Shape of the wavefront which produces a transverse cusp diffraction catastrophe

Abstract

Diffraction patterns characteristic of a transverse cusp are known to be observable in light scattered from drops [P. L. Marston, Opt. Lett. 10, 588–590 (1985)] or reflected from curved surfaces. It is to be expected that high‐frequency sound reflected from curved surfaces or refracted by inhomogeneities may also produce transverse cusps but to facilitate their description one must know the shape of the outgoing wave which propagates to produce such a cusp. A wave whose amplitude in the xy plane is exp[ik(a1x2 + a2y2x + a3y2 − ct)] is considered and the two‐dimensional diffraction integral which results from the Fresnel approximation of the Green's function is calculated. This diffraction integral reduces to one proportional to the Pearcey function P(X, Y) or to P* (X, Y) depending on the sign of a1 + (2z)−1, where is the distance from the xy plane. The real parameters X,Y in the one‐dimensional integral P depend on the aj, z, k, and the transverse coordinates in the observation plane. This transformation...