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Numerical Solution of Nekrasov's Equation in the Boundary Layer Near the Crest for Waves Near the Maximum Height
Author(s) -
ByattSmith J. G.
Publication year - 2001
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 - 10.1111/1467-9590.00171
Subject(s) - crest , mathematics , boundary (topology) , boundary layer , citation , combinatorics , history , art history , sociology , geometry , physics , mathematical analysis , computer science , library science , optics , mechanics
Nekrasov's integral equation describing water waves of permanent form,determines the angle phi that the wave surface makes with the horizontal. Theindependent variable s is a suitably scaled velocity potential, evaluated atthe free surface, with the origin corresponding to the crest of the wave. Forall waves, except for amplitudes near the maximum, phi satisfies the inequalitymod(phi) is less than pi/6. It has been shown numerically and analytically,that as the wave amplitude approaches its maximum, the maximum of phi canexceed pi/6 by about 1% near the crest. Numerical evidence suggested that thisoccurs in a small boundary layer near the crest where mod(phi(s)) rises rapidlyfrom zero and oscillates about pi/6, the number of oscillations increasing asthe maximum amplitude is approached. McLeod derived, from Nekrasov's equation,an integral equation for phi in the boundary layer, whose width tends to zeroas the maximum wave is approached. He also conjectured the asymptotic form ofthe oscillations of mod(phi(s)) about pi/6 as s tends to infinity. We solveMcLeod's boundary layer equation numerically and verify the asymptotic form ofphi.