Application of Differential Geometry to Acoustics: Development of a Generalized Paraxial Ray-Trace Procedure from Geodesic Deviation

摘要

In this report, the application of abstract differential geometry acoustics is explored. The results of this application are as follows: (1) a generalized paraxial ray-trace procedure valid for acoustic propagation in a random media with subsonic flow; (2) the demonstration of a continuum of equivalent paraxial systems related by a conformal transformation; and (3) a unified approach to treating problems in acoustics, which leads to generalized versions of Snell's law, Fermat's principle, and range- and travel-time integrals for layered media. The geodesic deviation vector is used to model beam deformation and provides one with an all-purpose tool for measuring geometric transmission loss and locating caustics within a ray skeleton without repeatedly solving the ray equations. When applied to layered media, the deviation vector is solved exactly. Compared to traditional approaches, the results are equivalent. However, the difference is vast when implemented in numerical ray-trace codes. Applications are made to several depth-dependent scenarios, including piecewise-linear sound-speed and fluid-velocity profiles for which the exact caustic structures are determined.