# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 4 Numerical Results and Discussion
The value \(q_{cr}=\sqrt{20}/3\) is determined as the value at which the derivative of the dispersion relation with respect to the wave number \(k\) equals zero. The value \(q_{cr}=2\) is the value when the dispersion relation itself equals 0. For the ice cover in the channel, only the wave with the lowest frequency has the same behavior. However, in a channel's case, these values for this wave are 2.12 and 2.477, respectively. This is explained by the influence of the channel walls. Dispersion relations in the channel are computed numerically from a matrix problem. Therefore, these values are also determined solely by numerical methods. In this work, we shall consider compressive stresses \(Q_{x}\) not exceeding the critical value \(2.477\sqrt{\rho_{IS}D}\). It is convenient to use scaled parameters \(\overline{q}_{x}\) and \(\overline{q}_{y}\), \(\overline{q}_{x}=q_{x}\sqrt{\beta}\), \(\overline{q}_{y}=q_{y}/\sqrt{\beta}\), to describe compression in the ice plate instead of nondimensional parameters \(q_{x}\) and \(q_{y}\). The phase (a) and group (b) speeds for the first hydroelastic wave with the lowest frequency for the considered channel are shown in Figure 2. The blue lines show the results for \(\overline{q}_{x}=0\), the red lines, for \(\overline{q}_{x}=1\), the purple lines, for \(\overline{q}_{x}=2.12\), and the black lines, for \(\overline{q}_{x}=2.477\). For the moving body, the phase speeds are most important. It can be observed from the plots that the speed \(U=3\) m/s is subcritical for the uncompressed and weakly compressed plate. However, for high compression (\(\overline{q}_{x}=2.12\) and \(\overline{q}_{x}=2.477\)), this speed becomes supercritical, and there is one long wave that will propagate behind the body, and one short wave that will propagate in front of it. The direction of propagation of these waves depends on the sign of the difference between the group speed and moving body speed, see [55]. Note that in the case of \(\overline{q}_{x}=2.477\), for a low speed of the body, the long waves will have a negative group speed. The accuracy of the calculations of the ice deflections depends on the number \(Nmod\) and the method of computing the integrals in (25). Numerical analysis showed that all functions of \(\zeta\) involved in the integrals decay rapidly and are smooth enough for all investigated sets of parameters. For visual accuracy of the integral's calculations, it is sufficient to take \(N_{g}=40\) and \(400\) steps in \(\zeta\) from 0 to \(N_{g}\). The convergence of the numerical solution is generally assessed by the norm \(max|w_{Nmod1}(x,y)-w(x,y)_{Nmod2}|\), where lower indexes correspond to a number of modes used in the calculations. Test calculations showed that to achieve convergence of the solution with visual accuracy, it is sufficient to take 10 modes. The convergence of the solution can also be affected by reducing the porosity parameter \(\alpha\). The effect of compressed ice along the \(x\) direction (direction of motion) is shown in Figure 3. The calculations were carried out for \(U=3\) m/s. This speed, for different values of compressibility \(q_{x}\), can be either subcritical, critical, or supercritical, see Figure 2. The dimensionless deflections along the central line of the channel are shown in Figure 3a, the  
Figure 2: Phase (**a**) and group (**b**) speeds for the first hydroelastic wave with the lowest frequency for the considered channel. The blue lines show the results for \(\overline{q}_{x}=0\), the red lines, for \(\overline{q}_{x}=1\), the purple lines, for \(\overline{q}_{x}=2.12\), and the black lines, for \(\overline{q}_{x}=2.477\). dimensionless strains are shown in Figure 3c along the central line of the channel, and in Figure 3d, along the walls. Results for \(\vec{q}_{x}=0\) are shown by black, for \(\vec{q}_{x}=1\), by green, \(\vec{q}_{x}=2.12\), by red, and \(\vec{q}_{x}=2.477\), by blue lines. The speed \(U=3\) m/s is subcritical for \(\vec{q}_{x}=0\) and \(\vec{q}_{x}=1\).