# Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice
## 1 Introduction
The study in [47] considered a problem that incorporated the effects of variable thickness and porosity of the ice cover. An external load was represented by an oscillating dipole along the principal axes in a frozen channel. The dipole of small radius was submerged to a sufficient depth to model the oscillations of a rigid sphere. The ice thickness was linear and symmetric about the center of the channel. The porosity of the ice cover was accounted for using the fluid penetration velocity into the ice plate within the Darcy model. The effects of porosity and symmetric linear thickness on the reaction of the ice cover was investigated separately, as well as the effect of thickness variation together with sufficiently high values of ice porosity. This article considers the motion of a submerged body along the central line of a channel, taking into account non-uniform compressibility. Non-uniform compressibility implies different compressive stresses along (longitudinal) and across (transverse) the channel. The submerged body is modeled as a dipole, whose potential in the channel is determined using the method of mirror images. The caused oscillations of the ice cover are damped either by accounting for porosity or by viscosity of ice. The problem is solved within the linear theory of hydroelasticity. The general formulation of the problem for heterogeneous ice and the formulation in the considered case of compressed ice are given in Section 2. In Section 3 the method of solution is presented. The numerical results are described in Section 4. The influence of inhomogeneous compression on ice deflections and strain is presented in the same Section. A brief comparison of the damping effects of viscosity and porosity is given in Section 4.1. The effect of symmetric linearly varying thickness of the channel ice cover on ice deflections and deformations in the case of longitudinal compression is investigated in Section 4.2. In problems involving the interaction of ice, liquid, structures, and external loads, mathematical models are quite extensive, with a significant number of equations, often being nonlinear in their complete formulation. For the construction of analytical or semi-analytical solutions, which is our focus, it is necessary to consider problem formulations under certain conditions. Fortunately, comparisons between experiments and computational results, as demonstrated in works such as those by [PERSON] and their research group [48], indicate that the linear theory of hydroelasticity and potential flow theory effectively describe the considered problems at realistic scales. The advantage of using linear theory and the solution method described in this article is that the solution of the problem for the ice and hydrodynamic parts is constructed simultaneously. The formulas obtained in the form of integrals for calculating ice deflections and maximum strains in the ice cover allow for a careful analysis of the behavior of ice in the considered ice-liquid-structure system. Since the results of modeling ice as thin plates showed good agreement with experiments, we believe that the main aspects of porous ice in the considered problem can be analyzed within the same theory. In this case, porous ice is understood as an elastic porous ice plate. As will be shown later, within such models, at least the damping effect of ice porosity is captured. Using full-fledged filtration models to describe the movement of fluid in porous ice is very challenging for the considered ice-fluid-structure interaction problems.