# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 4 Numerical Results and Discussion  
The described problem of the ice response with non-uniform compression caused by the moving submerged body is solved numerically for an ice plate with the following parameters: \(\rho_{i}=917\) kg/m\({}^{3}\), \(E=4.2\times 10^{9}\) N/m\({}^{2}\), \(\
u=0.3\), \(h_{i}=0.1\) m, \(b=10\) m, \(H=2\) m, \(1.8\cdot 10^{-3}\) kg/(m s), and \(K=0.7\cdot 10^{-8}\) m\({}^{2}\) until other values specified for any presented plot. The porosity parameter \(\alpha\) (dimensional) is described according to Equation (4). The nondimensional porosity parameter \(\delta\) is described according to Equation (17). The values of the permeability parameter \(K\) are discussed in the next subsection. Dimensionless strains and how they are calculated are described according to Equation (25). The dipole is located at the point \((0,0,-h/2)\) and moves with a speed \(U\), which varies in the calculations. The shape of the submerged object is determined by formulas (17). The terms describing the images from the channel walls decay with the increase in their number. In Equations (16) and (17), the number \(n\) corresponds to the number of images from the vertical walls, and the number \(m\) corresponds to the number of images from the horizontal channel walls. For each combination of \(m\) and \(n\), \(16\) images are calculated. The corresponding sums converge relatively quickly. Each term with a reflected potential in the final dipole potential \(\varphi^{D}\) behaves as \(O\left(\frac{1}{n^{2}m^{3}}\right)\) with a correction factor depending on \(h\). Along \(x\), the dipole potential behaves as \(O\left(\frac{1}{x^{2}}\right)\) and along \(y\), as \(O\left(\frac{1}{y^{3}}\right)\). The dipole potential was calculated with a limit on the number of terms in (16)and (17) by \(N_{1}\) and \(N_{2}\). The calculations showed that the relative difference in potentials calculated with \(N_{1}=N_{2}=10\) and \(N_{1}=N_{2}=11\) does not exceed \(10^{-2}\) percent near the dipole and \(2.6\) percent near the channel walls. Further dipole potential calculations were conducted for \(N_{1}=N_{2}=10\). The body speed \(U\) is selected based on the critical speeds of hydroelastic waves in a frozen channel, see [49]. For the given case, the characteristic length of the ice cover, \((D/\rho_{i}g)^{1/4}\), is \(2.48\) m. In the calculations, the primary variable parameters are the dimensionless compression parameters \(q_{x}\) and \(q_{y}\). Ice deflections resulting from the motion of the body are characterized by localized, rapidly decaying disturbances in the area directly above the body and by a set of hydroelastic waves propagating from the moving body at a specific velocity. The number of these waves is determined by the intersections of the horizontal line representing the body's speed with the phase speed lines of the hydroelastic waves within the channel. Although there are numerous waves within the channel, only a subset of them, contingent upon the body's speed, contributes to the ice cover's response. These waves can be arranged in increasing order of their frequencies. The compressive coefficients \(q_{x}\) and \(q_{y}\) in the plate equation can change the values of phase speeds, thus, the ratio of the body speed to the phase speeds in the case of compressed ice will be different. In the case of compressed ice, the characteristics of periodic hydroelastic waves were investigated in [53]. In the linear theory of hydroelasticity, if the maximum value of compressive stresses in the direction of wave propagation, in our case \(Q_{x}\), exceeds the value \(q_{cr}\sqrt{\rho_{i}gD}\), waves with negative frequency and negative group speeds will appear for some range of wave numbers (see [8; 9; 53; 54]). The lines of group speeds and frequency of the periodic wave touch zero line for some wave numbers \(k_{1,*}\) and \(k_{2,*}\) when \(q_{cr}=\sqrt{20}/3\) and \(q_{cr}=2\), respectively, for an unbounded ice cover. These values are determined from the dispersion relation. They can be obtained analytically.