# Dispersion relation of a medium

In a dispersive medium I have a wave equation: $$\frac{\partial^2E}{\partial z^2} + \eta \frac{\partial^4E}{\partial z^4} - \frac{1}{c^2}\frac{\partial^2E}{\partial t^2}=0.$$ How can I find the dispersion relation for this medium? Nothing else is given.

• Note that we use MathJax to typeset mathematics; you can find a good tutorial here. I have edited your first question for you $-$ please use this notation in future posts. (I have also changed your notation from $\frac{\delta^2E}{\delta z^2}$ and similar to the more usual $\partial$ notation, which is typeset as \partial. If you want to retain the original notation for some reason (?), feel free to roll it back. – Emilio Pisanty Oct 29 '19 at 11:03

$$\psi(x,t) = Ae^{i(kx-\omega(k)t)}$$
Or just put $$A\cos(kx-\omega(k)t)$$, it is up to you.
When you take derivatives you will find a relation between $$\omega$$ and $$k$$. Your equation contains dispersive terms, you will get an extra terms instead of getting non dispersive relation, which is $$\omega(k) = vk.$$