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A numerical solution to a free surface wave problem

The aim of this paper is to present a numerical method for the calculation of water waves in a semi-infinite wave tank. The motion of the fluid, which is assumed to be inviscid, is caused by the motion of part of the boundary of the basin. The correct numerical simulation of the interaction between the side walls and the fluid is an intrinsic feature of the problem. In the literature a series of theoretical results concerning free surface waves are presented. Most of these results deal with the fluid motion in unbounded domains or in domains bounded completely by a free surface. In these cases the boundary of the domain of fluid motion is smooth and solutions in classes of high regularity can be found. Classical existence results for a free surface problem for a viscous fluid were obtained by Solonnikov for a finite drop [19] and by Beale for an infinite domain [1]. Both authors use a mixed Euler–Lagrange formulation of the problem to transform it to a fixed domain. They apply fixed point principles to derive existence results using the smoothing effect of the friction term in the Navier–Stokes’ equations and a priori estimates for related linearized problems. Under certain compatibility and regularity assumptions for the initial condition and external forces, Solonnikov proved the existence of a