# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 5 Conclusions  
The problem of the motion of a submerged body along the central line of the frozen channel with non-uniformly compressed ice along the principal coordinates is considered. The body is moving at a constant speed. Cases with different longitudinal and transverse compressive stresses were studied, considering only positive stresses. The submerged body is modeled as a dipole, the potential of which in the channel is determined by the method of mirror images. The damping of oscillations was modeled by taking into account the porosity or viscosity of the ice. The problem is solved within the linear theory of hydroelasticity. By applying the Fourier transform along the channel, the problem is reduced to a problem of the profile of vibrations of compressed ice across the channel, which is solved by the method of normal modes. Classical normal modes for a beam of constant thickness were used, taking into account the clamped conditions at the channel walls. The main part of the article is devoted to the motion of the body along compressed ice. Alongside this, two additional problems are studied: a comparison of the damping effects of porosity and viscosity, and a study of the influence of variable ice thickness on the ice deflections and strains during considered motion. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\mathbf{n=1,\theta=0.25}\) & \(\mathbf{n=1,\theta=0.5}\) & \(\mathbf{n=2,\theta=0.25}\) & \(\mathbf{n=2,\theta=0.5}\) \\ \hline \(\overline{q}_{x}=1\) & 1.47 & 3.38 & 3.46 & 8.53 \\ \(\overline{q}_{x}=2\) & 1.28 & 2.85 & 2.91 & 7.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ratio of maximum strains, \(\max\overline{\epsilon}(x,y)/\max(\epsilon(x,y))\), where \(\overline{\epsilon}\) is the strains for the ice plate with variable thickness for corresponding cases described by (27). Figure 11: Dimensionless strains along the channel walls for \(\overline{q}_{x}=1\) (**a**) and \(\overline{q}_{x}=2\) (**b**). Legend is the same as in Figure 9. The results for constant ice thickness are shown by black lines, blue lines show results for \(\theta=0.25\) and \(n=1\), and red lines, for \(\theta=0.5\) and \(n=2\) in (27). Figure 10: Dimensionless strains along the central line of the channel for \(\overline{q}_{x}=1\) (**a**,**c**) and \(\overline{q}_{x}=2\) (**b**,**d**) and for \(\theta=0.25\) (**a**,**b**) and \(\theta=0.5\) (**c**,**d**). The results for constant ice thickness are shown by black lines, blue lines show the results for \(n=1\) and orange lines, for \(n=2\) in (27). Relating the problem of the body motion along the compressed ice, the following results were obtained:  
* The most important compressive stress is the stress in the direction of motion of the submerged body, in our case, the longitudinal stress. Increasing longitudinal stresses changes the phase speeds of hydroelastic waves, which form the ice response to the moving submerged body. These speeds become smaller as the longitudinal stresses increase. Therefore, the speed of the body, which was subcritical for uncompressed ice, may become critical or supercritical. For the considered channel, the speed \(U=3\) m/s was subcritical for uncompressed ice and compressed ice with a longitudinal stress of \(Q_{x}=\sqrt{\rho_{l}gD}\), and supercritical for \(Q_{x}=2.12\sqrt{\rho_{l}gD}\) and \(Q_{x}=2.47\sqrt{\rho_{l}gD}\). In general, for subcritical speeds, increasing longitudinal compression leads to a sequential increase in the ice deflections and strain amplitudes, followed by a change in the shape of the ice deflections from localized deflection above the body to a system of hydroelastic waves in front of and behind the body. * Transverse compressive stresses do not qualitatively change the shape of the ice deflections. Increasing transverse compressive stresses leads to an increase in ice deflection amplitudes above the body and a decrease in these amplitudes near the walls, narrowing the oscillation shape towards the center of the channel.