Modified Ray Theory for the Two-Turning-Point Problem

Abstract

A wavelength-dependent modified ray theory is developed to represent the reflected and transmitted fields for a harmonic point source located in an inhomogeneous medium giving rise to the two-turning-point, or split-beam, type of sound field. A generalization of the WKB procedure based on Weber's functions is used to construct approximate solutions that, unlike the ordinary WKB approximations, are valid throughout the two-turning-point region. These solutions are used for the construction of a contour-integral representation for the sound field of the point source. A stationary phase condition is invoked to develop, formally, a modified ray theory. For example, in a symmetric parabolic index of refraction n(z), the ordinary ray scheme is modified such that a ray with turning point at level z = z1 is reflected instead at some level z = zc. The cutoff level zc has the approximate value −(12)D[1+(z1/D)2], where the parameter D is wavelength dependent. The level z = zc can be interpreted as a reflection plane; there is some transmission from such rays, and it can be represented in the transmission region, beyond the second turning-point level, as originating from a transmission plane at level z = −zc.