The Numerical Solution of Nekrasov's Equation in the Boundary Layer near the Crest, for Waves near the Maximum Height

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.