Theory of the almost-highest wave: the inner solution

This paper investigates the flow near the summit of steep, progressive gravity wave when the crest is still rounded but the flow is approaching Stokes's corner flow. The natural length scale in the neighbourhood of the summit is seen to be l = q 2 /2 g , where g denotes gravity and q is the particle speed at the crest in a reference frame moving with the wave speed. We show that a class of self-similar smooth local flows exists which satisfy the free-surface condition and which tend to Stokes's corner flow when the radial distance r becomes large compared with l . The behaviour of the solution at large values of r / l is shown to depend on the roots of the transcendental equation \[ K tan h K = π/2{3}. \] The two real roots correspond to a damped oscillation of the free surface decaying like ( l/r ) ½ . The positive imaginary roots correspond to perturbations vanishing like higher negative powers of r . The complete flow is calculated by transforming the domain onto the interior of a circle in the complex plane and expanding the potential at the surface in a Fourier series. The computation is checked by an independent method, based on approximating the flow by a sequence of dipoles. The profile of the surface is found to intersect its asymptote at large values of r/l . This implies that the maximum slope slightly exceeds 30°. The computed value 30·37° is in close agreement with that obtained by extrapolating the maximum slopes of steep gravity waves, as calculated by previous authors. The vertical acceleration of a particle at the crest is 0·388 g . In the far field, however, the acceleration tends to the value ½ g corresponding to the Stokes corner flow.