the method of stationary phase
Many integrals in optics, acoustics and wave physics look like (integral of f(t) e^(i N phi(t)) dt) for large N, where the amplitude f is slowly varying and the phase N phi(t) spins around very fast. Where the phase spins fast, the integrand swings between positive and negative so quickly that neighboring contributions cancel almost completely. The only places that survive are where the spinning momentarily slows to a stop — where the phase is stationary. The method of stationary phase says: the whole integral is dominated by tiny windows around the points where phi'(t) = 0.
At such a stationary point t_0 the phase looks locally like phi(t_0) + (1/2) phi''(t_0)(t - t_0)^2, and the integral over that window evaluates to a Fresnel-type Gaussian. The leading contribution from each stationary point is f(t_0) e^(i N phi(t_0)) times the square root of (2 pi / (N |phi''(t_0)|)), times a phase e^(i (pi/4) sign of phi''(t_0)) that records whether the phase curves up or down. Summing over all stationary points gives the large-N asymptotics, with size of order 1 over the square root of N — much larger than the exponentially small leftovers from the rapidly oscillating regions.
Stationary phase is the oscillatory twin of steepest descent: where steepest descent has a real exponent that decays, stationary phase has a purely imaginary exponent that oscillates, and the two are unified by deforming the contour into the complex plane (a stationary point of phi is a saddle of i phi). It explains why you see a sharp caustic of bright light, why a rainbow has a definite angle, and why far-field wave patterns concentrate along rays of geometric optics. The caveat: it is again an asymptotic method; if two stationary points collide, the simple formula breaks and you must use a uniform approximation (an Airy function near a fold caustic).
For (integral from minus infinity to infinity of e^(i N (t^3/3 + x t)) dt) — the integral defining the Airy function — the phase phi(t) = t^3/3 + x t has phi'(t) = t^2 + x. When x < 0 there are two real stationary points at t = plus or minus the square root of -x, and stationary phase predicts an oscillating, decaying tail; when x > 0 there is no real stationary point and the function decays exponentially instead.
The Airy integral's two stationary points (for x < 0) give its oscillatory tail.
Stationary phase only sees real stationary points of the phase; an endpoint of the integration range or a point where the amplitude f is non-smooth also contributes, and those boundary terms are easy to forget.