# Coupled Mode Sound Propagation in Inhomogeneous Stratified Waveguides
## 1 Introduction
[PERSON] and [PERSON] [16] derived a coupled system of first-order differential equations for sound propagation in simple one-layer waveguides with range-dependent environments based on [PERSON]'s coupled mode method and the multimodal admittance method. We extend the method here to present an efficient coupled mode scheme for sound propagation and wave scattering problems in complex inhomogeneous stratified waveguides. This paper focuses on the two-way solutions in horizontally stratified waveguides with both bathymetric and volumetric variations. In particular, waveguides with penetrable scatterers are also inspected, modeled by horizontally stratified waveguides with intersecting interfaces. We note that the paper deals with acoustic waveguides in which the seabed is also modeled as a (layered) acoustic medium. In many applications, the modeling of seabed needs to account for the shear rigidity (seabed elasticity); see, e.g., [17] and the references cited therein. In an attempt to reduce the complexity involved in the computation of usual local modes and their derivatives for complex multilayered environments, the local modes for expansion of acoustic pressure in this paper are defined by the transverse eigenfunctions in a one-layer waveguide with constant depth equal to the local total depth of the multilayered waveguide and constant physical parameters. We note that such local modes and their derivatives can be analytically computed. A set of second-order coupled mode equations governing the modal amplitudes is derived by projecting the local mode onto the Helmholtz equation. The energy conservation is guaranteed by introducing interface matrices. Remarkably, we introduce a set of unknown quantities with respect to the modal amplitudes and their horizontal derivatives. Using this unknown vector, the second-order coupled mode equations are reduced into a set of first-order differential equations with respect to the modal amplitudes and the unknown quantities. This first-order coupled system preserves the energy conservation property, but no interface matrix is contained. The required coupling matrices can be computed with numerical integral methods since their integrands are analytical. Moreover, it is straightforward to implement the admittance method for numerically stable solutions. The use of the unknown vector substantially increases the efficiency of the method. Since a more realistic representation of the ocean bottom is a penetrable infinite half space, the application of our coupled mode method to waveguides with a penetrable infinite half space is studied on basis of the elimination of branch cut and branch line integrals [18]. Numerical results for sound propagation in inhomogeneous multilayered waveguides predicted by commercially available finite element software COMSOL multiphysics are also presented to validate the efficiency and accuracy of this method. The remainder of the paper is organized as follows: Section 2 presents the basic formulation of two-way and one-way asymptotic coupled mode methods for sound propagation in horizontally stratified inhomogeneous waveguides. The sound field expression generated by an arbitrary incident wave in such an environment is provided. By constructing mapping between incident waves and wave fields, optimal incident waves resulting in sound self-focusing at any position [19] in the multilayered waveguide are studied. Section 3 illustrates the sound fields predicted by our coupled mode method and COMSOL for a complex multilayered waveguide, and the sound self-focusing pattern in this numerical example is shown. Section 4 presents the extension of the present coupled mode formalism to waveguides with the effect of internal waves and a penetrable bottom. Section 5 gives the conclusions.