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The equation of state (ideal gas law) |
In an adiabatic process, pressure "P" as a function of density formula_20 can be linearized to |
where "C" is some constant. Breaking the pressure and density into their mean and total components and noting that formula_22: |
The adiabatic bulk modulus for a fluid is defined as |
Condensation, "s", is defined as the change in density for a given ambient fluid density. |
The continuity equation (conservation of mass) in one dimension is |
Where "u" is the flow velocity of the fluid. |
Again the equation must be linearized and the variables split into mean and variable components. |
Rearranging and noting that ambient density changes with neither time nor position and that the condensation multiplied by the velocity is a very small number: |
Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is: |
where formula_33 represents the convective, substantial or material derivative, which is the derivative at a point moving along with the medium rather than at a fixed point. |
Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation: |
Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in: |
Multiplying the first by formula_38, subtracting the two, and substituting the linearized equation of state, |
where formula_41 is the speed of propagation. |
Feynman provides a derivation of the wave equation for sound in three dimensions as |
where formula_43 is the Laplace operator, formula_1 is the acoustic pressure (the local deviation from the ambient pressure), and formula_6 is the speed of sound. |
A similar looking wave equation but for the vector field particle velocity is given by |
In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form |
and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity): |
The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time-dependence factor of formula_50 where formula_51 is the angular frequency. The explicit time dependence is given by |
where the asymptotic approximations to the Hankel functions, when formula_56, are |
Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist. |
Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone. The SI unit of sound pressure is the pascal (Pa). |
A sound wave in a transmission medium causes a deviation (sound pressure, a "dynamic" pressure) in the local ambient pressure, a "static" pressure. |
Sound pressure, denoted "p", is defined by |
In a sound wave, the complementary variable to sound pressure is the particle velocity. Together, they determine the sound intensity of the wave. |
"Sound intensity", denoted I and measured in W·m−2 in SI units, is defined by |
"Acoustic impedance", denoted "Z" and measured in Pa·m−3·s in SI units, is defined by |
"Specific acoustic impedance", denoted "z" and measured in Pa·m−1·s in SI units, is defined by |
The "particle displacement" of a "progressive sine wave" is given by |
It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave "x" are given by |
Taking the Laplace transforms of "v" and "p" with respect to time yields |
Since formula_18, the amplitude of the specific acoustic impedance is given by |
Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by |
When measuring the sound pressure created by a sound source, it is important to measure the distance from the object as well, since the sound pressure of a "spherical" sound wave decreases as 1/"r" from the centre of the sphere (and not as 1/"r"2, like the sound intensity): |
If the sound pressure "p"1 is measured at a distance "r"1 from the centre of the sphere, the sound pressure "p"2 at another position "r"2 can be calculated: |
The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity: |
The sound pressure may vary in direction from the centre of the sphere as well, so measurements at different angles may be necessary, depending on the situation. An obvious example of a sound source whose spherical sound wave varies in level in different directions is a bullhorn. |
Sound pressure level (SPL) or acoustic pressure level is a logarithmic measure of the effective pressure of a sound relative to a reference value. |
Sound pressure level, denoted "L""p" and measured in dB, is defined by |
The commonly used reference sound pressure in air is |
which is often considered as the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). The proper notations for sound pressure level using this reference are or , but the suffix notations , , dBSPL, or dBSPL are very common, even if they are not accepted by the SI. |
Most sound-level measurements will be made relative to this reference, meaning will equal an SPL of . In other media, such as underwater, a reference level of is used. These references are defined in ANSI S1.1-2013. |
The main instrument for measuring sound levels in the environment is the sound level meter. Most sound level meters provide readings in A, C, and Z-weighted decibels and must meet international standards such as IEC 61672-2013. |
The lower limit of audibility is defined as SPL of , but the upper limit is not as clearly defined. While ( or ) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere (i.e. if the thermodynamic properties of the air are disregarded, in reality the sound wave become progressively non-linear starting over 150 dB), larger sound waves can be present in other atmospheres or other media such as under water or through the Earth. |
Ears detect changes in sound pressure. Human hearing does not have a flat spectral sensitivity (frequency response) relative to frequency versus amplitude. Humans do not perceive low- and high-frequency sounds as well as they perceive sounds between 3,000 and 4,000 Hz, as shown in the equal-loudness contour. Because the frequency response of human hearing changes with amplitude, three weightings have been established for measuring sound pressure: A, B and C. A-weighting applies to sound pressures levels up to , B-weighting applies to sound pressures levels between and , and C-weighting is for measuring sound pressure levels above . |
In order to distinguish the different sound measures, a suffix is used: A-weighted sound pressure level is written either as dBA or LA. B-weighted sound pressure level is written either as dBB or LB, and C-weighted sound pressure level is written either as dBC or LC. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dBL or just L. Some sound measuring instruments use the letter "Z" as an indication of linear SPL. |
According to the inverse proportional law, when sound level "L""p"1 is measured at a distance "r"1, the sound level "L""p"2 at the distance "r"2 is |
The formula for the sum of the sound pressure levels of "n" incoherent radiating sources is |
in the formula for the sum of the sound pressure levels yields |
A Dynamical Theory of the Electromagnetic Field |
"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave. |
In part III of the paper, which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations which were to become known as Maxwell's equations, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in "On Physical Lines of Force". |
Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern vector notation. They actually only contain one of the original eight—equation "G" (Gauss's Law). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation "A") with Ampère's circuital law (equation "C"). This amalgamation, which Maxwell himself had actually originally made at equation (112) in "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's displacement current. |
Eighteen of Maxwell's twenty original equations can be vectorized into six equations, labeled (A) to (F) below, each of which represents a group of three original equations in component form. The 19th and 20th of Maxwell's component equations appear as (G) and (H) below, making a total of eight vector equations. These are listed below in Maxwell's original order, designated by the letters that Maxwell assigned to them in his 1864 paper. |
Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive media with permittivity "ϵ" and permeability "μ", although he also discussed the possibility of anisotropic materials. |
Gauss's law for magnetism () is not included in the above list, but follows directly from equation (B) by taking divergences (because the divergence of the curl is zero). |
Substituting (A) into (C) yields the familiar differential form of the . |
Equation (D) implicitly contains the Lorentz force law and the differential form of Faraday's law of induction. For a "static" magnetic field, formula_11 vanishes, and the electric field becomes conservative and is given by , so that (D) reduces to |
This is simply the Lorentz force law on a per-unit-charge basis — although Maxwell's equation (D) first appeared at equation (77) in "On Physical Lines of Force" in 1861, 34 years before Lorentz derived his force law, which is now usually presented as a supplement to the four "Maxwell's equations". The cross-product term in the Lorentz force law is the source of the so-called "motional emf" in electric generators (see also "Moving magnet and conductor problem"). Where there is no motion through the magnetic field — e.g., in transformers — we can drop the cross-product term, and the force per unit charge (called ) reduces to the electric field , so that Maxwell's equation (D) reduces to |
Taking curls, noting that the curl of a gradient is zero, we obtain |
which is the differential form of Faraday's law. Thus the three terms on the right side of equation (D) may be described, from left to right, as the motional term, the transformer term, and the conservative term. |
In deriving the electromagnetic wave equation, Maxwell considers the situation only from the rest frame of the medium, and accordingly drops the cross-product term. But he still works from equation (D), in contrast to modern textbooks which tend to work from Faraday's law (see below). |
The constitutive equations (E) and (F) are now usually written in the rest frame of the medium as and . |
Maxwell's equation (G), viewed in isolation as printed in the 1864 paper, at first seems to say that . However, if we trace the signs through the previous two triplets of equations, we see that what seem to be the components of are in fact the components of . The notation used in Maxwell's later "Treatise on Electricity and Magnetism" is different, and avoids the misleading first impression. |
In part VI of "A Dynamical Theory of the Electromagnetic Field", subtitled "Electromagnetic theory of light", Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force", which is defined as displacement current, to derive the electromagnetic wave equation. |
He obtained a wave equation with a speed in close agreement to experimental determinations of the speed of light. He commented, |
Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method which combines the corrected version of Ampère's Circuital Law with Faraday's law of electromagnetic induction. |
To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. Using (SI units) in a vacuum, these equations are |
If we take the curl of the curl equations we obtain |
where formula_20 is any vector function of space, we recover the wave equations |
is the speed of light in free space. |
Of this paper and Maxwell's related works, fellow physicist Richard Feynman said: "From the long view of this history of mankind – seen from, say, 10,000 years from now – there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electromagnetism." |
Albert Einstein used Maxwell's equations as the starting point for his special theory of relativity, presented in "The Electrodynamics of Moving Bodies", one of Einstein's 1905 "Annus Mirabilis" papers. In it is stated: |
Maxwell's equations can also be derived by extending general relativity into five physical dimensions. |
Electromagnetic or magnetic induction is the production of an electromotive force across an electrical conductor in a changing magnetic field. |
Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction. Lenz's law describes the direction of the induced field. Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism. |
Electromagnetic induction has found many applications, including electrical components such as inductors and transformers, and devices such as electric motors and generators. |
Electromagnetic induction was discovered by Michael Faraday, published in 1831. It was discovered independently by Joseph Henry in 1832. |
Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically. An exception was James Clerk Maxwell, who used Faraday's ideas as the basis of his quantitative electromagnetic theory. In Maxwell's model, the time varying aspect of electromagnetic induction is expressed as a differential equation, which Oliver Heaviside referred to as Faraday's law even though it is slightly different from Faraday's original formulation and does not describe motional EMF. Heaviside's version (see Maxwell–Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations. |
In 1834 Heinrich Lenz formulated the law named after him to describe the "flux through the circuit". Lenz's law gives the direction of the induced EMF and current resulting from electromagnetic induction. |
Faraday's law of induction and Lenz's law. |
Faraday's law of induction makes use of the magnetic flux ΦB through a region of space enclosed by a wire loop. The magnetic flux is defined by a surface integral: |
where "dA is an element of the surface Σ enclosed by the wire loop, B is the magnetic field. The dot product B·"dA corresponds to an infinitesimal amount of magnetic flux. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop. |
When the flux through the surface changes, Faraday's law of induction says that the wire loop acquires an electromotive force (EMF). The most widespread version of this law states that the induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux enclosed by the circuit: |
where formula_3 is the EMF and ΦB is the magnetic flux. The direction of the electromotive force is given by Lenz's law which states that an induced current will flow in the direction that will oppose the change which produced it. This is due to the negative sign in the previous equation. To increase the generated EMF, a common approach is to exploit flux linkage by creating a tightly wound coil of wire, composed of "N" identical turns, each with the same magnetic flux going through them. The resulting EMF is then "N" times that of one single wire. |
Generating an EMF through a variation of the magnetic flux through the surface of a wire loop can be achieved in several ways: |
In general, the relation between the EMF formula_5 in a wire loop encircling a surface Σ, and the electric field E in the wire is given by |
where "d"ℓ is an element of contour of the surface Σ, combining this with the definition of flux |
we can write the integral form of the Maxwell–Faraday equation |
It is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. |
Faraday's law describes two different phenomena: the "motional EMF" generated by a magnetic force on a moving wire (see Lorentz force), and the "transformer EMF" this is generated by an electric force due to a changing magnetic field (due to the differential form of the Maxwell–Faraday equation). James Clerk Maxwell drew attention to the separate physical phenomena in 1861. This is believed to be a unique example in physics of where such a fundamental law is invoked to explain two such different phenomena. |
Albert Einstein noticed that the two situations both corresponded to a relative movement between a conductor and a magnet, and the outcome was unaffected by which one was moving. This was one of the principal paths that led him to develop special relativity. |
The principles of electromagnetic induction are applied in many devices and systems, including: |
The EMF generated by Faraday's law of induction due to relative movement of a circuit and a magnetic field is the phenomenon underlying electrical generators. When a permanent magnet is moved relative to a conductor, or vice versa, an electromotive force is created. If the wire is connected through an electrical load, current will flow, and thus electrical energy is generated, converting the mechanical energy of motion to electrical energy. For example, the "drum generator" is based upon the figure to the bottom-right. A different implementation of this idea is the Faraday's disc, shown in simplified form on the right. |
When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, "d" ΦB / "d t". Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF. If the two ends of this loop are connected through an electrical load, current will flow. |
A current clamp is a type of transformer with a split core which can be spread apart and clipped onto a wire or coil to either measure the current in it or, in reverse, to induce a voltage. Unlike conventional instruments the clamp does not make electrical contact with the conductor or require it to be disconnected during attachment of the clamp. |
Faraday's law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage ℇ generated in the magnetic field "B" due to a conductive liquid moving at velocity "v" is thus given by: |
where ℓ is the distance between electrodes in the magnetic flow meter. |
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