The Stokes and Krasovskii Conjectures for the Wave of Greatest Height

摘要

It is shown that there exists a solution to Nekrasov's integral equation which describes a wave of greatest height and of permanent form moving on the surface of a non-viscous, irrotational, infinitely deep flow. It is also shown that this wave can be obtained as the limit, in a specified sense, of waves of almost extreme form. Stokes conjectured, almost 100 years ago, that in the extreme case the wave is sharply crested and the wave surface makes an angle of pi/6 with the horizontal at the crest, and Krasovskii conjectured that, for waves of non-extreme form, which are smooth-crested, the angle between the surface and the horizontal at no point exceeds pi/6, the latter belief being widely held until some recent numerical calculations cast some doubt upon it. While the present paper makes only partial progress towards deciding Stokes' conjecture, it does confirm the numerical evidence and prove that the Krasovskii conjecture is false for waves sufficiently close to the extreme form, the angle exceeding pi/6 in a boundary layer.