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

TL;DR

This paper introduces a linear ODE model to analyze parasitic capillary ripples on steep Stokes waves with surface tension, capturing their asymptotic behavior beyond all orders of small surface tension.

Contribution

A new linear ODE model is developed to efficiently study parasitic ripples, matching the asymptotic scaling of the fully nonlinear problem.

Findings

Model reproduces ripple scaling and behavior

Analytical confirmation of ripple characteristics

Potential for extension to viscous and time-dependent cases

Abstract

In this work, we develop a linear model 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.