Acoustic wave equation

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 p or particle velocity u as a function of position x and time t. 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]

The wave equation describing a standing wave field in one dimension (position ${\displaystyle x}$) is

$${\displaystyle p_{xx}-{\frac {1}{c^{2}}}p_{tt}=0,}$$

where ${\displaystyle p}$ is the acoustic pressure (the local deviation from the ambient pressure) and ${\displaystyle c}$ the speed of sound, using subscript notation for the partial derivatives.^[2]

Start with the ideal gas law

${\displaystyle P=\rho R_{\text{specific}}T,}$

where ${\displaystyle T}$ the absolute temperature of the gas and specific gas constant ${\displaystyle R_{\text{specific}}}$. Then, assuming the process is adiabatic, pressure ${\displaystyle P(\rho )}$ can be considered a function of density ${\displaystyle \rho }$.

The conservation of mass and conservation of momentum can be written as a closed system of two equations^[3] $${\displaystyle {\begin{aligned}\rho _{t}+(\rho u)_{x}&=0,\\(\rho u)_{t}+(\rho u^{2}+P(\rho ))_{x}&=0.\end{aligned}}}$$ This coupled system of two nonlinear conservation laws can be written in vector form as: $${\displaystyle q_{t}+f(q)_{x}=0,}$$ with $${\displaystyle q={\begin{bmatrix}\rho \\\rho u\end{bmatrix}}={\begin{bmatrix}q_{(1)}\\q_{(2)}\end{bmatrix}},\quad f(q)={\begin{bmatrix}\rho u\\\rho u^{2}+P(\rho )\end{bmatrix}}={\begin{bmatrix}q_{(2)}\\q_{(2)}^{2}/q_{(1)}+P(q_{(1)})\end{bmatrix}}.}$$

To linearize this equation, let^[4] $${\displaystyle q(x,t)=q_{0}+{\tilde {q}}(x,t),}$$ where ${\displaystyle q_{0}=(\rho _{0},\rho _{0}u_{0})}$ is the (constant) background state and ${\displaystyle {\tilde {q}}}$ is a sufficiently small perturbation, i.e., any powers or products of ${\displaystyle {\tilde {q}}}$ can be discarded. Hence, the taylor expansion of ${\displaystyle f(q)}$ gives: $${\displaystyle f(q_{0}+{\tilde {q}})\approx f(q_{0})+f'(q_{0}){\tilde {q}}}$$ where $${\displaystyle f'(q)={\begin{bmatrix}\partial f_{(1)}/\partial q_{(1)}&\partial f_{(1)}/\partial q_{(2)}\\\partial f_{(2)}/\partial q_{(1)}&\partial f_{(2)}/\partial q_{(2)}\end{bmatrix}}={\begin{bmatrix}0&1\\-u^{2}+P'(\rho )&2u\end{bmatrix}}.}$$ This results in the linearized equation $${\displaystyle {\tilde {q}}_{t}+f'(q_{0}){\tilde {q}}_{x}=0\quad \Leftrightarrow \quad {\begin{aligned}{\tilde {\rho }}_{t}+({\widetilde {\rho u}})_{x}&=0\\({\widetilde {\rho u}})_{t}+(-u_{0}^{2}+P'(\rho _{0})){\tilde {\rho }}_{x}+2u_{0}({\widetilde {\rho u}})_{x}&=0\end{aligned}}}$$ Likewise, small perturbations of the components of ${\displaystyle q}$ can be rewritten as: $${\displaystyle \rho u=(\rho _{0}+{\tilde {\rho }})(u_{0}+{\tilde {u}})=\rho _{0}u_{0}+{\tilde {\rho }}u_{0}+\rho _{0}{\tilde {u}}+{\tilde {\rho }}{\tilde {u}}}$$ such that $${\displaystyle {\widetilde {\rho u}}\approx {\tilde {\rho }}u_{0}+\rho _{0}{\tilde {u}},}$$ and pressure perturbations relate to density perturbations as: $${\displaystyle p=p_{0}+{\tilde {p}}=P(\rho _{0}+{\tilde {\rho }})=P(\rho _{0})+P'(\rho _{0}){\tilde {\rho }}+\dots }$$ such that: $${\displaystyle p_{0}=P(\rho _{0}),\quad {\tilde {p}}\approx P'(\rho _{0}){\tilde {\rho }},}$$ where ${\displaystyle P'(\rho _{0})}$ is a constant, resulting in the alternative form of the linear acoustics equations: $${\displaystyle {\begin{aligned}{\tilde {p}}_{t}+u_{0}{\tilde {p}}_{x}+K_{0}{\tilde {u}}_{x}&=0,\\\rho _{0}{\tilde {u}}_{t}+{\tilde {p}}_{x}+\rho _{0}u_{0}{\tilde {u}}_{x}&=0.\end{aligned}}}$$ where ${\displaystyle K_{0}=\rho _{0}P'(\rho _{0})}$ is the bulk modulus of compressibility. After dropping the tilde for convenience, the linear first order system can be written as: $${\displaystyle {\begin{bmatrix}p\\u\end{bmatrix}}_{t}+{\begin{bmatrix}u_{0}&K_{0}\\1/\rho _{0}&u_{0}\end{bmatrix}}{\begin{bmatrix}p\\u\end{bmatrix}}_{x}=0.}$$ While, in general, a non-zero background velocity is possible (e.g. when studying the sound propagation in a constant-strength wind), it will be assumed that ${\displaystyle u_{0}=0}$. Then the linear system reduces to the second-order wave equation: $${\displaystyle p_{tt}=-K_{0}u_{xt}=-K_{0}u_{tx}=K_{0}\left({\frac {1}{\rho _{0}}}p_{x}\right)_{x}=c_{0}^{2}p_{xx},}$$ with ${\displaystyle c_{0}={\sqrt {K_{0}/\rho _{0}}}}$ 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: $${\displaystyle q_{t}+Aq_{x}=0,}$$ with $${\displaystyle q={\begin{bmatrix}p\\u\end{bmatrix}},\quad A={\begin{bmatrix}0&K_{0}\\1/\rho _{0}&0\end{bmatrix}}.}$$ Conversely, given the second-order equation ${\displaystyle p_{tt}=c_{0}^{2}p_{xx}}$ a first-order system can be derived: $${\displaystyle q_{t}+{\hat {A}}q_{x}=0,}$$ with $${\displaystyle q={\begin{bmatrix}p_{t}\\-p_{x}\end{bmatrix}},\quad {\hat {A}}={\begin{bmatrix}0&c_{0}^{2}\\1&0\end{bmatrix}},}$$ where matrix ${\displaystyle A}$ and ${\displaystyle {\hat {A}}}$ are similar.^[5]

Provided that the speed ${\displaystyle c}$ is a constant, not dependent on frequency (the dispersionless case), then the most general solution is

${\displaystyle p=f(ct-x)+g(ct+x)}$

where ${\displaystyle f}$ and ${\displaystyle g}$ are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (${\displaystyle f}$) traveling up the x-axis and the other (${\displaystyle g}$) down the x-axis at the speed ${\displaystyle c}$. The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either ${\displaystyle f}$ or ${\displaystyle g}$ to be a sinusoid, and the other to be zero, giving

${\displaystyle p=p_{0}\sin(\omega t\mp kx)}$.

where ${\displaystyle \omega }$ is the angular frequency of the wave and ${\displaystyle k}$ is its wave number.

Feynman^[6] provides a derivation of the wave equation for sound in three dimensions as

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

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

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

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

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

${\displaystyle \nabla ^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=0}$

and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):

${\displaystyle \mathbf {u} =\nabla \Phi \;}$,

${\displaystyle p=-\rho {\partial \over \partial t}\Phi }$.

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 ${\displaystyle e^{i\omega t}}$ where ${\displaystyle \omega =2\pi f}$ is the angular frequency. The explicit time dependence is given by

${\displaystyle p(r,t,k)=\operatorname {Real} \left[p(r,k)e^{i\omega t}\right]}$

Here ${\displaystyle k=\omega /c\ }$ is the wave number.

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

${\displaystyle p(r,k)=AH_{0}^{(1)}(kr)+\ BH_{0}^{(2)}(kr)}$.

where the asymptotic approximations to the Hankel functions, when ${\displaystyle kr\rightarrow \infty }$, are

${\displaystyle H_{0}^{(1)}(kr)\simeq {\sqrt {\frac {2}{\pi kr}}}e^{i(kr-\pi /4)}}$

${\displaystyle H_{0}^{(2)}(kr)\simeq {\sqrt {\frac {2}{\pi kr}}}e^{-i(kr-\pi /4)}}$.

${\displaystyle p(r,k)={\frac {A}{r}}e^{\pm ikr}}$.

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.

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

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