text: Already in 1914, Wilton extended the Stokes expansion for deep-water surface gravity waves to tenth order, although introducing errors at the eight order. A fifth-order theory for finite depth was derived by De in 1955. For engineering use, the fifth-order formulations of Fenton are convenient, applicable to both Stokes first and second definition of phase speed (celerity). The demarcation between when fifth-order Stokes theory is preferable over fifth-order cnoidal wave theory is for Ursell parameters below about 40.
  Different choices for the frame of reference and expansion parameters are possible in Stokes-like approaches to the nonlinear wave problem. In 1880, Stokes himself inverted the dependent and independent variables, by taking the velocity potential and stream function as the independent variables, and the coordinates (x,z) as the dependent variables, with x and z being the horizontal and vertical coordinates respectively. This has the advantage that the free surface, in a frame of reference in which the wave is steady (i.e. moving with the phase velocity), corresponds with a line on which the stream function is a constant. Then the free surface location is known beforehand, and not an unknown part of the solution. The disadvantage is that the radius of convergence of the rephrased series expansion reduces.
  Another approach is by using the Lagrangian frame of reference, following the fluid parcels. The Lagrangian formulations show enhanced convergence, as compared to the formulations in both the Eulerian frame, and in the frame with the potential and streamfunction as independent variables.
  An exact solution for nonlinear pure capillary waves of permanent form, and for infinite fluid depth, was obtained by Crapper in 1957. Note that these capillary waves – being short waves forced by surface tension, if gravity effects are negligible – have sharp troughs and flat crests. This contrasts with nonlinear surface gravity waves, which have sharp crests and flat troughs.
  By use of computer models, the Stokes expansion for surface gravity waves has been continued, up to high (117th) order by . Schwartz has found that the amplitude a (or a) of the first-order fundamental reaches a maximum before the maximum wave height H is reached. Consequently, the wave steepness ka in terms of wave amplitude is not a monotone function up to the highest wave, and Schwartz utilizes instead kH as the expansion parameter. To estimate the highest wave in deep water, Schwartz has used Padé approximants and Domb–Sykes plots in order to improve the convergence of the Stokes expansion.
  Extended tables of Stokes waves on various depths, computed by a different method (but in accordance with the results by others), are provided in .
  Several exact relationships exist between integral properties – such as kinetic and potential energy, horizontal wave momentum and radiation stress – as found by . He shows, for deep-water waves, that many of these integral properties have a maximum before the maximum wave height is reached (in support of Schwartzs findings). , using a method similar to the one of Schwartz, computed and tabulated integral properties for a wide range of finite water depths (all reaching maxima below the highest wave height). Further, these integral properties play an important role in the conservation laws for water waves, through Noethers theorem.
  In 2005, Hammack, Henderson and Segur have provided the first experimental evidence for the existence of three-dimensional progressive waves of permanent form in deep water – that is bi-periodic and two-dimensional progressive wave patterns of permanent form. The existence of these three-dimensional steady deep-water waves has been revealed in 2002, from a bifurcation study of two-dimensional Stokes waves by Craig and Nicholls, using numerical methods.