Derivation of the Paraxial Ray Equations in Three Conformal Representations

TL;DR

This paper derives comprehensive paraxial ray equations for acoustics in inhomogeneous media using three conformal representations of the acoustic metric, making advanced geometric techniques more accessible for acoustic ray tracing.

Contribution

It provides the first complete derivation of paraxial ray equations in three conformal representations for acoustic media, bridging abstract geometric methods with practical acoustics applications.

Findings

Derived paraxial equations for three conformal acoustic metrics

Unified geometric framework for acoustic ray tracing

Enhanced understanding of acoustic wave propagation in inhomogeneous media

Abstract

Acoustic rays can be described by null geodesics of a pseudo-Riemannian manifold. This allows for the immediate generalization of paraxial ray systems via geodesic deviation. The technique has appeared in the literature for the past decade and has been used by the author in the development of dynamic ray trace codes. The techniques involved are powerful but abstract and frequently unfamiliar in the field of acoustics. This brief paper derives the complete paraxial system for the case of acoustics in the presence of an inhomogeneous isotopic time independent media described by a sound speed profile c(x,y,z). The derivation is performed for three distinct conformal representations of the acoustic metric.