Wave Propagation near a Smooth Caustic

A caustic is an envelope of a family of rays of geometrical optics. At a caustic, the usual equations of geometrical optics are not valid. We give a formal asymptotic series, valid both near and away from a smooth caustic, which satisfies the differential equation exactly. In the case of high-frequency oscillations, the solution is represented in terms of the Airy function and its derivative. There is a more general formulation (corresponding to a progressing wave expansion) in which the solution is expanded in terms of solutions of the Tricomi equation. Our procedure can be applied to general linear hyperbolic or time-reduced partial differential equations. The coefficients in our expansion are nearly the same as the coefficients which appear in ordinary geometrical optics; the only essential difference is that a factor which is singular at the caustic has been removed. Our terminology is explained in R. Courant, Methods of mathematical physics, Vol. II, Ch. VI. In order to illustrate our procedure, we consider the reduced wave equation Aw+few = 0, and we give only the leading term in the expansion. All of the essential features are illustrated in this special problem. We write the first term as