# 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.

## Definition in one dimension

The wave equation describing a standing wave field in one dimension (position $x$) is
$p_{xx}-{\frac {1}{c^{2}}}p_{tt}=0,$
where $p$ is the acoustic pressure (the local deviation from the ambient pressure) and $c$ the speed of sound, using subscript notation for the partial derivatives.

### Derivation

Start with the ideal gas law
    $P=\rho R_{\text{specific}}T,$
where $T$ the absolute temperature of the gas and specific gas constant $R_{\text{specific}}$. 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 equations ${\begin{aligned}\rho _{t}+(\rho u)_{x}&=0,\\(\rho u)_{t}+(\rho u^{2}+P(\rho ))_{x}&=0.\end{aligned}}$ This coupled system of two nonlinear conservation laws can be written in vector form as: $q_{t}+f(q)_{x}=0,$ with $q={\begin{bmatrix}\rho \\\rho u\end{bmatrix}}={\begin{bmatrix}q_{(1)}\\q_{(2)}\end{bmatrix}},\quad f(q)={\begin{bmatrix}\rho u\\\rho u^{2}+P(\rho )\end{bmatrix}}={\begin{bmatrix}q_{(2)}\\q_{(2)}^{2}/q_{(1)}+P(q_{(1)})\end{bmatrix}}.$
To linearize this equation, let $q(x,t)=q_{0}+{\tilde {q}}(x,t),$ where $q_{0}=(\rho _{0},\rho _{0}u_{0})$ is the (constant) background state and ${\tilde {q}}$ is a sufficiently small perturbation, i.e., any powers or products of ${\tilde {q}}$ can be discarded. Hence, the taylor expansion of $f(q)$ gives: $f(q_{0}+{\tilde {q}})\approx f(q_{0})+f'(q_{0}){\tilde {q}}$ where $f'(q)={\begin{bmatrix}\partial f_{(1)}/\partial q_{(1)}&\partial f_{(1)}/\partial q_{(2)}\\\partial f_{(2)}/\partial q_{(1)}&\partial f_{(2)}/\partial q_{(2)}\end{bmatrix}}={\begin{bmatrix}0&1\\-u^{2}+P'(\rho )&2u\end{bmatrix}}.$ This results in the linearized equation ${\tilde {q}}_{t}+f'(q_{0}){\tilde {q}}_{x}=0\quad \Leftrightarrow \quad {\begin{aligned}{\tilde {\rho }}_{t}+({\widetilde {\rho u}})_{x}&=0\\({\widetilde {\rho u}})_{t}+(-u_{0}^{2}+P'(\rho _{0})){\tilde {\rho }}_{x}+2u_{0}({\widetilde {\rho u}})_{x}&=0\end{aligned}}$ Likewise, small perturbations of the components of $q$ can be rewritten as: $\rho u=(\rho _{0}+{\tilde {\rho }})(u_{0}+{\tilde {u}})=\rho _{0}u_{0}+{\tilde {\rho }}u_{0}+\rho _{0}{\tilde {u}}+{\tilde {\rho }}{\tilde {u}}$ such that ${\widetilde {\rho u}}\approx {\tilde {\rho }}u_{0}+\rho _{0}{\tilde {u}},$ and pressure perturbations relate to density perturbations as: $p=p_{0}+{\tilde {p}}=P(\rho _{0}+{\tilde {\rho }})=P(\rho _{0})+P'(\rho _{0}){\tilde {\rho }}+\dots$ such that: $p_{0}=P(\rho _{0}),\quad {\tilde {p}}\approx P'(\rho _{0}){\tilde {\rho }},$ where $P'(\rho _{0})$ is a constant, resulting in the alternative form of the linear acoustics equations: ${\begin{aligned}{\tilde {p}}_{t}+u_{0}{\tilde {p}}_{x}+K_{0}{\tilde {u}}_{x}&=0,\\\rho _{0}{\tilde {u}}_{t}+{\tilde {p}}_{x}+\rho _{0}u_{0}{\tilde {u}}_{x}&=0.\end{aligned}}$ where $K_{0}=\rho _{0}P'(\rho _{0})$ is the bulk modulus of compressibility. After dropping the tilde for convenience, the linear first order system can be written as: ${\begin{bmatrix}p\\u\end{bmatrix}}_{t}+{\begin{bmatrix}u_{0}&K_{0}\\1/\rho _{0}&u_{0}\end{bmatrix}}{\begin{bmatrix}p\\u\end{bmatrix}}_{x}=0.$ 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 $u_{0}=0$. Then the linear system reduces to the second-order wave equation: $p_{tt}=-K_{0}u_{xt}=-K_{0}u_{tx}=K_{0}\left({\frac {1}{\rho _{0}}}p_{x}\right)_{x}=c_{0}^{2}p_{xx},$ with $c_{0}={\sqrt {K_{0}/\rho _{0}}}$ 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: $q_{t}+Aq_{x}=0,$ with $q={\begin{bmatrix}p\\u\end{bmatrix}},\quad A={\begin{bmatrix}0&K_{0}\\1/\rho _{0}&0\end{bmatrix}}.$ Conversely, given the second-order equation $p_{tt}=c_{0}^{2}p_{xx}$ a first-order system can be derived: $q_{t}+{\hat {A}}q_{x}=0,$ with $q={\begin{bmatrix}p_{t}\\-p_{x}\end{bmatrix}},\quad {\hat {A}}={\begin{bmatrix}0&c_{0}^{2}\\1&0\end{bmatrix}},$ where 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 is
    $p=f(ct-x)+g(ct+x)$
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, giving
    $p=p_{0}\sin(\omega t\mp kx)$.
where $\omega$ is the angular frequency of the wave and $k$ is its wave number.

## In three dimensions
### Equation

Feynman 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
    $\nabla ^{2}\mathbf {u} \;-{1 \over c^{2}}{\partial ^{2}\mathbf {u} \; \over \partial t^{2}}=0$.
In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form
    $\nabla ^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=0$
and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):
    $\mathbf {u} =\nabla \Phi \;$,
    $p=-\rho {\partial \over \partial t}\Phi$.

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

#### Cartesian coordinates

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

#### Cylindrical coordinates

    $p(r,k)=AH_{0}^{(1)}(kr)+\ BH_{0}^{(2)}(kr)$.
where the asymptotic approximations to the Hankel functions, when $kr\rightarrow \infty$, are
    $H_{0}^{(1)}(kr)\simeq {\sqrt {\frac {2}{\pi kr}}}e^{i(kr-\pi /4)}$
    $H_{0}^{(2)}(kr)\simeq {\sqrt {\frac {2}{\pi kr}}}e^{-i(kr-\pi /4)}$.

#### Spherical coordinates

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