Symbolic Computation of the Stokes Wave

Abstract

Despite their familiarity, our understanding of the dynamics of surface water waves is far from complete, mainly because of the nonlinearity of the basic equations. G.G. Stokes was the first to find a particular solution in the form of a perturbation series. His result has been improved upon only recently, when fast electronic computers allowed researchers to carry out Stokes’ program numerically to very high order, although numerical noise precludes drawing definitive conclusions for large amplitude waves. However, the development of modern symbolic computation systems has made it possible to obtain exact results, and we report in this paper on the use of two such systems (MAPLE and MACSYMA) for this problem. Central to the success of the approach is a new mathematical formulation, particularly suitable for symbolic computation.