On the Behavior Near the Crest of Waves of Extreme Form

Abstract

ABSTRACT. The angle </> 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 2'1T/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 11=1to 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 ILn are transcendental numbers. We are unable to evaluate the coefficients an explicitly. but define some in terms of global properties of </>. and the others in terms of earlier coefficients. It is proved in [8] that al < O. and follows here that a2 > O. 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.

An approximation for the highest gravity waves on water of finite depth

“…This hypothesis was justified rigorously in the papers by Toland (1978), Amick, Fraenkel & Toland (1982) and Plotnikov (1983). An asymptotic expansion in the vicinity of a singular point was found by Grant (1973), Norman (1974), Amick & Fraenkel (1987) and McLeod (1987).…”

Ludwig Edward Fraenkel. 28 May 1927—27 April 2019

“…Of his 31 papers before 1970, only three had a co-author, but from 1970 onward he had several significant collaborators: M. S. Berger* (16, 17, 20, 24); his brilliant student C. J. Amick who died tragically young (23, 27, 29, 30, 31); and, towards the end of his life, a significant collaboration with his near contemporary J. B. McLeod (FRS 1992), which led to two publications (35, 38) and to a copious collection of unpublished material that means significant additional progress had been made on the problem of wedge entry into water.…”

“…However the conjecture has been proved correct and Edward made several contributions, involving estimates of increasing sophistication, to this challenging problem. First, the Stokes conjecture was shown to be correct (27) and (Plotnikov 1982), independently, and an exotic asymptotic series that describes, to arbitrary order, the shape of an extreme wave close to its crest was justified (30). Then, 20 years later, in a solo tour de force he obtained the existence of an extreme wave for which the Stokes conjecture is automatically satisfied (40).…”

Extreme water-wave profile recovery from pressure measurements at the seabed

“…This work originated with Stokes (1880), who postulated that the wave of maximum amplitude has an included crest angle of , the crest being a stagnation point which, in the conformal plane, has a power singularity. It was shown by Grant (1973) that higher-order expansions in the vicinity of the crest involve non-algebraic powers (see also Norman 1974); this work was placed on a rigorous footing in Amick & Fraenkel (1987) and McLeod (1987). Performing the same local analysis (around the crest at where ) in the physical plane, one obtains after some algebra where is a strictly positive (dimensionless) constant (McLeod 1987), is a (freely choosable) characteristic wavenumber (e.g.…”