Fractional Fourier approximations for potential gravity waves on deep water

Abstract

Abstract. In the framework of the canonical model of hydrodynamics, where fluid is assumed to be ideal and incompressible, waves are potential, two-dimensional, and symmetric, the authors have recently reported the existence of a new type of gravity waves on deep water besides well studied Stokes waves (Lukomsky et al., 2002b). The distinctive feature of these waves is that horizontal water velocities in the wave crests exceed the speed of the crests themselves. Such waves were found to describe irregular flows with stagnation point inside the flow domain and discontinuous streamlines near the wave crests. In the present work, a new highly efficient method for computing steady potential gravity waves on deep water is proposed to examine the character of singularity of irregular flows in more detail. The method is based on the truncated fractional approximations for the velocity potential in terms of the basis functions 1/(1 - exp(y0 - y -  ix))n, y0 being a free parameter. The non-linear transformation of the horizontal scale x = c - g sin c, 0  < g  <  1,  is additionally applied to concentrate a numerical emphasis on the crest region of a wave for accelerating the convergence of the series. For lesser computational time, the advantage in accuracy over ordinary Fourier expansions in terms of the basis functions exp(n(y + ix))  was found to be from one to ten decimal orders for steep Stokes waves and up to one decimal digit for irregular flows. The data obtained supports the following conjecture: irregular waves to all appearance represent a family of sharp-crested waves like the limiting Stokes wave but of lesser amplitude.