Numerical solution of Nekrasov's equation in the boundary layer near the crest for waves near the maximum height

摘要

Nekrasov's integral equation describing water waves of permanent form, determines the angle phi (s) 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 (s) satisfies the inequality phi (s) < i>/6. It has been shown numerically and analytically that, as the wave amplitude approaches its maximum, the maximum of phi (s) 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 phi (s) rises rapidly from phi (0) = 0 and oscillates about pi /6, the number of oscillations increasing as the maximum amplitude is approached. McLeod derived, from Nekrasov's equation, the following integral equation [GRAPHICS] for phi (s) in the boundary layer, whose width tends to zero as the maximum wave is approached, He also conjectured that the asymptotic form of phi (s) as s --> infinity satisfies phi (s) = pi /6 [1 + As-1 sin(beta log s + c) + o(s(-1))], where A, beta, and c are constants with beta approximate to 0 . 71 satisfying the equation root3 beta tanh 1/2 pi beta = 1. We solve McLeod's boundary layer equation numerically and verify the above asymptotic form. [References: 6]