# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 2 Formulation of the Problem
To consider the unsteady problem, this formulation needs to be supplemented with initial conditions for \(w\), \(w_{t}\), and \(\varphi^{E}\) such as  
\[\rho_{i}h_{i}(y)w_{tt}+\left(1+\tau\frac{\partial}{\partial t}\right)\Lambda w +Q_{x}\frac{\partial^{2}w}{\partial x^{2}}+Q_{y}\frac{\partial^{2}w}{\partial y ^{2}}+Q_{xy}\frac{\partial^{2}w}{\partial x\partial y}=p(x,y,0,t), \tag{1}\]6  
and boundary conditions at the channel walls for the ice deflections. The described formulation problem uses general assumptions that are usually applied to describe gravity waves in a fluid, the linear equations of motion of the ice plate, and small dipole radius, and include the porosity of the plate through the simplified Darcy's law. The size of the considered region is not very small and the motion is not very fast, therefore, the effects of capillarity, vorticity, and fluid compression are not taken into account. The linear model is valid when the ice deflections have small curvature, \(w_{x}^{2}+w_{y}^{2}<<1\), and the amplitudes of hydroelastic waves are an order of magnitude smaller than their wavelength. Comparisons with experiments in ice tanks have shown good agreement between the calculations within the considered linear theory and experimental data, see [50]. This linear theory with a variable ice thickness \(h_{i}(y)\) is also valid in the leading order as long as \(h_{i}(y)<<1\), see [12]. The study of ice cover oscillations due to external loads within simplified models has been a subject of previous author's works. In the work by [PERSON] and [PERSON] [51], a non-stationary problem was investigated, where the ice was modeled as a thin viscoelastic plate with uniform thickness. The load was represented by a localized pressure moving at a constant speed along the channel's central line. Further investigations into viscoelastic ice plates were conducted by [PERSON] et al in [49], where a submerged body moving steadily through the channel served as the external force. A two-dimensional analysis of oscillations in thin poroelastic ice covers under the influence of oscillating external loads was carried out by [PERSON] et al. in [44]. Additionally, [PERSON] in [12] studied the problem of periodic hydroelastic waves in a channel covered with ice. Ice was modeled as a thin elastic plate with linearly changing ice thickness across the channel. The behavior of hydroelastic waves depending on the ice thickness slope parameter was investigated. The combination of viscosity and non-uniform ice thickness was studied in [52], where the ice thickness varied linearly and symmetrically across the channel. It was found that even a small variation in ice thickness significantly changes the characteristics of hydroelastic waves in the channel. In this article, we shall study ice cover oscillations caused by a moving body taking into account compressibility effects in combination with viscosity/porosity/variable ice thickness. The Motion of a submerged Body with Constant Speed in a Frozen Channel with Non-Uniform Compression of the Ice Cover  
The motion of the submerged body along the frozen channel with constant speed \(U(t)=U\) is considered. Accounting for the viscosity of the ice results in the attenuation of the ice deflections away from the moving body (see [49]). Additionally, considering porosity also leads to the damping of these oscillations, see [47], thus simultaneous consideration of these two parameters is not needed. For the considered problem, we will include only porosity for most of the calculations. At this stage of the solution, we shall consider a constant thickness of the ice cover, \(h_{i}(y)=h_{0}\), and zero combined compressions of the ice, \(Q_{xy}\). The porosity model is chosen without considering hydrostatic pressure, as done, for example, in [36]. The problems (1)-(10) are solved in dimensionless variables with the following scales: the length scale is \(b\) and the time scale is \(b/U\).