computational physics - Minimal Extension of Wave Equation to Include Dispersion

Let's say you are modeling some process with the wave equation $\frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = \nabla^{2}\psi$. You wish to improve your model by including dispersive effects, but you want your model to be as simple as possible for computational tractability.

What is the minimal appropriate model for phase velocity, say $c \approx c_{0} + c_{2}\omega^{2}$? and how should the wave equation be altered?

Answer

What you want to do is change the wave equation into a Klein-Gordon equation:

$$\frac {1}{c^2} \frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi + \alpha^2 \psi = 0,$$

where $\alpha$ is a constant of appropriate dimension and usually (in quantum theory) given by

$$\alpha=\frac {m c}{\hbar}.$$

Inserting an ansatz of the form

$$\psi=e^{i(kx-\omega t)}$$

yields the dispersion relation

$$\omega^2=c^2(k^2+\alpha^2),$$

from which one can deduce an expression for the phase velocity, given by

$$v_{phase}=c\sqrt{1+\frac{\alpha^2}{k^2}}.$$

You might consider reading these lecture notes for more insights.