What exactly is a dispersion relation?
It shows up all the time in my physics classes and I never fully understood its meaning.
Currently I am working on the linear chain model in crytsallography and after doing all the work of using F=ma and doing an ansatz with complex waves, all that shows up and seems to be of general interest is the following dispersion relation:
$$\omega=2\sqrt{\frac{k}{m}}\left\vert{\sin\frac{ka}{2}}\right\vert$$
I don't care that much for the specific problem at hand, I'd rather like to know why expressions like these show up so often and how I should read them?
Thanks for any help!
Many physical effects are modeled by systems of linear partial differential equations that have certain elementary plane-wave solutions in space-time: solutions of the form $ e^{i \vec k \cdot\vec x+\omega t}$. (This is especially true of partial differential equations that have constant coefficients.) Often there are boundary conditions or other compatibility conditions that impose one or more constraints on $\omega$ and $\vec k$. These are the dispersion relations.
Additional comments. One can visualize each such constraining relation as defining a surface in the frequency domain $(\vec k, \omega)$ that is dual to space-time. The intersection of all these constraints defines a special subset $S$ in that dual space. An arbitrary superposition of such plane-wave solutions selected from $S$ is also a solution (by linearity). Thus one is led to consider expressions such as $$(1)\qquad I(\vec x, t)=\int_{(\vec k, \omega)\in S} a(\vec k, \omega) e^{ i\vec k \cdot \vec x + \omega t}$$ which are (at least formally) solutions of the PDE system.
Typically the set $S$ is a smooth or piecewise smooth curved submanifold, and then one can give a rigorous definition of such superposition integrals.
Moreover, by approximating the curved object $S$ by an osculating polynomial approximation, one can sometimes construct explicit analytical approximations to true solutions that are asymptotically good in certain regimes (e.g. far-field regions in $(x,t)$ space).
To understand how the geometry of $S$ affects the integral (1) , it might be helpful to mentally transpose the roles of space-time and its dual space $(\vec k, \omega)$ when visualizing the comment that follows.
We can interpret (1) as the how much illumination from a plane wave directed along $<\vec x,t>$ is absorbed by the target surface $S$. Then we see that the net contribution to the integral (1) when a very high-frequency plane wave arriving from the direction $<x,t>$ encounters a small surface patch $\Delta S$ will generally be small (because high-frequency oscillations passing through $S$ mostly self-cancel.) The exception to this principle occurs when the plane wave-fronts hit $\Delta S$ "broadside". That is the direction $<x,t>$ is a normal vector to the patch $ \Delta S$. This is a so-called stationary phase condition, and is the basis for many asymptotic approximations in the far-field regions.
The modern theory of oscillatory integral operators has been built to create a rigorous foundation for such intuitive principles.