Transverse cusp caustics produced by reflection and transmission: The caustic surface and optical simulations

Abstract

High‐frequency sound reflected from curved surfaces or refracted by inhomogeneities may produce cusped caustics that open up roughly transverse to the direction of propagation. Though the better‐known longitudinal cusp or arête is produced by a concave cylindrical wave front, transverse cusps are produced by distinctly noncylindrical wave fronts of the form W(x,y) = − (a1x2 + a2y2x + a3y2), where a2≠0 and the distance z to the observation plane ≠ (−2a1)−1 [Marston, J. Acoust. Soc. Am. 81, 226–232 (1987)]. The wave front propagates in the z direction and in the orthogonal (u,v) observation plane the caustic has the form D(u − uc)3 = v2. The present research is concerned with the caustic surface generated by considering the z dependence of the caustic parameters D and uc. Reflection and transmission problems that lead to this W(x,y) in the paraxial approximation are discussed as well as a novel property of merging rays. Sound radiated by a point source so as to reflect from a surface whose height relative to the xy plane is of the form h1x2 + h2 y2x + h3 y2 produces a transverse cusp. The reflected wavefield is described by the Pearcey function. Transverse cusps were simulated by reflecting light from a surface of this form. [Work supported by ONR.]