# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 4 Numerical Results and Discussion
### A Compressed Viscoelastic Ice Plate  
The oscillations of a poroelastic plate, like those of a viscoelastic one, dampen as their locations move away from the moving body. In this section, a comparison of the effects of ice viscosity within the Kelvin-Voigt model [48] and ice porosity within the Meylan model [36] is conducted. We shall consider the motion of the submerged body under a compressed viscoelastic ice plate. To do this, we set \(\alpha=0\) and \(\tau\
eq 0\) in Equations (1)-(10). The same motion of a dipole with a constant speed \(U\) along the central line of the channel is studied. The solution method follows the steps described in Section 3. Changes will occur in the calculation of the added mass matrix and the resulting matrix equation. For the viscoelastic plate, the potential \(\varphi^{F}\) is sought in the form  
\[(\varphi^{E})^{F}=-i\xi\phi^{(1)},\]  
Figure 5: Dimensionless ice deflections (**a**) and dimensionless strains (**c**) along the central line of the channel, and dimensionless strains along the walls (**d**) and normalized deflections across the channel, \(x=0\), (**b**) for \(U=3\) m/s and \(q_{x}=1.4\). Results for \(\overline{q}_{y}=0\) are shown by black, for \(\overline{q}_{y}=1\), by green, \(\overline{q}_{y}=1.75\), by red, and \(\overline{q}_{y}=2.5\), by blue lines. Figure 6: Dimensionless ice deflections (**a**) and strains (**b**) along the central line of the channel for \(\overline{q}_{x}=2.477\). Results for \(U=0.2\) m/s are shown by black, \(U=0.15\) m/s, by green, \(U=0.1\) m/s, by red, and \(U=0.05\) m/s by blue lines. \[\phi_{yy}^{(1)}+\phi_{zz}^{(1)}=\xi^{2}\phi^{(1)},\] \[\phi_{y}^{(1)}=0\quad(y=\pm 1),\quad\phi_{z}^{(1)}=0\quad(z=-h), \quad\phi_{z}^{(1)}=w^{F}\quad(z=0).\]  
Similarly, the potential \(\phi^{(1)}\) is expanded into functions \(\phi_{n}\) with principal coordinates \(a_{n}\). The boundary problem for \(\phi_{n}\) is also adjusted to account for the new kinematic condition at \(z=0\). For the viscoelastic plate, the added mass matrix is computed more simply than for the poroelastic plate, and it will not have an imaginary part. The final matrix equation will take the form  
\[\Big{(}\mathbf{D}+(2\xi^{2}-q_{y})\mathbf{C}-\xi^{2}hFr^{2}\mathbf{M}^{(1)}-i \mathbf{Q}\Big{)}\vec{a}=\vec{P}, \tag{26}\]  
where \(\mathbf{Q}=\beta\xi\sigma(\mathbf{D_{1}}+2\xi^{2}\mathbf{C})\), \(\mathbf{D_{1}}=diag(\lambda_{1}^{4}+\xi^{4},...)\), \(\vec{P}=(P_{1},P_{2},...)^{T}\), \(\sigma=\tau U/b\) and  
\[M_{nm}^{(1)}=\int\limits_{-1}^{1}\phi_{n}\psi_{m}dy,\quad P_{m}=\int\limits_{ -1}^{1}P\psi_{m}dy.\]  
The solution of Equation (26) is obtained using the same algorithm as the solution of Equation (24). For the comparison of viscosity and porosity effects of ice, the ice deflections and strains were calculated for a compressed ice cover along the \(x\) direction with \(q_{x}=2.12\). The dipole speed was \(U=3\) m/s. The coefficient \(K\) in the considered model describes the porosity of the ice plate. In the works [38; 39], it is shown that the values of this coefficient in the range from \(10^{-13}\) m\({}^{2}\) to \(10^{-10}\) m\({}^{2}\) correspond to an approximate porosity of the ice plate from \(0.1\) to \(0.4\). The value \(K=10^{-12}\) m\({}^{2}\) was used in [42]. Relatively large values of \(K\) correspond to rubbish ice, \(10^{-8}\) m\({}^{2}\) in [41] and \(10^{-7}\) m\({}^{2}\) in [56]. Calculations have shown that the values of the porosity coefficient affect the damping of oscillations and the length of the observed oscillation area in the ice cover. The tendency of \(K\) to \(0\) leads the form of ice oscillations to a form of periodic waves, which will propagate to infinity, see [44]. The same asymptotic behavior of ice deflections is observed in the results obtained in this article.