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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)

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 1 , July 2017 , pp. 103 - 114
Copyright
© 2017 Australian Mathematical Society 

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