CaltechAUTHORS
  A Caltech Library Service

Nonlinear interactions of random waves, in a dispersive medium

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. http://resolver.caltech.edu/CaltechAUTHORS:20151214-114109805

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20151214-114109805

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
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1098/rspa.1966.0013DOIArticle
http://rspa.royalsocietypublishing.org/content/289/1418/301PublisherArticle
http://www.jstor.org/stable/2415123JSTORArticle
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.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)AFOSR 492-64
Record Number:CaltechAUTHORS:20151214-114109805
Persistent URL:http://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:14 Dec 2015 21:30

Repository Staff Only: item control page