Limiting conditions for cnoidal and Stokes waves

The second approximation to cnoidal waves is compared with the third approximation for Stokes waves of permanent form in water of finite depth. The comparison clearly indicates that cnoidal wave theory should not be applied to finite amplitude waves if their wavelengths are shorter than 5 times the depth. It is shown how the limiting heights of cnoidal waves are also related to the vanishing of the pressure gradient near the wave crest. The third approximation to Stokes waves in finite water depths is verified by the use of the classical small-perturbation expansion method which is best suited for small wave amplitudes. For finite amplitude waves the series expansion in terms of the infinitesimal-wave parameter is found to be most suitable for wavelengths shorter than 8 times the depth.