Cnoidal-type surface waves in deep water

Abstract

Two-dimensional potential flows due to progressive surface waves in deep water are considered. For periodic waves, only gravity is included in the dynamic boundary condition, but both gravity and surface tension are taken into account for solitary waves. The validity of the steady first-order cnoidal wave approximation, i.e. the periodic solution of KdV, is extended to infinite depth by renormalizations. This renormalized cnoidal wave (RCW) solution is expressed as a Fourier-Padé approximation. It is analytically simpler and more accurate than fifth-order Stokes approximations. It is also capable of describing the recently discovered sharp-crested wave. A sharp-crested wave is obtained when the fluid velocity at the crest is larger than the phase speed. When the wavelength is infinite, RCW yields an algebraic solitary wave. Depending on the surface tension, the solitary wave involves one or two interfaces: a wave of depression; a wave of depression with a pocket of air; a wave of elevation with a pocket of air. Solitary waves are found for all values of the surface tension. However, this does not necessarily mean that these waves are solutions of the exact equations. Moreover, RCW approximate solitary waves always present a dipole singularity. It is also shown that a cnoidal wave in deep water can be rewritten as a periodic distribution of dipoles, each dipole representing an algebraic solitary wave. This provides a new paradigm for descriptions of water wave phenomena.

Accurate simple approximation for the solitary wave

“…La méthode de renormalisation est d'abord brièvement décrite en Section 2 (voir [2][3][4] pour plus de détails et généralisations). On l'applique ensuite à la solution de Gwyther en Section 3.…”

“…Cette exemple démontre que la renormalisation est un outil simple et efficace pour améliorer les approximations de type eau peu profonde généralisées. Les améliorations apportées aux solutions KdV [2][3][4] ne sont donc pas le fruit d'un « heureux hasard », comme d'aucun pourrait légitimement le penser (discussions privées).…”

“…An unified description, which is also tractable, is clearly desirable. This can be achieved via renormalization of the first-order shallow water approximation [2][3][4]. The improved solution consequently derived is uniformly valid from long to short waves, the wavelength being compared to the mean water depth.…”

“…Une solution unifiée, valide sur tout le domaine de variation des paramètres, est souhaitable. Cela a été obtenu par renormalisation de la théorie de l'eau peu profonde du premier ordre [2][3][4]. Cette solution est valide tant pour les faibles profondeurs qu'en profondeur infinie.…”

On the interaction of deep water waves and exponential shear currents

“…This kind of problem was investigated by Fenton [4] who derived Stokes' fifth-order solution for deep-water waves. For shallow water waves with a uniform current, Clamond [5,6] and Clamond and Fructus [7] used the renormalization method to obtain accurate approximations. However, in practice, currents are mostly non-uniform and hence the vorticity appears.…”