On the unsteady motions of a heavy fluid at a sloping beach

Abstract

Waves on the surface of an infinitely deep fluid, and also on the surface of a fluid of finite constant depth, have been investigated in many papers. In this connection Fourier transformations were used for the most part for the solution to the problem. We note that in [1] and [2] a system of standing waves at beaches of one surface slope were investigated. In the work of Keldysh [3] a solution of the non-stationary problem for a beach with a slope angle of 45° was obtained with the help of an integration with respect to a parameter of the solution of the standing wave problem and the construction of an inversion formula. In an analogous manner the linear problem of the unsteady motions of an incompressible fluid at a sloping beach is considered below. A simpler inversion formula is obtained with the help of the generalized Fourier transformations. As an example the problem of Cauchy-Poisson waves at a shallow water approximation is examined.