Benney, D. J. and Saffman, P. G.
(1966)
*Nonlinear interactions of random waves, in a dispersive medium.*
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 289
(1418).
pp. 301-320.
ISSN 0080-4630.
doi:10.1098/rspa.1966.0013.
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## Abstract

A study is made of the way that the spectrum function of random, spatially homogeneous, dispersive waves varies slowly with time owing to weak nonlinear interactions between the waves. A continuous representation is used throughout and the slow variation is obtained with the aid of the multiple time scale method of nonlinear mechanics. It is shown that provided the dispersion equation satisfies a fairly general requirement (the non-existence of 'double resonance'), a closed integro-differential equation for the energy spectrum function can be obtained which describes asymptotically the transfer of energy between wave numbers on a time scale є^(-2) times a characteristic period of the waves, where the parameter є measures the relative order of the nonlinear terms. The equation is derived without imposing restrictions on the probability distribution of the waves, and in particular it is not found necessary to assume that the distribution is Gaussian. Nevertheless, the result is the same as if the distribution were Gaussian to zero order in e and this is true for arbitrary initial probability distributions. For a conservative system where total energy is conserved, the equation simplifies to a form previously derived by Litvak (1960), according to which transfer of energy takes place between resonant triads of wave numbers if they exist.

Item Type: | Article | ||||||||||||
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Additional Information: | © 1966 Royal Society. Communicated by G. K. Batchelor, F .R.S.-Received 28 January 1965. It is a pleasure to acknowledge many helpful discussions with Professor G. B. Whitham. During this work the first author was partially supported by a grant from the Air Force Office of Scientific Research, Contract No. AFOSR 492-64. | ||||||||||||

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Issue or Number: | 1418 | ||||||||||||

DOI: | 10.1098/rspa.1966.0013 | ||||||||||||

Record Number: | CaltechAUTHORS:20151214-114109805 | ||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20151214-114109805 | ||||||||||||

Official Citation: | Benney, D. J., & Saffman, P. G.. (1966). Nonlinear Interactions of Random Waves in a Dispersive Medium. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 289(1418), 301–320. | ||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||

ID Code: | 62873 | ||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||

Deposited By: | Ruth Sustaita | ||||||||||||

Deposited On: | 14 Dec 2015 21:30 | ||||||||||||

Last Modified: | 10 Nov 2021 23:08 |

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