An exact ray theoretical formulation of the Helmholtz equation

Abstract

Practical computational procedures for obtaining ray theoretical solutions to the inhomogeneous Helmhotz equation ∇2Ψ+k2Ψ=S(r,ω) resort to a well-known approximation. A computational method is presented that enables one to trace rays without resorting to the ray theory approximation, provided a solution to the Helmholtz equation is available by independent means. In other words, given a solution to the Helmholtz equation, the exact rays for that case can be computed. This ray theory therefore serves, not as a computational method, but as a new method of displaying solutions to the Helmholtz equation. Exact ray diagrams are constructed for several cases using this technique. The resulting ray diagrams usually bear little resemblance to the corresponding classical ray diagrams. It is shown that the discrepancy is attributable to the nature of the classical ray theory approximation, which proves in most cases not to be a small perturbation. Some of the properties of the exact rays that distinguish them from their classical counterparts are: (1) The ray trajectories depend on the source frequency and configuration and on the boundaries; (2) the exact rays intrude into shadow zones impenetrable by classical rays; (3) the field is finite at caustics; and (4) the exact rays never exhibit multipathing, which is the hallmark of classical ray diagrams. The contrasts between classical and exact ray theory are demonstrated and explained.