Asymptotic Evaluation of Integrals Related to Time-Dependent Fields Near Caustics

Abstract

Integrals which arise in the theory of time-dependent wave fields near caustics in nondispersive problems are much simpler than their time-harmonic counterparts because they involve delta functions rather than exponentials. Fields near smooth caustics, for instance, involve solutions of cubic equations, which can be expressed in elementary terms, instead of the Airy functions of the better known time-harmonic theory. The Pearcey functions, which arise in the theory of the cusped caustic, are similarly replaced by expressions involving solutions of quartic equations, etc. As is well known, each type of caustic is related to a particular type of catastrophe. The theory presented here deals conveniently with fields reducible to single integrals since the delta function and the integral will annihilate each other, leaving closed-form expressions; fields involving multiple integrals will be reduced somewhat in complexity, but not in general to closed form, and are not treated here.