the method of stationary phase
After you solve a wave-like PDE by transform, the answer is an inverse-transform integral: a sum over all frequencies of rapidly oscillating waves. To learn what the solution looks like far away and long after — its large-time behaviour — you must estimate this integral. But adding up wildly oscillating terms is delicate: almost everywhere the oscillations cancel. The method of stationary phase is the insight that the integral is dominated by just a few special frequencies, the ones where the oscillation momentarily stops racing.
Here is the picture. The inverse transform looks like integral of A(k) e^(i t phi(k)) dk for large t, where phi(k) is a phase (built from the dispersion relation). When t is large, e^(i t phi(k)) spins around extremely fast as k varies, so neighbouring contributions point in opposite directions and cancel — except where the phase is momentarily flat, that is, where phi'(k) = 0. At such a stationary point the oscillation pauses, nearby contributions add up in step instead of cancelling, and that one neighbourhood supplies essentially the whole integral. Expanding the phase to second order there gives a Gaussian-type integral you can do exactly, and the result is an asymptotic formula: the solution decays like a power of t (typically 1/sqrt(t) in one dimension) times an oscillation set by the stationary point.
The payoff is physical, not just technical. The stationary-phase condition phi'(k) = 0 is exactly the statement that an observer moving at a fixed velocity sees the wavenumber whose group velocity matches that velocity — so stationary phase is the rigorous reason a wave packet travels at the group velocity, and why dispersive waves spread out and decay in amplitude over time. It is the standard route to the large-time asymptotics of solutions of the Schrodinger equation, water waves, and other dispersive equations. The honest limit: it gives the leading large-t behaviour, not an exact answer, and it needs a genuine stationary point with non-vanishing second derivative; when phi'' also vanishes (coalescing stationary points) the simple 1/sqrt(t) law fails and you need finer asymptotics.
For the free Schrodinger equation the solution at position x and large time t is an integral with phase phi(k) = k(x/t) - omega(k)/... whose stationary point sits at the k with group velocity equal to x/t. Stationary phase then gives the familiar spreading wave packet decaying like 1/sqrt(t): the probability you find the particle near x is governed by the single wavenumber whose group velocity carries it there.
Only frequencies where the phase is momentarily flat survive — and that is the group-velocity frequency.
Stationary phase handles oscillatory integrals (real, rapidly spinning phase); when the exponent has a real decaying part instead you want its cousin, the method of steepest descent. The clean 1/sqrt(t) decay assumes a single, non-degenerate stationary point.