What is a dispersion relation?
Well, that's a bit of a tricky question from a standing start. Essentially, it's something that tells you about how a specific type of wave (like a sound waves, or an electromagnetic wave, etc.) will act in a certain medium. So, before going any further, what does one mean by a 'wave'? Here's some maths... (don't be alarmed!) A 'wave' is something which looks like this:
exp(i(k x - ω t))
What does that mean? Well, exp is an exponential. k is one over the wavelength, and ω is the frequency (or the frequency times a constant, strictly). If this doesn't make sense to you, keep reading; I'll try to provide some intuition through examples. A dispersion relation is an equation that will relate ω and k. i, here, is the square root of minus one, and its presence indicates we're looking for waves - things which go up and down.
So, why care about a dispersion relation? Well, in a lot of physical cases, you can't just bung in any values of ω and k - the actual stuff your wave is travelling through has something to say about that. Let's look at a concrete example.
A concrete example
Consider speaking down a long, rectangular air duct. Air ducts don't like to transmit every sound (making your voice sound 'flatter' and more boring, unless your voice makes the metal vibrate and add tinny sounds of its own). If we do some maths, we get the dispersion relation for a rectangular waveguide:
ω2 - cs2k2 = cs2 a-2(n2 + m2)
Here, the geometry of the problem has taken the usual dispersion relationship for speaking in air (the left hand side) and added in a constraint of these numbers n and m on the right - these numbers are integers, at least one.
That wasn't very interesting.
An interesting example - the Plateau-Rayleigh Instability
This is going to be more involved. See, earlier, when I said the presence if 'i' indicated we were looking for waves? We aren't necessarily going to find waves - at least not ones that travel. Some waves (called instabilities) don't travel at all, but just grow. Finding instabilities and understanding them is a fundamental part of applied mathematics.
Consider a cylinder of fluid, surrounded by nothing (air is close to being nothing). Imagine, for example, you've got some golden syrup, and you're pouring it nice and slow over some delicious waffles. Well, the vibrations from your hand (and the air currents in the room, and even thermal fluctuations) are going to make ti-iny waves on the surface of that cylinder. Let's see what happens.
For simplicity's sake, say that the long cylinder (which, without the waves, would be radius a) isn't moving, and initially has a tiny wave on its surface - to remind us that it's tiny, let's say ε is a really tiny thing, and so we write:
radius = a( 1 + ε exp(i(kx - ω t)))
All we've done is say, we have a cylinder; it has a little wave on its surface; for simplicity we're going to assume it's not initially moving (although it doesn't matter, actually). In principle, if we felt like doing some work, then we could apply our knowledge of fluid dynamics to remove everything from the problem except k and ω. Well, (skipping ahead) these waves aren't going to grow. We find that for real of k (here, you can pleasingly read 'real' to mean 'allowed'), ω is imaginary. For completeness, I include Lord Rayleigh's original result here (although it's nasty):
i ω = actually, this is too nasty to bother. Something 'real'
So what? You say. Well, this is a basic example of a stability argument. If ω is imaginary, i ω is real, and that means the size of our disturbance can grow. If our waves don't propagate, but stay where they are and grow exponentially with time, we know that a system is unstable (and won't occur in nature - or, not for long).
These instabilities need time to grow. Try pouring it from a metre onto a plate on the floor, and watch it separate into droplets before hitting the plate.
A dispersion relationship is something that tells us how waves behave in a medium or situation; it also tells us if a system is unstable.
At least, unless we have a continuous spectrum. But I don't understand them yet.