Acoustic wave equation

Remarks

It is obtained from the mass and momentum equations of an inviscid fluid and an isentropic equation of state. One keeps only the terms that are linear in the perturbations $p'$, $\rho' = \rho - \rho_0$ and the velocity $u$. The density perturbation $\rho' = p' / c_0^2$ and the velocity potential $\phi$, with $u = \nabla \phi$ and $p' = -\rho_0 \, \partial \phi / \partial t$, satisfy the same equation.

Example: the plane wave $p'(x, t) = A \cos(k \cdot x - \omega t)$ with $k \in \R^3$ is a solution exactly when $\omega^2 = c_0^2 \abs{k}^2$. Indeed $\partial_t^2 p' = -\omega^2 p'$ and $\Delta p' = -\abs{k}^2 p'$.

Non-example: $p'(x, t) = \cos(x_1) e^{-t}$ is not a solution, since $\partial_t^2 p' = p'$ while $c_0^2 \Delta p' = -c_0^2 p'$.

In one space dimension every solution has the form $p'(x, t) = F(x - c_0 t) + G(x + c_0 t)$, two waves that move in opposite directions at speed $c_0$.