that like Carnot, Kelvin worked under the assumption that heat is conserved ("the conversion of heat (or caloric) into mechanical effect is probably impossible"), and if heat goes into the heat engine, then heat must come out.
Kelvin, realizing after Joule's experiments that heat is not a conserved quantity but is convertible with mechanical work, modified his scale in the 1851 work An Account of Carnot's Theory of the Motive Power of Heat. In this work, he defined as follows: | Thermodynamic temperature | Wikipedia | 464 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
The above definition fixes the ratios between absolute temperatures, but it does not fix a scale for absolute temperature. For the scale, Thomson proposed to use the Celsius degree, that is, the interval between the freezing and the boiling point of water.
In 1859 Macquorn Rankine (1820–1872) proposed a thermodynamic temperature scale similar to William Thomson's but which used the degree Fahrenheit for its unit increment, that is, the interval between the freezing and the boiling point of water. This absolute scale is known today as the Rankine thermodynamic temperature scale.
Ludwig Boltzmann (1844–1906) made major contributions to thermodynamics between 1877 and 1884 through an understanding of the role that particle kinetics and black body radiation played. His name is now attached to several of the formulas used today in thermodynamics.
Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed in the 1930s that absolute zero was equivalent to −273.15 °C.
Resolution 3 of the 9th General Conference on Weights and Measures (CGPM) in 1948 fixed the triple point of water at precisely 0.01 °C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared "will be fixed at a later date". The implication is that if the value of absolute zero measured in the 1930s was truly −273.15 °C, then the triple point of water (0.01 °C) was equivalent to 273.16 K. Additionally, both the International Committee for Weights and Measures (CIPM) and the CGPM formally adopted the name Celsius for the degree Celsius and the Celsius temperature scale.
Resolution 3 of the 10th CGPM in 1954 gave the kelvin scale its modern definition by choosing the triple point of water as its upper defining point (with no change to absolute zero being the null point) and assigning it a temperature of precisely 273.16 kelvins (what was actually written 273.16 degrees Kelvin at the time). This, in combination with Resolution 3 of the 9th CGPM, had the effect of defining absolute zero as being precisely zero kelvins and −273.15 °C. | Thermodynamic temperature | Wikipedia | 472 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
Resolution 3 of the 13th CGPM in 1967/1968 renamed the unit increment of thermodynamic temperature kelvin, symbol K, replacing degree absolute, symbol . Further, feeling it useful to more explicitly define the magnitude of the unit increment, the 13th CGPM also decided in Resolution 4 that "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water".
The CIPM affirmed in 2005 that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin thermodynamic temperature scale would refer to water having an isotopic composition defined as being precisely equal to the nominal specification of Vienna Standard Mean Ocean Water.
In November 2018, the 26th General Conference on Weights and Measures (CGPM) changed the definition of the Kelvin by fixing the Boltzmann constant to when expressed in the unit J/K. This change (and other changes in the definition of SI units) was made effective on the 144th anniversary of the Metre Convention, 20 May 2019. | Thermodynamic temperature | Wikipedia | 233 | 41789 | https://en.wikipedia.org/wiki/Thermodynamic%20temperature | Physical sciences | Thermodynamics | Physics |
A time standard is a specification for measuring time: either the rate at which time passes or points in time or both. In modern times, several time specifications have been officially recognized as standards, where formerly they were matters of custom and practice. An example of a kind of time standard can be a time scale, specifying a method for measuring divisions of time. A standard for civil time can specify both time intervals and time-of-day.
Standardized time measurements are made using a clock to count periods of some period changes, which may be either the changes of a natural phenomenon or of an artificial machine.
Historically, time standards were often based on the Earth's rotational period. From the late 18 century to the 19th century it was assumed that the Earth's daily rotational rate was constant. Astronomical observations of several kinds, including eclipse records, studied in the 19th century, raised suspicions that the rate at which Earth rotates is gradually slowing and also shows small-scale irregularities, and this was confirmed in the early twentieth century. Time standards based on Earth rotation were replaced (or initially supplemented) for astronomical use from 1952 onwards by an ephemeris time standard based on the Earth's orbital period and in practice on the motion of the Moon. The invention in 1955 of the caesium atomic clock has led to the replacement of older and purely astronomical time standards, for most practical purposes, by newer time standards based wholly or partly on atomic time.
Various types of second and day are used as the basic time interval for most time scales. Other intervals of time (minutes, hours, and years) are usually defined in terms of these two. | Time standard | Wikipedia | 331 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
Terminology
The term "time" is generally used for many close but different concepts, including:
instant as an object – one point on the time axis. Being an object, it has no value;
date as a quantity characterising an instant. As a quantity, it has a value which may be expressed in a variety of ways, for example "2014-04-26T09:42:36,75" in ISO standard format, or more colloquially such as "today, 9:42 a.m.";
time interval as an object – part of the time axis limited by two instants. Being an object, it has no value;
duration as a quantity characterizing a time interval. As a quantity, it has a value, such as a number of minutes, or may be described in terms of the quantities (such as times and dates) of its beginning and end.
chronology, an ordered sequence of events in the past. Chronologies can be put into chronological groups (periodization). One of the most important systems of periodization is the geologic time scale, which is a system of periodizing the events that shaped the Earth and its life. Chronology, periodization, and interpretation of the past are together known as the study of history.
Definitions of the second
There have only ever been three definitions of the second: as a fraction of the day, as a fraction of an extrapolated year, and as the microwave frequency of a caesium atomic clock.
In early history, clocks were not accurate enough to track seconds. After the invention of mechanical clocks, the CGS system and MKS system of units both defined the second as of a mean solar day. MKS was adopted internationally during the 1940s.
In the late 1940s, quartz crystal oscillator clocks could measure time more accurately than the rotation of the Earth. Metrologists also knew that Earth's orbit around the Sun (a year) was much more stable than Earth's rotation. This led to the definition of ephemeris time and the tropical year, and the ephemeris second was defined as "the fraction of the tropical year for 1900 January 0 at 12 hours ephemeris time". This definition was adopted as part of the International System of Units in 1960. | Time standard | Wikipedia | 463 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
Most recently, atomic clocks have been developed that offer improved accuracy. Since 1967, the SI base unit for time is the SI second, defined as exactly "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom" (at a temperature of 0 K and at mean sea level). The SI second is the basis of all atomic timescales, e.g. coordinated universal time, GPS time, International Atomic Time, etc.
Current time standards
Geocentric Coordinate Time (TCG) is a coordinate time having its spatial origin at the center of Earth's mass. TCG is a theoretical ideal, and any particular realization will have measurement error.
International Atomic Time (TAI) is the primary physically realized time standard. TAI is produced by the International Bureau of Weights and Measures (BIPM), and is based on the combined input of many atomic clocks around the world, each corrected for environmental and relativistic effects (both gravitational and because of speed, like in GNSS). TAI is not related to TCG directly but rather is a realization of Terrestrial Time (TT), a theoretical timescale that is a rescaling of TCG such that the time rate approximately matches proper time at mean sea level.
Universal Time (UT1) is the Earth Rotation Angle (ERA) linearly scaled to match historical definitions of mean solar time at 0° longitude. At high precision, Earth's rotation is irregular and is determined from the positions of distant quasars using long baseline interferometry, laser ranging of the Moon and artificial satellites, as well as GPS satellite orbits.
Coordinated Universal Time (UTC) is an atomic time scale designed to approximate UT1. UTC differs from TAI by an integral number of seconds. UTC is kept within 0.9 second of UT1 by the introduction of one-second steps to UTC, the "leap second". To date these steps (and difference "TAI-UTC") have always been positive.
The Global Positioning System broadcasts a very precise time signal worldwide, along with instructions for converting GPS time (GPST) to UTC. It was defined with a constant offset from TAI: GPST = TAI - 19 s. The GPS time standard is maintained independently but regularly synchronized with or from, UTC time. | Time standard | Wikipedia | 485 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
Standard time or civil time in a time zone deviates a fixed, round amount, usually a whole number of hours, from some form of Universal Time, usually UTC. The offset is chosen such that a new day starts approximately while the Sun is crossing the nadir meridian. Alternatively the difference is not really fixed, but it changes twice a year by a round amount, usually one hour, see Daylight saving time.
Julian day number is a count of days elapsed since Greenwich mean noon on 1 January 4713 B.C., Julian proleptic calendar. The Julian Date is the Julian day number followed by the fraction of the day elapsed since the preceding noon. Conveniently for astronomers, this avoids the date skip during an observation night. Modified Julian day (MJD) is defined as MJD = JD - 2400000.5. An MJD day thus begins at midnight, civil date. Julian dates can be expressed in UT1, TAI, TT, etc. and so for precise applications the timescale should be specified, e.g. MJD 49135.3824 TAI.
Barycentric Coordinate Time (TCB) is a coordinate time having its spatial origin at the center of mass of the Solar System, which is called the barycenter.
Conversions
Conversions between atomic time systems (TAI, GPST, and UTC) are for the most part exact. However, GPS time is a measured value as opposed to a computed "paper" scale. As such it may differ from UTC(USNO) by a few hundred nanoseconds, which in turn may differ from official UTC by as much as 26 nanoseconds. Conversions for UT1 and TT rely on published difference tables which are specified to 10 microseconds and 0.1 nanoseconds respectively.
Definitions:
LS = TAI − UTC = leap seconds from USNO Table of Leap Seconds
DUT1 = UT1 − UTC published in IERS Bulletins or U.S. Naval Observatory EO
DTT = TT − TAI − 32.184 s published in BIPM's TT(BIPM) tables.
TCG is linearly related to TT as: TCG − TT = LG × (JD − 2443144.5) × 86400 seconds, with the scale difference LG defined as 6.969290134 exactly. | Time standard | Wikipedia | 490 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
TCB is a linear transformation of TDB and TDB differs from TT in small, mostly periodic terms. Neglecting these terms (on the order of 2 milliseconds for several millennia around the present epoch), TCB is related to TT by: TCB − TT = LB × (JD − 2443144.5) × 86400 seconds. The scale difference LB has been defined by the IAU to be 1.550519768e-08 exactly.
Time standards based on Earth rotation
Apparent solar time or true solar time is based on the solar day, which is the period between one solar noon (passage of the real Sun across the meridian) and the next. A solar day is approximately 24 hours of mean time. Because the Earth's orbit around the Sun is elliptical, and because of the obliquity of the Earth's axis relative to the plane of the orbit (the ecliptic), the apparent solar day varies a few dozen seconds above or below the mean value of 24 hours. As the variation accumulates over a few weeks, there are differences as large as 16 minutes between apparent solar time and mean solar time (see Equation of time). However, these variations cancel out over a year. There are also other perturbations such as Earth's wobble, but these are less than a second per year. | Time standard | Wikipedia | 281 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
Sidereal time is time by the stars. A sidereal rotation is the time it takes the Earth to make one revolution with rotation to the stars, approximately 23 hours 56 minutes 4 seconds. A mean solar day is about 3 minutes 56 seconds longer than a mean sidereal day, or more than a mean sidereal day. In astronomy, sidereal time is used to predict when a star will reach its highest point in the sky. For accurate astronomical work on land, it was usual to observe sidereal time rather than solar time to measure mean solar time, because the observations of 'fixed' stars could be measured and reduced more accurately than observations of the Sun (in spite of the need to make various small compensations, for refraction, aberration, precession, nutation and proper motion). It is well known that observations of the Sun pose substantial obstacles to the achievement of accuracy in measurement. In former times, before the distribution of accurate time signals, it was part of the routine work at any observatory to observe the sidereal times of meridian transit of selected 'clock stars' (of well-known position and movement), and to use these to correct observatory clocks running local mean sidereal time; but nowadays local sidereal time is usually generated by computer, based on time signals.
Mean solar time was a time standard used especially at sea for navigational purposes, calculated by observing apparent solar time and then adding to it a correction, the equation of time, which compensated for two known irregularities in the length of the day, caused by the ellipticity of the Earth's orbit and the obliquity of the Earth's equator and polar axis to the ecliptic (which is the plane of the Earth's orbit around the sun). It has been superseded by Universal Time. | Time standard | Wikipedia | 366 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
Greenwich Mean Time was originally mean time deduced from meridian observations made at the Royal Greenwich Observatory (RGO). The principal meridian of that observatory was chosen in 1884 by the International Meridian Conference to be the Prime Meridian. GMT either by that name or as 'mean time at Greenwich' used to be an international time standard, but is no longer so; it was initially renamed in 1928 as Universal Time (UT) (partly as a result of ambiguities arising from the changed practice of starting the astronomical day at midnight instead of at noon, adopted as from 1 January 1925). UT1 is still in reality mean time at Greenwich. Today, GMT is a time zone but is still the legal time in the UK in winter (and as adjusted by one hour for summer time). But Coordinated Universal Time (UTC) (an atomic-based time scale which is always kept within 0.9 second of UT1) is in common actual use in the UK, and the name GMT is often used to refer to it. (See articles Greenwich Mean Time, Universal Time, Coordinated Universal Time and the sources they cite.)
Versions of Universal Time such as UT0 and UT2 have been defined but are no longer in use. | Time standard | Wikipedia | 251 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
Time standards for planetary motion calculations
Ephemeris time (ET) and its successor time scales described below have all been intended for astronomical use, e.g. in planetary motion calculations, with aims including uniformity, in particular, freedom from irregularities of Earth rotation. Some of these standards are examples of dynamical time scales and/or of coordinate time scales. Ephemeris Time was from 1952 to 1976 an official time scale standard of the International Astronomical Union; it was a dynamical time scale based on the orbital motion of the Earth around the Sun, from which the ephemeris second was derived as a defined fraction of the tropical year. This ephemeris second was the standard for the SI second from 1956 to 1967, and it was also the source for calibration of the caesium atomic clock; its length has been closely duplicated, to within 1 part in 1010, in the size of the current SI second referred to atomic time. This Ephemeris Time standard was non-relativistic and did not fulfil growing needs for relativistic coordinate time scales. It was in use for the official almanacs and planetary ephemerides from 1960 to 1983, and was replaced in official almanacs for 1984 and after, by numerically integrated Jet Propulsion Laboratory Development Ephemeris DE200 (based on the JPL relativistic coordinate time scale Teph).
For applications at the Earth's surface, ET's official replacement was Terrestrial Dynamical Time (TDT), which maintained continuity with it. TDT is a uniform atomic time scale, whose unit is the SI second. TDT is tied in its rate to the SI second, as is International Atomic Time (TAI), but because TAI was somewhat arbitrarily defined at its inception in 1958 to be initially equal to a refined version of UT, TDT was offset from TAI, by a constant 32.184 seconds. The offset provided a continuity from Ephemeris Time to TDT. TDT has since been redefined as Terrestrial Time (TT). | Time standard | Wikipedia | 430 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
For the calculation of ephemerides, Barycentric Dynamical Time (TDB) was officially recommended to replace ET. TDB is similar to TDT but includes relativistic corrections that move the origin to the barycenter, hence it is a dynamical time at the barycenter. TDB differs from TT only in periodic terms. The difference is at most 2 milliseconds. Deficiencies were found in the definition of TDB (though not affecting Teph), and TDB has been replaced by Barycentric Coordinate Time (TCB) and Geocentric Coordinate Time (TCG), and redefined to be JPL ephemeris time argument Teph, a specific fixed linear transformation of TCB. As defined, TCB (as observed from the Earth's surface) is of divergent rate relative to all of ET, Teph and TDT/TT; and the same is true, to a lesser extent, of TCG. The ephemerides of Sun, Moon and planets in current widespread and official use continue to be those calculated at the Jet Propulsion Laboratory (updated as from 2003 to DE405) using as argument Teph. | Time standard | Wikipedia | 249 | 41799 | https://en.wikipedia.org/wiki/Time%20standard | Technology | Timekeeping | null |
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances (this can be as short as millimetres depending on frequency). However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.
Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas (they are then called feed lines or feeders), distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses. RF engineers commonly use short pieces of transmission line, usually in the form of printed planar transmission lines, arranged in certain patterns to build circuits such as filters. These circuits, known as distributed-element circuits, are an alternative to traditional circuits using discrete capacitors and inductors.
Overview
Ordinary electrical cables suffice to carry low frequency alternating current (AC), such as mains power, which reverses direction 100 to 120 times per second, and audio signals. However, they are not generally used to carry currents in the radio frequency range, above about 30 kHz, because the energy tends to radiate off the cable as radio waves, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as connectors and joints, and travel back down the cable toward the source. These reflections act as bottlenecks, preventing the signal power from reaching the destination. Transmission lines use specialized construction, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform impedance, called the characteristic impedance, to prevent reflections. Types of transmission line include parallel line (ladder line, twisted pair), coaxial cable, and planar transmission lines such as stripline and microstrip. The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the transmitted frequency's wavelength is sufficiently short that the length of the cable becomes a significant part of a wavelength. | Transmission line | Wikipedia | 484 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
At frequencies of microwave and higher, power losses in transmission lines become excessive, and waveguides are used instead, which function as "pipes" to confine and guide the electromagnetic waves. Some sources define waveguides as a type of transmission line; however, this article will not include them.
History
Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin, and Oliver Heaviside. In 1855, Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885, Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.
The four terminal model
For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadripole), as follows:
In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a two parameters called characteristic impedance, symbol Z0 and propagation delay, symbol . Z0 is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission. Propagation delay is proportional to the length of the transmission line and is never less than the length divided by the speed of light. Typical delays for modern communication transmission lines vary from to .
When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance equal to Z0, in which case the transmission line is said to be matched. | Transmission line | Wikipedia | 449 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating). The transmission line is modelled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.
The total loss of power in a transmission line is often specified in decibels per metre (dB/m), and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.
Propagation delay is often specified in units of nanoseconds per metre. While propagation delay usually depends on the frequency of the signal, transmission lines are typically operated over frequency ranges where the propagation delay is approximately constant.
Telegrapher's equations
The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage () and current () on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's equations.
The transmission line model is an example of the distributed-element model. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:
The distributed resistance of the conductors is represented by a series resistor (expressed in ohms per unit length).
The distributed inductance (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (in henries per unit length).
The capacitance between the two conductors is represented by a shunt capacitor (in farads per unit length).
The conductance of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (in siemens per unit length). | Transmission line | Wikipedia | 496 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
The model consists of an infinite series of the elements shown in the figure, and the values of the components are specified per unit length so the picture of the component can be misleading. , , , and may also be functions of frequency. An alternative notation is to use , , and to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.
The line voltage and the current can be expressed in the frequency domain as
(see differential equation, angular frequency ω and imaginary unit )
Special case of a lossless line
When the elements and are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the and elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:
These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.
General case of a line with losses
In the general case the loss terms, and , are both included, and the full form of the Telegrapher's equations become:
where is the (complex) propagation constant. These equations are fundamental to transmission line theory. They are also wave equations, and have solutions similar to the special case, but which are a mixture of sines and cosines with exponential decay factors. Solving for the propagation constant in terms of the primary parameters , , , and gives:
and the characteristic impedance can be expressed as
The solutions for and are:
The constants must be determined from boundary conditions. For a voltage pulse , starting at and moving in the positive direction, then the transmitted pulse at position can be obtained by computing the Fourier Transform, , of , attenuating each frequency component by , advancing its phase by , and taking the inverse Fourier Transform. The real and imaginary parts of can be computed as
with
the right-hand expressions holding when neither , nor , nor is zero, and with
where atan2 is the everywhere-defined form of two-parameter arctangent function, with arbitrary value zero when both arguments are zero. | Transmission line | Wikipedia | 498 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
Alternatively, the complex square root can be evaluated algebraically, to yield:
and
with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium. ( is usually negative, since and are typically much smaller than and , respectively, so is usually positive. is always positive.)
Special, low loss case
For small losses and high frequencies, the general equations can be simplified: If and then
Since an advance in phase by is equivalent to a time delay by , can be simply computed as
Heaviside condition
The Heaviside condition is .
If R, G, L, and C are constants that are not frequency dependent and the Heaviside condition is met,
then waves travel down the transmission line without dispersion distortion.
Input impedance of transmission line
The characteristic impedance of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.
The impedance measured at a given distance from the load impedance may be expressed as
,
where is the propagation constant and is the voltage reflection coefficient measured at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:
.
Input impedance of lossless transmission line
For a lossless transmission line, the propagation constant is purely imaginary, , so the above formulas can be rewritten as
where is the wavenumber.
In calculating the wavelength is generally different inside the transmission line to what it would be in free-space. Consequently, the velocity factor of the material the transmission line is made of needs to be taken into account when doing such a calculation.
Special cases of lossless transmission lines
Half wave length
For the special case where where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that
for all This includes the case when , meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.
Quarter wave length
For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes | Transmission line | Wikipedia | 503 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
Matched load
Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that
for all and all .
Short
For the case of a shorted load (i.e. ), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)
Open
For the case of an open load (i.e. ), the input impedance is once again imaginary and periodic
Matrix parameters
The simulation of transmission lines embedded into larger systems generally utilize admittance parameters (Y matrix), impedance parameters (Z matrix), and/or scattering parameters (S matrix) that embodies the full transmission line model needed to support the simulation.
Admittance parameters
Admittance (Y) parameters may be defined by applying a fixed voltage to one port (V1) of a transmission line with the other end shorted to ground and measuring the resulting current running into each port (I1, I2) and computing the admittance on each port as a ratio of I/V The admittance parameter Y11 is I1/V1, and the admittance parameter Y12 is I2/V1. Since transmission lines are electrically passive and symmetric devices, Y12 = Y21, and Y11 = Y22.
For lossless and lossy transmission lines respectively, the Y parameter matrix is as follows:
Impedance parameters
Impedance (Z) parameter may defines by applying a fixed current into one port (I1) of a transmission line with the other port open and measuring the resulting voltage on each port (V1, V2) and computing the impedance parameter Z11 is V1/I1, and the impedance parameter Z12 is V2/I1. Since transmission lines are electrically passive and symmetric devices, V12 = V21, and V11 = V22.
In the Y and Z matrix definitions, and . Unlike ideal lumped 2 port elements (resistors, capacitors, inductors, etc.) which do not have defined Z parameters, transmission lines have an internal path to ground, which permits the definition of Z parameters.
For lossless and lossy transmission lines respectively, the Z parameter matrix is as follows:
Scattering parameters
Scattering (S) matrix parameters model the electrical behavior of the transmission line with matched loads at each termination. | Transmission line | Wikipedia | 498 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
For lossless and lossy transmission lines respectively, the S parameter matrix is as follows, using standard hyperbolic to circular complex translations.
Variable definitions
In all matrix parameters above, the following variable definitions apply:
= characteristic impedance
Zp = port impedance, or termination impedance
= the propagation constant per unit length
= attenuation constant in nepers per unit length
= wave number or phase constant radians per unit length
= frequency radians / second
= Speed of propagation
= wave length in unit length
L = inductance per unit length
C = capacitance per unit length
= effective dielectric constant
= 299,792,458 meters / second = Speed of light in a vacuum
Coupled transmission lines
Transmission lines may be placed in proximity to each other such that they electrically interact, such as two microstrip lines in close proximity. Such transmission lines are said to be coupled transmission lines. Coupled transmission lines are characterized by an even and odd mode analysis. The even mode is characterized by excitation of the two conductors with a signal of equal amplitude and phase. The odd mode is characterized by excitation with signals of equal and opposite magnitude. The even and odd modes each have their own characteristic impedances (Zoe, Zoo) and phase constants (). Lossy coupled transmission lines have their own even and odd mode attenuation constants (), which in turn leads to even and odd mode propagation constants ().
Coupled matrix parameters
Coupled transmission lines may be modeled using even and odd mode transmission line parameters defined in the prior paragraph as shown with ports 1 and 2 on the input and ports 3 and 4 on the output,
..
Practical types
Coaxial cable | Transmission line | Wikipedia | 346 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them.
In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable other transverse modes can propagate. These modes are classified into two groups, transverse electric (TE) and transverse magnetic (TM) waveguide modes. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.
The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried long distance telephone connections.
Planar lines
Planar transmission lines are transmission lines with conductors, or in some cases dielectric strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Several forms of planar transmission lines exist.
Microstrip
A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure.
Stripline
A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.
Coplanar waveguide | Transmission line | Wikipedia | 503 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
A coplanar waveguide consists of a center strip and two adjacent outer conductors, all three of them flat structures that are deposited onto the same insulating substrate and thus are located in the same plane ("coplanar"). The width of the center conductor, the distance between inner and outer conductors, and the relative permittivity of the substrate determine the characteristic impedance of the coplanar transmission line.
Balanced lines
A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits. There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.
Twisted pair
Twisted pairs are commonly used for terrestrial telephone communications. In such cables, many pairs are grouped together in a single cable, from two to several thousand. The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.
Star quad
Star quad is a four-conductor cable in which all four conductors are twisted together around the cable axis. It is sometimes used for two circuits, such as 4-wire telephony and other telecommunications applications. In this configuration each pair uses two non-adjacent conductors. Other times it is used for a single, balanced line, such as audio applications and 2-wire telephony. In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.
When used for two circuits, crosstalk is reduced relative to cables with two separate twisted pairs.
When used for a single, balanced line, magnetic interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers.
The combined benefits of twisting, balanced signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as microphone cables, even when installed very close to a power cable. The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.
Twin-lead | Transmission line | Wikipedia | 449 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
Twin-lead consists of a pair of conductors held apart by a continuous insulator. By holding the conductors a known distance apart, the geometry is fixed and the line characteristics are reliably consistent. It is lower loss than coaxial cable because the characteristic impedance of twin-lead is generally higher than coaxial cable, leading to lower resistive losses due to the reduced current. However, it is more susceptible to interference.
Lecher lines
Lecher lines are a form of parallel conductor that can be used at UHF for creating resonant circuits. They are a convenient practical format that fills the gap between lumped components (used at HF/VHF) and resonant cavities (used at UHF/SHF).
Single-wire line
Unbalanced lines were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of single-wire earth return in use in many locations.
General applications
Signal transfer
Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss. One familiar example is the down lead from a TV or radio aerial to the receiver.
Transmission line circuits
A large variety of circuits can also be constructed with transmission lines including impedance matching circuits, filters, power dividers and directional couplers.
Stepped transmission line
A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be . The input impedance can be obtained from the successive application of the chain relation
where is the wave number of the -th transmission line segment and is the length of this segment, and is the front-end impedance that loads the -th segment.
Because the characteristic impedance of each transmission line segment is often different from the impedance of the fourth, input cable (only shown as an arrow marked on the left side of the diagram above), the impedance transformation circle is off-centred along the axis of the Smith Chart whose impedance representation is usually normalized against . | Transmission line | Wikipedia | 479 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
Approximating lumped elements
At higher frequencies, the reactive parasitic effects of real world lumped elements, including inductors and capacitors, limits their usefulness. Therefore, it is sometimes useful to approximate the electrical characteristics of inductors and capacitors with transmission lines at the higher frequencies using Richards' Transformations and then substitute the transmission lines for the lumped elements.
More accurate forms of multimode high frequency inductor modeling with transmission lines exist for advanced designers.
Stub filters
If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.
Wideband filters can be achieved using multiple stubs. However, this is a somewhat dated technique. Much more compact filters can be made with other methods such as parallel-line resonators.
Pulse generation
Transmission lines are used as pulse generators. By charging the transmission line and then discharging it into a resistive load, a rectangular pulse equal in length to twice the electrical length of the line can be obtained, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed power sources for radar transmitters and other devices.
Sound
The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are called acoustic transmission lines. | Transmission line | Wikipedia | 442 | 41811 | https://en.wikipedia.org/wiki/Transmission%20line | Technology | Signal transmission | null |
The troposphere is the lowest layer of the atmosphere of Earth. It contains 80% of the total mass of the planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the planetary surface of the Earth, the average height of the troposphere is in the tropics; in the middle latitudes; and in the high latitudes of the polar regions in winter; thus the average height of the troposphere is .
The term troposphere derives from the Greek words tropos (rotating) and sphaira (sphere) indicating that rotational turbulence mixes the layers of air and so determines the structure and the phenomena of the troposphere. The rotational friction of the troposphere against the planetary surface affects the flow of the air, and so forms the planetary boundary layer (PBL) that varies in height from hundreds of meters up to . The measures of the PBL vary according to the latitude, the landform, and the time of day when the meteorological measurement is realized. Atop the troposphere is the tropopause, which is the functional atmospheric border that demarcates the troposphere from the stratosphere. As such, because the tropopause is an inversion layer in which air-temperature increases with altitude, the temperature of the tropopause remains constant. The layer has the largest concentration of nitrogen.
Structure | Troposphere | Wikipedia | 293 | 41822 | https://en.wikipedia.org/wiki/Troposphere | Physical sciences | Atmosphere: General | Earth science |
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