## 2 Statement of the Problem and Solution Representation
First, it can be seen from the matrix form of solution (4) that all the new effects arising from the propagation of sound in a randomly inhomogeneous medium are determined by the matrix exponential (as a function of \(r\)). Therefore, these effects (changes in the laws of decay of sound intensity with distance, or in the signal transmission losses) quantitatively weakly depend on the horizons of the location of the radiation point (\(z_{0}\)) and the observation point (\(z\)). Secondly, the form of solution (4) is optimal in terms of calculations. The matrix exponential does not contain additional parameters, such as the initial amplitudes of modes \(\varphi_{m}(0,z_{0})\) in [12; 13; 14; 15; 16]. In addition, no special computational schemes are required to compute matrix exponential in (4). At the same time, such schemes are necessary when solving differential equations for the mode amplitudes in the framework of the well-known approximations, such as Wentzel-Kramers-Brillouin (WKB) and the mode parabolic equation (MPE). Note that solution (4) for mode amplitudes corresponds to the approximation of a one-way propagation (OW) [17] for boundary value problem (2)-(3) to the original Equation (1). If we neglect the backscattered field, then this OW approximation directly follows from the causal equations of the first order, derived previously. They are equivalent to the original boundary Equations (1)-(3). The causal equations in question were obtained by the method of differentiation with respect to the parameter (imbedding method) in the works [12; 14; 18; 19] for a 2D-inhomogeneous marine environment and in [20] for 3D inhomogeneities. The imbedding method for wave problems is well described in [21]. Solution (4) takes into account the scattering of modes at any angles, but not exceeding \(90^{\circ}\). In what follows, neglecting backscattering, we call it the exact solution, or OW. If the density of the medium does not change in the horizontal direction, then the matrix \(V\) (\(r\)) in Equation (4) becomes skew-symmetric: \(V_{mn}(r)=-V_{mm}(r)\), \(V_{nn}=0\). If inhomogeneities of the medium change smoothly and the scattering angles are small, so that \(\kappa(r)V(r)\kappa^{-1}(r)\approx V(r)\), then the WKB approximation in the horizontal direction can be obtained from Equation(4), as well as the approximation of the MPE [10; 17; 22; 23]. As shown by our calculations, the WKB and MPE solutions for the waveguide models considered below also provide a sufficient approximation to the exact solution. By calculating the pressure field \(p(r,z)\) according to Equations (2)-(4) for each random realization \(c_{1}(z,r),\rho_{1}\) from an ensemble of \(N\) realizations, it is easy to obtain the change in the average intensity, or the average function of transmission loss for sound propagation along the path in a randomly inhomogeneous waveguide:  
\[\langle I\rangle=\Big{\langle}|p|^{2}\Big{\rangle}=\sum_{n}\Big{\langle}|G_{n} |^{2}|\varphi_{n}|^{2}\Big{\rangle}+\sum_{(n\
eq m)}\langle G_{n}G_{m}^{*}( \varphi_{n}\varphi_{m}^{*})\rangle \tag{5}\]  
where angled brackets denote statistical averaging. Similarly, according to well-known formulas, other statistical characteristics of the field and intensity are calculated. In this work, we particularly need an expression for the second normalized statistical moment \(S=(\langle I^{2}\rangle-\langle I\rangle^{2})^{1/2}/\langle I\rangle\) (\(S^{2}\) is known as the scintillation index), which describes the intensity fluctuations during sound propagation in the waveguide.