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.

University Technology Commercialization in Taiwan: National Taiwan University (NTU) and National Taiwan University of Science and Technology (NTUST)

“…In this section, we present a derivation of fifth-order Stokes wave solutions in a linear shear current, based on the framework of Zhao & Liu [15]. The governing equations and the boundary conditions are first presented, being followed by the detailed perturbation method solution process.…”

“…Analytical Stokes wave solutions provide information on wave properties and velocity fields for ocean and coastal engineering applications, enabling extensive studies of wave dynamics. Since the pioneering work of Stokes [10], subsequent researchers have presented numerous perturbation solutions [11][12][13][14][15], which generates extensive studies on the nonlinear characteristics of waves [16][17][18][19]. Based on the stream function expansions, Fenton [12] introduced a set of fifthorder solutions, which has been widely utilized in coastal and ocean engineering community as a reliable solution.…”

“…Based on the stream function expansions, Fenton [12] introduced a set of fifthorder solutions, which has been widely utilized in coastal and ocean engineering community as a reliable solution. However, Zhao & Liu [15] recently conducted a detailed investigation into the fifth-order Stokes wave solutions. They pointed that the assumption made by Fenton [12], requiring the wave height of the solutions be twice pf the wave amplitude of the first order and first harmonic solution (H = 2A), was non-physical.…”

“…They pointed that the assumption made by Fenton [12], requiring the wave height of the solutions be twice pf the wave amplitude of the first order and first harmonic solution (H = 2A), was non-physical. Zhao & Liu [15] employed an alternative method by introducing a perturbation expansion to the wave speed and derived a new set of fifth-order Stokes wave solutions for pure waves.…”

“…Therefore, to better understand wave properties in a linear shear current and better serve engineering applications, a more physical and accurate fifth-order solution is still in wanting. In this paper, based on the perturbation method proposed by Zhao & Liu [15] for deriving the fifth-order Stokes wave solutions, a set of fifth-order Stokes wave solutions for waves in a linear shear current is proposed. In section 2, the governing equations, boundary conditions and the perturbation method solution process are briefly summarized.…”