The asymptotic behavior near the crest of waves of extreme form

Abstract

The angle which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle 2 3 π \frac {2} {3}\pi . Amick and Fraenkel have investigated the asymptotic behavior of the free surface near the crest and obtained an asymptotic expansion for this behavior, but are unable to say whether the leading term in this expansion has a nonzero coefficient (and so whether it is in fact the leading term or not). The present paper shows that the coefficient is nonzero and determines its sign.