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

Abstract 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 = A cosh ( 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 / ( 3 sinh 1 2 ) = 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.