Analytical solution of the problem of small forced oscillations of the ideal fluid

The study aims to determine the shape of the free surface of an ideal liquid vibrating under variable external pressure in a rectangular vessel. The two-dimensional problem has been solved analytically. The motion of the ideal fluid has been simulated by the solution of Laplace’s equation for the fluid velocity potential in the flow region. On the solid boundary, the impermeability conditions have been assigned; on the free surface, the Cauchy–Lagrange integral. The equation was solved using the variable separation method. The free surface condition has been transformed in the case of small oscillations and varying pressure on the free surface and it has been used to determine the velocity potential as a function of time. The implementation of this condition required solving the system of linear second-order differential equations. The same condition gives the formula for determining the shape of the free surface of liquid at any time instant in the form of deviation of free surface points from the equilibrium position. The method described in this paper has been applied in the cases with the external pressure varying harmonically and acting on the restricted part of the free surface during a limited time. The obtained results have been compared with the currently available solutions of similar problems.