On the behavior near the crest of waves of extreme form

Abstract:

The angle $\phi$ which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. 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 $2\pi /3$. In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion \[ ( * )\qquad \phi (s) = \frac {\pi } {6} + \sum \limits _{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text {as}} s \downarrow 0\], to arbitrary order; the coordinate $s$ is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents ${\mu _n}$ are transcendental numbers. We are unable to evaluate the coefficients ${a_n}$ explicitly, but define some in terms of global properties of $\phi$, and the others in terms of earlier coefficients. It is proved in [8] that ${a_1} < 0$, and follows here that ${a_2} > 0$. The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of $n$.