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Published online by Cambridge University Press:  12 January 2024

Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103 West Bengal, India; e-mail:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103 West Bengal, India; e-mail:


A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

Research Article
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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