## 3 Results of Numerical Simulation  
Below are examples of statistical simulation for Gaussian and non-Gaussian fluctuations in the speed of sound. As a model for calculations, a shallow water three-layer waveguide with a regular thermocline and two-dimensional fluctuations of the speed of sound (Figure 1) was taken. Fluctuations are described by the exponential correlation function \(B_{\varepsilon}(r_{1}-r_{2},z_{1}-z_{2})\), given above. The waveguide characteristics are as follows: Depth \(H=50\) m, surface layer \(15\) m thick, where \(c_{0}(z)=\langle c(z)\rangle=1525\) m/s, and intensity of fluctuations \(\langle(\delta c/c_{0})^{2}\rangle=10^{-6}\), bottom layer \(10\) m thick, where \(c_{0}(z)=1500\) m/s and \(\langle(\delta c/c_{0})^{2}\rangle=10^{-6}\), a layer of linear thermocline \(25\) m thick with \(1500\) m/s \(\leq c_{0}(z)\leq 1525\) m/s, and \(\langle(\delta c/c_{0})^{2}\rangle=10^{-5}\). \(\rho_{1}\),\(c_{1}\), \(\beta_{1}\) were the parameters of a homogeneous liquid bottom were used different for the representativeness of the research. These hydrological conditions are typical (with the exception of constant depth) for the autumn observation period in the shelf areas of the Sea of Japan, when a pronounced thermocline region is formed in the water layer. Active hydrodynamic processes occur at these horizons, randomly disturbing the average sound velocity profile. The considered formulation of the problem corresponds to the passage of a sound signal along the propagation path of internal waves (major mode), or other hydrodynamic disturbances. For calculations, a sound frequency of \(500\) Hz was chosen as the reference. In this situation, 4-11 propagating (trapped) modes are formed in the waveguide, depending on the bottom penetrability, or only leaky modes are present. In the presence of a thermocline, the behavior of horizontal wave numbers \(\kappa_{m}(r)\) (eigenvalues of modes) on the complex plane (\(Re(\kappa_{m})\), \(Im(\kappa_{m})\)), is characterized by substantial non-monotonicity, unlike a  
Figure 1: Random shallow water waveguide. The left graph: \(r\)=19900 m, 20 random realizations of a sound speed profile from the ensemble of \(1000\). The right graph: 5 random realizations of a sound speed profile along the propagation path at the horizon \(H\)-\(z=25\) m.  
homogeneous water layer (Pekeris waveguide model), for which the consistent growth of \(\kappa_{m}\) on the complex plane is typical. The presence of fluctuations enhances non-monotony: in many individual realizations, the modes with higher numbers (2 nd-5 th) can propagate even better than the 1 st one. This feature leads to the rapid development of strong fluctuations of the field in the waveguide, established in [11; 14; 22; 32], which is manifested in a significant dispersion of the levels of individual intensity realizations. Average sound field intensity \(\langle I\rangle=\langle\left\langle p\right|^{2}\rangle\) was calculated by averaging over an ensemble of 1000-2000 random realizations. The following three examples show the effect of Gaussian fluctuations in the speed of sound. In Figure 2 there is a graph of transmission losses for a waveguide with bottom parameters \(c_{1}=1600\) m/s, \((\rho_{1}/\rho_{0})=2\), \(\beta_{1}=0.01\). In such a waveguide, there are 11 propagating modes that fluctuate and couple with each other. This example is illustrative, since the model bottom chosen is sufficiently rigid, with the result that mode attenuation with a distance becomes noticeable at relatively large distances \(r>10\)-15 km. The curves in the graphs are given starting from a distance of 200 m from the source. It is clearly seen that, compared with the deterministic model, there is a noticeable slowdown in the decay of the average intensity of the sound field with a distance. As a result, the level of average intensity gradually rises above the level of averaged (over spatial oscillations) intensity corresponding to deterministic propagation conditions (dashed curve).