## On Gaussian Beams Described by Jacobi's Equation    [PDF]

Steven Thomas Smith
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) It is shown that the \v{C}erven\'y equations for the amplitude and phase are equivalent to the classical Jacobi Equation of differential geometry. The \v{C}erven\'y equations describe Gaussian beams using Hamilton-Jacobi theory, whereas the Jacobi Equation expresses how Gaussian and Riemannian curvature determine geodesic flow on a Riemannian manifold. 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.
View original: http://arxiv.org/abs/1304.1931