Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T05:03:15.026Z Has data issue: false hasContentIssue false
  • English
  • Français

On the resonant interaction between a surface wave and a weak surface current

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  26 February 2010

K. Stewartson
K. Stewartson
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT
Get access

Abstract

An internal wave motion, below a layer of uniform fluid, induces a weak current on the free surface in the form of a long wave with phase velocity cI. A uniform progressive train of surface waves, whose wave-length is much shorter than that of the current is incident on it from infinity and undergoes modification. In particular, when the group velocity cg of the progressive wave is equal to cI, the resonance takes place and then, even though the amplitude of the current is small, the interaction builds up near a number of its wavelengths until the train of surface waves is significantly modified. The equations governing the modifications are derived, using the method of multiple scales, and the roles of the Döppler shift and the radiation stress in resonant situations are elucidated. Three-dimensional interactions are discussed and an analogy is drawn between the fundamental equation describing the interactions and Schrödinger's equation.

MSC classification

Secondary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Type
Research Article
Information
Mathematika , Volume 24 , Issue 1 , June 1977 , pp. 37 - 49
Copyright
Copyright © University College London 1977

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

Benney, D. J. and Roskes, G. J.. Studies in Applied Mathematics, 48 (1969), 377.CrossRefGoogle Scholar
Chu, V. H. and Mei, C. C.. J. Fluid Mech., 41 (1970), 873.CrossRefGoogle Scholar
Chu, V. H. and Mei, C. C.J. Fluid Mech., 47 (1971), 337.CrossRefGoogle Scholar
Davey, A. and Stewartson, K.. Proc. Roy. Soc. London A, 338 (1974), 101.Google Scholar
Gargett, A. E. and Hughes, B. A.. J. Fluid Mech., 52 (1972), 179.CrossRefGoogle Scholar
Hasimoto, H. and Ono, H.. J. Phys. Soc. Japan, 33 (1972), 805.CrossRefGoogle Scholar
Hayes, W. D.. Proc. Roy. Soc. London A, 332 (1973), 199.Google Scholar
Hughes, B. A.. J. Fluid Mech., 74 (1976), 667.CrossRefGoogle Scholar
Landau, L. D. and Lifschitz, E. M.. Quantum Mechanics: Non-Relativistic Theory (Pergamon Press). (1958).Google Scholar
Lewis, J. E.Lake, B M. and Ko, D. R. S.. J. Fluid Mech., 63 (1974), 773.CrossRefGoogle Scholar
Higgins, M. S. Longuet and Stewart, R. W.. J. Fluid Mech., 8 (1960), 565.CrossRefGoogle Scholar
Higgins, M. S. Longuet and Stewart, R. W.. J. Fluid Mech., 10 (1961), 529.CrossRefGoogle Scholar
Phillips, O. M.. Izv. Atmospheric and Ocean Physics, 9 (1973), 954.Google Scholar
Thomson, J. A. and West, B. J.. J. Physical Oceanography, 5 (1975), 736.2.0.CO;2>CrossRefGoogle Scholar
Whitham, G. B.. Linear and Non-Linear Waves (Wiley). (1974).Google Scholar