Solution of the Problem of Submerged Body Oscillations in a Frozen Channel with Linearly Varying Ice Thickness

The paper considers the problem of hydroelastic waves generated by a submerged body undergoing vertical translational oscillations in a rectangular frozen channel of finite depth and width. The ice is modeled as a thin viscoelastic plate with its thickness varying linearly across the channel. The edges of the plate are frozen to the channel walls. The deflection of the ice cover is described within the framework of linear elasticity theory. The case of the symmetric thickness variation of the ice cover with respect to the channel's central line is studied. The liquid beneath the plate is inviscid and incompressible. The fluid flow induced by the plate deflection is potential. The oscillating submerged body is modeled as a three-dimensional dipole, which generates flow and pressure corresponding to a rigid sphere when oscillating in an unbounded fluid. The radius of the sphere is related to the dipole's velocity and its intensity. The dipole velocity varies periodically, resulting in changes in its intensity, while maintaining the body's shape unchanged. The velocity potential of the dipole placed in a rectangular channel with rigid walls is obtained using the method of image reflections.