A New Approach to Stokes Wave Theory

Abstract

Stokes wave theories to third-order approximation have widely been employed to calculate wave properties for waves propagating over finite depths of water in most engineering applications. However, different and often inconsistent expressions of wave variables can be observed from the numerous theories available. Examinations of the usual Stokes wave theories are on the so-called Stokes definitions of wave celerity and Bernoulli constant, as well as on the physical explanations of some theoretical problems involved. A new Stokes wave theory to third order approximation is derived by applying only necessary conditions and assumptions, without using the definitions of wave celerity. The resulting mathematical formulations for some pertinent wave variables are presented. Comparison is made between the new third-order approximation and the usual ones derived from using the Stokes definitions of wave celerity, showing different expressions of wave celerity, horizontal water particle velocity, and mass transport velocities. It is found that the mass transport velocity exists in the Eulerian description as well as the usual Stokes drift in the Lagrangian description.