Three-dimensional problem of unsteady wave motions of a liquid in a region of variable depth

Abstract

The three-dimensional problem of unsteady wave motions of a liquid above a plane inclined floor in the framework of a linear dispersion model was considered for the first time in [1] for the particular case β=π/4, where β is the angle of inclination of the floor plane to the free surface of the liquid. The class of exact self-similar solutions of the problem for β=π/2(2m + 1), m=0, 1, 2,..., for the case of an initial perturbation of a free surface of a special type which is constant in the direction of the normal to the shoreline was found in paper [2]. The present paper is devoted to the investigation of wave motions of a liquid due to an initial perturbation of arbitrary form for the angles of inclination of the floor assumed in [2]. A complete system of eigenfunctions corresponding to the continuous and discrete spectra is found. The theorem of the expansion of an arbitrary absolutely integrable function with respect to the boundary values of the eigenfunctions is proved. An exact solution of the problem is obtained and its asymptotic analysis is carried out.