# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 4 Numerical Results and Discussion
In this range, there is an increase in the amplitudes of ice deflections and strains in the local area above the moving body. Hydroelastic waves do not propagate from the moving body. For \(\vec{q}_{x}=2.12\) and \(\vec{q}_{x}=2.477\), this speed is supercritical. Analysis of phase speeds predicts the appearance of one short wave in front of the body and one long wave behind it. These waves are observed in the plots. The speed \(U=3\) m/s is closest to critical in the case \(\vec{q}_{x}=2.12\). So the ice deflections and strains in this case are the largest among all shown. Strains along the central line of the channel are 1.5-2 times larger than strains along the walls for all considered cases. Normalized deflections across the channel, at \(x=0\), are shown in Figure 3b. It is evident that for subcritical cases, the primary contribution to the ice deflections is made by the first normal vibration mode, \(\psi_{1}(y)\), while for supercritical cases, the primary contribution is made by the first and second normal vibrations modes, \(\psi_{1}(y)\) and \(\psi_{2}(y)\). The variation in compressive stresses across the channel, \(q_{y}\), does not lead to noticeable changes in the phase speeds of hydroelastic waves in a channel. Therefore, in the case of compression-only along the \(y\) direction, the shape of the ice deflections should not significantly change compared to uncompressed ice. The effect of compressibility across the channel is shown in Figure 4 for the subcritical speed \(U=3\) m/s. The supercritical case will be shown later in combination with longitudinal compression. Dimensionless ice deflections are shown in Figure 4a,b, and dimensionless strains along the central line and across the channel are shown in (c,d), respectively. Results for \(\vec{q}_{y}=0\) are shown by black, for \(\vec{q}_{y}=1\), by green, \(\vec{q}_{y}=1.75\), by red, and \(\vec{q}_{y}=2.5\), by blue lines. For the considered range of compressive stresses \(q_{y}\), the deflection is localized above the dipole, and no propagating waves appear. An increase in transverse compressive stresses leads to an increase in deflection amplitudes and a slight increase in strain amplitudes above the submerged body and along the central line of the channel. In contrast, strains decrease at the channel walls. The shape of the ice oscillations across the channel becomes narrower with increasing \(q_{y}\). Figure 3: 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. 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 simultaneous effect of compressive stresses along and across the channel is shown in Figure 5. In the presence of non-zero longitudinal compressive stresses, the addition of transverse stresses amplifies the effects of the longitudinal ones. Therefore, if a certain motion speed \(U\) was subcritical even for longitudinally compressed ice, it may become critical or supercritical with the presence of transverse stresses as well. Dimensionless ice deflections along the central line of the channel are shown in Figure 3a, dimensionless strains are shown in Figure 3c, along the central line of the channel, and in Figure 3d, along the walls. Calculations of the ice deflections and strains in the case of non-zero compressive stresses in both directions were performed for a fixed value of \(\overline{q}_{x}=1.4\) and the speed of the body \(U=3\) m/s. 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. The speed \(U=3\) m/s is subcritical for the first three cases and supercritical for the last case.