Analytic derivation of the wave profile and phase speed of sixth order Stokes waves in deep water

The wave profile for steady, surface gravity Stokes waves in deep water is investigated. The expression for wave profile for sixth order is derived analytically by substituting the Taylor series approximations for the variables in the free boundary conditions. The wave potential is represented by Fourier series and the coefficients in the series are written as perturbation expansions in terms of a parameter which increases with wave height. The expressions are numerically studied and analyzed graphically. The sixth order phase speed increases with increasing wave steepness δ. It is observed that the wave profile and the phase speed for the sixth order are higher than those of lower orders