Advertisement

On the Changes in Phase Speed of One Train of Water Waves in the Presence of Another

  • S. J. Hogan
  • Idith Gruman
  • M. Stiassnie

Abstract

We present calculations of the change in phase speed of one train of water waves in the presence of another. We use a general method, based on Zakharov’s (1968) integral equation. In the important case of gravity-capillary waves, we present the correct form of the Zakharov kernel. This is used to find the expressions for the changes in phase-speed. These results are then checked using a perturbation method based on that of Longuet-Higgins and Phillips (1962). Agreement to 6 significant digits has been obtained between the calculations based on these two distinct methods. Full numerical results in the form of polar diagrams over a wide range of wavelengths, away from conditions of triad resonance, are provided.

Keywords

Surface Tension Gravity Wave Water Wave Wave Train Phase Speed 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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, 177–191CrossRefGoogle Scholar
  2. Hogan, S.J. 1988 The superharmonic normal mode instabilities of nonlinear deep water capillary waves. J. Fluid Mech. 190, 165–177.CrossRefGoogle Scholar
  3. Hogan, S.J., Gruman, I., and Stiassnie, M. 1988 On the changes in phase speed of one train of water waves in the presence of another. J. Fluid Mech. 192, 97–114.CrossRefGoogle Scholar
  4. Longuet-Higgins, M.S. & Phillips, O.M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12, 333–336CrossRefGoogle Scholar
  5. McGoldrick, L.F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305–331CrossRefGoogle Scholar
  6. Stiassnie, M. and Shemer, L. 1984 On modification of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143, 47–67CrossRefGoogle Scholar
  7. Stokes, G.G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455Google Scholar
  8. Yuen, H.C. and Lake, B.M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22 67–229CrossRefGoogle Scholar
  9. Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. appl. Mech. tech. Phys. 2, 190–194Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • S. J. Hogan
    • 1
  • Idith Gruman
    • 2
  • M. Stiassnie
    • 3
  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Department of Civil EngineeringTechnion HaifaIsrael
  3. 3.School of MathematicsUniversity of BristolBristolUK

Personalised recommendations