## 3 Model of a Stochastic Waveguide  
To carry out a numerical analysis, we further reference the parameters that, according to [9; 10], are characteristic of the shelf zones of the Russian Arctic seas, particularly the Kara Sea (in the absence of ice cover on the surface in summer). It is important to note that the presence of ice cover is insignificant for the purposes of our study. This study is associated with the influence of fluctuations of parameters in bottom sediments. It is well known that the presence of ice on the sea surface leads to a faster decay of the signal during propagation [1; 4; 7]. As a result, the transmission loss curves will be located lower than without considering the presence of ice on the surface. We consider a shallow-water waveguide with depths of 30 m and 40 m with a horizontal surface and bottom, having homogeneous profiles of sound speed \(c=1460\) m/s and density \(\rho=1\) g/cm\({}^{3}\), in which a tonal sound signal with a frequency of 250 Hz propagates. The bottom is an absorbing liquid half-space of unconsolidated sediments with a refractive index at the water-bottom interface \(n=(c/c_{1})(1+i\beta_{1})\), \(\beta_{1}=0.02\) (see Figure 1). The parameters of the bottom sediments, \(c_{1}(z,r)\), \(\rho_{1}(r)\), randomly vary along the signal path. In this case, the impedance function \(g_{m}(r)\) in the boundary condition (3) is determined by the local values of the comparison waveguides: \(g_{m}(r)=-i\rho_{1}(r)\rho^{-1}[k^{2}-\kappa_{m}{}^{2}(r)]^{-1/2}\). In this work, following the measurement data given in [9; 10], we put \(\rho_{1}(r)=\langle\rho_{1}\rangle=1.85\) g/cm\({}^{3}\). We also took into account the fact that, as calculations show, random variations in density \(\delta\rho_{1}(r)\), \(\rho_{1}(r)=\langle\rho_{1}\rangle+\delta\rho_{1}(r)\), have a much weaker effect on sound propagation than fluctuations in the speed of sound. This fact is well known from the theory (see, for example, [17]). The variations in density in bottom sediments can be neglected if not too low radiation frequencies are examined (\(f=2\pi\omega>1\) Hz) and there are no large-amplitude jumps of \(\mid\delta\rho_{1}/\langle\rho_{1}\rangle\mid\) in the medium. For the speed of sound in the bottom sediments, we first considered the random process \(c_{1}(r)=\langle c_{1}\rangle+\delta c_{1}(r)\) (see Figure 2), setting it with a Gaussian probability distribution with an exponential correlation function: \(B_{c1}(r_{2}-r_{1})=\sigma_{c1}{}^{2}\) exp \((-\mid r_{2}-r_{1}\mid/L_{r})\). Based on the processing data [9; 10], as well as information from [24], the characteristic scale \(L_{r}\) of the inhomogeneities vary is further chosen to be 1 km, and the intensity of fluctuations \(\sigma_{c1}{}^{2}=\langle(\delta c_{1}/\langle c_{1}\rangle)^{2}\rangle=1.7 \cdot 10^{-3}\) (corresponds to the amplitude \(\mid\delta c_{1}\mid\approx 60\) m/s). Note that although the amplitude of fluctuations is small, \(\sigma_{c1}\ll 1\), for bottom sediments, it is more than an order of magnitude higher than the values used in modeling the fluctuations of the speed of sound in water, which are caused, for example, by background internal waves [22; 24; 25]. From Figure 2, it is seen that the signal propagating along the path passes through the bottom areas with inhomogeneities of the \"soft\" (\(c_{1}<c\)) and \"rigid\" (\(c_{1}>c\)) types, which is typical for the shelf of the Arctic seas. In this case, inhomogeneities of the \"soft\" type with \(c_{1}<c\) (including \(c_{1}<<c\)) are due to the presence of gas saturation in sediments [26]. In this case, when \(c_{1}<c\), even in the absence of absorption \(\beta_{1}\), there are no propagating (trapped) modes in the waveguide, and all modes are to be leaky. In the second case, for \(c_{1}>c\), depending on the degree of \"rigidness\" of the bottom, the first several modes at a frequency of 250 Hz are propagating and weakly attenuating (for \(\beta_{1}\
eq 0\)).