The ratio  of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:
In deep water, for large kh the ratio  has the asymptote
For long waves, i.e. small kh, the ratio  behaves as
or, in terms of the wave height  and wavelength :
with 
Here  is the Ursell parameter (or Stokes parameter). For long waves () of small height H, i.e. , second-order Stokes theory is applicable. Otherwise, for fairly long waves () of appreciable height H a cnoidal wave description is more appropriate. According to Hedges, fifth-order Stokes theory is applicable for , and otherwise fifth-order cnoidal wave theory is preferable.