# Exact form of the damped wave equation

The undamped wave equation has the standard form

$$\begin{equation*} \frac{\partial^2 \psi}{\partial t^2}=c^2\nabla^2\psi \end{equation*}$$

while the damped wave equation is frequenly found written in this way

$$\begin{equation*} \frac{\partial^2 \psi}{\partial t^2}=c^2\nabla^2\psi + \nu\frac{\partial \psi}{\partial t} \end{equation*}$$

but in a few other occasions you can find this form

$$\begin{equation*} \frac{\partial^2 \psi}{\partial t^2}=\nabla^2 \left(c^2\psi + \nu\frac{\partial \psi}{\partial t}\right) \end{equation*}$$

From the mathemathical point of view these two equations are very different, but since they are both called damped wave equation I expect them to describe similar phenomena. Can you please explain to me which are the phenomena described by the one and those described by the other?

Both equations describe damping. The second variant though creates a wavenumber dependend attenuation. If you Fourier-transform the equation, you'll see that the imaginary damping term in the second one reads $$\mathrm{i}\nu k^2\omega$$ while the $$k^2$$ is absent in the first variant. Hence the first variant describes a situation in which each eigenmode is damped equally strongly, while the second describes stronger damping at shorter wavelenght.