The almost-highest wave: a simple approximation

The crest of a steep, symmetric gravity wave is shown to be closely approximated by the expression \[ x+iy = {α +γ iχ}{(β + iχ)1/3}, \] where x, y are co-ordinates in the vertical plane, χ is the complex velocity potential and α, β, γ are certain constants. This expression is asymptotically correct both for small and for large values of |χ|; and the free surface agrees with the exact profile calculated by Longuet-Higgins & Fox (1977) everywhere to within 1·5 per cent. The pressure at the surface is constant to within 5 per cent.