Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-29T04:55:12.312Z Has data issue: false hasContentIssue false

Resonant interactions between waves. The case of discrete oscillations

Published online by Cambridge University Press:  28 March 2006

F. P. Bretherton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge

Abstract

The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.

Type
Research Article
Copyright
© 1964 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

Ball, K. F. 1964 J. Fluid Mech. 19, 465.
Benney, D. J. 1962 J. Fluid Mech. 14, 577.
Bretherton, F. P. 1963 Scientific Paper 63-158-130-P6, Westinghouse Research Laboratories, Pittsburgh 35, Pa.
Hasselmann, K. 1962 J. Fluid Mech. 12, 481.
Longuet-Higgins, M. S. 1962 J. Fluid Mech. 12, 321.
Longuet-Higgins, M. S. & Phillips, O. M. 1962 J. Fluid Mech. 12, 333.
Minorsky, N. 1962 Non-linear Oscillations, p. 329. New York: Van Nostrand.
Phillips, O. M. 1960 J. Fluid Mech. 9, 193.
Proudman, I. & Pearson, J. R. A. 1957 J. Fluid Mech. 2, 237.