text: According to Stokess third-order theory, the free surface elevation η, the velocity potential Φ, the phase speed (or celerity) c and the wave phase θ' are, for a progressive surface gravity wave on deep water – i.e. the fluid layer has infinite depth:
  where
  * x is the horizontal coordinate;
  * z is the vertical coordinate, with the positive z-direction upward – opposing to the direction of the Earths gravity – and z' = 0 corresponding with the mean surface elevation;
  * t is time;
  * a is the first-order wave amplitude;
  * k is the angular wavenumber,  with λ being the wavelength;
  * ω is the angular frequency,  where τ is the period, and
  * g is the strength of the Earth's gravity, a constant in this approximation.
  The expansion parameter ka is known as the wave steepness. The phase speed increases with increasing nonlinearity ka of the waves. The wave height H, being the difference between the surface elevation η at a crest and a trough, is:
  Note that the second- and third-order terms in the velocity potential Φ are zero. Only at fourth order do contributions deviating from first-order theory – i.e. Airy wave theory – appear. Up to third order the orbital velocity field u = ∇Φ consists of a circular motion of the velocity vector at each position (x,z). As a result, the surface elevation of deep-water waves is to a good approximation trochoidal, as already noted by .
  Stokes further observed, that although (in this Eulerian description) the third-order orbital velocity field consists of a circular motion at each point, the Lagrangian paths of fluid parcels are not closed circles.  This is due to the reduction of the velocity amplitude at increasing depth below the surface. This Lagrangian drift of the fluid parcels is known as the Stokes drift.