Reduction for the nonlinear problem of fluid waves to a system of integro-differential equations with an oceanographical application

A numerical procedure for the solution of the nonlinear problem of irrotational wave propagation inside a finite or an infinite homogeneous fluid mass is proposed. This procedure is applied to calculate the fluid gravity waves resulting from certain prescribed varying pressure applied to the free surface of an infinite fluid mass with finite or infinite depth. These waves also are calculated analytically within the frame of the linear theory of motion. The variation of the fluidʹs constant depth, for this application is found to have no influence on the resulting flow. A slight agreement between the numerical solution of the nonlinear problem and the analytical solution of the corresponding linearized problem is noticed in a narrow interval of time following the start of the motion. In the course of time, a significant divergence between the two theories is found, and the nonlinear theory is therefore indispensable for the theoretical prediction of this phenomenon. The proposed procedure can be applied to problems with more complicated geometry.