INTRODUCTION.

Analytical methods of modelling water waves of small but finite height are based on the linear theory and improved with weakly nonlinear theories (West, 1981). An alternative is to develop, with computer assistance, water wave models which are nonlinear in their lowest approximation and are valid for a range of heights up to the onset of wave breaking (Schwarz & Fenton, 1982). The present approach falls into the latter category, and is concerned with investigating wave geometries which occur locally in deep water.

Water waves propagating from a surface disturbance are subject to dispersion modified by nonlinear wave interactions. This property suggests that the numerical resolution into Fourier components of the nonlinear equations governing the evolution of a water wave system models the dispersion and its modification, and is therefore a natural method for investigating water wave properties. Fornberg S Whitham (1978) used this approach in studying certain nonlinear model equations for wave phenomena. It is applied here to Laplace's equation with the nonlinear boundary conditions appropriate to irrotational gravity wave propagation in deep water.

Analytical solutions in the form of perturbation expansions exist for two dimensional water waves of permanent shape in deep water (Stokes waves) for which the dispersion and nonlinear modification are in balance. A number of computer-based methods have been used (Schwartz S Fenton, 1982, §2) to extend the calculations up to the highest waves of permanent form. The present method is demonstrated first (§2) for the calculation of two dimensional permanent waves. Three dimensional permanent waves have been found recently as perturbations to two dimensional permanent waves. The present method allows calculations of three dimensional waves independently of two dimensional waves, and one such example is presented below (§3).

Waves on the ocean surface often occur locally as a wave group with an envelope that changes slowly as the waves propagate. Analytical solutions exist for weakly nonlinear wave groups of permanent envelope in two and three dimensions. The present method is applied to the calculation of wave groups of permanent envelope in two dimensions (§4) and in three dimensions (§5), in both cases without the restrictions on wave height which are needed for the analytical solutions.