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The instability and breaking of deep-water waves

Published online by Cambridge University Press:  20 April 2006

W. K. Melville
W. K. Melville
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, and Massachusetts Institute of Technology, Cambridge
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Abstract

An experimental study of the evolution to breaking of a nonlinear deep-water wave train is reported. Two distinct regimes are found. For ak [les ] 0·29 the evolution is sensibly two-dimensional with the Benjamin-Feir instability leading directly to breaking as found by Longuet-Higgins & Cokelet (1978). The measured side-band frequencies agree very well with those predicted by Longuet-Higgins (1978b). It is found that the evolution of the spectrum is not restricted to a few discrete frequencies but also involves a growing continuous spectrum, and the description of the evolution as a recurrence phenomenon is incomplete. It is found that the onset of breaking corresponds to the onset of the asymmetric development of the side bands about the fundamental frequency and its higher harmonics. This asymmetric evolution, which ultimately leads to the shift to lower frequency first reported by Lake et al. (1977), is interpreted in terms of Longuet-Higgins’ (1978b) breaking instability. For ak [ges ] 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 recent results of McLean et al. (1981).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 115 , February 1982 , pp. 165 - 185
Copyright
© 1982 Cambridge University Press

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References

Banner, M. L. & Phillips, O. M. 1974 On small-scale breaking waves. J. Fluid Mech. 65, 647–657.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains in deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
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, 177–191.Google Scholar
Flick, R. E., Lowe, R. L., Freilich, M. H. & Boylls, J. C. 1979 Coastal and laboratory wave-staff system. In Proc. Oceans 79, I.E.E.E. and Mar. Tech. Soc., pp. 623–625.
Holyer, J. 1979 Large-amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.Google Scholar
Lake, B. M. & Yuen, H. C. 1977 A note on some nonlinear water-wave experiments and the comparison of data with theory. J. Fluid Mech. 83, 75–81.Google Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2: Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Lighthill, M. J. 1967 Some special cases treated by the Whitham theory. Proc. R. Soc. Lond. A 299, 28–53.Google Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
Longuet-Higgins, M. S. 1980 Modulation of the amplitude of steep wind waves. J. Fluid Mech. 99, 705–713.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
Mclean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional instability of finite-amplitude water waves. Phys. Rev. Lett. 46, 817–820.Google Scholar
Melville, W. K. 1977 Wind stress and roughness length over breaking waves. J. Phys. Ocean 7, 702–710.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297, 459–475.Google Scholar
Peregrine, D. H. & Thomas, A. P. 1979 Finite-amplitude deep-water waves on currents. Phil. Trans. R. Soc. Lond. A 292, 371–390.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Saffman, P. G. & Yuen, H. C. 1980 Bifurcation and symmetry breaking in nonlinear dispersive waves. Phys. Rev. Lett. 44, 1097–1100.Google Scholar
Saffman, P. G. & Yuen, H. C. 1981 Three-dimensional deep-water waves: calculation of steady symmetric wave pattern. Submitted to J. Fluid Mech.Google Scholar
Stokes, G. G. 1880 Supplement to a paper on the theory of oscillatory waves. Mathematical and Physical Papers, vol. 1, pp. 314–326. Cambridge University Press.
Su, M. Y. 1980 Experiments on water-wave breaking on deep water. Part I: Three-dimensional subharmonic instability (unpublished manuscript).
Su, M. Y. 1981 Three-dimensional deep-water waves. Laboratory experiments on spilling breakers. Submitted to J. Fluid Mech.Google Scholar
Whitham, G. B. 1965 A general approach to linear and nonlinear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273–283.Google Scholar