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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)

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 2 , October 2016 , pp. 143 - 161
Copyright
© 2016 Australian Mathematical Society 

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