The Problem of Plates Hit with a Water Layer and the Method of Point Potentials

Abstract

The problem of the impact of an absolutely solid plate on the surface of an ideal fluid layer is considered. The problem is formulated for the velocity potential as a boundary value problem for the Laplace equation in a layer and half-space (M.V. Keldysh – two-dimensional case, I.I. Vorovich, V.I. Yudovich – round disk case in ${\bf R}^3$). In the works of the mentioned authors, a flat plate and a flat bottom were considered. This made it possible to apply the Fourier transform, obtain an integral equation for the potential, and, using the expansion of the solution in special functions, calculate some basic hydrodynamic values.

To solve this problem, an algorithm is proposed in which the most difficult step is to solve a mixed boundary-value problem for the Laplace equation with a given boundary value on the plate surface.

For the numerical solution of this problem, the method of point potentials is used, which is also convenient for curvilinear boundaries. An approximate solution is represented as a linear combination of point potentials. To determine its coefficients, a variational problem is constructed, the solution of which reduces to a system of linear algebraic equations.

For a flat plate and a flat bottom, the results obtained are compared with the known ones. The results of solving the problem with a convex plate and a curved bottom are presented.