Doubly focused backscattering from finite targets in an Airy caustic formed by a curved reflecting surface

Caustics can be formed in the water column when sound scatters off a curved-reflecting surface such as the ocean floor or surface. The simplest caustic is an Airy caustic formed by the merging of two rays. Small targets lying in or near Airy caustics have backscattered echoes that can be focused both to the target and upon return. For a point target, the doubly focused backscattering amplitude is proportional to the square of an Airy function whose argument depends on the target location through the changes in relative return times of contributing rays. For a finite sized target, the symmetry is broken and the amplitude unfolds into a hyperbolic umbilic catastrophe. The arguments for the hyperbolic umbilic function are calculated using the relative return times of transient echoes. These doubly focused echoes can lead to amplitudes larger than that of direct or singly focused echoes (echoes which focus once, either to the target or upon return). Experiments using a cylindrical half-pipe as a reflecting surface confirm these predictions.