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 a medium having a maximum in sound velocity and giving rise to the two-turning-point problem. In a stratified inhomogeneous medium with index of refraction n(z) a function of z only, the wave equation can be separated. The undifferentiated term in the separated z equation has a factor (n2−a2) in its coefficient, where a is the separation constant. The geometrical-optics approximation fails in the neighborhood of the zeros (turning points) of this coefficient. The two-turning-point problem arises when n(z) has a minimum and there are two z values at which the coefficient is zero. A generalization of the WKB procedure based on Weber's functions is used to construct approximate solutions valid throughout the two-turning-point regions. These solutions are then applied to the construction of a contour integral representation for the field of a harmonic point source. A stationary phase condition on the integrand is invoked to develop, formally, a modified ray theory to represent the field in reflection and transmission regions. There are elements in the phase that depend on the separation constant a and lead to significant modifications of the ray scheme based on the stationary phase condition.