Propagation over Underwater Obstructions

Abstract

The paper gives a geometrical construction showing the effect of absorbing geological obstructions (ridges, seamounts, etc.) in long-range propagation in a sound channel. It applies the general methods recently developed by G. Raisbeck and the author [cf. “Some Mathematical and Computational Contributions to Underwater Sound Propagation,” 78th Meeting of the Acoustical Society of America, paper 6H1; J. Acoust. Soc. Amer. 47, 100 (1970)]. The ray equations in Hamiltonian form are studied in their phase-space (manifold of positions-and-directions). The standard single-valued integral invariants of such a system are used to examine the power flux and, in the cases of the various types of spreading (e.g., cylindrical), to obtain a graphical representation. For this purpose, the set of rays (curves in phase-space) from the emitter are cut by a family of surfaces in this space, thus defining an area-preserving surface transformation on a prototype of the surface, the “surface of section” (introduced into dynamics by H. Poincaré and G. D. Birkhoff). The ergodic mixing produced by this transformation simplifies the distribution of power, particularly after partial removal by obstructions cutting some of the rays. [This work was supported by the Naval Ordnance Systems Command.]