Extending the two-dimensional fluid parabolic equation to three dimensions and solving via a split-step Pade approach

Abstract

The development of three-dimensional solutions using fluid parabolic equations is an active area of research in underwater acoustics. Several approaches for solving the time-harmonic three-dimensional parabolic equation have been attempted: such as an N x 2D approach developed by Perkins and Baer; a higher-order Fourier transform approach developed by Lin and Duda; and a finite-difference approach by Sturm and Fawcett. In this work, a split-step Pade solution to the fluid parabolic equation in three dimensions is considered. The exponential square-root operator in a fluid parabolic equation is approximated by a Pade approximation (i.e., a rational-linear approximation). The approximated parabolic equation can be solved numerically, and the solution developed includes physical effects in the range (x), depth (z), and transverse (y). The model developed is compared against the numerical solution of the two-dimensional split-step Pade approach in the same environment. The approach used in this paper can be extended to include a sediment layer and variations in a sediment layer. Additionally, the 3D model can be improved by increasing computational efficiency.