the dispersion relation

Plug a single pure wave into a linear PDE and ask: for this wave to be a genuine solution, how must its frequency relate to its wavelength? The answer is a formula linking the temporal frequency omega to the spatial wavenumber k, and that formula is the dispersion relation. It is the fingerprint of the equation: read it and you know how fast waves of each wavelength travel and whether the medium spreads a pulse out or keeps it together.

Here is the mechanism, and it falls right out of the transform machinery. Substitute the trial wave u(x, t) = e^(i(k x - omega t)) into the PDE. Every x-derivative brings down a factor i*k and every t-derivative a factor -i*omega — this is just the differentiation rule for transforms in disguise. The PDE then reduces to a purely algebraic equation relating omega and k, with no derivatives left: that equation, solved for omega as a function of k, is omega = omega(k). For the wave equation u_tt = c^2 u_xx you get omega^2 = c^2 k^2, so omega = c|k| — every wavelength travels at the same speed c. For the free Schrodinger equation you get omega proportional to k^2, and for the heat equation omega = -i k^2, an imaginary frequency that signals decay rather than oscillation.

The shape of omega(k) sorts the world into two kinds. If omega is exactly proportional to k (a straight line through the origin, as for the simple wave equation), all wavelengths move in lockstep and a pulse keeps its shape — non-dispersive. If omega(k) is curved, different wavelengths move at different speeds, so a pulse made of many wavelengths gradually spreads and reshapes — dispersive, and that is literally where the word comes from. From this one curve you read off both the phase velocity and the group velocity, and an imaginary part of omega tells you about growth or damping (instability or diffusion). It is the single most informative thing you can extract from a constant-coefficient linear PDE.