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

Nekrasov's integral equation describing water waves of permanent form, determines the angle φ (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, φ (s) satisfies the inequality |φ (s) | < π/6. It has been shown numerically and analytically, that as the wave amplitude approaches its maximum, the maximum of |φ (s) | can exceed π/6 by about 1% near the crest. Numerical evidence suggested that this occurs in a small boundary layer near the crest where |φ(s)| rises rapidly from |φ (0) | = 0 and oscillates about π/6, the number of oscillations increasing as the maximum amplitude is approached. McLeod derived, from Nekrasov's equation, the following integral equation φ (s) = 1 3π ∞ 0 sin φ (t) 1 + t 0 sin φ (τ) dτ log s − t s + t dt for φ (s) in the boundary layer, whose width tends to zero as the maximum wave is approached. He also conjectured that the asymptotic form of φ (s) as s → ∞ satisfies φ (s) = π 6 1 + As −1 sin (β log s + c) + o(s −1) , where A, β and c are constants with β ≈ 0 · 71 satisfying the equation √ 3β tanh 1 2 πβ = 1. We solve McLeod's boundary layer equation numerically and verify the above asymptotic form.