## 2 Statement of the Problem and Solution Representation  
A monochromatic sound field of circular frequency \(\omega\) in a shallow sea range-dependent waveguide is described by linear equations of acoustics with suitable boundary conditions, which are set on the basis of the continuity of the sound pressure and velocity components when crossing the interfaces. For media with losses, it is also assumed that the condition of limiting absorption is satisfied. At a constant density \(\rho\) in water for the acoustic pressure \(p\), the acoustic equations are reduced to the Helmholtz equation of the form (implicit time factor \(e^{-i\omega t}\) is assumed):  
\[\left(r^{-1}\frac{\partial}{\partial r}(r\frac{\partial}{\partial r})+\frac{ \partial^{2}}{\partial z^{2}}+\frac{\omega^{2}}{c^{2}}\right)p(r,z)=-\frac{ \delta(r)\delta(z-z_{0})}{2\pi r}, \tag{1}\]  
where (\(r\),\(z\)) are the coordinates of the cylindrical system, \(r\) is directed horizontally and the point radiation source is located at the point (\(r=0\), \(z=z_{0}\)) (axially symmetric formulation of the problem), and \(c\) is the speed of sound in water. The boundary condition on the sea surface \(p(r\),\(0)=0\), and the condition at the bottom (\(z=H\)) corresponds to the continuity of pressure and the velocity component that is normal to the boundary \(H\). In the wave zone of the source, the field \(p(r\),\(z)\) is sought using the expansion in local modes of an irregular waveguide:  
\[p(r,z)=\sum_{m}G_{m}(r)\varphi_{m}(r,z);\ \ \ \frac{\partial^{2}}{\partial z^{2}} \varphi_{m}(r,z)+\left[k^{2}(r,z)-\kappa_{m}^{2}(r)\right]\varphi_{m}(r,z)=0 \tag{2}\]In Equation (2), \(k=\omega/c\), \(\kappa_{m}(r)\) are the eigenvalues, \(m=1,2,\ldots,\varphi_{m}\) are the eigenfunctions of the Sturm-Liouville problem, which, on the surface and at the bottom of the ocean, satisfy the following boundary conditions \((\varphi^{\prime}_{m}=\partial\varphi_{m}(r,z)/\partial z)\):  
\[\varphi_{m}(r,0)=0,\,\varphi_{m}(r,H)+g_{m}(r)\varphi^{\prime}_{m}(r,H)=0 \tag{3}\]  
The function \(g_{m}(r)\) in Equation (3) characterizes the impedance of the penetrable bottom, and it is a random function due to fluctuations in the speed of sound \(c_{1}\) and density \(\rho_{1}\) in the bottom sediments. This circumstance carries novelty to the mathematical formulation of the boundary value problem since the influence of random fluctuations of the bottom impedance in the horizontal direction on the propagation of sound in the water layer has not been studied previously. Thus, it is obvious from Equations (2) and (3) that the eigenfunctions and eigenvalues, and with them the local modes of the waveguide, will also be random functions of \(r\). It can be shown that in an irregular waveguide in the forward scattering approximation, the amplitudes of modes \(G_{m}(r)\) are determined by the following solution presented in matrix form:  
\[G(r)=\{G_{m}(r)\}=A(r)exp\left\{\int\limits_{0}^{r}\left[i\kappa(\xi)-\left( \kappa(\xi)V(\xi)\kappa^{-1}(\xi)-V^{T}(\xi)\right)/2\right]dg\right\}b(0), \tag{4}\]  
where \(\kappa(r)\) is the diagonal matrix of eigenvalues \(\{\kappa_{m}(r)\}\), \(\kappa_{m}\), \(r\gg 1\), \(A(r)=(i/8\pi r)^{1/2}\kappa^{-1/2}(r)\kappa^{-1/2}(0)\); \(b(0)=\{\varphi_{m}(0,z_{0})\kappa_{m}{}^{1/2}(0)\}\) is the column vector of the initial amplitudes of modes; and exp \(\{...\}\) is the matrix exponential. In Equation (4), \(V(r)\) is a matrix with elements \(V_{mn}\) (\(r\)) = \(\int\limits_{0}^{\infty}\frac{\varphi_{m}(r,z)}{\rho(r,z)}\frac{\partial \varphi_{n}(r,z)}{\partial r}dz\), and \(V^{T}(r)\) is a transposed matrix \(V\). The latter matrices describe mode coupling due to horizontal changes caused by fluctuations of the speed of sound and density in the bottom sediments. The analytical matrix form of solution (4) has a number of advantages over the recording of the solution for individual mode amplitudes, which was used earlier in [12; 13; 14; 15; 16].