APPLICATION OF FOURIER METHODS TO WATER-WAVES IN SMALL DEPTHS

Fourier approximation methods for solving symmetrical water waves in v ery small depths (large Ursell number) are improved in two respects. F irst, by using a simple and effective initial estimate of the solution , which avoids the appearance of multiple crested solutions and allows direct convergence for large wave amplitudes, with a substantial redu ction of the computing time in comparison with the usual stepping meth od. Second, by introducing a method which permits a clustering around the crest of the points on the free surface at which the equations are to be satisfied. Both improvements are based on the application of th e so-called WPC approach (Wave Plus Current), which consists of repres enting a very long wave by a shorter wave prolonged with a uniform cur rent. This approach has an error which is shown to decrease exponentia lly with the square root of the Ursell number. Fourier methods are the n extended to allow very accurate calculations for almost the entire r ange of wave parameters, the only exceptions being near breaking waves . (C) 1997 Elsevier Science Limited.