# Coupled Mode Sound Propagation in Inhomogeneous Stratified Waveguides
## 1 Introduction  
Sound propagation in shallow water in the presence of inhomogeneities has been studied for several decades, conventionally modeled by the nonseparable Helmholtz equation in horizontally stratified inhomogeneous waveguides [1, 2, 3, 4, 5]. The step-wise coupled mode method is commonly used for sound propagation in waveguides in which a step-wise variation of the properties with range occurs [3, 4]. However, for the sound propagation problem in waveguides with continuously varying properties, the stepwise approximation of the properties renders the numerical solution procedure cumbersome [6]. The continuous coupled mode method is a classical approach to sound propagation problems in range-dependent waveguides, originally presented by [PERSON] [1] and [PERSON] [2]. They expanded the sound field in terms of usual local modes with range-dependent modal amplitudes and derived a system of coupled mode equations containing two coupled coefficients. [PERSON] and [PERSON] indicated that the Pierce-Milder coupled mode equations do not conserve the energy among modes because the usual local modes satisfy different boundary conditions than the acoustic pressure [7]. [PERSON] derived a coupled system of differential equations containing two coupling matrices and two interface matrices, and treated this coupled mode system by a finite difference scheme [8]. It is noted that, the coupling matrices and interface matrices, which require a knowledge of derivatives of usual local modes, can only be approximately computed [9]. [PERSON]'s approach considers the backscattering effect from range-dependent environments and full coupling betweenmodes, and conserves the energy among modes. However, the complexity involved in computing coupling matrices and interface matrices along with the complex numerical scheme for solving differential equations makes this approach impractical, especially in long range and high frequency applications, where the computational requirements increase substantially. In order to reduce computational complexity, several asymptotic coupled mode methods have been developed. The adiabatic coupled mode method neglects mode coupling among modes and provides accurate predictions when the range dependence of a medium is sufficiently weak [5; 10]. The one-way approximation coupled mode method is efficient for sound propagation in waveguides where the backscattering effect can be ignored [5; 11]. However, asymptotic coupled mode models can provide very poor solutions when the range dependence of a medium is significant. Thus, it is still very important to study an efficient two-way coupled mode method for sound propagation in complex underwater waveguides. The multimodal admittance method presented by [PERSON] [12; 13] is a two-way coupled mode method for wave propagation in waveguides with range-dependent cross-sections. This method basically consists in rewriting the Helmholtz equation into a set of first-order evolution equations and then projecting onto the usual local modes to obtain a set of first-order differential equations governing the modal amplitudes of the wave fields. In the case where these differential equations are to be integrated numerically, the problem of numerical divergence owing to the presence of evanescent modes appears. The admittance matrix, namely, the Dirichlet-to-Neumann operator in modal domain, is introduced to avoid numerical problems. Using the admittance matrix, first-order differential equations are reformulated into the Riccati equation governing the admittance matrix and the first-order evolution equation governing the modal amplitudes of acoustic pressure. Both the Riccati equation and the evolution equation can be computed with classical numerical schemes [12; 13; 14; 15]. This method is efficient and numerically stable.