Behavior of Modified Ray Theory

Abstract

Both modified ray theory [E. L. Murphy, J. Acoust. Soc. Amer. 47, 899–908 (1970) and J. A. Davis, J. Acoust. Soc. Amer. 52, 137 (1972)] and ordinary ray theory can be obtained from the method of stationary phase. Use of a WKB approximation leads to two points of stationary phase and the result is ray theory. Use of a generalized WKB approximation leads to an infinite number of stationary phase points and the result of using the first two is modified ray theory. The degree to which modified ray theory accurately includes frequency dependent diffraction effects depends on the contribution to the field by the remaining points of stationary phase. In sound propagation problems where there is a shadow region, the ray theory field may or may not contain a caustic, whereas the modified ray theory field always contains a caustic and in some cases a cusped or higher-order caustic. For ranges much less than the shadow boundary range the modified ray theory field approaches the ray theory field. For ranges much greater than the shadow range, the modified ray theory field (suitably extended to take the caustic into account) approaches either (1) the ray theory field (also extended), or (2) the field due to the first mode, depending on whether or not the problem parameters are such that a ray theory caustic is present. For ranges in the vicinity of the shadow range, the behavior can be very complicated.