Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-17T12:01:13.057Z Has data issue: false hasContentIssue false
  • English
  • Français

On modifications of the Zakharov equation for surface gravity waves

Published online by Cambridge University Press:  20 April 2006

Michael Stiassnie  and
Lev Shemer
Michael Stiassnie
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology Permanent address: Technion, Israel Institute of Technology, Haifa 32000, Israel.
Lev Shemer
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

The Zakharov integral equation for surface gravity waves is modified to include higher-order (quintet) interactions, for water of constant (finite or infinite) depth. This new equation is used to study some aspects of class I (4-wave) and class II (5-wave) instabilities of a Stokes wave.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 143 , June 1984 , pp. 47 - 67
Copyright
© 1984 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

Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Dysthe, K. B. 1980 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481–500.Google Scholar
Herterich, K. & Hasselmann, K. 1980 A similarity relation for the nonlinear energy transfer in a finite-depth gravity-wave spectrum. J. Fluid Mech. 97, 215–224.Google Scholar
Iusim, R. & Stiassnie, M. 1982 Note on a modification of the nonlinear Schrödinger equation for waves moving over an uneven bottom. Progress Rep. Dept Civ. Engng Technion.
Janssen, P. A. E. M. 1983 On a fourth-order envelope equation for deep-water waves. J. Fluid Mech. 126, 1–11.Google Scholar
Longuet-Higgins, M. S. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum: a simplified model. Proc. R. Soc. Lond. A 347, 311–328.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. & Phillips, O. M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12, 333–336.Google Scholar
McLean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
McLean, J. W. 1982b Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.Google Scholar
Skjelbreia, L. & Hendrickson, J. 1961 Fifth order gravity wave theory. In Proc. 7th Conf. Coastal Engng, vol. 1, pp. 184–196.
Stiassnie, M. 1984 Note on the modified nonlinear Schrödinger equation for deep water waves. Wave Motion (in press).
Su, M.-Y. 1982 Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric wave patterns. J. Fluid Mech. 124, 73–108.Google Scholar
Su, M.-Y., Bergin, M., Marler, P. & Myrick, R. 1982 Experiments on nonlinear instabilities and evolution of steep gravity-wave trains. J. Fluid Mech. 124, 45–72.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. (Engl. Transl.) 9, 190–194.Google Scholar
Zakharov, V. E. & Kharitonov, V. G. 1970 Instability of monochromatic waves on the surface of a liquid of arbitrary depth. J. Appl. Mech. Tech. Phys. (Engl. Transl.) 11, 741–751.Google Scholar