Acoustic wave equation explained

In physics, the acoustic wave equation is a second-order partial differential equation that governs the propagation of acoustic waves through a material medium resp. a standing wavefield. The equation describes the evolution of acoustic pressure or particle velocity as a function of position and time . A simplified (scalar) form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions. For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article or the survey paper.^[1]

Definition in one dimension

The wave equation describing a standing wave field in one dimension (position

x ) is

$$p\_\; -\; \backslash frac\; p\_\; =0,$$

where

p is the acoustic pressure (the local deviation from the ambient pressure), and where c is the speed of sound.^[2]

Derivation

Start with the ideal gas law

P = \rho R _{specificT,} where T the absolute temperature of the gas and specific gas constant R _{specific} .Then, assuming the process is adiabatic, pressure P(\rho) can be considered a function of density \rho .

The conservation of mass and conservation of momentum can be written as a closed system of two equationsThis coupled system of two nonlinear conservation laws can be written in vector form as:$$q\_t\; +\; f(q)\_x\; =\; 0,$$with$$q\; =\; \backslash begin\backslash rho\; \backslash \backslash \; \backslash rho\; u\backslash end\; =\; \backslash beginq\_\; \backslash \backslash \; q\_\backslash end,\; \backslash quad\; f(q)\; =\; \backslash begin\backslash rho\; u\; \backslash \backslash \; \backslash rho\; u^2\; +\; P(\backslash rho)\backslash end=\backslash begin\; q\_\; \backslash \backslash \; q\_^2/q\_\; +\; P(q\_)\backslash end.$$

To linearize this equation, let$$q(x,t)\; =\; q\_0\; +\; \backslash tilde(x,t),$$where

q _{0} = ( \rho _{0} , \rho _{0} u _{0)} is the (constant) background state and \tilde{q} is a sufficiently small pertubation, i.e., any powers or products of \tilde{q} can be discarded. Hence, the taylor expansion of f(q) gives:$$f(q\_0\; +\; \backslash tilde)\; \backslash approx\; f(q\_0)\; +\; f\text{'}(q\_0)\backslash tilde$$where This results in the linearized equationLikewise, small pertubations of the components of q can be rewritten as:$$\backslash rho\; u\; =\; (\backslash rho\_0\; +\backslash tilde)(u\_0\; +\backslash tilde)\; =\; \backslash rho\_0\; u\_0\; +\; \backslash tildeu\_0\; +\; \backslash rho\_0\; \backslash tilde\; +\; \backslash tilde\backslash tilde$$such that$$\backslash widetilde\; \backslash approx\; \backslash tildeu\_0\; +\; \backslash rho\_0\; \backslash tilde,$$and pressure pertubations relate to density pertubations as:$$p\; =\; p\_\; +\; \backslash tilde=\; P(\backslash rho\_0\; +\; \backslash tilde)\; =\; P(\backslash rho\_)\; +\; P\text{'}(\backslash rho\_)\backslash tilde\; +\; \backslash dots$$such that:$$p\_0\; =\; P(\backslash rho\_0),\; \backslash quad\; \backslash tilde\backslash approx\; P\text{'}(\backslash rho\_0)\backslash tilde,$$ where P'(\rho _{0)} is a constant, resulting in the alternative form of the linear acoustics equations:where K _{0} = \rho _{0} P'(\rho _{0)} is the bulk modulus of compressibility. After dropping the tilde for convenience, the linear first order system can be written as:While, in general, a non-zero background velocity is possible (e.g. when studying the sound propagation in a constant-strenght wind), it will be assumed that u _{0} =0 . Then the linear system reduces to the second-order wave equation:$$p\_\; =\; -K\_0\; u\_\; =\; -K\_0\; u\_\; =\; K\_0\backslash left(\backslash fracp\_x\backslash right)\_x\; =\; c\_^2\; p\_,$$with c _{0} = \sqrt{K _{0/\rho} _{0}} the speed of sound.

Hence, the acoustic equation can be derived from a system of first-order advection equations that follow directly from physics, i.e., the first integrals:$$q\_t\; +\; Aq\_x\; =\; 0,$$withConversely, given the second-order equation

p _{tt} = c ^{2} _{0} p _{xx} a first-order system can be derived:$$q\_t\; +\; \backslash hatq\_x\; =\; 0,$$withwhere matrix A and \hat{A} are similar.

Solution

Provided that the speed

c is a constant, not dependent on frequency (the dispersionless case), then the most general solution isp = f(c t - x) + g(c t + x)

where

f and g are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (f ) traveling up the x-axis and the other (g ) down the x-axis at the speed c . The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either f or g to be a sinusoid, and the other to be zero, givingp = p _{0} \sin(\omega t \mp kx) .

where

\omega is the angular frequency of the wave and k is its wave number.

In three dimensions

Equation

Feynman^[3] provides a derivation of the wave equation for sound in three dimensions as

\nabla ^{2} p - { 1 \over c ^{2}} { \partial ^{2} p \over \partial t ^{2} } = 0 ,

where

\nabla ^{2} is the Laplace operator, p is the acoustic pressure (the local deviation from the ambient pressure), and c is the speed of sound.

A similar looking wave equation but for the vector field particle velocity is given by

\nabla ^{2} u - { 1 \over c ^{2}} { \partial ^{2} u \over \partial t ^{2} } = 0 .

In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form

\nabla ^{2} \Phi - { 1 \over c ^{2}} { \partial ^{2} \Phi \over \partial t ^{2} } = 0 and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):u = \nabla \Phi ,p = - \rho {\partial \over \partial t}\Phi .

Solution

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

e ^{i\omega} where \omega = 2 \pi f is the angular frequency. The explicit time dependence is given byp(r,t,k) = \operatorname{Real}\left[p(r,k) e ^{i\omega} \right] Here k = \omega/c is the wave number.

Cartesian coordinates

p(r,k) = Ae ^{\pm} .

Cylindrical coordinates

p(r,k) = AH ^{( 1 )} _{0} (kr) + BH ^{( 2 )} _{0} (kr) .

where the asymptotic approximations to the Hankel functions, when

kr → inf ty , areH ^{( 1 )} _{0} (kr) \simeq \sqrt{ (2)/(\pi kr) }e^H ^{( 2 )} _{0} (kr) \simeq \sqrt{ (2)/(\pi kr) }e^.

Spherical coordinates

p(r,k) = (A)/(r) e ^{\pm} .

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.

See also

• Acoustics
• Acoustic attenuation
• Acoustic theory
• Wave equation
• One-way wave equation
• Differential equations
• Thermodynamics
• Fluid dynamics
• Pressure
• Ideal gas law

References

• Book: LeVeque, Randall J. . Finite Volume Methods for Hyperbolic Problems . Cambridge University Press . 2002 . 978-0-521-81087-6 . 10.1017/cbo9780511791253.

Notes and References

1. S. P. Näsholm and S. Holm, "On a Fractional Zener Elastic Wave Equation," Fract. Calc. Appl. Anal. Vol. 16, No 1 (2013), pp. 26-50, DOI: 10.2478/s13540-013--0003-1 Link to e-print
2. [Richard Feynman]
3. [Richard Feynman]