Free oscillations of an ideal fluid in a rectangular vessel with a horizontal permeable membrane

The study aims to determine the shape of the free surface of an ideal fluid executing free oscillations in a rectangular vessel divided by a horizontal partially permeable membrane. The plane problem is solved analytically. The motion of the ideal fluid is simulated by solving Laplace's equation for the fluid velocity potential in the region occupied by fluid. Two regions are considered: the area between the bottom of the vessel and membrane and the area between membrane and free surface. On the solid boundary, the impermeability conditions are used; on the free surface, the Cauchy–Lagrange integral. On the membrane, the boundary condition is set as follows: the normal velocity of the fluid near membrane in both regions is proportional to the difference in the values of fluid velocity potential on either side of membrane. In the considered regions, the Laplace's equation is solved using the method of separation of variables. To determine the dependence of the velocity potential on time, the boundary condition on the free surface is used, which is transformed for the case of small oscillations. This same condition yields the formula for determining the shape of the free surface of fluid at any time instant in terms of deviation of the free surface points from equilibrium position. The method described in this paper has been applied for the cases of different permeability of horizontal membrane. The obtained results have been compared with currently available solutions to similar problems.