The free-surface boundary conditions  and  apply at the yet unknown free-surface elevation . They can be transformed into boundary conditions at a fixed elevation  by use of Taylor series expansions of the flow field around that elevation.
Without loss of generality the mean surface elevation – around which the Taylor series are developed – can be taken at . This assures the expansion is around an elevation in the proximity of the actual free-surface elevation. Convergence of the Taylor series for small-amplitude steady-wave motion was proved by .
The following notation is used: the Taylor series of some field  around  – and evaluated at  – is:
with subscript zero meaning evaluation at , e.g.: .
Applying the Taylor expansion to free-surface boundary condition  in terms of the potential Φ gives:
showing terms up to triple products of η, Φ and u, as required for the construction of the Stokes expansion up to third-order ((ka)). Here, ka is the wave steepness, with k a characteristic wavenumber and a a characteristic wave amplitude for the problem under study. The fields η, Φ and u are assumed to be (ka).
The dynamic free-surface boundary condition  can be evaluated in terms of quantities at  as:
The advantages of these Taylor-series expansions fully emerge in combination with a perturbation-series approach, for weakly non-linear waves .