Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T17:11:10.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution of weakly nonlinear short waves riding on long gravity waves

Published online by Cambridge University Press:  26 April 2006

Jun Zhang and
W. K. Melville
Jun Zhang
Affiliation:
Ocean Engineering Program, Department of Civil Engineering. Texas A & M University, College Station, TX 77843, USA
W. K. Melville
Affiliation:
R. M. Parsons Laboratory, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get accessRights & Permissions [Opens in a new window]

Abstract

A nonlinear Schrödinger equation, describing the evolution of a weakly nonlinear short gravity wavetrain riding on a longer finite-amplitude gravity wavetrain, is derived. This equation is then used to predict the steady envelope of the short wavetrain relative to the long wavetrain. It is found that approximate analytical solutions agree very well with numerical solutions over a realistic range of wave steepness. The solutions are compared with corresponding studies of the modulation of linear short waves by Longuet-Higgins & Stewart (1960) and Longuet-Higgins (1987). We find that the effect of the nonlinearity of the short waves is to increase the modulation of their wavenumber, significantly reduce the modulation of their amplitude, and reduce the modulation of their slope when compared with the predictions of Longuet-Higgins (1987) for linear short waves on finite-amplitude long waves. The question of the stability of these steady solutions remains open but may be addressed by solutions of this nonlinear Schrödinger equation.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 214 , May 1990, pp. 321 - 346
Copyright
© 1990 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.)

Article purchase

Temporarily unavailable

References

Allan, T. D. 1983 Satellite Microwave Sensing. John Wiley & Sons.
Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in inhomogeneous moving media.. Proc. R. Soc. Lond. A 302, 529–554.Google Scholar
Henyey, F. S., Creamer, D. B., Dysthe, K. B., Schult, R. L. & Wright, J. A. 1988 The energy and action of small waves riding on large wave. J. Fluid Mech. 189, 443–462.Google Scholar
Hogan, S. J. 1980 Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96, 417–445.Google Scholar
Hogan, S. J. 1981 Some effects of surface tension on steep water waves. Part 3. J. Fluid Mech. 110, 381–410.Google Scholar
Keller, W. C. & Wright, J. W. 1975 Microwave scattering and the straining of wind-generated waves. Radio Science 10, 139–147.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonic.. Proc. R. Soc. Lond. A 350, 526–562.Google Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293–306.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. I. A numerical method of computation.. Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1978 The deformation of steep surface waves on water. II. Growth of normal mode instabilities.. Proc. R. Soc. Lond. A 364, 1–28.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565–583.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stress in water waves, a physical discussion with application. Deep Sea Res. 11, 529–562.Google Scholar
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves.. Proc. R. Soc. Lond. A 406, 115–125.Google Scholar
Oppenheim, A. V. & Schafer, R. W. 1975 Digital Signal Processing. Prentice-Hall.
Oppenheim, A. V., Willsky, A. S. & Young, I. T. 1985 Signal and Systems. Prentice-Hall.
Phillips, O. M. 1981 The dispersion of short wavelets in presence of dominant long wave. J. Fluid Mech. 107, 465–485.Google Scholar
Phillips, O. M. 1984 On the response of the short ocean wave components at a fixed wavenumber to ocean current variations. J. Phys. Oceanogr. 14, 1425–1433.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Stewart, R. H. 1985 Methods of Satellite Oceanography. University of California Press.
Valenzuela, G. R. & Wright, J. W. 1979 Modulation of short gravity capillary waves by longer periodic flows – a higher order theory. Radio Science 14, 1099–1110.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar
Zhang, J. 1987 Nonlinear interaction between surface water waves. ScD thesis, MIT.