Exponential asymptotics for steady parasitic capillary ripples on steep gravity waves

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. (2021), J. Fluid Mech., vol 922] 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 Poincar\'e expansion is insufficient for their description. Here, we shall 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 analysis relies crucially upon the derivation and analysis of singularities in the analytic continuation of the classic 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 to 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 (1), 1963, pp. 138-159].