## 3 Results of Numerical Simulation
\(H\)-\(z_{0}=42\) m. Blue curves: \(H\)-\(z=42\) m, red curves: \(H\)-\(z=26\) m. Thin dash curves correspond to adiabatic approximation, thick dash curves represent a deterministic waveguide (\(\varepsilon=0\)). the picture is similar, only the discrepancy with Gaussian average intensity curves increases towards attenuation of propagation losses to a distance of 30 km, additional attenuation is 20 dB or more for different observation horizons. The following example demonstrates the situation with a log-normal probability distribution of the fluctuations \(\varepsilon_{2}(r)\). Log-normal distributions are quite often encountered in statistical wave  
Figure 5: Transmission loss for a waveguide with Gaussian fluctuations \(\varepsilon\) and with non-Gaussian fluctuations in the form (14) with telegraph process \(\varepsilon_{2}(r)\); \(L_{z}\) = 20 m, \(L_{r}\) = 4 km. Parameters of the bottom: \(c_{1}\) = 1530 m/s, \((\rho_{1}/\rho_{0})\) = 1.5, \(\beta_{1}\) = 0.01. \(H\)-\(z_{0}\) = 42 m. Blue curves: \(H\)-\(z\) = 42 m; red curves: \(H\)-\(z\) = 26 m. Bold curves are the telegraph process \(\varepsilon_{2}(r)\), dash curves are the adiabatic approximation, thin curves are similar to ones in Figure 3, correspond to Gaussian fluctuations. Figure 6: Transmission loss for a waveguide with Gaussian fluctuations \(\varepsilon\) and with non-Gaussian fluctuations in the form (14) with telegraph process \(\varepsilon_{2}(r)\), \(L_{z}\) = 20 m, \(L_{r}\) = 4 km. Parameters of the bottom: \(c_{1}\) = 1200 m/s, \((\rho_{1}/\rho_{0})\) = 1.5, \(\beta_{1}\) = 0.01. \(H\)-\(z_{0}\) = 42 m. Blue curves: \(H\)-\(z\) = 42 m; red curves: \(H\)-\(z\) = 26 m. Bold curves are the telegraph process \(\varepsilon_{2}(r)\), dash curves are the adiabatic approximation, thin curves are similar to ones in Figure 4, correspond to Gaussian fluctuations. problems [2; 33]. Main feature of the such distribution in comparison with Gaussian one is the asymmetry and the presence of slowly falling tails, which leads to the appearance in the ensemble of random realizations of rare but strong emissions in intensity. The probability density of the generated process \(\varepsilon_{2}\) is shown in Figure 7. As before, \(\varepsilon_{2}\) has zero mean value and is described by an exponential correlation function. The median of the distribution is non-zero, but the distribution parameters are chosen so that the fluctuations are adequate in intensity to those observed in the experiment and correspond to the values described above. As applied to the propagation of low-frequency sound on the sea shelf, log-normal fluctuations in the speed of sound can be viewed as the result of the influence of transient hydrodynamic perturbations leading to the formation of background internal waves (formed on the tails of numerous collapses of nonlinear internal waves), which are commonly described by Gaussian processes. From the transmission loss curves given below, it can be seen that the effect of reducing the decay of the average intensity is preserved also with log-normal fluctuations in the speed of sound. Compared with Gaussian fluctuations (thin curves in Figure 8), at distances of 10-20 km the effect can be 2-4 dB, depending on the observation horizon. Figure 7: Probability density of the log-normal distribution of fluctuations \(\varepsilon_{2}(r)\), c=0.45/\(\sigma_{2}\).