# Coupled Mode Sound Propagation in Inhomogeneous Stratified Waveguides
## 2 Method
### Two-Dimensional Two-Way Coupled Mode Equations
Then, a projection of the Helmholtz equation onto \(\psi_{m}(z;x)\) gives  
\[\int_{0}^{h_{J}(x)}\psi_{m}^{*}\left[\frac{\partial}{\partial x}\left(\frac{1 }{\rho}\frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial z}\left( \frac{1}{\rho}\frac{\partial p}{\partial z}\right)+\frac{\omega^{2}}{\rho c^{2 }}p\right]dz=0. \tag{7}\]  
By substituting Equation (4) into Equation (7) and applying [PERSON]'s approach [8], one can obtain second-order coupled-mode equations  
\[\sum_{n}A_{mn}P_{n}^{\prime\prime}+\sum_{n}(B_{mn}+2C_{mn}+D_{mn})P_{n}^{ \prime}+\sum_{n}(E_{mn}+F_{mn}+G_{mn}+K_{mn}-L_{mn})P_{n}=0, \tag{8}\]  
where \(A_{mn}\), \(K_{mn}\), and \(L_{mn}\) are correction coefficients resulting from the use of unusual eigenfunctions \(\psi_{n}(z;x)\); \(B_{mn}\), \(C_{mn}\), \(F_{mn}\), and \(G_{mn}\) are mode coupling coefficients; and \(D_{mn}\) and \(E_{mn}\) are interface coefficients given by, respectively,  
\[A_{mn}=\int_{0}^{h_{J}(x)}\frac{1}{\rho_{1}}\psi_{m}^{*}\psi_{n}dz+\sum_{j=1}^{ J-1}\int_{h_{j}(x)}^{h_{j+1}(x)}\left(\frac{1}{\rho_{j+1}}-\frac{1}{\rho_{1}} \right)\psi_{m}^{*}\psi_{n}dz, \tag{9}\]  
\[B_{mn}=\int_{0}^{h_{J}(x)}\frac{\partial}{\partial x}\left(\frac{1}{\rho_{1}} \right)\psi_{m}^{*}\psi_{n}dz+\sum_{j=1}^{J-1}\int_{h_{j}(x)}^{h_{j+1}(x)} \frac{\partial}{\partial x}\left(\frac{1}{\rho_{j+1}}-\frac{1}{\rho_{1}} \right)\psi_{m}^{*}\psi_{n}dz, \tag{10}\]  
\[C_{mn}=\int_{0}^{h_{J}(x)}\frac{1}{\rho_{1}}\psi_{m}^{*}\frac{\partial\psi_{n} }{\partial x}dz+\sum_{j=1}^{J-1}\int_{h_{j}(x)}^{h_{j+1}(x)}\left(\frac{1}{ \rho_{j+1}}-\frac{1}{\rho_{1}}\right)\psi_{m}^{*}\frac{\partial\psi_{n}}{ \partial x}dz, \tag{11}\]\[D_{mn}=\frac{1}{\rho_{j}(h_{I})}h_{I}^{\prime}(x)\psi_{m}^{*}(h_{I})\psi_{n}(h_{I}) -\sum_{j=1}^{I-1}\Biggl{(}\frac{1}{\rho_{j+1}(h_{j})}-\frac{1}{\rho_{j}(h_{j})} \Biggr{)}h_{I}^{\prime}\psi_{m}^{*}(h_{j})\psi_{n}(h_{j}), \tag{12}\]  
\[E_{mn}=\frac{1}{\rho_{j}(h_{I})}h_{I}^{\prime}(x)\psi_{m}^{*}(h_{I})\frac{ \partial\psi_{n}(h_{I})}{\partial x}-\sum_{j=1}^{I-1}\Biggl{(}\frac{1}{\rho_{j +1}(h_{j})}-\frac{1}{\rho_{j}(h_{j})}\Biggr{)}h_{I}^{\prime}\psi_{m}^{*}(h_{j} )\frac{\partial\psi_{n}(h_{j})}{\partial x}, \tag{13}\]  
\[F_{mn}=\int_{0}^{h_{I}(x)}\frac{1}{\rho_{1}}\psi_{m}^{*}\frac{\partial^{2} \psi_{n}}{\partial x^{2}}dz+\sum_{j=1}^{I-1}\int_{h_{j}(x)}^{h_{j+1}(x)} \Biggl{(}\frac{1}{\rho_{j+1}}-\frac{1}{\rho_{1}}\Biggr{)}\psi_{m}^{*}\frac{ \partial^{2}\psi_{n}}{\partial x^{2}}dz, \tag{14}\]  
\[G_{mn}=\int_{0}^{h_{I}(x)}\frac{\partial}{\partial x}\Biggl{(}\frac{1}{\rho} \Biggr{)}\psi_{m}^{*}\frac{\partial\psi_{n}}{\partial x}dz+\sum_{j=1}^{I-1} \int_{h_{j}(x)}^{h_{j+1}(x)}\frac{\partial}{\partial x}\Biggl{(}\frac{1}{ \rho_{j+1}}-\frac{1}{\rho_{1}}\Biggr{)}\psi_{m}^{*}\frac{\partial\psi_{n}}{ \partial x}dz, \tag{15}\]  
\[\int_{0}^{h_{J}(x)}\psi_{m}^{*}\left[\frac{\partial}{\partial x}\left(\frac{1 }{\rho}\frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial z}\left( \frac{1}{\rho}\frac{\partial p}{\partial z}\right)+\frac{\omega^{2}}{\rho c^{2 }}p\right]dz=0. \tag{7}\]0  
\[\int_{0}^{h_{J}(x)}\psi_{m}^{*}\left[\frac{\partial}{\partial x}\left(\frac{1 }{\rho}\frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial z}\left( \frac{1}{\rho}\frac{\partial p}{\partial z}\right)+\frac{\omega^{2}}{\rho c^{2 }}p\right]dz=0. \tag{7}\]1  
Since \(\psi_{n}(z;x)\) satisfy different boundary and interface conditions than acoustic pressure when the bottom boundary and interfaces are range dependent, interface coefficients \(D_{mn}\) and \(E_{mn}\) are introduced in the derivation to guarantee energy conservation among modes by properly applying the boundary and interface conditions. Details of the derivation of \(D_{mn}\) and \(E_{mn}\) are presented in Appendix A. The interface coefficients can be analytically computed. Other coefficients can be numerically computed by numerical integration method. In this paper, we use the Clenshaw-Curtis method [20] to calculate the correction and coupling coefficients.