## 2 Formulation of the Statistical Problem and Some Analytical Results
In this representation \(V_{mn}(r)=\int\limits_{0}^{H}\frac{\varphi_{m}(r,z)}{\rho(r,z)}\frac{\partial \varphi_{m}(r,z)}{\partial r}dz\) is the element of an skew-symmetric matrix, \(V_{mn}\)\((r)=-V_{mn}\)\((r)\), \(V_{mn}=0\), describing mode coupling due to horizontal variations caused by fluctuations in the speed of sound. The neglect of backscattering in the problem under consideration is justified due to the smallness of \(\varepsilon(r,z)\)[27; 28]. To perform analytical estimates, we use an equation that obeys the local eigenvalues \(\kappa_{m}(r)\), as well as an expression for the mode coupling matrix, written via derivatives of the waveguide parameters [23; 24; 25; 30]:  
\[\begin{split}\frac{\partial}{\partial r}\kappa_{m}^{2}(r)& =\int\limits_{0}^{H}dz\varphi_{m}^{2}(r,z)\frac{\partial}{\partial r }k^{2}(r,z)=\int\limits_{0}^{H}dz\varphi_{m}^{2}(r,z)k_{0}{}^{2}(z)\frac{ \partial}{\partial r}\varepsilon(r,z  
Similarly, from (3) follows an approximate expression for the mode coupling matrix:  
\[V_{mn}(r)\approx\left[\kappa_{n}^{2}(r)-\kappa_{m}^{2}(r)\right]^{-1}\int\limits_ {0}^{H}dz\varphi_{0m}(z)\varphi_{0n}(z)k_{0}^{2}(z)\frac{\partial\varepsilon(r,z )}{\partial r}. \tag{6}\]  
For the mode amplitudes \(G_{m}(r)\) solution (2), taking into account (5) and (6), is written (\(a_{n}\)/\(a_{m}\approx 1\)):  
\[ \begin{array}{c}G_{m}(r)\approx A_{m}exp\{i\kappa_{0m}r+i(2\kappa_{0m})^{-1} \int\limits_{0}^{H}\varphi_{0m}^{2}(z)k_{0}^{2}(z)dz\int\limits_{0}^{r} \varepsilon(\varepsilon_{z}z)d\xi+\\ \sum\limits_{n}\frac{1}{2\kappa_{0n}}\int\limits_{0}^{H}\varphi_{0m}(z) \varphi_{0n}(z)k_{0}^{2}(z)\frac{\varepsilon(r,z)}{(\kappa_{0n}-\kappa_{0m})} dz\}.\end{array}  \tag{7}\]  
The second term in the exponent (7) contains the integral of \(r\) from the inhomogeneities and, therefore, characterizes the effect of fluctuations of the speed of sound on the acoustic field accumulating as the distance increases. In the third term related to the coupling of modes, there is no such integral. This implies that the coupling of modes is some variable-sign addition to the solution, weakly dependent on distance. This suggests that in this situation the adiabatic approximation to the solution for mode amplitudes  
\[G_{m}(r)=A_{m}(r)exp\{i\int\limits_{0}^{r}\kappa_{m}(\xi)d\xi\}\] ( \[2^{\prime}\] )  
describes the main statistical effects, which are just accumulating with distance. Calculations in many cases confirm this conclusion. We note that in the framework of the theory of [7; 8; 9; 13; 14; 15; 16; 17; 20] instead of (\(1^{\prime}\)), the solution is sought through the eigenvalues and eigenfunctions of the unperturbed waveguide for \(\varepsilon(r,z)=0\)) \(\kappa_{0m}\), \(\varphi_{0m}\). In addition, for the mode amplitudes \(G_{m}(r)\) and mode coupling matrix \(V_{mn}\) (\(r\)) approximate equations are used:  
\[ \begin{array}{c}\frac{\partial}{\partial r}G_{m}(r)=i\kappa_{0m}G_{m}(r)+i \sum\limits_{n=1}^{N}V_{mn}(r)G_{n}(r),\kappa_{0m}(r)=\kappa_{0m}^{(1)}(r)+i \kappa_{0m}^{(2)}(r),\\ V_{mn}(r)=0.5(\kappa_{0m}^{(1)}\kappa_{0n}^{(1)})^{-1/2}\int\limits_{0}^{H} \varphi_{0m}(z)\varphi_{0n}(z)k_{0}^{2}(z)\varepsilon(r,z)dz\end{array}  \tag{8}\]  
In the first order of the perturbation method with respect to \(\varepsilon\), the diagonal term of the matrix \(V\) in (8) gives a solution that is close to adiabatic (the first term of the exponent in (2), based on (4), (5) \(\kappa_{m}(r)=\kappa_{0m}+\delta\kappa_{m}(r),\delta\kappa_{m}(r)\approx \frac{1}{2\kappa_{0m}}\int\limits_{0}^{H}\varphi_{0m}^{2}(z)k_{0}^{2} \varepsilon(r,z)dz\). However, as shown in [31], for actual parameters of a shallow sea waveguide with a not too rigid bottom, the difference between solution (8) and more accurate (2) can be significant and grows with distance. The second statistical moment of the pressure field (average intensity), represented by incoherent and coherent sums, has the form:  
\[