Acoustic wave equation

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.

Definition in one dimension

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.

Derivation

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 $${\displaystyle \begin{array}{cc}{\rho }_t+(\rho u{)}_x& =0,\\ (\rho u{)}_t+(\rho u^2+P(\rho ){)}_x& =0.\end{array}}$$ This coupled system of two nonlinear conservation laws can be written in vector form as: $${\displaystyle q_t+f(q{)}_x=0,}$$ with $${\displaystyle q=\left[\begin{array}{c}\rho \\ \rho u\end{array}\right]=\left[\begin{array}{c}{q}_{(1)}\\ q_{(2)}\end{array}\right],f(q)=\left[\begin{array}{c}\rho u\\ \rho u^2+P(\rho )\end{array}\right]=\left[\begin{array}{c}{q}_{(2)}\\ q_{(2)}^2/q_{(1)}+P(q_{(1)})\end{array}\right].}$$

To linearize this equation, let $${\displaystyle q(x,t)=q_0+\tilde{q}(x,t),}$$ 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: $${\displaystyle f(q_0+\tilde{q})\approx f(q_0)+f^{\prime }(q_0)\tilde{q}}$$ where $${\displaystyle f^{\prime }(q)=\left[\begin{array}{cc}\mathrm{\partial }{f}_{(1)}/\mathrm{\partial }{q}_{(1)}& \mathrm{\partial }{f}_{(1)}/\mathrm{\partial }{q}_{(2)}\\ \mathrm{\partial }{f}_{(2)}/\mathrm{\partial }{q}_{(1)}& \mathrm{\partial }{f}_{(2)}/\mathrm{\partial }{q}_{(2)}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -u^2+P^{\prime }(\rho )& 2u\end{array}\right].}$$ This results in the linearized equation $${\displaystyle {\tilde{q}}_t+f^{\prime }(q_0){\tilde{q}}_x=0\iff \begin{array}{cc}{\tilde{\rho }}_t+(\widetilde{\rho u}{)}_x& =0\\ (\widetilde{\rho u}{)}_t+(-u_0^2+P^{\prime }({\rho }_0)){\tilde{\rho }}_x+2u_0(\widetilde{\rho u}{)}_x& =0\end{array}}$$ 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: $${\displaystyle p=p_0+\tilde{p}=P({\rho }_0+\tilde{\rho })=P({\rho }_0)+P^{\prime }({\rho }_0)\tilde{\rho }+\dots }$$ such that: $${\displaystyle p_0=P({\rho }_0),\tilde{p}\approx P^{\prime }({\rho }_0)\tilde{\rho },}$$ where ${\displaystyle P^{\prime }({\rho }_0)}$ is a constant, resulting in the alternative form of the linear acoustics equations: $${\displaystyle \begin{array}{cc}{\tilde{p}}_t+u_0{\tilde{p}}_x+K_0{\tilde{u}}_x& =0,\\ {\rho }_0{\tilde{u}}_t+{\tilde{p}}_x+{\rho }_0{u}_0{\tilde{u}}_x& =0.\end{array}}$$ where ${\displaystyle K_0={\rho }_0{P}^{\prime }({\rho }_0)}$ is the bulk modulus of compressibility. After dropping the tilde for convenience, the linear first order system can be written as: $${\displaystyle {\left[\begin{array}{c}p\\ u\end{array}\right]}_t+\left[\begin{array}{cc}{u}_0& K_0\\ 1/{\rho }_0& u_0\end{array}\right]{\left[\begin{array}{c}p\\ u\end{array}\right]}_x=0.}$$ 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: $${\displaystyle q_t+A{q}_x=0,}$$ with $${\displaystyle q=\left[\begin{array}{c}p\\ u\end{array}\right],A=\left[\begin{array}{cc}0& K_0\\ 1/{\rho }_0& 0\end{array}\right].}$$ 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 $${\displaystyle q=\left[\begin{array}{c}{p}_t\\ -p_x\end{array}\right],\hat{A}=\left[\begin{array}{cc}0& c_0^2\\ 1& 0\end{array}\right],}$$ where matrix ${\displaystyle A}$ and ${\displaystyle \hat{A}}$ are similar.

Solution

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.