## 3 Model of a Stochastic Waveguide
To find the local eigenvalues \(\kappa_{m}(r)\) and eigenfunctions \(\varphi_{m}(r)\), the reference to the Pekeris cut on the complex plane of \(\kappa_{m}\) was carried out. Therefore, the required number of propagating and leaky modes was taken into account in the sum (2) [17; 27]. As a rule, with the waveguide hydrology and sound frequency described above, it was sufficient to use no more than 4-6 modes of different types in the calculations for the purpose of our statistical modeling. It should be emphasized that despite the uniform nature of the sound velocity and density profiles in the water layer, the random waveguide as a whole, being a water-bottom sediment fluctuating system, is not the traditional Pekeris model [27]. This is obviously a much more complex 2D waveguide of a shallow sea (see also Section 5). Figure 1: Average profiles of the speed of sound \(\langle c(z)\rangle=\langle c_{1}(z)\rangle=1460\) m/s and density \(\langle\rho(z)\rangle\) in the waveguide with the liquid bottom. Dotted lines show the bottom boundary location for two examples. Figure 2: Three arbitrary random realizations of the speed of sound in the bottom half-space from the ensemble of \(N\) realizations, \(L_{r}=1\) km. Statistical Analysis of the Sound Transmission Loss in a Shallow Sea: The Presence of Bottom Sediments with Fluctuating \(\mathbf{c_{1}(r)}\)  
Figures 3 and 4 contain the results of statistical modeling of the average intensity (5) in a shallow-water waveguide with a highly penetrable bottom (Figures 1 and 2). Average intensity curves are given in dB relative to the intensity level in the free field at a distance of 1 m from the source. In this case, as was shown in [16; 28] using the example of volume fluctuations of the speed of sound in the water layer, one should expect the greatest statistical effect during the sound signal propagation. This is because the wavenumbers of even the lowest (small grazing angles) modes along the path can have a relatively large imaginary part (at \(c_{1}\approx c\)), which is significantly influenced by random inhomogeneities. Over the course of the numerical simulation, an ensemble of realizations \(N\) from 300 to 1000 was considered. It is clearly seen from Figures 3 and 4 that fluctuations in the bottom impedance, as well as fluctuations in the speed of sound in the water layer [16; 28], led to a significant slowdown in the average intensity decrease along the path, which is well noticeable even for small distances. For example, in the presence of fluctuations in the speed of sound in the bottom sediments for a waveguide with the depth of 30 m, the level of average intensity at a distance of 3 km is 10 dB higher than for a deterministic (when \(\delta c_{1}=0\)) waveguide, and at a distance of 5 km, this difference is about 20 dB. However, in the situation under consideration, this effect is noticeably less pronounced than in the presence of fluctuations in the speed of sound in the water column \(\delta c\), despite the relative smallness of such \(\delta c\). In the latter case, as was shown in [28], the slowing down of the average intensity decay at a distance of 10 km can reach 100 dB (for the frequency of 500 Hz and \(H=50\) m). This fact is of a general nature [24], since inhomogeneities in the bottom sediments have a weaker effect on the values of the modal wave numbers of the waveguide \(\kappa_{m}(r)\), particularly on the imaginary parts \(\kappa_{m}(r)\) describing acoustic energy losses. The modal wavenumbers \(\kappa_{m}(r)\) determine the fluctuations of the signal intensity in individual realizations and, therefore, the statistical effect of transmission loss reduction, shown in Figures 3 and 4. In terms of physics, the fact of a decrease in the transmission loss of sound can be explained by the appearance of local fluctuation waveguides along the propagation path, with varying degrees of focusing of the acoustic energy.