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Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves

Published online by Cambridge University Press:  30 March 2022

Josh Shelton Open the ORCID record for Josh Shelton [Opens in a new window]  and
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]
Josh Shelton*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Philippe H. Trinh*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

In this paper, we develop an asymptotic theory for steadily travelling gravity–capillary waves under the small-surface tension limit. In an accompanying work (Shelton et al., J. Fluid Mech., vol. 922, 2021) it was demonstrated that solutions associated with a perturbation about a leading-order gravity wave (a Stokes wave) contain surface-tension-driven parasitic ripples with an exponentially small amplitude. Thus, a naive Poincaré expansion is insufficient for their description. Here, we develop specialised methodologies in exponential asymptotics for derivation of the parasitic ripples on periodic domains. The ripples are shown to arise in conjunction with Stokes lines and the Stokes phenomenon. The resultant analysis associates the production of parasitic ripples to the complex-valued singularities associated with the crest of a steep Stokes wave. A solvability condition is derived, showing that solutions of this type do not exist at certain values of the Bond number. The asymptotic results are compared with full numerical solutions and show excellent agreement. The work provides corrections and insight of a seminal theory on parasitic capillary waves first proposed by Longuet-Higgins (J. Fluid Mech., vol. 16, issue 1, 1963, pp. 138–159).

JFM classification

Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 939 , 25 May 2022 , A17
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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