## 3 Model of a Stochastic Waveguide
The degree of focusing by these local fluctuation waveguides is determined by fluctuations of the imaginary parts of the eigenvalues \(\kappa_{m}(r)\). The larger the value of these imaginary parts \(\text{Im}(\kappa_{m}(r))\), the more they fluctuate, and in the statistical mean, there is a decrease in the transmission loss of a signal. It is seen from Figure 4 that an increase in the waveguide depth with the same values of other parameters weakens the above-described effect of slowing down the decay of the average intensity. This is explained by the decrease in the imaginary parts of \(\kappa_{m}(r)\) of the lowest-number modes with increasing depth. It is these modes that determine the attenuation of the acoustic field in the sea with increasing distance from the source (compare the deterministic curves in Figures 3 and 4). It should be noted that in the situations examined, the adiabatic approximation describes the average transmission loss well. The curves in Figures 3 and 4 for OW and \"adiabatic\" differ by fractions of a decibel, so they are hardly distinguishable on the graphs. This is due to the overwhelming contribution of the first mode in most realizations. With distance from the source, this mode is the least attenuated, and the necessary condition for the adiabatic approximation, \(L_{r}\gg\mid\kappa_{1}(r)-\kappa_{n}(r)\mid^{-1}\), is satisfied. Nevertheless, in individual intensity realizations, the result of mode coupling can be noticeably manifested. First of all, this refers to those realizations that describe a sufficiently fast decay of the signal intensity along the propagation path. In Figures 3 and 4, it is clearly seen that the curves of statistical modeling at \(r>1.5\)-2 km are quite smooth. This is due to two circumstances. The first is the statistical averaging of the intensity over the ensemble of realizations. Second, the step \(\Delta r\) of calculating the mode amplitudes by formula (4) was chosen much less than the characteristic scale (correlation radius) of inhomogeneities \(\Delta r\ll L_{r}\). At the same time, it was comparable to the scale of the interference structure of the field \(\Delta r\sim\mid\kappa_{m}(r)-\kappa_{m}(r)\mid^{-1}\). the water layer of the shallow sea, this fact of the appearance of strong fluctuations with increasing distance is known from [24; 25; 29] and was also noted in [16; 30]. The appearance of strong intensity fluctuations in the waveguide means an increase in the spread of the intensity levels as a function of the distance [30] in individual realizations. For example, the range of this spread in intensity levels for the situation presented in Figures 3 and 5 at a distance of 5 km from the source is 53 dB, and at a distance of 10 km, it reaches 103 dB. Above, a model of liquid bottom sediments with a horizontal scale of sound velocity fluctuations \(L_{r}=1\) km was considered. The scale of fluctuations in depth \(L_{z}\) was assumed to be large enough (theoretically, infinite). It is also possible to consider randomly stratified bottom sediments with a limited \(L_{z}\) value. Qualitatively, the pattern of sound propagation in the water column changes little. The quantitative differences in the intensity decay in this case are considered in the next section. Statistical Analysis of Sound Transmission Loss in the Waveguide: Stratified Bottom Sediments with Random \(c_{1}\)(\(z\),\(r\))  
For the speed of sound in the bottom sediments, consider a two-dimensional Gaussian random field \(c_{1}\)(\(z\),\(r\)) = \(\langle c_{1}\rangle+\delta c_{1}\)(\(z\),\(r\)) with an exponential correlation function: \(B_{c1}\)(\(z_{2}-z_{1}\), \(r_{2}-r_{1}\)) = \(\sigma_{c1}\)\({}^{2}\) exp(\(-\left|\,z_{2}-z_{1}\,\right|/L_{z}-\left|\,r_{2}-r_{1}\,\right|/L_{r}\)). Let us assume that in the vertical direction, fluctuations occur in the bottom sedimentary layer 15 m thick (Figure 6), and below, there is a half-space with \(c_{1}=c=1460\) m/s. This thickness of the bottom sedimentary layer was not chosen by chance.