On an approximation to the limiting Stokes wave in deep water

A simple but highly accurate approximation to the form and speed of a limiting Stokes wave in deep water was recently given by Rainey and Longuet-Higgins [R.C.T. Rainey, M.S. Longuet-Higgins, A close one-term approximation to the highest Stokes wave on deep water, Ocean Eng. 33 (2006) 2012–2024]. An expression for the corresponding velocity field in the interior of the fluid was also suggested. Here this expression is examined numerically and it is shown that there is no singularity in the interior, but that there is a very weak singularity at the wave crest. The nature of the singularity is exhibited by use of a property of Stokes’s 120° corner-flow, namely that q2/grq2/gr is everywhere equal to 1, where q is the particle speed, g denotes gravity, and r is the radial distance from the wave crest.