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?


1 Answer 1


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


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


You might consider reading these lecture notes for more insights.

  • 1
    $\begingroup$ I'm completely surprised that this can be done while keeping the equation linear! $\endgroup$
    – user14717
    Commented Sep 1, 2013 at 23:49
  • $\begingroup$ Sometimes the solution to a problem is simpler than expected. ;) $\endgroup$ Commented Sep 2, 2013 at 0:14

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