Geometrically Derived Ray-Theory Results and Direct Verification of the Pekeris Solution for Unbounded Constant-Gradient Media

This paper directly verifies the Pekeris solution to the point-source Helmholtz equation for an unbounded constant-gradient medium using only elementary vector calculus. Self-contained geometrical derivations of ray-theory results for such a medium are presented: (1) ray-path location and travel time as a function of source location, ray start angle, and ray angle; (2) the wavefront equation as a function of source location and travel time; (3) the wavefront location and ray angle along a ray as a function of source location, ray start angle, and travel time; and (4) source angle and receiver angle as a function of source location and receiver location. A short mathematical derivation gives the travel time between two points for a given source location and a given receiver location. In some cases, the form of the results seems to be simpler than that of the equivalent results previously given in the literature.