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Resonant interactions between waves. The case of discrete oscillations

Published online by Cambridge University Press:  28 March 2006

F. P. Bretherton
F. P. Bretherton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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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
Information
Journal of Fluid Mechanics , Volume 20 , Issue 3 , November 1964 , pp. 457 - 479
Copyright
© 1964 Cambridge University Press

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References

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