  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.
