On Stokes wave solutions

Abstract

Using a perturbation method with the aid of MATLAB ® Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton's, and Skjelbreia and Hendrickson's solutions.