Instabilities of finite-amplitude water waves

A numerical investigation of normal-mode perturbations of a finite-amplitude Stokes wave has revealed regions of instability lying near resonance curves given by the linear-dispersion relation. It is found that, for small amplitude, the dominant instability is two-dimensional (of Benjamin-Fier type) but, for larger amplitudes, the dominant instability becomes a three-dimensional perturbation. Results are compared with recent experimental observations of steep wave trains.

[1]  O. Phillips.  On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions , 1960, Journal of Fluid Mechanics.

[2]  M. J. Lighthill,et al.  Contributions to the Theory of Waves in Non-linear Dispersive Systems , 1965 .

[3]  T. Brooke Benjamin,et al.  The disintegration of wave trains on deep water Part 1. Theory , 1967, Journal of Fluid Mechanics.

[4]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[5]  L. Schwartz.  Computer extension and analytic continuation of Stokes’ expansion for gravity waves , 1974, Journal of Fluid Mechanics.

[6]  G. Dagan.  Taylor instability of a non-uniform free-surface flow , 1975, Journal of Fluid Mechanics.

[7]  E. Cokelet,et al.  Steep gravity waves in water of arbitrary uniform depth , 1977, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[8]  Michael Selwyn Longuet-Higgins,et al.  The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  M. Longuet-Higgins.  Some New Relations Between Stokes's Coefficients in the Theory of Gravity Waves , 1978 .

[10]  D. Hasselmann.  The high wavenumber instabilities of a Stokes wave , 1979 .

[11]  P. Saffman.  Long wavelength bifurcation of gravity waves on deep water , 1980, Journal of Fluid Mechanics.

[12]  P. Saffman,et al.  A new type of three-dimensional deep-water wave of permanent form , 1980, Journal of Fluid Mechanics.

[13]  B. Chen,et al.  Numerical Evidence for the Existence of New Types of Gravity Waves of Permanent Form on Deep Water , 1980 .

[14]  Bruce M. Lake,et al.  Stability of weakly nonlinear deep-water waves in two and three dimensions , 1981, Journal of Fluid Mechanics.

[15]  Henry C. Yuen,et al.  Three-Dimensional Instability of Finite-Amplitude Water Waves , 1981 .