Imaged travel time surface of the transverse cusp caustic produced by reflection from a curved surface and the shape of reflected transients.

Abstract

As a curved wavefront propagates in a uniform medium it may form caustics. Catastrophe theory can be used to classify the structurally stable caustics, where the initial local shape of the wave front determines the classification. These caustics have an associated travel time surface that describes the merging and disappearance of rays as the caustic is crossed. The travel time surface is defined by the travel time of signals propagating along rays from the wave front to the observer. Experiments were performed to image an acoustical transverse cusp diffraction catastrophe along with its associated travel time surface. The finite difference in arrival times of the acoustical signals allows the resolution of the travel time surface. Reflection of a spherical source signal from a curved surface in water was used to produce an initial wave front with the general local shape that produces a transverse cusp. Calculations of the expected travel time surface compared well to images cut through the experimental surface. Acoustical signals traveling along rays that touch the caustic are shown to have the Hilbert transform shape of the incident signal. Those signals that do not touch the caustic have the same general shape as the incident signal. [Work supported by ONR.]