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Stability of weakly nonlinear deep-water waves in two and three dimensions

Published online by Cambridge University Press:  20 April 2006

Donald R. Crawford ,
Bruce M. Lake ,
Philip G. Saffman  and
Henry C. Yuen
Donald R. Crawford
Affiliation:
Fluid Mechanics Department, TRW Defense and Space Systems Group, One Space Park, Redondo Beach, CA 90278
Bruce M. Lake
Affiliation:
Fluid Mechanics Department, TRW Defense and Space Systems Group, One Space Park, Redondo Beach, CA 90278
Philip G. Saffman
Affiliation:
Fluid Mechanics Department, TRW Defense and Space Systems Group, One Space Park, Redondo Beach, CA 90278 Applied Mathematics, California Institute of Technology, Pasadena, CA 91125.
Henry C. Yuen
Affiliation:
Fluid Mechanics Department, TRW Defense and Space Systems Group, One Space Park, Redondo Beach, CA 90278
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Abstract

The stability of a weakly nonlinear wave train on deep water to two- and three-dimensional modulations is investigated using an improved approximation due to Zakharov (1968). The results are expressible in simple analytical forms, and show good quantitative agreement with available experimental data and exact numerical calculations over a broad range of wave steepness in the unidirectional case.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 105 , April 1981 , pp. 177 - 191
Copyright
© 1981 Cambridge University Press

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References

Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. Roy. Soc. A 299, 59–75.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wavetrains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Benney, D. J. 1962 Nonlinear gravity wave interactions. J. Fluid Mech. 14, 577–584.Google Scholar
Benney, D. J. & Roskes, G. 1969 Wave instabilities. Stud. Appl. Math. 48, 377–385.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. Roy. Soc. A 286, 183–230.Google Scholar
Crawford, D. R., Saffman, P. G. & Yuen, H. C. 1980 Evolution of a random inhomogeneous field of nonlinear deep water gravity waves. J. Wave Motion 2, 1–16.Google Scholar
Davey, A. 1972 The propagation of a weak nonlinear wave. J. Fluid Mech. 53, 769–781.Google Scholar
Hasimoto, H. & Ono, H. 1972 Nonlinear modulation of gravity waves. J. Phys. Soc. Japan 33, 805–811.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. Roy. Soc. A 299, 28–53.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. Roy. Soc. A 360, 489–505.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. (submitted).Google Scholar
Martin, D. U., Saffman, P. G. & Yuen, H. C. 1980 Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation. J. Wave Motion (to appear).Google Scholar
Martin, D. U. & Yuen, H. C. 1980 Quasi-recurring energy leakage in the two-space dimensional nonlinear Schrödinger equation. Phys. Fluids (in press).Google Scholar
Peregrine, D. H. & Thomas, G. P. 1979 Finite-amplitude deep-water waves on currents. Phil. Trans. Roy. Soc. A 292, 371–390.Google Scholar
Saffman, P. G. & Yuen, H. C. 1978 Stability of a plane soliton to infinitesimal two-dimensional perturbations. Phys. Fluids 21, 1450–1451.Google Scholar
Saffman, P. G. & Yuen, H. C. 1980a Bifurcation and symmetry breaking in nonlinear dispersive waves. Phys. Rev. Lett. 44, 1097–1100.Google Scholar
Saffman, P. G. & Yuen, H. C. 1980b A new type of three-dimensional deep-water wave of permanent form. J. Fluid Mech. 101, 797–808.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yuen, H. C. & Ferguson, W. E. 1978a Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids 21, 1275–1278.Google Scholar
Yuen, H. C. & Ferguson, W. E. 1978b Fermi-Pasta-Ulam recurrence in the two-space dimensional nonlinear Schrödinger equation. Phys. Fluids 21, 2116–2118.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 2, 86–94. (Translated in J. Appl. Mech. Tech. Phys. 2, 190–194.)Google Scholar
Zakharov, V. E. & Rubenchik, A. M. 1973 Instability of waveguides and solitons in nonlinear media. Zh. Eksp. Teor. Fiz. 65, 997–1011. (Translated in Sov. Phys. J. Exp. Theor. Phys. 38, 1974, 494–500.)Google Scholar
Zakharov, V. E. & Shabat, A. B. 1971 Exact theory of two-dimensional self-focusing and one-dimensional self-modulating waves in nonlinear media. Zh. Eksp. Teor. Fiz. 61, 118–134. (Translated in Sov. Phys. J. Exp. Theor. Phys. 34, 64–69, 1972.)Google Scholar