# 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?

## 1 Answer

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.

• Thank you for your answer, it is already useful, but from a physical point of view the waves are always better described by the second equation and the first one is just an approximation or there are cases in which one is correct and the the other is not? For example sound waves, EM waves, beam vibrations etc. are subjected to different kind of damping? – Alessandro Zunino Dec 13 '18 at 16:42
• @AlessandroZunino I'll have to look that up to be honest. But it's an interesting question. – Nephente Dec 13 '18 at 17:23