Nonlinear effects in surface gravity waves

• Main Author: Fox, Michael John Haviland
• Published: University of Cambridge 1977
• Subjects: 530.15
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.455850

In the first section of this dissertation the resonant transfer of energy within a continuous spectrum of water waves is considered.� Using a transfer equation derived by LonguetHigg: ins (1975) for narrow spectra, and choosing suitable simple forms for these spectra, we make accurate evaluations of the energy transfer rate as a function of the two dimensional wavenumber. This is found to have four positive maxima surrounding the peak of the spectrum, separated in general by regions of negative or reduced positive transfer. Integration of tha two dimensional transf'er function gives a transfer f1.mction for the frequency spectrum; this in all cases h as a negative trough near the peak rrequency. Comparisons are made with the less accurate frequency spectrum transfer rates computed by Sell and Hasselmann (1972) for two spectra observed in the North Sea during the JONSWAP project. Qualitative agreement is found for the narrower of the two. The tendency toward positive transfer at the peak of broader spectra found by Sell and Hasselmann is not reproduced here. Secondly, we develop a new method of calculating steep, steady, progressive surface waves. It was suggested by Longuet-Higgins that there might be a self-similar family of free surface flows of decreasing length scale representing the flow near the crest of such waves as they approach the limiting form with 0 Stokes' 120 angle. This family of flows is calculated by transforming the domain onto the interior of a circle in the complex plarie and expanding the surface coordinates in a Fourier series. The computation is found to give good agreement with an independent method of Longuet-Higgins. The maximum slope of the free surface slightly exceeds 30�, the computed value of 30.37� being in close agreement with an extrapolation of the results of previous authors for steep _gravity waves. This flow is then matched asymptotically to an outer solution representing the flow in the rest of tbe wave . The calculations are performed both for .steep solitary waves and for progressive waves in water of infinite depth. The results confirm that the highest solitary waves in a given depth of water do not have the greatest speed, energy or momentum. Similar results hold for waves in deep water. In fact the curves of speed against wave height are found to contain both maxima and minima.