Hostname: page-component-cb9f654ff-k7rjm Total loading time: 0 Render date: 2025-08-24T00:08:10.955Z Has data issue: false hasContentIssue false
  • English
  • Français

Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Published online by Cambridge University Press:  17 February 2009

A. D. D. Craik
A. D. D. Craik
Affiliation:
Mathematical Institute, University of St. Andrews, St. Andrews Fife KY16 9SS, Scotland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.

The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 4 , April 1988 , pp. 430 - 439
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Craik, A. D. D., ‘Wave Interactions and Fluid Flows’ (Cambridge Univ. Press, 1986).CrossRefGoogle Scholar
[2]Crawford, D. R., Saffman, P. G., and Yuen, H. C., “Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves”, Wave Motion 2 (1980), 1–16.CrossRefGoogle Scholar
[3]Crawford, D. R., Lake, B. N., Saffman, P. G., and Yuen, H. C., “Stability of weakly nonlinear deep-water waves in two and three dimensions”, J. Fluid Mech. 105 (1981), 177–191.CrossRefGoogle Scholar
[4]Grimshaw, R., “The modulation of short gravity waves by long waves or currents”, J. Austral. Math. Soc. Ser. B 29 (1988), 410–429.CrossRefGoogle Scholar
[5]Longuet-Higgins, M. S., and Stewart, R. W., “Changes in the form of short gravity waves on long waves and tidal currents”, J. Fluid Mech. 8 (1960), 565–583.CrossRefGoogle Scholar
[6]Yuen, H. C., and Lake, B. M., “Nonlinear dynamics of deep-water gravity waves”, Adv. in Appl. Mech. 22 (1982), 67–229.CrossRefGoogle Scholar
[7]Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys. 2 (1968), 190–194. [Translation of Zh. Prikl. Mekh. i. Tekhn. Fiz. 9 (2), 86–94].Google Scholar