Gravity-capillary solitary waves in water of infinite depth and related free-surface flows

Abstract

Two-dimensional free-surface flows due to a pressure distribution moving at a constant velocity U at the surface of a fluid of infinite depth are considered. Both gravity g and surface tension T are included in the dynamic boundary condition. The velocity U is assumed to be smaller than (4gT/ρ)¼, so that there are no waves in the far field. Here ρ is the density of the fluid. The problem is solved numerically by a boundary integral equation technique. It is shown that for some values of U, four different flows are possible. Three of these flows are interpreted as perturbations of solitary waves in water of infinite depth. It is found that both elevation and depression solitary waves are possible in water of infinite depth. The numerical results for depression waves confirm and extend the solutions previously computed by Longuet-Higgins (1989).