Ray Theory of the General Epstein Profile

Abstract

This paper presents the detailed mathematical properties and the ray-theory ranges, intensities, and travel times of the five-parameter Epstein profile with the parameters assuming all values capable of producing real velocities. Although examples represent the underwater-sound situation, results apply to any ray theory. The wealth of profile forms produces special cases requiring intricate analysis, but leads to more complete understanding of field theory. Analyses of rays that become horizontal at velocity extrema show little dependence on profile symmetry for the case of a velocity maximum, but a marked dependence for the case of a velocity minimum. This last is a point of significance apparently neglected in the literature covering ray-theory channels. Very steep, channeled rays, which become horizontal at very high (infinite) velocities, are shown to have a loop range πC(dZ/dC), where the velocity-slope product is evaluated as a limit at the infinite velocity. This expression, valid for profiles other than the Epstein profile, was suggested by examination of the hyperbolic cosine profile, which is a degenerate case of the Epstein profile.