Theory of spatially periodic progressive (propagat ing with constant velocity without change of the amplitude) waves in two dimensional (2D) potential flow of an ideal incompressible fluid with free surface in gravitational field was founded in pioneering works

675 Theory of spatially periodic progressive (propagat ing with constant velocity without change of the amplitude) waves in two dimensional (2D) potential flow of an ideal incompressible fluid with free surface in gravitational field was founded in pioneering works by Stokes [1, 2] and developed further by Michell [3], Nekrasov [4, 5], and many others (see, e.g., book [6] by Sretenskii for review of older works as well as [7– 14] and there in for more recent progress). There are two major approaches to analyze the Stokes wave, both originally developed by Stokes. The first approach is the perturbation expansion in amplitude of Stokes wave called by the Stokes expansion. That approach is very effective for small amplitudes but converges very slowly (or does not converge at all, depending on the formulation) as the wave approaches to the maximum height Hmax (also called by the wave of the greatest height or the limiting Stokes wave) which is defined at the distance from the crest to the trough of Stokes wave over a spatial period λ. The sec ond approach is to consider a limiting Stokes wave, which is the progressive wave with the highest nonlin earity. Stokes found that the limiting Stokes wave has the sharp angle of 2π/3 radians on the crest [15]; i.e., the surface is non smooth (has a jump of slope) at that spatial point. That corner singularity explains a slow convergence of Stokes expansion as H Hmax.