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CURRENT-MODIFIED EVOLUTION EQUATION FOR A BROADER BANDWIDTH CAPILLARY–GRAVITY WAVE PACKET

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  08 November 2016

SUMA DEBSARMA Open the ORCID record for SUMA DEBSARMA [Opens in a new window]  and
K. P. DAS
SUMA DEBSARMA*
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700009, India email sdappmath@caluniv.ac.in, kalipada_das@yahoo.com
K. P. DAS
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata 700009, India email sdappmath@caluniv.ac.in, kalipada_das@yahoo.com
*
✉sdappmath@caluniv.ac.in
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Abstract

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We derive a higher order nonlinear evolution equation for a broader bandwidth three-dimensional capillary–gravity wave packet, in the presence of a surface current produced by an internal wave. Instead of a set of coupled equations, a single nonlinear evolution equation is obtained by eliminating the velocity potential for the wave-induced slow motion. Finally, the equation is expressed in an integro-differential equation form, similar to Zakharov’s integral equation. Using the evolution equation derived here, we show that the two sidebands of a surface capillary–gravity wave get excited as a result of resonance with an internal wave, all propagating in the same direction. It is also shown that surface waves can grow exponentially with time at the expense of the energy of the internal wave.

Keywords

capillary–gravity waveinternal wavesideband instabilitysurface current

MSC classification

Primary: 76B07: Free-surface potential flows
Secondary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B45: Capillarity (surface tension)
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 2 , October 2016 , pp. 143 - 161
Copyright
© 2016 Australian Mathematical Society 

References

Bakhanov, V. V., Kemarskaya, O. N. and Pozdnyakova, V. I. et al. , “Evolution of surface waves of finite amplitude in field of inhomogeneous current”, Int. Geosci. Remote Sens. Sympos. Proc. 1 (1996) 609–611; doi:10.1109/IGARSS.1996.516418.Google Scholar
Chen, F. F., Introduction to plasma physics and controlled fusion, Volume 1: Plasma physics, 2nd edn (Plenum Press, New York, 1984).CrossRefGoogle Scholar
Davey, A. and Stewartson, K., “On three dimensional packets of surface waves”, Proc. R. Soc. Lond. Ser. A 338 (1974) 101–110; doi:10.1098/rspa.1974.0076.Google Scholar
Debsarma, S. and Das, K. P., “A higher order nonlinear evolution equation for much broader bandwidth gravity waves in deep water”, Int. J. Appl. Mech. Eng. 12 (2007) 557–563.Google Scholar
Dhar, A. K. and Das, K. P., “A fourth order evolution equation for deep water surface gravity waves in the presence of wind blowing over water”, Phys. Fluids 2 (1990) 778–783; doi:10.1063/1.857731.CrossRefGoogle Scholar
Donato, A. N., Peregrine, D. H. and Stocker, J. R., “The focussing of surface waves by internal waves”, J. Fluid Mech. 384 (1999) 27–58; doi:10.1017/S0022112098003917.CrossRefGoogle Scholar
Dysthe, K. B., “Note on a modification to the nonlinear Schrödinger equation for application to deep water waves”, Proc. R. Soc. Lond. Ser. A 369 (1979) 105–114; doi:10.1098/rspa.1979.0154.Google Scholar
Dysthe, K. B. and Das, K. P., “Coupling between a surface-wave spectrum and an internal wave: modulational interaction”, J. Fluid Mech. 104 (1981) 483–503; doi:10.1017/S0022112081003017.CrossRefGoogle Scholar
Hasselmann, K., “A criterion for nonlinear wave stability”, J. Fluid Mech. 30 (1967) 737–739; doi:10.1017/S0022112067001739.CrossRefGoogle Scholar
Hogan, S. J., “Fourth order evolution equation for deep water gravity–capillary waves”, Proc. R. Soc. Lond. Ser. A 402 (1985) 359–372; doi:10.1098/rspa.1985.0122.Google Scholar
Janssen, P. A. E. M., “On a fourth-order envelope equation for deep-water waves”, J. Fluid Mech. 126 (1983) 1–11; doi:10.1017/S0022112083000014.CrossRefGoogle Scholar
Jonsson, I. G., “Wave–current interactions”, in: The sea (eds. B. Le Méhauté and D. M. Hanes, Ocean Engineering Science 9A, 65–120) (John Wiley & Sons, New York, 1990).Google Scholar
McLean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. and Yuen, H. C., “Three-dimensional instability of finite amplitude water waves”, Phys. Rev. Lett. 46 (1981) 817–820; doi:10.1103/PhysRevLett.46.817.CrossRefGoogle Scholar
Peregrine, D. H., “Interaction of water waves and currents”, Adv. Appl. Mech. 16 (1976) 9–117; doi:10.1016/S0065-2156(08)70087-5.CrossRefGoogle Scholar
Ruban, V. P., “On the nonlinear Schrödinger equation for waves on a nonuniform current”, JETP Lett. 95 (2012) 486–491; doi:10.1134/S002136401209010X.CrossRefGoogle Scholar
Stiassnie, M., “Note on the modified nonlinear Schrödinger equation for deep water waves”, Wave Motion 6 (1984) 431–433; doi:10.1016/0165-2125(84)90043-X.CrossRefGoogle Scholar
Stocker, J. R. and Peregrine, D. H., “The current modified nonlinear Schrödinger equation”, J. Fluid Mech. 399 (1999) 335–353; doi:10.1017/S0022112099006618.CrossRefGoogle Scholar
Stocker, J. R. and Peregrine, D. H., “Three-dimensional surface waves propagating over long internal waves”, Eur. J. Mech. B Fluids 18 (1999) 545–559; doi:10.1016/S0997-7546(99)80049-1.Google Scholar
Toffoli, A., Waseda, T., Houtani, H., Kinoshita, T., Collins, K., Proment, D. and Onorato, M., “Excitation of rogue waves in a variable medium: an experimental study on the interaction of water waves and currents”, Phys. Rev. E 87 (2013) 051201 (1–4); doi:10.1103/PhysRevE.87.051201.CrossRefGoogle Scholar
Trulsen, K. and Dysthe, K. B., “A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water”, Wave Motion 24 (1996) 281–289; doi:10.1016/S0165-2125(96)00020-0.CrossRefGoogle Scholar
Trulsen, K., Kliakhandler, I., Dysthe, K. B. and Velarde, M. G., “On weakly nonlinear modulation of waves on deep water”, Phys. Fluids 12 (2000) 2432–2437; doi:10.1063/1.1287856.CrossRefGoogle Scholar
Trulsen, K. and Mei, C. C., “Double reflection of capillary/gravity waves by a non-uniform current: a boundary-layer theory”, J. Fluid Mech. 251 (1993) 239–271; doi:10.1017/S0022112093003404.CrossRefGoogle Scholar
Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of deep fluid”, J. Appl. Mech. Tech. Phys. 9 (1968) 190–194; doi:10.1007/BF00913182.CrossRefGoogle Scholar