A modification to the Schrödinger equation for broader bandwidth gravity-capillary waves on deep water with depth-uniform current

Authors

DOI:

https://doi.org/10.21914/anziamj.v64.17255

Keywords:

nonlinear Schrödinger equation, gravity-capillary waves, depth-uniform current, broader bandwidth, modulational instability

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.

 

doi: 10.1017/S1446181123000020

Author Biographies

S. Halder, Indian Institute of Engineering Science and Technology, Shibpur

Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, India.

Dr. Asoke Kumar Dhar, Indian Institute of Engineering Science and Technology, Shibpur

Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, India.

Published

2023-03-19

Issue

Section

Articles for Printed Issues