# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 4 Numerical Results and Discussion
### A Compressed Poroelastic Ice Plate with Variable Ice Thickness
The final matrix equation will take the form  
\[\big{(}(1-\xi^{2}q_{x})\mathbf{I}-q_{y}\mathbf{C}-mh^{2}Fr^{2} \xi^{2}\mathbf{K_{10}}+\beta\Big{[}\mathbf{K_{34}}-2\xi^{2}\mathbf{K_{32}}+ \xi^{4}\mathbf{K_{30}}+6(\mathbf{K_{23}}-\xi^{2}\mathbf{K_{21}})\] \[+3(\mathbf{L_{12}}-\xi^{2}\
u\mathbf{L_{10}})\Big{]}+\xi hFr^{2}( \mathbf{M}^{(1)}-i\mathbf{M}^{(2)})\big{)}\vec{a}=\vec{P}^{(1)}+i\vec{P}^{(2)}, \tag{28}\]  
where \(\mathbf{K_{nm}}=\{K_{nm}\}\), \(\mathbf{L_{nm}}=\{L_{nm}\}\) and  
\[K_{10}=\int\limits_{-1}^{1}h_{i}\psi_{n}\psi_{m}dy,\quad K_{34}=\int\limits_{ -1}^{1}h_{i}^{3}\psi_{n}^{\prime\prime\prime\prime}\psi_{m}dy,\quad K_{32}= \int\limits_{-1}^{1}h_{i}^{3}\psi_{n}^{\prime\prime}\psi_{m}dy,\]  
\[K_{30}=\int\limits_{-1}^{1}h_{i}^{3}\psi_{n}\psi_{m}dy,\quad K_{23}=\int \limits_{-1}^{1}h_{i}^{2}h_{i}^{\prime}\psi_{n}^{\prime\prime\prime}\psi_{m}dy,\quad K_{21}=\int\limits_{-1}^{1}h_{i}^{2}h_{i}^{\prime}\psi_{n}^{\prime}\psi _{m}dy,\]  
Figure 7:0  
The matrix Equation (28) is not symmetric. Calculations were performed for \(\theta=0.25\) and \(\theta=0.5\), and \(n=1\) and \(n=2\) in (27). The thickness scale was chosen so that its average value was equal to 0.1 m. Two cases of compressibility variation were investigated, \(\overline{q}_{x}=1\) and \(\overline{q}_{x}=2\), for the supercritical speed of motion \(U=7\) m/s and permeability parameter \(K=0.7\cdot 10^{-8}\) m\({}^{2}\). For this speed, the appearance of a short wave in front of the moving body is expected. The convergence of the results was checked using the norm \(|(w_{N_{1}}-w_{N_{2}})/w_{N_{1}}|\) calculated at each point on the computational grid in the plane \((x,y)\), where \(N_{1}\) and \(N_{2}\) are the numbers of considered classic normal modes \(\psi_{n}(y)\). The calculations showed that for \(N_{1}=60\) and \(N_{2}=80\), this relative difference does not exceed 0.2 percent along the central line of the channel. The absolute difference (with a maximum value of dimensionless deflections around 100, see Figure 9 below) over the entire ice cover does not exceed 0.04. Further increasing the number of modes does not lead to a noticeable increase in these differences. The relatively small value of this number is associated with the consideration of small inclination angles for symmetric thickness. More complex shapes of ice thickness will require a greater number of modes, and in general, it is better to use normal modes for an elastic beam with variable thickness, see [12]. For comparison, to achieve the same accuracy in the case of a constant thickness of ice, for the considered parameters of the problem, it is sufficient to use 10-15 modes. Thus, calculations in the case of variable thickness were performed with \(Nmod=60\) modes. This value is sufficient to obtain deflections and strains with visual accuracy. The ice deflections along the central line of the channel are shown in Figure 7:1 The results for a constant ice thickness are shown by the black lines, the blue lines show the results for \(\theta=0.25\) and \(n=1\), and the red lines, for \(\theta=0.5\) and \(n=2\). In this and the following figures, the left side corresponds to the case \(\overline{q}_{x}=1\), and the right side corresponds to \(\overline{q}_{x}=2\). The shape of the ice deflections qualitatively remains the same. An increase in the variation in symmetric ice thickness leads to a slight increase in ice deflection amplitudes. A variable ice thickness can lead to changes in the parameters of hydroelastic waves, see [12]. In the case under consideration, the linear ice thickness variation with a small parameter \(\theta=0.25\) does not lead to a significant change in the wavelength. Quadratic thickness variation with \(\theta=0.5\) leads to a 9% decrease in the wavelength. The strains along the central line of the channel are shown in Figure 7:2 The results for a constant ice thickness are shown by the black lines, the blue lines show the results for \(\theta=0.25\), and the red lines, for \(\theta=0.5\).