Abstract
Below, we present the analytic solution of a two-dimensional problem of hyperelasticity describing the coupled oscillations of a liquid in a rectangular channel and a plane membrane located on the free surface of the liquid, The Bubnov--Galerkin method is used to construct an approximate solution to this problem. The efficacy of the analytic solution is made evident from a comparison of the two approaches. 1, Formulation of the Problem. We will examine a rectangular channel filled with an ideal incompressible fluid to the depth h. We will assume that the free surface of the liquid is covered by a plane membrane that is fastened to the walls of the channel. We choose a coordinate system Oxyz such that the Ox axis is directed along the channel and the Oz axis is directed upward and coincides with the axis of symmetry of its cross section. We place the origin of the system Oxyz in the plane of the membrane and we designate the width of the channel as b. We will examine plane membranes in a "membrane - liquid" mechanical system. In the given formulation, the membrane will be subjected to a transverse load from the side of the liquid. This load will be constant along the channel, which allows us to describe the motion of the membrane by means of the following equation:
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