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Wave interactions - the evolution of an idea

Published online by Cambridge University Press:  20 April 2006

O. M. Phillips
O. M. Phillips
Affiliation:
Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, Maryland 21218
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Abstract

This essay gives a personal and possibly incomplete history of the way in which the simple idea of weak resonant wave interactions grew to find application to a variety of phenomena in several contexts. The development involved incremental steps by many people in the past twenty years, gaining simplicity with maturity. The final stage seems to be approaching when the limits of usefulness of the idea are beginning to become apparent.

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Type
Research Article
Information
Journal of Fluid Mechanics , Volume 106 , May 1981, pp. 215 - 227
Copyright
© 1981 Cambridge University Press

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References

Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of non-linear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647–664.Google Scholar
Ball, K. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465–478.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.Google Scholar
Benney, D. T. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14, 577–584.Google Scholar
Benney, D. J. & Newell, A. C. 1967 The propagation of non-linear wave envelopes. J. Math. Phys. 46, 133–145.Google Scholar
Brekhovskikh, L. M., Goncharov, V. V., Kurtipov, V. M. & Nangol'nykh, K. A. 1972 Resonant excitation of internal waves by non-linear interaction of surface waves. Izv. Akad. Nauk SSSR, Fiz-Atmos. Okeana, 8, 192–203. (Engl. trans.: Atmos. Oceanic Phys. 8, 192–197.)Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. A 302, 529.Google Scholar
Chu, V. H. & Mei, C. C. 1970 On slowly varying Stokes waves. J. Fluid Mech. 41, 837–887.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. Roy. Soc. A 338, 101–110.Google Scholar
Davis, R. E. & Acrivos, A. 1967 The stability of oscillatory internal waves. J. Fluid Mech. 30, 723–736.Google Scholar
Gargett, A. E. & Hughes, B. A. 1972 On the interaction of surface and internal waves. J. Fluid Mech. 52, 179–192.Google Scholar
Garrett, C. & Munk, W. H. 1972 Space-time scales of internal waves. Geophys. Fluid Dyn. 3, 225–264.Google Scholar
Hasimoto, H. & Ono, H. 1972 Non-linear modulation of gravity waves. J. Phys. Soc. Japan 33, 805.Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity wave spectrum. Part 1. J. Fluid Mech. 12, 481–500. Part 2 (1963) Ibid, 15, 273–281. Part 3, (1963) Ibid, 15, 385–398.Google Scholar
Hasselmann, K. 1967 A criterion for non-linear wave stability. J. Fluid Mech. 30, 737–739.Google Scholar
Haurwitz, B., Stommel, H. & Munk, W. H. 1959 On thermal unrest in the ocean. Rossby Mem. Vol. pp. 74–94. New York: Rockefeller Inst. Press.
Kim, Y. Y. & Hanratty, T. J. 1971 Weak quadratic interactions of two-dimensional waves. J. Fluid Mech. 50, 107–132.Google Scholar
Lewis, J. E., Lake, B. M. & Ko, D. R. S. 1974 On the interaction of internal and surface gravity waves. J. Fluid Mech. 63, 773–800.Google Scholar
Lighthill, M. J. 1950 Aerodynamic noise, or, turbulence as a source of sound. Aero. Res. Counc., London. Rep. no. FM 1467 10 Aug. 1950.
Lighthill, M. J. 1952 On sound generated aerodynamically I. General theory. Proc. Roy. Soc. A 211, 564–587.Google Scholar
Lighthill, M. J. 1954 On sound generated aerodynamically II. Turbulence as a source of sound. Proc. Roy. Soc. A 222, 1–32.Google Scholar
Long, R. R. 1953a A laboratory model resembling the ‘Bishop Wave’ phenomenon. Bull. Amer. Met. Soc. 34, 205–211.Google Scholar
Long, R. R. 1953b Some aspects of the flow of stratified fluids. I. A theoretical investigation. Tellus 5, 42–57.Google Scholar
Long, R. R. 1954 Some aspects of the flow of stratified fluids. II. Experiments with a two-fluid system. Tellus 6, 97–115.Google Scholar
Long, R. R. 1955 Some aspects of the flow of stratified fluids. III. Continuous density gradients. Tellus 7, 342–357.Google Scholar
Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12, 321–332.Google Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. Part I. Superharmonics. Proc. Roy. Soc. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. & Phillips, O. M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12, 333–336.Google Scholar
Longuet-Higgins, M. S. & Smith, N. D. 1966 An experiment on third order resonant wave interactions. J. Fluid Mech. 25, 417–435.Google Scholar
Martin, S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single internal waves. J. Fluid Mech. 53, 17–44.Google Scholar
McComas, C. H. 1977 Non-linear interactions of internal gravity waves. J. Geophys. Res. 82, 1397–1412.Google Scholar
McEwan, A. D. 1971 Degeneration of resonantly excited internal gravity waves. J. Fluid Mech. 50, 431–448.Google Scholar
McGoldrick, L. F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305–332.Google Scholar
McGoldrick, L. F. 1970 An experiment on second order capillary-gravity resonant interactions. J. Fluid Mech. 40, 251–271.Google Scholar
McGoldrick, L. F. 1972 On the rippling of small waves: a harmonic non-linear nearly resonant interaction. J. Fluid Mech. 52, 725–751.Google Scholar
McGoldrick, L. F., Phillips, O. M., Huang, N. & Hodgson, T. 1966 Measurements on resonant wave interactions. J. Fluid Mech. 25, 437–456.Google Scholar
Nayfeth, A. H. 1971 Third harmonic resonance in the interaction of capillary and gravity waves. J. Fluid Mech. 48, 385–395.Google Scholar
Peierls, R. E. 1929 Zur kinetischen Theorie der Wärmeleitungen in Kristallen. Ann. Phys. 3 1055–1101.Google Scholar
Perry, R. B. & Schimke, G. R. 1965 Large amplitude internal waves observed off the north-west coast of Sumatra. J. Geophys. Res. 70, 2319–2324.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part I. J. Fluid Mech. 9, 193–217. Part II (1961) ibid. 11, 143–455.Google Scholar
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. (in the press).Google Scholar
Watson, K. M., West, B. J. & Cohen, B. I. 1976 Coupling of surface and internal gravity waves: a Hamiltonian model. J. Fluid Mech. 77, 185–208.Google Scholar
Whitham, G. B. 1960 A note on group velocity. J. Fluid Mech. 9, 347–352.Google Scholar
Whitham, G. B. 1962 Mass, momentum and energy flux in water waves. J. Fluid Mech. 12, 135–147.Google Scholar
Whitham, G. B. 1965 A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273–283.Google Scholar
Yuen, H. C. & Lake, B. M. 1975 Non-linear deep water waves: theory and experiment. Phys. Fluids 18, 956–960.Google Scholar
Yuen, H. C. & Lake, B. M. 1978 Non-linear wave concepts applied to deep water waves. In Solitons in Action, pp. 89–126. Academic Press.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Fiz. 9, 86. (Engl. transl. J. Appl. Mech. Tech. Phys. 2, 190).Google Scholar
Zakharov, V. E. & Shabat, A. B. 1971 Exact theory of two-dimensional self-modulating waves in non-linear media. Zh. Eksp. Teor. Fiz. 61, 118. Also Soviet J. Exp. Theor. Phys. 34 (1972), 62–69.Google Scholar