A close one-term approximation to the highest Stokes wave on deep water

A remarkably accurate approximation to the profile of a limiting progressive gravity wave in water of infinite depth is given by the expression(0.1)y/L=Acosh(x/L),where L is the wavelength, x and y are horizontal and vertical coordinates and A is a constant given by(0.2)A=1/(3sinh12)=1.1080.This determines the wave steepness (H/L) as 0.14140 a proportional error of less than 0.3% (about 10 times closer than previous approximations) and the phase speed c/(gL)1/2 as 0.43511, which is accurate to within 0.2%. The entire surface profile is accurate to less than 0.7%. The corresponding particle velocities are found by a straightforward numerical integration. It is shown that this type of approximation cannot be made exact by the introduction of further parameters.