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

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