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FIFTH-ORDER EVOLUTION EQUATION OF GRAVITY–CAPILLARY WAVES

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  09 August 2017

DIPANKAR CHOWDHURY Open the ORCID record for DIPANKAR CHOWDHURY [Opens in a new window]  and
SUMA DEBSARMA Open the ORCID record for SUMA DEBSARMA [Opens in a new window]
DIPANKAR CHOWDHURY*
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India email dipankar.chowdhury05@rediffmail.com, suma_debsarma@rediffmail.com
SUMA DEBSARMA
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India email dipankar.chowdhury05@rediffmail.com, suma_debsarma@rediffmail.com
*
✉dipankar.chowdhury05@rediffmail.com
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Abstract

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We extend the evolution equation for weak nonlinear gravity–capillary waves by including fifth-order nonlinear terms. Stability properties of a uniform Stokes gravity–capillary wave train is studied using the evolution equation obtained here. The region of stability in the perturbed wave-number plane determined by the fifth-order evolution equation is compared with that determined by third- and fourth-order evolution equations. We find that if the wave number of longitudinal perturbations exceeds a certain critical value, a uniform gravity–capillary wave train becomes unstable. This critical value increases as the wave steepness increases.

Keywords

evolution equationgravity–capillary wavemodulational instabilitywind effect

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 59 , Issue 1 , July 2017 , pp. 103 - 114
Copyright
© 2017 Australian Mathematical Society 

References

Crawford, D. R., Lake, B. M., Saffman, P. G. and Yuen, H. C., “Stability of weakly nonlinear deep-water waves in two and three dimensions”, J. Fluid Mech. 105 (1981) 177–191; doi:10.1017/S0022112081003169.Google Scholar
Debsarma, S. and Das, K. P., “Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water”, ANZIAM J. 43 (2002) 513–524; doi:10.1017/S1446181100012116.CrossRefGoogle Scholar
Debsarma, S. and Das, K. P., “A higher order nonlinear evolution equation for broader bandwidth gravity waves in deep water”, Phys. Fluids 17 (2005), 1–8; doi:10.1063/1.2046714.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
Djordjevic, V. D. and Redekopp, L. G., “On two-dimensional packets of capillary-gravity waves”, J. Fluid Mech. 79 (1977) 703–714; doi:10.1017/S0022112077000408.CrossRefGoogle Scholar
Dysthe, K. B., “Note on a modification to the nonlinear Schrodinger 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
Gramstad, O. and Trulsen, K., “Hamiltonian form of the modified nonlinear Schrodinger equation for gravity waves on arbitrary depth”, J. Fluid Mech. 670 (2011) 404–426; doi:10.1017/S0022112010005355.Google Scholar
Hogan, S. J., “Fourth order evolution equation for deep gravity-capillary waves”, Proc. R. Soc. Lond. A 402 (1985) 359–372; doi:10.1098/rspa.1985.0122.Google Scholar
Hogan, S. J., Gruman, I. and Stiassnie, M., “On the changes in phase speed of one train of water waves in the presence of another”, J. Fluid Mech. 192 (1988) 97–114; doi:10.1017/S0022112088001806.Google Scholar
King, F. W., Hilbert transforms (Cambridge University Press, Cambridge, 2009).Google Scholar
Krasitskii, V. P., “On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves”, J. Fluid Mech. 272 (1994) 1–20; doi:10.1017/S0022112094004350.Google Scholar
Krasitskii, V. P. and Kalmykov, V. A., “Four-wave reduced equations for surface gravity waves”, Izv. Atmos. Phys. 29 (1993) 222–228.Google Scholar
Longuet-Higgins, M. S., “The instability of gravity waves of finite amplitude in deep water, I. Superharmonics”, Proc. R. Soc. Lond. A 360 (1978a) 471–488; doi:10.1098/rspa.1978.0080.Google Scholar
Longuet-Higgins, M. S., “The instability of gravity waves of finite amplitude in deep water, II. Subharmonics”, Proc. R. Soc. Lond. A 360 (1978b) 489–505; doi:10.1098/rspa.1978.0081.Google Scholar
McGoldrick, L. F., “On Wilton’s ripple: a special case of resonant interactions”, J. Fluid Mech. 42 (1970) 193–200; doi:10.1017/S0022112070001179.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/Phys.Rev.Lett.46.817.CrossRefGoogle Scholar
McLean, J. W., “Instabilities of finite amplitude water waves”, J. Fluid Mech. 114 (1982) 315–330; doi:10.1017/S0022112082000172.Google Scholar
Trulsen, K. and Dysthe, K. B., “A modified nonlinear Schrodinger equation for border bandwidth gravity waves on deep water”, Wave Motion 24 (1996) 281–289; doi:10.1016/S0165-2125(96)00020-0.Google 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.Google Scholar
Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys. 2 (1968) 190–194; doi:10.1007/BF00913182.Google Scholar
Zhang, J. and Melville, W. K., “On the stability of weakly-nonlinear gravity-capillary waves”, Wave Motion 8 (1986) 439–454; doi:10.1016/0165-2125(86)90029-6.CrossRefGoogle Scholar
Zhang, J. and Melville, W. K., “3-dimensional instabilities of nonlinear gravity capillary waves”, J. Fluid Mech. 174 (1987) 187–208; doi:10.1017/S0022112087000090.Google Scholar