Exact form of the damped wave equation

Question

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?

Answer 1

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.