Transverse cusp diffraction catastrophes: Some pertinent wave fronts and a Pearcey approximation to the wave field

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Abstract
Diffraction patterns characteristic of transverse cusps are known to be observable in light reflected from curved surfaces or scattered from liquid drops. It is anticipated that transverse cusps may be produced when high-frequency sound is reflected from (or radiated by) certain curved surfaces or is refracted by inhomogeneities. An explicit description is given of a wave which propagates to produce a transverse cusp; the amplitude in the xy plane is exp[ik(g−ct)] with g=a1x2+a2y2x+a3y2, a2≠0. Propagation of this wave in a homogeneous medium is shown to yield a shear-free transverse cusped caustic which locates a transition in the number of rays which contribute to the amplitude. The Fresnel approximation of the two-dimensional diffraction integral is evaluated. The diffracted wave field is proportional to the Pearcey function P(X,Y) or to P*(X,Y), depending on the sign of a1+(2z)−1, where z is the distance from the xy plane. The real parameters X, Y depend on the aj, z, k, and the transverse coordinates in the observation plane. The stationary-phase points for the diffraction integral are discussed. The problem considered is distinct from that of acoustical longitudinal cusps which unfold along the propagation direction.

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