The Interaction between Long and Short Wind-Generated Waves

  • M. T. Landahl
  • J. A. Smith
  • S. E. Widnall


The interaction of wind-induced short (capillary) and long (gravity) surface waves of disparate scales is analyzed with the use of an expansion procedure accounting for the lowest-order effects of finite short-wave amplitude on the growth of a small-amplitude long wave. Interactions in both water and air are considered. In the water there is transfer of momentum from the short to the long waves arising from the modulation of the short-wave growth rate by the long wave; this may be interpreted as the effect of the shortwave surface-stress modulations in phase with the long-wave surface elevation and is consistent with earlier treatments employing energy considerations. The interaction in the air is treated with an in viscid model. This shows that the modulation of the short-wave Reynolds stresses by the long wave can give rise to a phase shift between long-wave pressure and surface slope which may significantly increase the rate of transfer of momentum from the wind to the long wave.


Short Wave Critical Layer Longe Wave Stoke Drift Wave Slope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Benney, D. J. (1976): Significant interaction between small and large scale surface waves. Stud. Appl Math. 55, 93.MathSciNetGoogle Scholar
  2. Davis, R. E. (1972): On prediction of the turbulent flow over a wavy boundary. J. Fluid Mech. 52, 287.MATHCrossRefGoogle Scholar
  3. Garrett, C., and J. Smith (1976): On the interaction between long and short surface waves. J. Phys. Oceanogr. 6, 926.CrossRefGoogle Scholar
  4. Gent, P. R., and P. A. Taylor (1976): A numerical model of the air flow above water waves. J. Fluid Mech. 77, 105.MATHCrossRefGoogle Scholar
  5. Hasselmann, K. (1962): On the non-linear energy transfer in a gravity wave spectrum. Part 1. J. Fluid Mech. 12, 481.MathSciNetMATHCrossRefGoogle Scholar
  6. Hasselmann, K. (1963): On the non-linear energy transfer in a gravity wave spectrum. Part 2. J. Fluid Mech. 15, 273; Part 3 15, 385.MathSciNetMATHCrossRefGoogle Scholar
  7. Hasselmann, K. (1971): On the mass and momentum transfer between short gravity waves and larger-scale motions. J. Fluid Mech. 50, 189.MATHCrossRefGoogle Scholar
  8. Landahl, M. T., S. E. Widnall, and L. Hultgren (1979): An interaction mechanism between large and small scales for wind-generated water waves. Proceedings, Twelfth Symposium on Naval Hydrodynamics, National Academy of Sciences, 541.Google Scholar
  9. Larson, T. R., and J. Wright (1975): Wind generated gravity-capillary waves: Laboratory measurements of temporal growth rates using microwave backscatter. J. Fluid Mech. 70, 417.CrossRefGoogle Scholar
  10. Lin, C. C. (1955): The Theory of Hydrodynamic Stability, Cambridge University Press, London.MATHGoogle Scholar
  11. Longuet-Higgins, M. S. (1969a): A non-linear mechanism for the generation of sea waves. Proc. Soc. London Ser. A 311, 371.CrossRefGoogle Scholar
  12. Longuet-Higgins, M. S. (1969b): Action of a variable stress at the surface of water waves. Phys. Fluids 12, 737.CrossRefGoogle Scholar
  13. Manton, M. J. (1972): On the generation of sea waves by a turbulent wind. Boundary-Layer Meteorol. 2, 348.CrossRefGoogle Scholar
  14. Miles, J. W. (1957): On the generation of waves by shear flow. J. Fluid Mech. 3, 185.MathSciNetMATHCrossRefGoogle Scholar
  15. Miles, J. W. (1959): On the generation of waves by shear flows. Part 2. J. Fluid Mech. 6, 568.MathSciNetMATHCrossRefGoogle Scholar
  16. Miles, J. W. (1962): On the generation of waves by shear flows. Part 4. J. Fluid Mech. 13, 433.MathSciNetMATHCrossRefGoogle Scholar
  17. Phillips, O. M. (1957): On the generation of waves by turbulent wind. J. Fluid Mech. 2, 417.MathSciNetMATHCrossRefGoogle Scholar
  18. Plant, W. J., and J. W. Wright (1977): Growth and equilibrium of short gravity waves in a wind wave tank. J. Fluid Mech. 82, 767.CrossRefGoogle Scholar
  19. Reichardt, H. (1951): Vollständige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen. Z. Angew. Math. Mech. 31, 208.MATHCrossRefGoogle Scholar
  20. Townsend, A. A. (1972): Flow in a deep turbulent boundary layer over a surface distorted by water waves. J. Fluid Mech. 55, 719.MATHCrossRefGoogle Scholar
  21. Townsend, A. A. (1980): The response of sheared turbulence to additional distortion. J. Fluid Mech. 81, 171.CrossRefGoogle Scholar
  22. Valenzuela, G. R. (1976): The growth of gravity-capillary waves in a coupled shear flow. J. Fluid Mech. 76, 229.MATHCrossRefGoogle Scholar
  23. Valenzuela, G. R., and M. B. Laing (1972): Nonlinear energy transfer in gravity-capillary wave spectra, with application. J. Fluid Mech. 54, 507.MATHCrossRefGoogle Scholar
  24. Valenzuela, G. R., and J. W. Wright (1976): Growth of waves by modulated wind stress. J. Geophys. Res. 81, 5795.CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • M. T. Landahl
    • 1
  • J. A. Smith
    • 1
  • S. E. Widnall
    • 1
  1. 1.Department of Aeronautics and AstronauticsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations