Gaussian Beams, Ray-centered Coordinate System, Parabolic Equation Method, p–q Equations

Abstract

The method of Gaussian beam tracing, a high-frequency asymptotic procedure for computing acoustic fields is exposited. The explanation is made according to two papers, Porter and Bucker (1986) and Cerveny et al. (1981). This article cumulates results from references including details of calculation processes. It is not just a translation of these papers but confers the full mastership on the Gaussian beam method. A mathematical view is somewhat emphasized, however, some illustrations aid for our understanding the theory. The Gaussian beam method associates with each classical ray a beam with a Gaussian intensity profile normal to the ray. The concept of the theory has this simplicity, while lots of mathematical works are needed to solve the governing equation. First of all, the Helmholtz equation governing the pressure due to a monochromatic point source is transformed from Cartesian coordinates into ray-centered curvilinear coordinates. A parabolic approximation is then applied to the Helmholtz equation, and a first order ordinary differential equations governing the beam properties are derived. There exists an additional difficulty of matching the beam phase described in the curvilinear coordinates at a planar interface. We strongly recommend reading this article following the above papers.