Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface in water

Abstract

The transient response of a wave front that forms a transverse cusp caustic is studied. The travel time surface of a simple cuspoid caustic is known to have the general shape of the singular surface of the next higher order cuspoid catastrophe. Though the transverse cusp wave front considered here is curved in two dimensions, it is also shown to have a travel time surface with the general shape of the singular surface of the swallow tail catastrophe. Ultrasonic experiments to image the travel time surface of the transverse cusp caustic were performed. The imaged travel time surface has the shape of the swallow tail singular surface imposed on a slowly varying travel time surface due to the spherical nature of the source. Imaged cuts through the travel time surface compare very well to calculated travel time curves. The calculated curves are a superposition of a smooth background contribution and a contribution due to the shape of the reflecting surface. It is the second contribution that describes the merging and disappearance of rays as the cusp caustic is crossed. There are no adjustable parameters used in the comparison. It is possible to infer at least qualitative information about the reflecting surface from the imaged travel time surface.