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

Josh Shelton*, Philippe H. Trinh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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
Pages (from-to)318-342
Number of pages25
JournalIMA Journal of Applied Mathematics
Volume89
Issue number2
Early online date13 Jun 2024
DOIs
Publication statusPublished - 13 Jun 2024

Keywords

  • Exponential asymptotics
  • Gravity-capillary waves
  • Parasitic capillary ripples
  • Free-surface flows

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