Elementary source-type solutions of the reduced wave equation

A class of exact and elementary solutions of the reduced wave equation exhibiting source-type behavior is obtained which considerably enlarges the catalogue of known elementary solutions for layered media, and which provides the first such example known for a medium possessing variability both in range (x) and depth (y). The medium, in general, is an unbounded one in which the square of the refractive index (n) is specified as an arbitrary biquadratic form in x and y. We are then able to obtain an elementary solution for a line source extending at right angles to the x–y plane. When n is independent of x (layered medium), we show that the solution for point-source excitation can be obtained in a similar form. We treat in some detail the cases: (i) n2 = 1−y2, corresponding to a sound-channel profile, which provides an example of cusped-caustic formation in the ray interpretation; and (ii) n2 = 1+y2, corresponding to a sound-barrier profile, which provides an example of the phenomenon known as ’’tunneling.’’ The latter can be analyzed in terms of Keller’s ’’complex rays.’’