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In this paper, we examine the formation of small capillary waves (parasiticripples) on the surface of steep steadily-travelling gravity waves. Previously,authors have developed ad-hoc analytical procedures for describing theformation of such parasitic ripples in potential flows; however, it has notbeen clear whether the small-surface tension limit is well-posed -- that is,whether it is possible for an appropriate travelling gravity-capillary wave tobe continuously deformed to the classic Stokes wave in the limit of vanishingsurface tension. The work of Chen & Saffman (1980) had suggested smoothcontinuation was not possible. In this paper, we numerically explore the lowsurface tension limit of the steep gravity-capillary travelling-wave problem.Our results allow for a classification of the bifurcation structure thatarises, and serve to unify a number of previous numerical studies. Crucially,we demonstrate that different choices of solution amplitude can lead to subtlerestrictions on the continuation procedure; the use of wave energy as anamplitude condition allows solution branches to be continuously deformed to thezero surface tension limit.

On the structure of parasitic gravity-capillary waves in the small surface tension limit