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The energy and action of small waves riding on large waves

Published online by Cambridge University Press:  21 April 2006

Frank S. Henyey ,
Dennis B. Creamer ,
Kristian B. Dysthe ,
Roy L. Schult  and
Jon A. Wright
Frank S. Henyey
Affiliation:
Center for Studies of Nonlinear Dynamics
Affiliated with the University of California, San Diego.
, La Jolla Institute, 10280 N. Torrey Pines Road, Suite 260, La Jolla, CA 92037, USA
Dennis B. Creamer
Affiliation:
Center for Studies of Nonlinear Dynamics
Affiliated with the University of California, San Diego.
, La Jolla Institute, 10280 N. Torrey Pines Road, Suite 260, La Jolla, CA 92037, USA
Kristian B. Dysthe
Affiliation:
Center for Studies of Nonlinear Dynamics
Affiliated with the University of California, San Diego.
, La Jolla Institute, 10280 N. Torrey Pines Road, Suite 260, La Jolla, CA 92037, USA
Roy L. Schult
Affiliation:
Center for Studies of Nonlinear Dynamics
Affiliated with the University of California, San Diego.
, La Jolla Institute, 10280 N. Torrey Pines Road, Suite 260, La Jolla, CA 92037, USA
Jon A. Wright
Affiliation:
Center for Studies of Nonlinear Dynamics
Affiliated with the University of California, San Diego.
, La Jolla Institute, 10280 N. Torrey Pines Road, Suite 260, La Jolla, CA 92037, USA
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Abstract

We derive the dynamics of small waves riding on larger waves using a canonical, Hamiltonian formulation. The small waves are treated linearly and their energy is derived to all orders in the scale separation between the waves. Our results are similar to those of Longuet-Higgins (1987), but we have extended his calculations to include gravity-capillary waves and to allow for a more general, two-dimensional, large-wave field. Our result for the small-wave Hamiltonian is expressed in both Eulerian (horizontal) coordinate system and a non-inertial system determined by the large wave's surface. On further assuming scale separation between the small and large waves the averaged Lagrangian equations and the action density are derived. Action conservation is explicitly demonstrated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 189 , April 1988 , pp. 443 - 462
Copyright
© 1988 Cambridge University Press

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