A generalized impedance method for application of the parabolic approximation to underwater acoustics

Abstract

The parabolic equation (PE) has become an important computational tool for approximating underwater acoustic path losses in inhomogeneous environments. The numerical solution of this equation is performed on a grid over the propagation medium which includes the water column, and has also included a homogeneous "false bottom" below the actual ocean bottom. This situation is viewed as one in which the propagation medium can be divided into a horizontally inhomogeneous upper layer and a horizontally homogeneous lower layer. The pressure field in the latter layer is found using standard techniques valid for homogeneous media, and in the former layer is found using an implicit finite difference scheme. These fields and their normal derivatives are matched along the interface between the layers. The result is referred to as a generalized impedance boundary condition (GIBC), which serves to eliminate the need for a false bottom. Predictions obtained using the standard parabolic equation and a completely homogeneous bottom medium indicate improved numerical efficiency, and excellent agreement with other methods. The improved efficiency can be potentially significant for three-dimensional propagation problems.

Sound propagation over impedance discontinuities with the parabolic approximation

“…The first dealt with transparent boundary conditions to the Schroedinger equation, 14 while the other addresses the lower boundary condition when using the PE to model underwater acoustic propagation over extensive layers of sediment. 15 Both papers used slightly different formulations, but arrived at essentially equivalent results by requiring that the propagation domain be bounded by a homogeneous half-space into which the sound was transmitted and upon which was imposed a radiation condition. As discussed in Refs.…”

Transparent Boundary Condition for the Parabolic Equation Modeled by the 4RW

“…If these boundary conditions exactly simulate an unbounded exterior region as far as the interior problem is concerned, they are termed as exact or transparent boundary conditions. Transparent boundary conditions can be devised for the original partial differential equation or to its discretized version, and this has been done by several works in the past: [2], [4], [9], [10], [13], and [1] over several realizations of Brownian motion or its discrete counterpart of random walk traversing the problem domain [6]. A direct application of stochastic techniques to the PE as done in [3] necessitates analytical continuation of fields and boundary data, only possible for very limited problems, that is highly undesirable.…”

Dirichlet-to-Neumann Maps for Unbounded Wave Guides

“…The relation between the DtN method and the mode-matching method has been established by Astley [1]. Other schemes that use DtN-related ideas for various problems and configurations, can also be found in [3,7,10,12,13,24,39,42,44,45,55].…”