page_title: Stokes wave
section_title: Stokes' approach to the non-linear wave problem
hierarchical_section_title: Stokes wave / Overview / Stokes' approach to the non-linear wave problem
caption_attribution_description: English: Comment by the author on Flickr: A boat on the Maas-Waalkanaal, Nijmegen, The Netherlands. I had to run to make this shot. A lot of levelpumping in PS.
context_page_description: In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth.
  This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for non-linear wave motion.
  Stokes' wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of coastal and offshore structures, in order to determine the wave kinematics. The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude. In such shallow water, a cnoidal wave theory often provides better periodic-wave approximations.
  While, in the strict sense, Stokes wave refers to progressive periodic waves of permanent form, the term is also used in connection with standing waves and even for random waves.
context_section_description: 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 non-linear 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 non-linear 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 non-linear waves is that the phase speed of non-linear 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.
caption: Waves in the Kelvin wake pattern generated by a ship on the Maas–Waalkanaal in The Netherlands. The transverse waves in this Kelvin wake pattern are nearly plane Stokes waves.
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