A model ODE for the exponential asymptotics of nonlinear parasitic capillary ripples

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In this work, we develop a linear model ordinary differential equation (ODE) to study the parasitic capillary ripples present on steep Stokes waves when a small amount of surface tension is included in the formulation. Our methodology builds upon the exponential asymptotic theory of Shelton & Trinh (J. Fluid Mech., vol. 939, 2022, A17), who demonstrated that these ripples occur beyond-all-orders of a small-surface-tension expansion. Our model equation, a linear ODE forced by solutions of the Stokes wave equation, forms a convenient tool to calculate numerical and asymptotic solutions. We show analytically that the parasitic capillary ripples that emerge in solutions to this linear model have the same asymptotic scaling and functional behaviour as those in the fully nonlinear problem. It is expected that this work will lead to the study of parasitic capillary ripples that occur in more general formulations involving viscosity or time-dependence.

Original languageEnglish
JournalIMA Journal of Applied Mathematics
Early online date6 Jun 2024
Publication statusE-pub ahead of print - 6 Jun 2024

Data Availability Statement

The MATLAB code and data underlying this article is available from the corresponding author on reasonable request; a simplified version of this is provided in Appendix C.

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