Numerical Evidence For The Existence of A New Type of Steady Gravity Waves On Deep Water (original) (raw)

Egs General Assembly Conference Abstracts, 2002

Abstract

A new type of steady steep two-dimensional irrotational symmetric periodic gravity waves on inviscid incompressible fluid of infinite depth is revealed. The Bernoulli equation is quadratic in velocity and admits two values of the particle speed at the crest. The first one is smaller than the wave phase speed c and corresponds to the Stokes branch. The second one is greater than c and should correspond to a new type of waves. These waves are found to have sharper crests in comparison with the Stokes waves of the same wavelength and steepness. Because of this we named them "the spike waves". To give numerical evidence we used several independent methods: (i) Fourier approximations. The elevation (x) of a free surface and the velocity po- tential (x, y) are looked for in the form of series in terms of the basis functions exp(inx) and exp n(y + ix) , respectively, x and y being the horizontal and vertical coordinates, and n being a positive integer. (ii) Fourier approximations with singularities. The basis functions for the velocity po- n tential are 1/ 1 - exp(y0 - y - ix) , y0 being a free parameter. Herein a countable number of isolated singular points at y = y0 are to be located outside the flow do- main for calculating potential periodic progressive waves. Otherwise, the local vortex structures may be studied. (iii) Inverse Fourier approximations. The velocity potential and the stream function are used as independent variables. The spatial coordinates are expanded into series in terms of the functions exp n(-/c + i/c) . For the almost-highest spike waves, the routine may converge to different sets of val- ues for the unknowns depending on starting conditions. This non-uniqueness is pos- sibly caused by the superharmonic or crest instabilities, which were investigated in detail by Longuet-Higgins et al. for the Stokes branch. The results obtained for the almost-highest Stokes waves are in sufficient agreement with the high accuracy calcu- lations by the program of Prof. M. Tanaka, which he kindly placed at our disposal. This research has been supported by INTAS Grant 99-1637.

Mohammed Debiane hasn't uploaded this paper.

Let Mohammed know you want this paper to be uploaded.

Ask for this paper to be uploaded.