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The Stokes and Krasovskii Conjectures for the Wave of Greatest Height
Author(s) -
McLeod J. B.
Publication year - 1997
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00051
Subject(s) - mathematics , conjecture , limit (mathematics) , conservative vector field , mathematical analysis , function (biology) , boundary (topology) , wave equation , mathematical physics , geometry , pure mathematics , physics , mechanics , compressibility , biology , evolutionary biology
The integral equation: φ μ (s) = (1/3 π)∫ π 0 ((sin φ μ (t))/(μ −1 + ∫ t 0 sin φ μ (u) d u )) (log((sin½( s + t ))/ (sin½( s − t )))d t was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φ μ giving the angle that the wave surface makes with the horizontal. The wave of greatest height is the singular case μ=∞, and it is shown that there exists a solution φ ∞ to the equation in this case and that it can be obtained as the limit (in a specified sense) as μ→∞ of solutions for finite μ. Stokes conjectured that φ ∞ ( s )→⅙π as s ↓0, so that the wave is sharply crested in the limit case; and Krasovskii conjectured that sup s ∈[0,π] φ μ ( s )≤⅙π for all finite μ. Stokes' conjecture was finally proved by Amick, Fraenkel, and Toland, and the present article shows Krasovskii's conjecture to be false for sufficiently large μ, the angle exceeding ⅙π in what is a boundary layer.