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.

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.

Start with the ideal gas law

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