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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.

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  • $\begingroup$ 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. $\endgroup$ – Emilio Pisanty Oct 29 '19 at 11:03
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Since you are seeking dispersion relation, you can insert your solution into your wave equation. You can assume your wave function is

$$ \psi(x,t) = Ae^{i(kx-\omega(k)t)} $$

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 $$ \omega(k) = vk. $$

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