# Coupled Mode Sound Propagation in Inhomogeneous Stratified Waveguides
## 4 Application to Waveguides with Penetrable Infinite Half Space and Internal Wave  
In this section, we apply the coupled mode formalism to sound propagation problems in a multilayered waveguide, considering the effect of a penetrable infinite half space and internal waves, which is a common scenario in real ocean environments. In previous sections, we assumed that the bottom is acoustically rigid. However, a more realistic representation of the ocean bottom is an infinite half space that allows for energy penetrating into the half space. The rigid bottom assumption neglects one or more components of sound fields depending on the choice of branch cut, for instance, leaky modes plus an integral along the Pekeris cut for Pekeris branch cut, or an integral of the continuous spectrum along the EJP cut for EJP branch cut. Difficulties in the application of bounded coupled mode models to a Pekeris waveguide consist of the discrete approximation of continuous spectrum. An available algorithm eliminates the branch cut and branch line integrals by introducing a small complex sound speed gradient in the half space [18]. The branch cut contribution is represented by discrete modes. This method has the advantage that leaky modes do not increase in amplitude with increasing depth. Since branch line modes make a minor contribution to the sound field in the water column, it is possible to ignore the branch line modes in numerical implementation [32]. Therefore, it provides a straightforward way to incorporate with the couple mode algorithms. We extend this approach here to coupled mode sound propagation in multilayered waveguides. We consider the waveguide shown in Figure 4.  
A two-layer system of sound speed in the water column is used. The upper mixed layer has sound speed \(c_{ml}\) and water density \(\rho_{1}\). The lower water layer has sound speed \(c_{1}\) and the same density \(\rho_{1}\). The shape of the internal solitary wave is formulated by  
\[h_{in}(x)=h+\Delta l\text{sech}^{2}\bigg{(}\frac{x-x_{\text{turtype}}}{L} \bigg{)}, \tag{35}\]  
where \(\Delta h\) is the amplitude of the internal wave, \(x_{\text{turtype}}\) denotes the wave center, and \(L\) is the width parameter [33]. The ocean bottom is characterized by a penetrable infinite half space with sound speed \(c_{2}\) and density \(\rho_{2}\). To apply the coupled mode formalism, the penetrable  
Figure 3: (Color online) Acoustic waves self-focusing in a two-layer range-dependent waveguide with a penetrable scatterer. The focus is at \((450,20)\) m. (**a**) Sound field. (**b**) Acoustic pressure modulus of the optimal incident wave. (**c**) Acoustic pressure modulus along \(z\)-direction at \(x=450\) m.  
infinite half space is approximately reproduced by a bounded layer. The corresponding sound speed \(\tilde{c}_{2}\) has a small gradient,  
\[\tilde{c}_{2}=\frac{c_{2}}{1+\mathrm{i}\alpha/\alpha_{c}}, \tag{36}\]  
where the attenuation \(\alpha\) is in Gaussian form, \(\alpha=\alpha_{0}\mathrm{e}^{-(z-h_{2}(x))^{2}/(2\lambda_{2}^{2})}\), \(\alpha_{0}\) is in the unit of \(\mathrm{dB}/\lambda_{2}\), \(\lambda_{2}=c_{2}/f\), and \(\alpha_{c}=40\pi\mathrm{log}_{10}\mathrm{e}\). The thickness of the lower bounded layer, \(h_{2}-h_{1}(x)\), is meant to be larger than \(3\lambda_{2}\), such that attenuation increases gently in the beginning, then grows rapidly with increasing depth, and finally smoothly reaches its maximum at \(z=h_{2}\). We use such a form of \(\alpha\) in an attempt to reduce the spurious reflection that can return to the water column and provide accurate solutions in the water column using an acceptable truncation depth. Figure 5 shows an example of computation of the sound field excited by a point source in the waveguide shown in Figure 4. The physical parameters are \(\rho_{1}=1000\) kg/m\({}^{3}\), \(c_{ml}=1530\) m/s, \(c_{1}=\left(1490-0.1(z-h_{in}(x))\right)\) m/s, \(c_{2}=1700\) m/s, and \(\rho_{2}=1500\) kg/m\({}^{3}\).