On Gaussian Beams Described by Jacobi's Equation

TL;DR

This paper introduces four novel theoretical results linking Gaussian beam propagation in acoustics to differential geometry, specifically Riemannian curvature, providing new formulas, models, and an intrinsic approach for analyzing acoustic channels.

Contribution

It develops a new intrinsic formulation of Gaussian beams using Jacobi's equation, connecting acoustic propagation to Riemannian curvature and introducing new models and formulas.

Findings

Derived a new formula for convergence zone distances: π(c/c'')^{1/2}.

Established a fundamental link between Gaussian beams and intrinsic Gaussian curvature.

Compared intrinsic and extrinsic methods, highlighting advantages of covariant geometric approaches.

Abstract

Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) A new version of the \v{C}erven\'y equations for the amplitude and phase of Gaussian beams is developed by applying the equivalence of Hamilton-Jacobi theory with Jacobi's equation that connects Riemannian curvature to geodesic flow. Thus the paper makes a fundamental connection between Gaussian beams and an acoustic channel's so-called intrinsic Gaussian curvature from differential geometry. (2) A new formula $\pi(c/c")^{1/2}$ for the distance between convergence zones is derived and applied to several well-known profiles. (3) A class of "model spaces" are introduced that connect the acoustics of ducting/divergence zones with the channel's Gaussian curvature $K=cc"-(c')^2$. The "model" SSPs yield constant Gaussian curvature in which the geometry of ducts corresponds to great circles on a sphere and convergence zones correspond to antipodes. The distance between caustics $\pi(c/c")^{1/2}$ is equated with an ideal hyperbolic cosine SSP duct. (4) An "intrinsic" version of \v{C}erven\'y's formulae for the amplitude and phase of Gaussian beams is derived that does not depend on an "extrinsic" arbitrary choice of coordinates such as range and depth. Direct comparisons are made between the computational frameworks used by the three different approaches to Gaussian beams: Snell's law, the extrinsic Frenet-Serret formulae, and the intrinsic Jacobi methods presented here. The relationship of Gaussian beams to Riemannian curvature is explained with an overview of the modern covariant geometric methods that provide a general framework for application to other special cases.