Nonlinear radial oscillations of a fluid in a parabolic basin with regard for the external action

Abstract

Within the framework of the theory of long waves, we find a class of exact analytic solutions of the problem of description of nonlinear axially symmetric oscillations of a fluid in a parabolic basin with regard for the action of stationary radial bulk forces. The radial projection of the velocity of these oscillations (seiches) is a linear function of the radial coordinate, whereas the azimuthal velocity and displacements of the free surface of the fluid are polynomials in the radial coordinate with time-dependent coefficients. The method of finding solutions is based on the exact replacement of the original problem by a system of ordinary differential and algebraic equations. The action of the bulk forces may result either in the increase in the frequency of oscillations of the fluid and in the decrease in this frequency and affect the motion of the water edge, the characteristics of waves, and the velocity field.