On nonlinearly detuned third harmonic ripples between two stratified fluids
Author:
Mark C. W. Jones
Journal:
Quart. Appl. Math. 59 (2001), 241-267
MSC:
Primary 76E17; Secondary 35Q55, 76B55, 76B70
DOI:
https://doi.org/10.1090/qam/1827813
MathSciNet review:
MR1827813
Full-text PDF Free Access
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Abstract: The problem of two semi-infinite fluids in uniform horizontal motion parallel to their interface is studied. Attention is focused on the interfacial disturbances that are caused by the interaction between a fundamental mode and its third harmonic. A series expansion for the disturbance profile is obtained in which the leading-order amplitudes are assumed to be slowly varying functions in time and space. By use of this expression we are able to derive a pair of coupled nonlinear Schrödinger-type equations which model the evolution of the interface. Solutions to this system are found and thus we are able to describe the possible wave profiles, which turn out to be tripleor quintuple-dimpled. We also find that at perfect resonance three profiles are always possible but that at near-resonance there may be one or three profiles depending on the values of the parameters present in the problem.
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A. H. Nayfeh, Finite amplitude surface waves in a liquid layer, J. Fluid Mech. 40, 671–684 (1970)
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A. H. Nayfeh and W. S. Saric, Nonlinear waves in a Kelvin-Helmholtz flow, J. Fluid Mech. 55, 311–327 (1972)
M. Perlin and J. Hammack, Experiments on ripple instabilities. Part 3. Resonant quartets of the Benjamin-Feir type, J. Fluid Mech. 229, 229–268 (1991)
M. Perlin, D. M. Henderson, and J. Hammack, Experiments on ripple instabilities. Part 2. Selective amplification of resonant triads, J. Fluid Mech. 219, 51–80 (1990)
W. S. Saric and B. W. Marshall, An experimental investigation of the stability of a thin liquid layer adjacent to a supersonic stream, J. Amer. Inst, of Aeronautics and Astronautics 9, 1546–1553 (1971)
J. R. Wilton, On ripples, Philos. Mag. 29, 688–700 (1915)
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2, 190–194 (1968)
T. B. Benjamin and J. E. Feir, The disintegration of wavetrains on deep water. Part 1. Theory, J. Fluid Mech. 27, 417–430 (1967)
V. Bontozoglou and T. J. Hanratty, Capillary-gravity Kelvin-Helmholtz waves close to resonance, J. Fluid Mech. 217, 71–91 (1990)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, 1961
P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves, J. Fluid Mech. 265, 303–343 (1994)
P. Christodoulides and F. Dias, Stability of capillary-gravity waves between two bounded fluids, Physics of Fluids 12, 3013–3027 (1995)
A. D. D. Craik, Wave interactions and fluid flows, Cambridge University Press, 1985
A. Davey and K. Stewartson, On three dimensional packets of surface waves, Proc. Roy. Soc. Lond. A 338, 101–110 (1974)
L. E. Dickson, New Course in the Theory of Equations, Wiley, New York, 1939
V. E. Djordjevic and L. E. Redekopp, On two dimensional packets of capillary-gravity waves, J. Fluid Mech. 79, 703–714 (1977)
J. L. Hammack and D. M. Henderson, Resonant interactions among surface water waves, Ann. Rev. Fluid Mech. 25, 55–97 (1987)
H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Soc. of Japan 33, 805–811 (1972)
D. M. Henderson and J. L. Hammack, Experiments on ripple instabilities. Part 1. Resonant triads, J. Fluid Mech. 184, 15–41 (1987)
M. C. W. Jones, On the stability of a third harmonic resonant wavetrain, Stability and Applied Analysis of Continuous Media 2, 323–338 (1992)
M. C. W. Jones, On coupled differential equations which model the evolution of interacting capillary-gravity wave modes and related questions of stability, IMA J. Appl. Math. 50, 13–28 (1993)
M. C. W. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction, J. Fluid Mech. 341, 295–315 (1997)
L. F. McGoldrick, On Wilton’s ripples: A special case of resonant interactions, J. Fluid Mech. 42, 193–200 (1970)
L. F. McGoldrick, On the rippling of small waves: A harmonic nonlinear nearly resonant interaction, J. Fluid Mech. 52, 725–751 (1972)
A. H. Nayfeh, Finite amplitude surface waves in a liquid layer, J. Fluid Mech. 40, 671–684 (1970)
A. H. Nayfeh, Triple- and quintuple-dimpled wave profiles in deep water, Phys. Fluids 13, 545–550 (1970)
A. H. Nayfeh, Third harmonic resonance in the interaction of capillary and gravity waves, J. Fluid Mech. 48, 384–395 (1971)
A. H. Nayfeh and S. D. Hassan, The method of multiple scales and nonlinear dispersive waves, J. Fluid Mech. 48, 463–475 (1971)
A. H. Nayfeh and W. S. Saric, Nonlinear waves in a Kelvin-Helmholtz flow, J. Fluid Mech. 55, 311–327 (1972)
M. Perlin and J. Hammack, Experiments on ripple instabilities. Part 3. Resonant quartets of the Benjamin-Feir type, J. Fluid Mech. 229, 229–268 (1991)
M. Perlin, D. M. Henderson, and J. Hammack, Experiments on ripple instabilities. Part 2. Selective amplification of resonant triads, J. Fluid Mech. 219, 51–80 (1990)
W. S. Saric and B. W. Marshall, An experimental investigation of the stability of a thin liquid layer adjacent to a supersonic stream, J. Amer. Inst, of Aeronautics and Astronautics 9, 1546–1553 (1971)
J. R. Wilton, On ripples, Philos. Mag. 29, 688–700 (1915)
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2, 190–194 (1968)
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Article copyright:
© Copyright 2001
American Mathematical Society
