Doubly-focused echos from spheres unfold into a hyperbolic umbilic diffraction catastrophe

Abstract

An underwater spherical target resides in an Airy field formed by reflection off a curved surface representing the sea floor or sea surface. In prior work [B. Dzikowicz and P. L. Marston, J. Acoust. Soc Am. 110, 2778 (2001)] direct returns of a tone burst from the surface reflection focused toward the target were shown to have a dependence on the target position described by an Airy function. The return echo can also be focused again by the surface onto the source and receive transducer. This gives the square of an Airy function for the case of a point target. With a finite sized target (as in the experiment) this goes over to a hyperbolic umbilic catastrophe with symmetric arguments. The arguments of the hyperbolic umbilic function are derived from only the relative return times of a transient pulse. Experiment confirms the predicted merging of transient echoes in the time domain, as well as the hyperbolic umbilic diffraction integral amplitudes for a tone burst. This method would allow for the observation of a target at a greater distance in the presence of a focusing surface. [Research supported by ONR.]