Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T21:56:51.506Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

Boltzmann, L. 1966 Ueber die mechanische Bedeutung des zweiten Hauptsatzes der Wär-methorie auf allgemeine mechanische. Proc. Kon. Wien. 53, 195–220.Google Scholar
Bretherton, F. & Garrett, C. 1968 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302, 529–554.Google Scholar
Clausius, R. 1870 Ueber die Zurückführung des zweiten Hauptsatzes der mechanischen Wärmetheorie auf algemeine mechanische Principien. Poggendorf Ann. 142. 433–461.Google Scholar
Courant, R. & Hilbert, D. 1937 Methods of Mathematical Physics, pp. 238–240. Interscience.
Ehrenfest, P. 1959 Collected Scientific Papers (ed. M. J. Klein). North-Holland; Interscience.
Furry, W. H. 1946 Two notes on phase-integral methods. Phys. Rev. 71, 360.Google Scholar
Garrett, C. & Smith, J. 1976 On the interaction between long and short surface waves. J. Phys. Oceanogr. 6, 925–930.Google Scholar
Goldstein, H. 1950 Classical Mechanics. Addison-Wesley.
Kruskal, M. 1962 Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. J. Math. Phys. 3, 806–828.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1976 Mechanics 3rd edn. pp. 154–162. Pergamon.
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293–306.Google Scholar
Miles, J. 1981 Hamiltonian formulations for surface waves. Appl. Sci. Res. 37, 103–110.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave J. Fluid Mech. 107, 465–485.Google Scholar
Seliger, R. L. & Whitham, G. B. 1968 Variational principles in continuum mechanics. Proc. R. Soc. Lond. A 305, 1–25.Google Scholar
Sudarshan, E. C. G. & Makunda, N. 1974 Classical Dynamics: A Modern Perspective. Wiley.
West, B. J. 1981 Deep Water Gravity Waves, p. 33. Springer.
Whitham, G. K. 1974a Linear and Nonlinear Waves. Wiley.
Whitham, G. B. 1974b Dispersive waves and variational principles. In: Nonlinear Waves (ed. S. Leibovich and A. R. Seebass), pp. 139–169. Cornell University Press.