Generalized paraxial ray tracing derived from Riemannian geometry

Abstract

In 1973 R. White demonstrated that acoustic rays in a generic environment could be identified with the null geodesics of a pseudo-Riemannian manifold. A general set of paraxial (dynamic) ray equations, suitable for three dimensional ray tracing in a generic environment, is derived from the geodesic deviation equation used in general relativity and the geometric transmission loss of a ray bundle modeled from this equation. (When fluid motion is removed the paraxial ray procedure used in seismology emerges from the formalism.) The results are applied to time independent layered media where it is found that the standard ray integrals used in underwater acoustics as well as a generalized version of Snell’s law, originally derived by Kornhauser, emerge naturally as conserved quantities related to symmetries of the metric. Finally, when the results are applied to torsion free rays the deviation equation reduces to a scalar equation and the sectional curvature reduces to a simple expression depending on the derivatives of the sound speed and fluid velocity as well as the ray parameters allowing one to determine ray divergence or the development of focal points simply by checking the sign of a single term in the equation.