Advertisement

The Instability and Breaking of a Deep-Water Wave Train

  • W. K. Melville

Abstract

We report measurements of the surface displacement and fluid velocity in a nonlinear deep- water wave train as it evolved to breaking. Two distinct regimes are found. For ak 0.29 the evolution is sensibly two-dimensional with the Benjamin-Feir instability leading directly to breaking as found by Longuet-Higgins and Cokelet (1978). The measured frequency of the most unstable sideband agrees very well with the predictions of Longuet-Higgins (1978). The surface displacement spectrum is not restricted to a few discrete frequencies but also involves a growing continuous spectrum. Within the accuracy of the measurements, the onset of breaking corresponds to the onset of the asymmetric development of the sidebands about the fundamental frequency. It is suggested that the asymmetric evolution, which ultimately leads to the shift to lower frequency (Lake et al, 1977), may be related to Longuet-Higgins’s (1978) breaking instabihty. For ak ⩾ 0.31 a full three-dimensional instability dominates the Benjamin-Feir instability and leads rapidly to breaking. Preliminary measurements of this instability agree very well with the results of McLean et al. (1981). Continuous measurements of the velocity field at the fluctuating surface were made with a laser anemometer and show significant differences between the velocity field in unbroken and breaking waves. In the unbroken waves the measured velocity agrees very well with that inferred by the measured surface displacement. In breaking waves the velocity in the spilling region is comparable to the phase speed of the waves while the perturbation to the surface displacement is small. With the exception of the velocity measurements, this work is reported by Melville (1982). The velocity measuring technique and analyzed data will be reported elsewhere (Melville and Rapp, 1985).

Keywords

Wave Train Surface Displacement Breaking Wave Surface Velocity Field Measured Surface Displacement 
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.

References

  1. Benjamin, T. B., and J. E. Feir (1967): The disintegration of wave trains on deep water. Part I. Theory. J. Fluid Mech. 27, 417–430.MATHCrossRefGoogle Scholar
  2. Lake, B. M., H. C. Yuen, H. Rungaldier, and W. E. Ferguson (1977): Nonlinear deep-water waves: Theory and experiment. Part 2. Evolution of a continuous wavetrain. J. Fluid Mech. 83, 49–74.CrossRefGoogle Scholar
  3. Lo, E. and 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.MATHCrossRefGoogle Scholar
  4. Longuet-Higgins, M. S. (1978): The instabihties of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. London Ser. A 360, 489–505.MathSciNetMATHCrossRefGoogle Scholar
  5. Longuet-Higgins, M. S., and E. D. Cokelet (1978): The deformation of steep surface waves on water. II. Growth of normal-mode instabilities. Proc. R. Soc. London Ser. A 364, 1–28.MathSciNetMATHCrossRefGoogle Scholar
  6. McLean, J. W., Y. C. Ma, D. U. Martin, P. G. Saffman, and H. C. Yuen (1981): Three-dimensional instability of finite-amplitude water waves.Phys. Rev. Lett. 46, 817–820.MathSciNetCrossRefGoogle Scholar
  7. Melville, W. K. (1982): The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.CrossRefGoogle Scholar
  8. Melville, W. K. and Rapp, R. J. (1985); The surface velocity field in steep and breaking waves J. Fluid Mech. (submitted).Google Scholar

Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • W. K. Melville
    • 1
    • 2
  1. 1.Institute of Geophysics and Planetary PhysicsUniversity of CaliforniaSan DiegoUSA
  2. 2.Department of Civil EngineeringMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations