Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-20T04:32:24.932Z Has data issue: false hasContentIssue false
  • English
  • Français

The propagation of short surface waves on longer gravity waves

Published online by Cambridge University Press:  21 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK, and Institute of Oceanographic Sciences, Wormley, Surrey, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

To understand the imaging of the sea surface by radar, it is useful to know the theoretical variations in the wavelength and steepness of short gravity waves propagated over the surface of a train of longer gravity waves of finite amplitude. Such variations may be calculated once the orbital accelerations and surface velocities in the longer waves have been accurately determined – a non-trivial computational task.

The results show that the linearized theory used previously for the longer waves is generally inadequate. The fully nonlinear theory used here indicates that for longer waves having a steepness parameter AK = 0.4, for example, the short-wave steepness can be increased at the crests of the longer waves by a factor of order 8, compared with its value at the mean level. (Linear theory gives a factor less than 2.)

The calculations so far reported are for free, irrotational gravity waves travelling in the same or directly opposite sense to the longer waves. However, the method of calculation could be extended without essential difficulty so as to include effects of surface tension, energy dissipation due to short-wave breaking, surface wind-drift currents, and to arbitrary angles of wave propagation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 177 , April 1987 , pp. 293 - 306
Copyright
© 1987 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

Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in homogeneous moving media. Proc. R. Soc. Lond. A 302, 529–554.Google Scholar
Dysthe, K., Henyey, F. S., Longuet-Higgins, M. S. & Schult, R. L. 1987 The double orbiting pendulum: an analogue to interacting gravity waves. Proc. R. Soc. Lond. A (submitted).Google Scholar
Garrett, C. J. F. 1967 The adiabatic invariant for wave propagation in a nonuniform moving medium. Proc. R. Soc. Lond. A 299, 26–27.Google Scholar
Kenyon, K., Sheres, D & Bernstein, R. 1983 Short waves on long waves. J. Geophys. Res. 88, 7589–7596.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuet-Higgins, M. S. 1985a Bifurcation in gravity waves. J. Fluid Mech. 151, 457–475.Google Scholar
Longuet-Higgins, M. S. 1985b Asymptotic behaviour of the coefficients in Stokes's series for surface gravity waves. I.M.A. J. Appl. Maths 34, 267–277.Google Scholar
Longuet-Higgins, M. S. 1985c Accelerations in steep gravity waves. J. Phys. Oceanogr. 15, 1570–1579.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1977 Theory of the almost-highest wave: the inner solution. J. Fluid Mech. 80, 721–741.Google Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.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. 1961 The changes in amplitude of short gravity waves on steady, non-uniform currents. J. Fluid Mech. 10, 529–549.Google Scholar
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
Williams, J. M. 1981 Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139–188.Google Scholar