A fundamental problem in finding solutions for surface gravity waves is that boundary conditions have to be applied at the position of the free surface, which is not known beforehand and is thus a part of the solution to be found.
Sir George Stokes solved this nonlinear wave problem in 1847 by expanding the relevant potential flow quantities in a Taylor series around the mean (or still) surface elevation. As a result, the boundary conditions can be expressed in terms of quantities at the mean (or still) surface elevation (which is fixed and known).
Next, a solution for the nonlinear wave problem (including the Taylor series expansion around the mean or still surface elevation) is sought by means of a perturbation series – known as the Stokes expansion – in terms of a small parameter, most often the wave steepness. The unknown terms in the expansion can be solved sequentially. Often, only a small number of terms is needed to provide a solution of sufficient accuracy for engineering purposes. Typical applications are in the design of coastal and offshore structures, and of ships.
Another property of nonlinear waves is that the phase speed of nonlinear waves depends on the wave height. In a perturbation-series approach, this easily gives rise to a spurious secular variation of the solution, in contradiction with the periodic behaviour of the waves. Stokes solved this problem by also expanding the dispersion relationship into a perturbation series, by a method now known as the Lindstedt–Poincaré method.