Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-23T08:44:42.868Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-order modulation effects on solitary wave envelopes in deep water

Published online by Cambridge University Press:  21 April 2006

T. R. Akylas
T. R. Akylas
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The envelope equation of Dysthe (1979), which provides an extension of the nonlinear Schrödinger equation (NLS) to fourth order in wave steepness, is used to discuss higher-order modulation effects on the long-time evolution of solitary wave envelopes in deep water. The Dysthe equation admits solitary-wave solutions, similar to those of the NLS. Using perturbation methods, it is shown that an initial disturbance in the form of a solitary wave group of the NLS evolves to a solitary wave of the Dysthe equation having lower peak amplitude and moving with higher speed than the original wave; the increase in wave speed is caused by a downshift in wavefrequency. Asymptotic expressions are derived for this amplitude decrease and frequency downshift, which are consistent with numerical and experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 198 , January 1989 , pp. 387 - 397
Copyright
1989 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

Dysthe, K. B.: 1979 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
Feir, J. E.: 1967 Discussion: some results from wave pulse experiments. Proc. R. Soc. Lond. A 299, 1–11.Google Scholar
Janssen, P. A. E. M.: 1983 On a fourth-order envelope equation for deep-water waves. J. Fluid Mech. 126, 1–11.Google Scholar
Kodama, Y. & Ablowitz, M. J., 1981 Perturbations of solitons and solitary waves. Stud. Appl. Maths 65, 225–245.Google Scholar
Lo, E. Y.: 1986 Long-time evolution of surface waves in coastal waters. Doctoral dissertation Department of Civil Engineering, MIT.
Lo, E. & Mei, C. C., 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–416.Google Scholar
Roskes, G. J.: 1977 Fourth order envelope equation for nonlinear dispersive gravity waves. Phys. Fluids 20, 1576–1577.Google Scholar
Su, M. Y.: 1982 Evolution of groups of gravity waves with moderate to high steepness. Phys. Fluids 25, 2167–2174.Google Scholar
Yuen, H. C. & Lake, B. M., 1982 Nonlinear dynamics of deep water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar