## 3 Results of Numerical Simulation
Moreover, the stronger this rise is expressed, the larger the horizontal scale \(L_{r}\) of the correlation of inhomogeneities, in full accordance with the above analytical estimates for Gaussian fluctuations. So, for the scale \(L_{r}=50\) km, the level rise reaches 23 dB to a distance of 50 km, for the scale \(L_{r}=1\) km, the level rise at a distance of 50 km is 3-7 dB depending on the observation horizon. We emphasize that the transmission losses for the scale \(L_{r}=50\) km are presented purely for demonstration purposes, to confirm the analytical estimates. Typical horizontal scales for modes of internal wave perturbations are within the range of 1-10 km [8; 9; 14; 18]. Based on the estimates made (9)-(12), it is also obvious that the effect is stronger, the greater the intensity of fluctuations, and the greater the modal absorption coefficient of sound by the bottom, since in this case the imaginary parts of the eigenvalues fluctuate more strongly (see (4) and (5)). The latter circumstance takes place, in particular, when the radiation  
Figure 2: Transmission losses reported relative to intensity in a free field at a distance of 1 m from the source. Lower dashed curve corresponds to the intensity, averaged over spatial oscillations, for a deterministic waveguide model (\(\varepsilon=0\)). Blue curve demonstrates presence of two-dimensional Gaussian fluctuations of the speed of sound \(\varepsilon(r,z)\) in the waveguide with the scale \(L_{r}=50\) km; red curve is \(L_{r}=5\) km, black curve is \(L_{r}=1\) km. Vertical scale is \(L_{z}=10\) m; \(H\)–\(z=26\) m, \(H\)–\(z_{0}=42\) m.  
frequency increases. In contrast, lowering the frequency of the sound reduces all statistical effects. Another important parameter on which the magnitude of the effects under consideration depends is the degree of penetrability of the waveguide bottom for sound waves, which is determined by the value of the refractive index at the water-bottom interface (\(c_{0}(0)/c_{1}(1+i\beta_{1})\)[26]. Above, a fairly rigid bottom boundary was considered. We now turn to waveguides with greater bottom penetrability, and we will call the bottom boundary \"rigid\" if \(c_{0}(0)<c_{1}\), and \"soft\" in the contrary case \(c_{0}(0)>c_{1}\). Figure 3 demonstrates the transmission losses in the waveguide with the parameters: \(c_{1}=1530\) m/s, (\(\rho_{1}/\rho_{0}\)) = 1.5, \(\beta_{1}=0.01\). In this case, four propagating modes are excited in the unperturbed water layer, and four leaky modes were additionally taken into account in the calculations (the reference was carried out to the Pekeris cut on the complex plane of \(\kappa\)). The imaginary parts of the horizontal wave numbers \(\kappa_{m}\) of modes for a given waveguide are noticeably superior to those of the previous waveguide, so the sound field decays with distance much faster. The source location corresponds to 8 m from the bottom, where the maximum of the first eigenfunction of the deterministic waveguide is located; two observation horizons are considered: 8 m and 24 m from the bottom. Figure 3 shows that the slowdown in the decay of the average intensity is manifested already at fairly close distances of 8-10 km from the source, and to a distance of 30 km it reaches 20-23 dB. In Figure 4 there is a graph of transmission losses in the waveguide with the \"soft\" bottom: \(c_{0}(0)>c_{1}=1200\) m/s, (\(\rho_{1}/\rho_{0}\)) = 1.5, \(\beta_{1}=0.01\). For this waveguide, no propagating mode is excited within the water layer, the calculations took into account eight leaky modes, which form the acoustic field in the deterministic waveguide at distances from the source of interest for statistical analysis. In this case, the sound field decays even faster than in the previous example (dashed curve in Figure 4), and the absorption value in the bottom \(\beta_{1}\) does not play a practical role and can be set equal to zero.