Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T21:26:42.779Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions

Published online by Cambridge University Press:  20 April 2006

J. D. Fenton  and
M. M. Rienecker
J. D. Fenton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W., Australia 2033
M. M. Rienecker
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W., Australia 2033
Get access Rights & Permissions [Opens in a new window]

Abstract

A numerical method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface and variable bed topography. It is applied to the unsteady motion of non-breaking water waves of arbitrary magnitude over a horizontal bed. All horizontal variation is approximated by truncated Fourier series. This and finite-difference representation of the time variation are the only necessary approximations. Although the method loses accuracy if the waves become sharp-crested at any stage, when applied to non-breaking waves the method is capable of high accuracy.

The interaction of one solitary wave overtaking another was studied using the Fourier method. Results support experimental evidence for the applicability of the Korteweg-de Vries equation to this problem since the waves during interaction are long and low. However, some deviations from the theoretical predictions were observed - the overtaking high wave grew significantly at the expense of the low wave, and the predicted phase shift was found to be only roughly described by theory. A mechanism is suggested for all such solitary-wave interactions during which the high and fast rear wave passes fluid forward to the front wave, exchanging identities while the two waves have only partly coalesced; this explains the observed forward phase shift of the high wave.

For solitary waves travelling in opposite directions, the interaction is quite different in that the amplitude of motion during interaction is large. A number of such interactions were studied using the Fourier method, and the waves after interaction were also found to be significantly modified - they were not steady waves of translation. There was a change of wave height and propagation speed, shown by the present results to be proportional to the cube of the initial wave height but not contained in third-order theoretical results. When the interaction is interpreted as a solitary wave being reflected by a wall, third-order theory is shown to provide excellent results for the maximum run-up at the wall, but to be in error in the phase change of the wave after reflection. In fact, it is shown that the spatial phase change depends strongly on the place at which it is measured because the reflected wave travels with a different speed. In view of this, it is suggested that the apparent time phase shift at the wall is the least-ambiguous measure of the change.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 118 , May 1982 , pp. 411 - 443
Copyright
© 1982 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abe, K. & Inoue, O. 1980 Fourier expansion solution of the Korteweg-de Vries equation. J. Comp. Phys. 34, 202–210.Google Scholar
Bona, J. L., Pritchard, W. G. & Scott, L. R. 1980 Solitary wave interaction. Phys. Fluids 23, 438–441.Google Scholar
Brennen, C. 1971 Some numerical solutions of unsteady free surface wave problems using the Lagrangian description of the flow. In Proc. 2nd Int. Conf. Numerical Methods in Fluid Dynamics, Berkeley, 1970 (ed. M. Holt), Lecture Notes in Physics, vol. 8, pp. 403–409. Springer.
Byatt-Smith, J. G. B. 1971 An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49, 625–633.Google Scholar
Chan, R. K. C. & Street, R. L. 1970 A computer study of finite-amplitude water waves. J. Comp. Phys. 6, 68–94.Google Scholar
Fenton, J. D. 1972 A ninth-order solution for the solitary wave. J. Fluid Mech. 53, 257–271.Google Scholar
Fenton, J. D. & Mills, D. A. 1977 Shoaling waves: numerical solution of exact equations. In Proc. I.U.T.A.M. Symp. Waves on water of variable depth, Canberra (ed. D. G. Provis & R. Radok), Lecture Notes in Physics, vol. 64, pp. 94–101. Springer.
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave behaviour. Phil. Trans. R. Soc. Land. A 289, 373–404.Google Scholar
Hirota, R. 1971 Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water I. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Maxworthy, T. 1976 Experiments on collisions between solitary waves. J. Fluid Mech. 76, 177–185.Google Scholar
Multer, R. H. 1973 Exact nonlinear model of wave generator. J. Hyd. Div. A.S.C.E. 99, 31–46.Google Scholar
Oikawa, M. & Yajima, N. 1973 Interactions of solitary waves - a perturbation approach to nonlinear systems. J. Phys. Soc. Japan 34, 1093–1099.Google Scholar
Orszag, S. A. 1971 Numerical simulation of incompressible flows within simple boundaries: accuracy. J. Fluid Mech. 49, 75–112.Google Scholar
Rienecker, M. M. & Fenton, J. D. 1981 A Fourier approximation method for steady water waves. J. Fluid Mech. 104, 119–137.Google Scholar
Su, C. H. & Mirie, R. M. 1980 On head-on collisions between two solitary waves. J. Fluid Mech. 98, 509–525.Google Scholar
Weidman, P. D. & Maxworthy, T. 1978 Experiments on strong interactions between solitary waves. J. Fluid Mech. 85, 417–431.Google Scholar
Whitney, A. K. 1971 The numerical solution of unsteady free surface flows by conformal mapping. In Proc. 2nd Int. Conf. Numerical Methods in Fluid Dynamics, Berkeley, 1970 (ed. M. Holt), Lecture Notes in Physics, vol. 8, pp. 458–462. Springer.