The Airy function Ai(x) solves y'' = xy. Near a caustic (rainbow, lens focus), the intensity goes as Ai²(ξ) where ξ is scaled angle from the rainbow. Geometric optics (ray theory) predicts infinite intensity at the rainbow angle — Airy's wave theory gives the correct finite oscillating pattern with supernumerary arcs.
Airy (1838) solved the rainbow by applying the wave equation at the caustic. The first zero of Ai(x) is at x ≈ −2.338. The rainbow angle (41.5°) is the caustic — ray theory's "infinity" is wave theory's oscillating envelope.