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A MODIFICATION TO THE SCHRÖDINGER EQUATION FOR BROADER BANDWIDTH GRAVITY-CAPILLARY WAVES ON DEEP WATER WITH DEPTH-UNIFORM CURRENT

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  03 March 2023

SOURAV HALDER Open the ORCID record for SOURAV HALDER [Opens in a new window]  and
ASOKE KUMAR DHAR Open the ORCID record for ASOKE KUMAR DHAR [Opens in a new window]
SOURAV HALDER*
Affiliation:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India; e-mail: asoke.dhar@gmail.com
ASOKE KUMAR DHAR
Affiliation:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India; e-mail: asoke.dhar@gmail.com
*
e-mail: souravhalder76@gmail.com
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Abstract

We derive a nonlinear Schrödinger equation for the propagation of the three-dimensional broader bandwidth gravity-capillary waves including the effect of depth-uniform current. In this derivation, the restriction of narrow bandwidth constraint is extended, so that this equation will be more appropriate for application to a realistic sea wave spectrum. From this equation, an instability condition is obtained and then instability regions in the perturbed wavenumber space for a uniform wave train are drawn, which are in good agreement with the exact numerical results. As it turns out, the corrections to the stability properties that occur at the fourth-order term arise from an interaction between the mean flow and the frequency-dispersion term. Since the frequency-dispersion term, in the absence of depth-uniform current, for pure capillary waves is of opposite sign for pure gravity waves, so too are the corrections to the instability properties.

Keywords

nonlinear Schrödinger equationgravity-capillary wavesdepth-uniform currentbroader bandwidthmodulational instability

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 64 , Issue 3 , July 2022 , pp. 292 - 313
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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