Observations of transverse cusp diffraction catastrophes produced by reflecting long and short ultrasonic bursts from a curved metal surface in water

Abstract

Transverse cusp caustics are predicted to be produced when sound or light from a point source is reflected by a smooth curved surface having the general shape h(x,y) = h1x2 + h2xy2 + h3y2 + h4x + h5y (with h2≠0) [P. L. Marston, in Acoustical Imaging (Plenum, New York, 1988), Vol. 16, p. 579]. The transverse cusp curve partitions space into regions where either one or three rays contribute to the wave field. The diffraction pattern exhibited by a cusp caustic is proportional to the Pearcey function. Experiments were conducted that produced a transverse cusp using a focused source of 1‐MHz sound and a smooth curved metal surface as a reflector. The shape of the reflector was measured to determine the parameters in h(x,y) for that particular surface. Long tone bursts were used to simulate a steady‐state signal in order to image the acoustical wave field in a plane distant from the reflector. Experimental and calculated diffraction patterns are in fair agreement when the measured values of h1 and h2 were used in the calculation. Single‐cycle bursts were used to show the transition from the three‐ray to the one‐ray region across the cusp curve. Temporal records manifest the transition expected for slices through the imaged wave field. An optical source and receiver replaced the acoustical source and receiver to image the corresponding optical wave field. A distinct optical cusp curve was visible as expected for the short wavelength limit. [Work supported by ONR.]