Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-29T01:35:08.026Z Has data issue: false hasContentIssue false

Modulation of three resonating gravity–capillary waves by a long gravity wave

Published online by Cambridge University Press:  26 April 2006

Karsten Trulsen
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Chiang C. Mei
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract

We consider a resonant triad of gravity–capillary waves, riding on top of a much longer gravity wave. The long-wave phase is assumed to vary on the same scale as the slow modulation of the short waves. Envelope equations are first deduced in the Lagrangian description. By perturbation analysis for a weak long wave, we then find that the long wave can resonate the natural modulation oscillations of the triad envelope, giving rise to various bifurcations in the Poincaré map. Numerical integration for a stronger long wave reveals that chaos can emerge from these bifurcations. The bifurcation criterion of Chen & Saffman (1979) for collinear Wilton's ripples is generalized to arbitrary non-collinear triads, and is found to play an important role as a criterion for the onset of chaotic behaviour.

Type
Research Article
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banerjee, P. P. & Korpel, A. 1982 Subharmonic generation by resonant three-wave interaction of deep-water capillary waves. Phys. Fluids 25, 19381943.Google Scholar
Case, K. M. & Chiu, S. C. 1977 Three-wave resonant interactions of gravity-capillary waves. Phys. Fluids 20, 742745.Google Scholar
Chen, B. & Saffman, P. G. 1979 Steady gravity-capillary waves in deep water-I. Weakly nonlinear waves. Stud. Appl. Maths 60, 183210.Google Scholar
Christodoulides, P. & Dias, F. 1994 Resonant capillary-gravity interfacial waves. J. Fluid Mech. 265, 303343.Google Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Craik, A. D. D. 1986 Exact solutions of non-conservative equations for three-wave and second-harmonic resonance. Proc. R. Soc. Lond. A 406, 112.Google Scholar
Craik, A. D. D. 1988 Interaction of a short-wave field with a dominant long wave in deep water:Derivation from Zakharov's spectral formulation. J. Austral. Soc. B 29, 430439.Google Scholar
Grimshaw, R. 1988 The modulation of short gravity waves by long waves or currents. J. Austral. Math. Soc. B 29, 410429.Google Scholar
Harrison, W. J. 1909 The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Math. Soc. 7, 107121.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30, 737739.Google Scholar
Henderson, D. M. & Hammack, J. L. 1987 Experiments on ripple instabilities. Part 1. Resonant triads. J. Fluid Mech. 184, 1541.Google Scholar
Henderson, D. M. & Hammack, J. L. 1993 Resonant interactions among surface water waves. Ann. Rev. Fluid Mech. 25, 5597.Google Scholar
Henyey, F. S., Creamer, D. B., Dysthe, K. B., Schult, R. L. & Wright, J. A. 1988 The energy and action of small waves riding on large waves. J. Fluid Mech. 189, 443462.Google Scholar
Janssen, P. E. M. 1986 The period-doubling of gravity-capillary waves. J. Fluid Mech. 172, 531546.Google Scholar
Janssen, P. E. M. 1987 The initial evolution of gravity-capillary waves. J. Fluid Mech. 184, 581597.Google Scholar
Jones, M. C. W. 1992 Nonlinear stability of resonant capillary-gravity waves. Wave Motion 15, 267283.Google Scholar
Kaup, D. J. 1981 The solution of the general initial value problem for the full three dimensional three-wave resonant interaction. Physica 3D, 374395.Google Scholar
Longuet-Higgins, M. S. 1987 The propagation of short surface waves on longer gravity waves. J. Fluid Mech. 177, 293306.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565583.Google Scholar
Ma, Y.-C. 1982 Weakly nonlinear steady gravity-capillary waves. Phys. Fluids 25, 945948.Google Scholar
Mcgoldrick, L. F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305331.Google Scholar
Mcgoldrick, L. F. 1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251271.Google Scholar
Meiss, J. D. & Watson, K. M. 1978 Discussion of some weakly nonlinear systems in continuum mechanics. In Topics in Nonlinear Dynamics (ed. S. Jorna), Vol 46, pp. 296323. American Institute of Physics.
Naciri, M. & Mei, C. C. 1992 Evolution of a short surface wave on a very long surface wave of finite amplitude. J. Fluid Mech. 235, 415452.Google Scholar
Perlin, M. & Hammack, J. L. 1991 Experiments on ripple instabilities. Part 3. Resonant quartets of the Benjamin-Feir type. J. Fluid Mech. 229, 229268.Google Scholar
Perlin, M., Henderson, D. M. & Hammack, J. L. 1990 Experiments on ripple instabilities. Part 2. Selective amplification of resonant triads. J. Fluid Mech. 219, 5180.Google Scholar
Perlin, M. & Ting, C.-L. 1992 Steep gravity-capillary waves within the internal resonance regime. Phys. Fluids A 4, 24662478.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193217.Google Scholar
Phillips, O. M. 1981 The dispersion of short wavelets in the presence of a dominant long wave. J. Fluid Mech. 107, 465485.Google Scholar
Reeder, J. & Shinbrot, M. 1981 On Wilton ripples, I: Formal derivation of the phenomenon. Wave Motion 3, 115135.Google Scholar
Shyu, J.-H. & Phillips, O. M. 1990 The blockage of gravity and capillary waves by longer waves and currents. J. Fluid Mech. 217, 115141.Google Scholar
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309, 551575.Google Scholar
Strizhkin, I. I. & Raletnev, V. I. 1986 Experimental studies of three- and four-wave resonant interactions of surface sea waves. Izv. Atmos. Ocean. Phys. 22(4), 311314.Google Scholar
Vyshkind, S. Y. & Rabinovich, M. I. 1976 The phase stochastization mechanism and the structure of wave turbulence in dissipative media. Sov. Phys. JETP 44, 292299.Google Scholar
Wersinger, J.-M., Finn, J. M. & Ott, E. 1980 Bifurcation and “strange” behavior in instability saturation by nonlinear three-wave mode coupling. Phys. Fluids 23, 11421154.Google Scholar
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688700.Google Scholar
Woodruff, S. L. & Messiter, A. F. 1994 A perturbation analysis of an interaction between long and short surface waves. Stud. Appl. Maths 92, 159189.Google Scholar
Zhang, J. & Melville, W. K. 1990 Evolution of weakly nonlinear short waves riding on long gravity waves. J. Fluid Mech. 214, 321346.Google Scholar
Zhang, J. & Melville, W. K. 1992 On the stability of weakly nonlinear short waves on finite-amplitude long gravity waves. J. Fluid Mech. 243, 5172.Google Scholar