dorsal/arxiv
View SchemaThe Numerical Solution of Nekrasov's Equation in the Boundary Layer near the Crest, for Waves near the Maximum Height
| Authors | J. G. Byatt-Smith |
|---|---|
| Categories | |
| ArXiv ID | physics/0011076 |
| URL | https://arxiv.org/abs/physics/0011076 |
Abstract
Nekrasov's integral equation describing water waves of permanent form, determines the angle phi that the wave surface makes with the horizontal. The independent variable s is a suitably scaled velocity potential, evaluated at the free surface, with the origin corresponding to the crest of the wave. For all waves, except for amplitudes near the maximum, phi satisfies the inequality mod(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 can exceed pi/6 by about 1% near the crest. Numerical evidence suggested that this occurs in a small boundary layer near the crest where mod(phi(s)) rises rapidly from zero and oscillates about pi/6, the number of oscillations increasing as the maximum amplitude is approached. McLeod derived, from Nekrasov's equation, an integral equation for phi in the boundary layer, whose width tends to zero as the maximum wave is approached. He also conjectured the asymptotic form of the oscillations of mod(phi(s)) about pi/6 as s tends to infinity. We solve McLeod's boundary layer equation numerically and verify the asymptotic form of phi.
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"abstract": "Nekrasov\u0027s integral equation describing water waves of permanent form,\ndetermines the angle phi that the wave surface makes with the horizontal. The\nindependent variable s is a suitably scaled velocity potential, evaluated at\nthe free surface, with the origin corresponding to the crest of the wave. For\nall waves, except for amplitudes near the maximum, phi satisfies the inequality\nmod(phi) is less than pi/6. It has been shown numerically and analytically,\nthat as the wave amplitude approaches its maximum, the maximum of phi can\nexceed pi/6 by about 1% near the crest. Numerical evidence suggested that this\noccurs in a small boundary layer near the crest where mod(phi(s)) rises rapidly\nfrom zero and oscillates about pi/6, the number of oscillations increasing as\nthe maximum amplitude is approached. McLeod derived, from Nekrasov\u0027s equation,\nan integral equation for phi in the boundary layer, whose width tends to zero\nas the maximum wave is approached. He also conjectured the asymptotic form of\nthe oscillations of mod(phi(s)) about pi/6 as s tends to infinity. We solve\nMcLeod\u0027s boundary layer equation numerically and verify the asymptotic form of\nphi.",
"arxiv_id": "physics/0011076",
"authors": [
"J. G. Byatt-Smith"
],
"categories": [
"physics.flu-dyn"
],
"title": "The Numerical Solution of Nekrasov\u0027s Equation in the Boundary Layer near the Crest, for Waves near the Maximum Height",
"url": "https://arxiv.org/abs/physics/0011076"
},
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