Expansions for the shape of maximum amplitude Stokes waves

Abstract

Grant (1973) showed that the expansion giving the profile of a steady Stokes wave near a 120° corner was more complicated than had previously been assumed. This paper gives further terms in such an expansion, and shows that generating them cannot introduce transcendental quantities beyond those noted by Grant.

On the Behavior Near the Crest of Waves of Extreme Form

“…When Il n $. {/31' /32' /33"" }, the corresponding coefficient an is determined by the previous coefficients a 1 , ... , an-I; this agrees with Norman's results in [9]. The unpleasant possibility that all the an = 0 (not disproved in the present paper, although contrary to numerical evidence in [ The step from (1.5) to (1.6) points the way to the long inductive proof, in §4, of the main results of the paper, which appear in Theorem 4.5 and Corollaries 4.6 and 4.7.…”

“…Grant pointed out that the second term must have an exponent that is irrational and" probably transcendental"; he concluded that" the structure near the corner is considerably more complicated than has been assumed in the past". Norman [9] contemplated terms beyond the two considered by Grant; inferred the nature of all the exponents; introduced the assumption that the numbers Pi' defined after (1.3) below, are linearly independent over the rationals; and established certain relationships between the coefficients of the series. (However, it seems that no coefficient after the first can be calculated by a merely local analysis.)…”

“…All these calculations were heuristic rather than rigorous; in particular, Grant [4] and Norman [9] were concerned only to find analytic functions z(X) that satisfy the nonlinear boundary condition of the problem in an asymptotic sense as <l > ~ 0. In this paper we proceed from (1.1) and (1.2) to prove that, subject to an assumption about a question in number theory to be explained presently,…”

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).…”

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.…”