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{{short description|Nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth}}
[[File:Stokes3 wave.svg|thumb|right|300px|Surface elevation of a deep water wave according to [[George Gabriel Stokes|Stokes]]' third-order theory. The wave steepness is: ''ka''&nbsp;=&nbsp;0.3, with ''k'' the [[wavenumber]] and ''a'' the wave [[amplitude]]. Typical for these [[surface gravity wave]]s are the sharp [[crest (physics)|crest]]s and flat [[trough (physics)|trough]]s.]]
[[File:JACOEL-Wave Tow Tank-Image16.png|thumb|right|300px|Model testing with periodic waves in the wave–tow tank of the Jere A. Chase Ocean Engineering Laboratory, [[University of New Hampshire]].]]
[[File:Undular bore Araguari River-Brazil-USGS-bws00026.jpg|thumb|right|300px|[[Undular bore]] and [[whelp (tidal bore)|whelp]]s near the mouth of [[Araguari River (Amapá)|Araguari River]] in north-eastern Brazil. View is oblique toward mouth from airplane at approximately {{convert|100|ft|m|abbr=on}} altitude.<ref>Figure 5 in: {{citation| author1=Susan Bartsch-Winkler |author2=David K. Lynch |publisher=[[United States Geological Survey|U. S. Geological Survey]] |type=Circular 1022 |year=1988 |title=Catalog of worldwide tidal bore occurrences and characteristics |journal=USGS Report |page=12 |bibcode=1988usgs.rept...12B |url=https://pubs.er.usgs.gov/#search:advance/page=1/page_size=100/advance=undefined/series_cd=CIR/report_number=1022:0}}</ref> The undulations following behind the bore front appear as slowly [[modulation|modulated]] Stokes waves.]]

In [[fluid dynamics]], a '''Stokes wave''' is a [[nonlinear system|nonlinear]] and [[periodic function|periodic]] [[surface wave]] on an [[inviscid flow|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 nonlinear wave motion.

'''Stokes's wave theory''' is of direct practical use for waves on intermediate and deep water. It is used in the design of [[coastal management|coastal]] and [[offshore construction|offshore structures]], in order to determine the wave [[kinematics]] ([[free surface]] elevation and [[flow velocity|flow velocities]]). The wave kinematics are subsequently needed in the [[design process]] to determine the [[wave load]]s on a structure.<ref>{{Citation | publisher = Elsevier | isbn = 9780080445687 | last = Chakrabarti | first = S.K. | title = Handbook of Offshore Engineering | year = 2005 | page = 235 }}</ref> For long waves (as compared to depth) – 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 a progressive periodic wave of permanent form, the term is also used in connection with [[standing wave]]s<ref>{{Citation | doi = 10.1017/S0022112073000364  | volume = 60 | issue = 3 | pages = 593–604 | last = Grant | first = M.A. | title = Standing Stokes waves of maximum height | journal = Journal of Fluid Mechanics | year = 1973 | bibcode = 1973JFM....60..593G | s2cid = 123179735 }}</ref> and even random waves.<ref>{{Citation | publisher = Elsevier | isbn = 9780080443126 | last = Ochi | first = Michel K. | title = Hurricane-generated seas | year = 2003 | page = 119 }}</ref><ref>{{Citation | doi = 10.1029/JC085iC03p01548 | volume = 85 | issue = C3 | pages = 1548–1552 | last = Tayfun | first = M.A. | title = Narrow-band nonlinear sea waves | journal = Journal of Geophysical Research | year = 1980 |bibcode = 1980JGR....85.1548T }}</ref>

==Examples==

The examples below describe Stokes waves under the action of gravity (without [[surface tension]] effects) in case of pure wave motion, so without an ambient mean current.

==={{anchor|Third-order Stokes wave on deep water}} Third-order Stokes wave on deep water===

[[File:Stokes3 wave definitions.svg|thumb|right|300px|Third-order Stokes wave in deep water under the action of gravity. The wave steepness is: ''ka''&nbsp;=&nbsp;0.3.]]
[[File:Stokes3 harmonics.svg|thumb|right|300px|The three [[harmonic]]s contributing to the surface elevation of a deep water wave, according to Stokes's third-order theory. The wave steepness is: ''ka''&nbsp;=&nbsp;0.3. For visibility, the vertical scale is distorted by a factor of four, compared to the horizontal scale.<br>''Description:''
* the dark blue line is the surface elevation of the 3rd-order Stokes wave,
* the black line is the [[fundamental frequency|fundamental]] wave component, with wavenumber ''k'' ([[wavelength]] λ, {{math|1=''k'' = 2''π'' / ''λ''}}),
* the light blue line is the harmonic at 2&nbsp;''k'' (wavelength {{1/2}}&nbsp;λ), and
* the red line is the harmonic at 3&nbsp;''k'' (wavelength {{1/3}}&nbsp;λ).]]

According to Stokes's third-order theory, the [[free surface]] elevation ''η'', the [[velocity potential]] Φ, the [[phase speed]] (or celerity) ''c'' and the wave [[phase (waves)|phase]] ''θ'' are, for a [[wave propagation|progressive]] [[surface gravity wave]] on deep water – i.e. the fluid layer has infinite depth:<ref name="Dingemans">{{citation | title=Water wave propagation over uneven bottoms | journal=NASA Sti/Recon Technical Report N | first=M.W. | last=Dingemans | year=1997 | series=Advanced Series on Ocean Engineering | volume=13 | isbn=978-981-02-0427-3 | oclc=36126836 | pages=171–184, §2.8 | bibcode=1985STIN...8525769K }}</ref>
<math display="block">\begin{align}
  \eta(x,t)   =& a \left\{ 
                          \left[1 - \tfrac{1}{16} (ka)^2 \right]  \cos  \theta 
                          + \tfrac12 (k a)\,   \cos 2\theta 
                          + \tfrac38 (k a)^2\, \cos 3\theta 
                    \right\}
               + \mathcal{O}\left( (ka)^4 \right),
  \\
  \Phi(x,z,t) =& a \sqrt{\frac{g}{k}}\, \text{e}^{kz}\, \sin \theta
               + \mathcal{O}\left( (ka)^4 \right),
  \\
  c           =& \frac{\omega}{k} = \left( 1 + \tfrac12 (ka)^2 \right)\, \sqrt{\frac{g}{k}}
               + \mathcal{O}\left( (ka)^4 \right), \text{ and}
  \\
  \theta(x,t) =& kx - \omega t,
\end{align}</math>
where
* ''x'' is the horizontal coordinate;
* ''z'' is the vertical coordinate, with the positive ''z''-direction upward – opposing to the direction of the [[Earth's gravity]] – and ''z''&nbsp;=&nbsp;0 corresponding with the [[average|mean]] surface elevation;
* ''t'' is time;
* ''a'' is the first-order wave [[amplitude]];
* ''k'' is the [[angular wavenumber]], {{math|1=''k'' = 2''π'' / ''λ''}} with ''λ'' being the [[wavelength]];
* ''ω'' is the [[angular frequency]], {{math|1=''ω'' = 2''π'' / ''τ''}} where ''τ'' is the [[period (physics)|period]], and
* ''g'' is the [[field strength|strength]] of the Earth's gravity, a [[constant (mathematics)|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 (physics)|crest]] and a [[trough (physics)|trough]], is:<ref>{{Citation | publisher = World Scientific | isbn = 9789812561428 | last = Svendsen | first = I.A. | title = Introduction to nearshore hydrodynamics | year = 2006 | page = 370 }}</ref>
<math display="block">H = 2a\, \left( 1 + \tfrac38\, k^2 a^2 \right).</math>

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.<ref name="Dingemans"/> Up to third order the orbital velocity [[vector field|field]] '''''u'''''&nbsp;=&nbsp;'''∇'''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 wave|trochoidal]], as already noted by {{harvtxt|Stokes|1847}}.<ref name="Toba">{{Citation | publisher = Springer | isbn = 9781402011719 | last = Toba | first = Yoshiaki | title = Ocean–atmosphere interactions | year = 2003 | pages = 27–31 }}</ref>

Stokes further observed, that although (in this [[Lagrangian and Eulerian specification of the flow field|Eulerian]] description) the third-order orbital velocity field consists of a circular motion at each point, the [[Lagrangian and Eulerian specification of the flow field|Lagrangian]] paths of [[fluid parcel]]s 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]].<ref name="Toba"/>

===Second-order Stokes wave on arbitrary depth===
[[File:Stokes3 amplitude double frequency.svg|thumb|right|300px|The ratio {{math|1=''{{mathcal|S}}'' = ''a''<sub>2</sub> / ''a''}} of the amplitude ''a''<sub>2</sub> of the [[harmonic]] with twice the wavenumber (2&nbsp;''k''), to the amplitude ''a'' of the [[fundamental frequency|fundamental]], according to Stokes's second-order theory for surface gravity waves. On the horizontal axis is the relative water depth ''h''&nbsp;/&nbsp;λ, with ''h'' the mean depth and λ the [[wavelength]], while the vertical axis is the Stokes parameter ''{{mathcal|S}}'' divided by the wave steepness ''ka'' (with {{math|1=''k'' = 2''π'' / ''λ''}}).<br>''Description:''
* the blue line is valid for arbitrary water depth, while
* the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and
* the dash-dot green line is the asymptotic limit for deep water waves.]]
The surface elevation ''η'' and the velocity potential Φ are, according to Stokes's second-order theory of surface gravity waves on a fluid layer of [[average|mean]] depth ''h'':<ref name="Dingemans"/><ref name="Whitham_13.13"/>
<math display="block">\begin{align}
    \eta(x,t) =& 
      a \left\{ 
            \cos\, \theta
            + ka\, \frac{3 - \sigma^2}{4\, \sigma^3}\, \cos\, 2\theta
          \right\}
      + \mathcal{O} \left( (ka)^3 \right),
    \\
      \Phi(x,z,t) =&
      a\, \frac{\omega}{k}\, \frac{1}{\sinh\, kh}
          \\ & \times
          \left\{\cosh\, k(z+h)
            \sin\, \theta
            + ka\, \frac{3 \cosh\, 2k(z+h)}{8\, \sinh^3\, kh}\, \sin\, 2\theta
          \right\}
    \\ &
      - (ka)^2\, \frac{1}{2\, \sinh\, 2kh}\, \frac{g\, t}{k}
      + \mathcal{O} \left( (ka)^3 \right),
    \\
      c =& \frac{\omega}{k} = \sqrt{\frac{g}{k}\, \sigma}
                            + \mathcal{O} \left( (ka)^2 \right),
    \\
      \sigma =& \tanh\, kh
      \quad \text{and} \quad 
      \theta(x,t) = k x - \omega t.
\end{align}</math>

Observe that for finite depth the velocity potential Φ contains a linear drift in time, independent of position (''x'' and ''z''). Both this temporal drift and the double-frequency term (containing sin&nbsp;2θ) in Φ vanish for deep-water waves.

==== {{anchor|Stokes and Ursell parameters}} Stokes and Ursell parameters ====

{{further|Ursell parameter}}
The ratio ''{{mathcal|S}}'' of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:<ref name="Dingemans"/>
<math display="block">\mathcal{S} = ka\, \frac{3 - \tanh^2\, kh}{4\, \tanh^3\, kh}.</math>

In deep water, for large ''kh'' the ratio ''{{mathcal|S}}'' has the [[asymptote]]
<math display="block">\lim_{kh \to \infty} \mathcal{S} = \frac{1}{2}\, ka.</math>

For long waves, i.e. small ''kh'', the ratio ''{{mathcal|S}}'' behaves as
<math display="block">\lim_{kh \to 0} \mathcal{S} = \frac{3}{4}\, \frac{ka}{(kh)^3},</math>
or, in terms of the wave height {{math|1=''H'' = 2''a''}} and wavelength {{math|1=''λ'' = 2''π'' / ''k''}}:
<math display="block">\lim_{kh \to 0} \mathcal{S} 
= \frac{3}{32\,\pi^2}\, \frac{H\, \lambda^2}{h^3}
= \frac{3}{32\,\pi^2}\, \mathcal{U},</math>
with <math display="block"> \mathcal{U} \equiv \frac{H\, \lambda^2}{h^3}.</math>

Here ''{{mathcal|U}}'' is the [[Ursell parameter]] (or Stokes parameter). For long waves ({{math|''λ'' ≫ ''h''}}) of small height ''H'', i.e. {{math|1=''{{mathcal|U}}'' ≪ 32π<sup>2</sup>/3 ≈ 100}}, second-order Stokes theory is applicable. Otherwise, for fairly long waves ({{math|1=''λ'' > 7''h''}}) of appreciable height ''H'' a [[cnoidal wave]] description is more appropriate.<ref name="Dingemans"/> According to Hedges, fifth-order Stokes theory is applicable for {{math|1=''{{mathcal|U}}'' < 40}}, and otherwise fifth-order [[cnoidal wave]] theory is preferable.<ref name="Hedges"/><ref name="Fenton"/>

===={{anchor|Third-order dispersion relation}} Third-order dispersion relation====
[[File:Stokes3 nonlin celerity.svg|thumb|right|300px|Nonlinear enhancement of the [[phase velocity|phase speed]] {{math|1=''c'' = ''ω'' / ''k''}} – according to Stokes's third-order theory for [[surface gravity wave]]s, and using Stokes's first definition of celerity –  as compared to the linear-theory phase speed ''c''<sub>0</sub>. On the horizontal axis is the relative water depth ''h''&nbsp;/&nbsp;λ, with ''h'' the mean depth and λ the [[wavelength]], while the vertical axis is the nonlinear phase-speed enhancement {{math|(''c'' − ''c''<sub>0</sub>) / ''c''<sub>0</sub>}} divided by the wave steepness ''ka'' squared.<br>''Description:''
* the solid blue line is valid for arbitrary water depth,
* the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and
* the dash-dot green line is the asymptotic limit for deep water waves.]]
For Stokes waves under the action of gravity, the third-order [[dispersion (water waves)|dispersion relation]] is – according to [[#Stokes's first definition of wave celerity|Stokes's first definition of celerity]]:<ref name="Whitham_13.13">{{harvtxt|Whitham|1974|pages=471–476, §13.13}}</ref>

<math display="block">\begin{align}
  \omega^2 &= \left( gk\, \tanh\, kh \right)\;
    \left\{
      1 
      + \frac{9 - 10\, \sigma^2 + 9\, \sigma^4}{8\, \sigma^4}\, ( ka )^2
    \right\}+ \mathcal{O}\left( (ka)^4 \right),
    \\
    &
    \qquad \text{with}
  \\
  \sigma &= \tanh\, kh.
\end{align}</math>

This third-order dispersion relation is a direct consequence of avoiding [[secular term]]s, when inserting the second-order Stokes solution into the third-order equations (of the perturbation series for the periodic wave problem).

In deep water (short wavelength compared to the depth):
<math display="block">\lim_{kh \to \infty} \omega^2 = gk\, \left\{ 1 + \left( ka \right)^2 \right\} + \mathcal{O}\left( (ka)^4 \right),</math>
and in shallow water (long wavelengths compared to the depth):
<math display="block">
  \lim_{kh \to 0} \omega^2 = 
    k^2\, gh\, \left\{ 1 + \frac98\, \frac{\left( ka \right)^2}{\left( kh \right)^4} \right\}
    + \mathcal{O}\left( (ka)^4 \right).
</math>

As [[#Stokes and Ursell parameters|shown above]], the long-wave Stokes expansion for the dispersion relation will only be valid for small enough values of the Ursell parameter: {{math|1=''{{mathcal|U}}'' ≪ 100}}.
{{Clear}}

==Overview==

===Stokes's approach to the nonlinear wave problem===
[[File:Wavemaker.jpg|thumb|right|300px|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.]]
[[File:Wea00810.jpg|thumb|right|300px|[[NOAA]] ship ''Delaware II'' in bad weather on [[Georges Bank]]. While these ocean waves are [[random]], and not Stokes waves (in the strict sense), they indicate the typical sharp [[crest (physics)|crest]]s and flat [[trough (physics)|trough]]s as found in nonlinear surface gravity waves.]]

A fundamental problem in finding solutions for surface gravity waves is that [[boundary condition]]s 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.<ref>{{harvtxt|Stokes|1847}}</ref> 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.<ref name="Dingemans"/><ref name="Toba"/> Often, only a small number of terms is needed to provide a solution of sufficient accuracy for engineering purposes.<ref name="Fenton">{{harvtxt|Fenton|1990}}</ref> Typical applications are in the design of [[coastal management|coastal]] and [[offshore construction|offshore structures]], and of [[naval architecture|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 (water waves)|dispersion relationship]] into a perturbation series, by a method now known as the [[Lindstedt–Poincaré method]].<ref name="Dingemans"/>

===Applicability===
[[File:Water wave theories.svg|thumb|right|300px|Validity of several theories for periodic water waves, according to Le Méhauté (1976).<ref name="Le Méhauté">{{citation | first=B. | last=Le Méhauté | year=1976 | title=An introduction to hydrodynamics and water waves | publisher=Springer | isbn=978-0387072326 }}</ref> The light-blue area gives the range of validity of [[cnoidal wave]] theory; light-yellow for [[Airy wave theory]]; and the dashed blue lines demarcate between the required order in Stokes's wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order [[stream function|stream-function]] theory, for high waves (''H''&nbsp;>&nbsp;{{1/4}}&nbsp;H<sub>breaking</sub>).]]

''Stokes's wave theory'', when using a low order of the perturbation expansion (e.g. up to second, third or fifth order), is valid for nonlinear waves on intermediate and deep water, that is for [[wavelength]]s (''λ'') not large as compared with the mean depth (''h''). In [[waves and shallow water|shallow water]], the low-order Stokes expansion breaks down (gives unrealistic results) for appreciable wave amplitude (as compared to the depth). Then, [[Boussinesq approximation (water waves)|Boussinesq approximation]]s are more appropriate. Further approximations on Boussinesq-type (multi-directional) wave equations lead – for one-way wave propagation – to the [[Korteweg–de Vries equation]] or the [[Benjamin–Bona–Mahony equation]]. Like (near) exact Stokes-wave solutions,<ref>{{Citation | doi = 10.1098/rspa.1974.0166 | volume = 340 | issue = 1623 | pages = 471–493 | last1 = Longuet-Higgins | first1 = M.S. | author1-link=Michael Longuet-Higgins | last2 = Fenton | first2 = J.D. | title = On the mass, momentum, energy and circulation of a solitary wave. II | journal = [[Proceedings of the Royal Society A]] | year = 1974 |bibcode = 1974RSPSA.340..471L | s2cid = 124253945 }}</ref> these two equations have [[solitary wave (water waves)|solitary wave]] ([[soliton]]) solutions, besides periodic-wave solutions known as [[cnoidal wave]]s.<ref name="Fenton"/>

===Modern extensions===

Already in 1914, Wilton extended the Stokes expansion for deep-water surface gravity waves to tenth order, although introducing errors at the eight order.<ref>{{harvtxt|Wilton|1914}}</ref> A fifth-order theory for finite depth was derived by De in 1955.<ref>{{harvtxt|De|1955}}</ref> For engineering use, the fifth-order formulations of Fenton are convenient, applicable to both Stokes [[#Stokes first definition|first]] and [[#Stokes second definition|second]] definition of phase speed (celerity).<ref>{{harvtxt|Fenton|1985}}, also (including corrections) in {{harvtxt|Fenton|1990}}</ref> The demarcation between when fifth-order Stokes theory is preferable over fifth-order [[cnoidal wave]] theory is for [[Ursell parameter]]s below about 40.<ref name="Hedges">{{citation | doi = 10.1680/iwtme.1995.27656 | volume = 112 | issue = 2 | pages = 111–114 | first = T.S. | last = Hedges | title = Regions of validity of analytical wave theories | journal = Proceedings of the Institution of Civil Engineers - Water Maritime and Energy | year = 1995 }}</ref><ref name="Fenton"/>

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.<ref name="Stokes_1880"/> 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.<ref name="Drennan">{{Citation | doi = 10.1007/BF00946637 | volume = 43 | issue = 2 | pages = 367–384 | first1 = W.M. | last1 = Drennan | first2 = W.H. | last2 = Hui | first3 = G. | last3 = Tenti | title = Accurate calculations of Stokes water waves of large amplitude | journal = Zeitschrift für Angewandte Mathematik und Physik | year = 1992 |bibcode = 1992ZaMP...43..367D | s2cid = 121134205 }}</ref>

Another approach is by using the [[Lagrangian and Eulerian specification of the flow field|Lagrangian frame of reference]], following the [[fluid parcel]]s. The Lagrangian formulations show enhanced convergence, as compared to the formulations in both the [[Lagrangian and Eulerian specification of the flow field|Eulerian frame]], and in the frame with the potential and streamfunction as independent variables.<ref name="Buldakov_et_al_2006">{{Citation | title=New asymptotic description of nonlinear water waves in Lagrangian coordinates |first1=E.V. |last1=Buldakov |first2=P.H. |last2=Taylor |first3=R. |last3=Eatock Taylor |year=2006 |journal=Journal of Fluid Mechanics |volume=562 |pages=431–444 |doi=10.1017/S0022112006001443 |doi-broken-date=2 December 2024 |bibcode = 2006JFM...562..431B |citeseerx=10.1.1.492.5377 |s2cid=29506471 }}</ref><ref name="Clamond_2007">{{Citation | doi=10.1017/S0022112007007811 |volume=589 |pages=433–454 | last=Clamond |first=D. |title=On the Lagrangian description of steady surface gravity waves |journal=Journal of Fluid Mechanics |year=2007 |bibcode = 2007JFM...589..433C |citeseerx=10.1.1.526.5643 |s2cid=123255841 }}</ref>

An exact solution for nonlinear pure [[capillary wave]]s 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.<ref>{{harvtxt|Crapper|1957}}</ref>

[[File:Stokes wave energy deep water.svg|thumb|right|300px|Several integral properties of Stokes waves on deep water as a function of wave steepness.<ref>This figure is a remake and adaptation of Figure 1 in {{harvtxt|Schwartz|Fenton|1982}}</ref> The wave steepness is defined as the ratio of [[wave height]] ''H'' to the [[wavelength]] λ. The wave properties are made [[dimensionless quantity|dimensionless]] using the [[wavenumber]] {{math|1=''k'' = 2π / ''λ''}}, [[Earth's gravity|gravitational acceleration]] ''g'' and the fluid [[density]] ''ρ''.<br>
Shown are the [[kinetic energy]] density ''T'', the [[potential energy]] density ''V'', the total energy density {{math|1=''E'' = ''T'' + ''V''}}, the horizontal wave [[momentum]] density ''I'', and the relative enhancement of the [[phase speed]] ''c''. Wave energy densities ''T'', ''V'' and ''E'' are integrated over depth and averaged over one wavelength, so they are energies per unit of horizontal area; the wave momentum density ''I'' is similar. The dashed black lines show 1/16&nbsp;(''kH'')<sup>2</sup> and 1/8&nbsp;(''kH'')<sup>2</sup>, being the values of the integral properties as derived from (linear) [[Airy wave theory]]. The maximum wave height occurs for a wave steepness {{math|''H'' / ''λ'' ≈ 0.1412}}, above which no periodic surface gravity waves exist.<ref name="Schwartz & Fenton">{{harvtxt|Schwartz|Fenton|1982}}</ref><br>
Note that the shown wave properties have a maximum for a wave height less than the maximum wave height (see e.g. {{harvnb|Longuet-Higgins|1975}}; {{harvnb|Cokelet|1977}}).]]

By use of computer models, the Stokes expansion for surface gravity waves has been continued, up to high (117th) order by {{harvtxt|Schwartz|1974}}. Schwartz has found that the amplitude ''a'' (or ''a''<sub>1</sub>) of the first-order [[fundamental frequency|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 approximant]]s and [[Domb–Sykes plot]]s 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 {{harvs|txt|last=Williams|year=1981|year2=1985}}.

Several exact relationships exist between integral properties – such as [[kinetic energy|kinetic]] and [[potential energy]], horizontal wave [[momentum]] and [[radiation stress]] – as found by {{harvtxt|Longuet-Higgins|1975}}. 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 Schwartz's findings). {{harvtxt|Cokelet|1978}}, 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 law (physics)|conservation law]]s for water waves, through [[Noether's theorem]].<ref>{{citation | title = Hamiltonian structure, symmetries and conservation laws for water waves | first1 = T.B. | last1 = Benjamin | author1-link = T. Brooke Benjamin | first2 = P.J. | last2 = Olver | author2-link = Peter J. Olver | year = 1982 | journal = Journal of Fluid Mechanics | volume = 125 | pages = 137–185 | doi = 10.1017/S0022112082003292 | doi-broken-date = 2 December 2024 |bibcode = 1982JFM...125..137B | s2cid = 11744174 }}</ref>

In 2005, Hammack, [[Diane Henderson|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.<ref>{{Citation| volume = 532 | pages = 1–52 | last1 = Hammack | first1 = J.L.| first2 = D.M. | last2 = Henderson |author2-link = Diane Henderson | first3 = H. | last3 = Segur | title = Progressive waves with persistent two-dimensional surface patterns in deep water | journal = Journal of Fluid Mechanics | year = 2005 | doi = 10.1017/S0022112005003733 | doi-broken-date = 2 December 2024 |bibcode = 2005JFM...532....1H | s2cid = 53416586 }}</ref> 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.<ref>{{Citation| doi = 10.1016/S0997-7546(02)01207-4| volume = 21| issue = 6| pages = 615–641| last1 = Craig| first1 = W.| first2 = D.P. | last2 = Nicholls| title = Traveling gravity water waves in two and three dimensions| journal = European Journal of Mechanics B | year = 2002|bibcode = 2002EuJMB..21..615C }}</ref>

===Convergence and instability===

====Convergence====
Convergence of the Stokes expansion was first proved by {{harvtxt|Levi-Civita|1925}} for the case of small-amplitude waves – on the free surface of a fluid of infinite depth. This was extended shortly afterwards by {{harvtxt|Struik|1926}} for the case of finite depth and small-amplitude waves.<ref>{{Citation
| publisher = Birkhäuser
| isbn = 9780817643232
| last = Debnath
| first = L.
| title = Nonlinear partial differential equations for scientists and engineers
| year = 2005
| pages=181 & 418–419
}}</ref>

Near the end of the 20th century, it was shown that for finite-amplitude waves the convergence of the Stokes expansion depends strongly on the formulation of the periodic wave problem. For instance, an inverse formulation of the periodic wave problem as used by Stokes – with the spatial coordinates as a function of [[velocity potential]] and [[stream function]] – does not converge for high-amplitude waves. While other formulations converge much more rapidly, e.g. in the [[Lagrangian and Eulerian specification of the flow field|Eulerian frame of reference]] (with the velocity potential or stream function as a function of the spatial coordinates).<ref name="Drennan" />

====Highest wave====
[[File:Stokes wave max height.svg|thumb|right|300px|Stokes waves of maximum [[wave height]] on deep water, under the action of gravity.]]

The maximum wave steepness, for periodic and propagating deep-water waves, is {{math|1=''H'' / ''λ'' = 0.1410633 ± 4 · 10<sup>−7</sup>}},<ref>{{Citation
| doi = 10.1111/sapm.12128
| last1 = Dyachenko
| first1=S.A.
| last2 = Lushnikov
| first2 = P.M.
| last3 = Korotkevich
| first3 =A.O.
| title = Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation
| journal = Studies in Applied Mathematics
| volume = 137
| number = 4
| pages = 419–472
| year = 2016
| url = https://onlinelibrary.wiley.com/doi/abs/10.1111/sapm.12128
| arxiv = 1507.02784
| s2cid = 52104285
}}</ref> so the wave height is about one-seventh ({{sfrac|1|7}}) of the wavelength λ.<ref name="Schwartz & Fenton"/> And surface gravity waves of this maximum height have a sharp [[crest (physics)|wave crest]] – with an angle of 120° (in the fluid domain)  – also for finite depth, as shown by Stokes in 1880.<ref name="Stokes_1880">{{harvtxt|Stokes|1880b}}</ref>

An accurate estimate of the highest wave steepness in deep water ({{math|1=''H'' / ''λ'' ≈ 0.142}}) was already made in 1893, by [[John Henry Michell]], using a numerical method.<ref>{{Citation
| doi = 10.1080/14786449308620499
| volume = 36
| issue = 222
| pages = 430–437
| last = Michell
| first = J.H.
| author-link = John Henry Michell
| title = The highest waves in water
| journal = Philosophical Magazine |series=Series 5
| year = 1893
| url = https://zenodo.org/record/1431203
}}</ref> A more detailed study of the behaviour of the highest wave near the sharp-cornered crest has been published by Malcolm A. Grant, in 1973.<ref>{{Citation
| doi = 10.1017/S0022112073001552
| volume = 59
| issue = 2
| pages = 257–262
| last = Grant
| first = Malcolm A.
| title = The singularity at the crest of a finite amplitude progressive Stokes wave
| journal = Journal of Fluid Mechanics
| year = 1973
|bibcode = 1973JFM....59..257G | s2cid = 119356016
}}</ref> The existence of the highest wave on deep water with a sharp-angled crest of 120° was proved by [[John Toland (mathematician)|John Toland]] in 1978.<ref>{{Citation
| doi = 10.1098/rspa.1978.0178
| volume=363
| issue=1715
| pages=469–485
| last=Toland
| first=J.F.
| author-link=John Toland (mathematician)| title=On the existence of a wave of greatest height and Stokes's conjecture
| journal=[[Proceedings of the Royal Society A]]
| year=1978
|bibcode = 1978RSPSA.363..469T | s2cid=120444295
}}</ref> The convexity of η(x) between the successive maxima with a sharp-angled crest of 120° was independently proven by C.J. Amick et al. and Pavel I. Plotnikov in 1982
.<ref>{{Citation
| volume=57
| pages=41–76
| last=Plotnikov 
| first=P.I.
| author-link=Pavel I. Plotnikov (mathematician)
| title=A proof of the Stokes conjecture in the theory of surface waves.
| journal=Dinamika Splosh. Sredy [in Russian]
| year=1982
}}
:Reprinted in: {{citation
| doi = 10.1111/1467-9590.01408
| first= P.I.
| last= Plotnikov 
| year= 2002
| volume=3
| issue= 2
| pages= 217–244
| title=A proof of the Stokes conjecture in the theory of surface waves.
| journal=[[Studies in Applied Mathematics]] }}
</ref><ref>{{Citation
| doi = 10.1007/BF02392728
| volume=148 
| pages=193–214
| last1=Amick
| first1=C.J.
| last2=Fraenkel
| first2=L.E.
| last3=Toland 
| first3=J.F.
| title=On the Stokes conjecture for the wave of extreme form
| journal=[[Acta Mathematica]]
| year=1982| doi-access=free
}}</ref>

The highest Stokes wave – under the action of gravity – can be approximated with the following simple and accurate representation of the [[free surface]] elevation ''η''(''x'',''t''):<ref name="Rainey & Longuet-Higgins">{{citation |first1=R.C.T. |last1=Rainey |first2=M.S. |last2=Longuet-Higgins |author2-link=Michael S. Longuet-Higgins |year=2006 |title=A close one-term approximation to the highest Stokes wave on deep water |journal=Ocean Engineering |volume=33 |issue=14–15 |pages=2012–2024 |doi=10.1016/j.oceaneng.2005.09.014 |bibcode=2006OcEng..33.2012R }}</ref>
<math display="block"> \frac{\eta}{\lambda} = A\, \left[ \cosh\, \left( \frac{x-ct}{\lambda} \right) - 1 \right], </math>
with <math display="block"> A = \frac{1}{\sqrt{3}\, \sinh \left( \frac12 \right)} \approx 1.108, </math>
for <math display="block">-\tfrac 1 2\,\lambda \le (x-ct) \le \tfrac12\, \lambda,</math>

and shifted horizontally over an [[integer]] number of wavelengths to represent the other waves in the regular wave train. This approximation is accurate to within 0.7% everywhere, as compared with the "exact" solution for the highest wave.<ref name="Rainey & Longuet-Higgins"/>

Another accurate approximation – however less accurate than the previous one – of the fluid motion on the surface of the steepest wave is by analogy with the swing of a [[pendulum]] in a [[grandfather clock]].<ref>{{Citation
| doi = 10.1063/1.862789
| volume = 22
| issue = 9
| pages = 1828–1829
| last = Longuet-Higgins
| first = M.S.
| author-link = Michael S. Longuet-Higgins
| title = Why is a water wave like a grandfather clock?
| journal = Physics of Fluids
| year = 1979
|bibcode = 1979PhFl...22.1828L | doi-access = free
}}</ref>

Large library of Stokes waves computed with high precision for the case of infinite depth, represented with high accuracy (at least 27 digits after decimal point) as a [[Padé approximant]] can be found at StokesWave.org<ref>{{Citation
| url = http://stokeswave.org/
| last1 = Dyachenko
| first1 = S.A.
| last2 = Korotkevich
| first2 = A.O.
| last3 = Lushnikov
| first3 = P.M.
| last4 = Semenova
| first4 = A.A.
| last5 = Silantyev
| first5 = D.A.
| title = StokesWave.org
| year = 2013–2022}}</ref>

====Instability====
In deeper water, Stokes waves are unstable.<ref name="Craik_WIFF">For a review of the instability of Stokes waves see e.g.:<br> {{Citation
| title = Wave interactions and fluid flows
| first = A.D.D.
| last = Craik
| publisher = Cambridge University Press
| year = 1988
| isbn = 978-0-521-36829-2
| pages = 199–219 }}</ref>  This was shown by [[T. Brooke Benjamin]] and Jim E. Feir in 1967.<ref>{{citation
| doi = 10.1017/S002211206700045X
| volume = 27
| issue = 3
| pages = 417–430
| last1 = Benjamin
| first1 = T. Brooke
| author1-link = T. Brooke Benjamin
| first2 = J.E.
| last2 = Feir
| title = The disintegration of wave trains on deep water. Part 1. Theory
| journal = Journal of Fluid Mechanics
| year = 1967
|bibcode = 1967JFM....27..417B | s2cid = 121996479
}}</ref><ref>{{Cite journal
 |doi          = 10.1016/j.physd.2008.12.002
 |volume       = 238
 |issue        = 5
 |pages        = 540–548
 |last1        = Zakharov
 |first1       = V.E.
 |author1-link = Vladimir E. Zakharov
 |first2       = L.A.
 |last2        = Ostrovsky
 |title        = Modulation instability: The beginning
 |journal      = Physica D
 |year         = 2009
 |url          = http://people.math.umass.edu/~kevrekid/math697wa/sdarticle_ZO.pdf
 |bibcode      = 2009PhyD..238..540Z
}}</ref> The [[Benjamin–Feir instability]] is a side-band or modulational instability, with the side-band modulations propagating in the same direction as the [[carrier wave]]; waves become unstable on deeper water for a relative depth {{math|''kh'' > 1.363}} (with ''k'' the [[wavenumber]] and ''h'' the mean water depth).<ref>{{Citation
| doi = 10.1098/rspa.1967.0123
| volume = 299
| issue = 1456
| pages = 59–76
| last = Benjamin
| first = T.B.
| author-link = T. Brooke Benjamin
| title = Instability of periodic wavetrains in nonlinear dispersive systems
| journal = [[Proceedings of the Royal Society A]]
| year = 1967
|bibcode = 1967RSPSA.299...59B | s2cid = 121661209
}} Concluded with a discussion by [[Klaus Hasselmann]].</ref> The Benjamin–Feir instability can be described with the [[nonlinear Schrödinger equation]], by inserting a Stokes wave with side bands.<ref name="Craik_WIFF"/> Subsequently, with a more refined analysis, it has been shown – theoretically and experimentally – that the Stokes wave and its side bands exhibit [[Fermi–Pasta–Ulam–Tsingou recurrence]]: a cyclic alternation between modulation and demodulation.<ref>{{Citation
| doi = 10.1017/S0022112077001037
| volume = 83
| issue = 1
| pages = 49–74
| last1 = Lake
| first1 = B.M.
| first2 = H.C.
| last2 = Yuen
| first3 = H.
| last3 = Rungaldier
| first4 = W.E.
| last4 = Ferguson
| title = Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train
| journal = Journal of Fluid Mechanics
| year = 1977
|bibcode = 1977JFM....83...49L | s2cid = 123014293
}}</ref>

In 1978 [[Michael Longuet-Higgins|Longuet-Higgins]], by means of numerical modelling of fully non-linear waves and modulations (propagating in the carrier wave direction), presented a detailed analysis of the region of instability in deep water: both for superharmonics (for perturbations at the spatial scales smaller than the wavelength <math>\lambda</math>) <ref>{{Citation
| doi = 10.1098/rspa.1978.0080
| volume = 360
| issue = 1703
| pages = 471–488
| last = Longuet-Higgins
| first = M.S.
| author-link = Michael Longuet-Higgins
| title = The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics
| journal = [[Proceedings of the Royal Society A]]
| year = 1978
|bibcode = 1978RSPSA.360..471L | s2cid = 202575377
}}</ref> and subharmonics (for perturbations at the spatial scales larger than <math>\lambda</math>).<ref>{{Citation
| doi = 10.1098/rspa.1978.0080
| volume = 360
| issue = 1703
| pages = 489–505
| last = Longuet-Higgins
| first = M.S.
| author-link = Michael Longuet-Higgins
| title = The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics
| journal = [[Proceedings of the Royal Society A]]
| year = 1978
|bibcode = 1978RSPSA.360..471L | s2cid = 202575377
}}</ref>
With increase of Stokes wave's amplitude, new modes of superharmonic instability appear. Appearance of a new branch of instability happens when the energy of the wave passes extremum. Detailed analysis of the mechanism of appearance of the new branches of instability has shown that their behavior follows closely a simple law, which allows to find with a good accuracy instability growth rates for all known and predicted branches.<ref>{{Citation
| doi = 10.1111/sapm.12535
| last1 = Korotkevich
| first1 = A.O.
| last2 = Lushnikov
| first2 = P.M.
| last3 = Semenova
| first3 = A.
| last4 = Dyachenko
| first4 = S.A.
| title = Superharmonic instability of stokes waves
| journal = Studies in Applied Mathematics
| year = 2022
| volume = 150
| pages = 119–134
| url = https://onlinelibrary.wiley.com/doi/abs/10.1111/sapm.12535
| arxiv = 2206.00725| s2cid = 249282423
}}</ref>
In Longuet-Higgins studies of two-dimensional wave motion, as well as the subsequent studies of three-dimensional modulations by McLean et al., new types of instabilities were found – these are associated with [[resonance|resonant]] wave interactions between five (or more) wave components.<ref>{{citation
| doi = 10.1103/PhysRevLett.46.817
| volume = 46
| issue = 13
| pages = 817–820
| first1 = J.W. | last1 = McLean
| first2 = Y.C. | last2 = Ma
| first3 = D.U. | last3 = Martin
| first4 = P.G. | last4 = Saffman |author4-link = Philip Saffman
| first5 = H.C. | last5 = Yuen
| title = Three-dimensional instability of finite-amplitude water waves
| journal = Physical Review Letters
| year = 1981
|bibcode = 1981PhRvL..46..817M | url = http://authors.library.caltech.edu/10155/1/MCLprl81.pdf}}</ref><ref>{{Citation
| doi = 10.1017/S0022112082000172
| volume = 114
| pages = 315–330
| last = McLean
| first = J.W.
| title = Instabilities of finite-amplitude water waves
| journal = Journal of Fluid Mechanics
| year = 1982
| doi-broken-date = 2 December 2024
|bibcode = 1982JFM...114..315M | s2cid = 122511104
}}</ref><ref name="Dias_Kharif">{{harvtxt|Dias|Kharif|1999}}</ref>

=={{anchor|Stokes expansion}} Stokes expansion==

===Governing equations for a potential flow===

In many instances, the oscillatory flow in the fluid interior of surface waves can be described accurately using [[potential flow]] theory, apart from [[boundary layer]]s near the free surface and bottom (where [[vorticity]] is important, due to [[viscosity|viscous effects]], see [[Stokes boundary layer]]).<ref name="Phillips"/> Then, the [[flow velocity]] '''u''' can be described as the [[gradient]] of a [[velocity potential]] <math>\Phi</math>:
{{NumBlk||<math display="block">\mathbf{u} = \boldsymbol{\nabla} \Phi.</math>|{{EquationRef|A}}}}

Consequently, assuming [[incompressible flow]], the velocity field '''u''' is [[divergence-free]] and the velocity potential <math>\Phi</math> satisfies [[Laplace's equation]]<ref name="Phillips"/>
{{NumBlk||<math display="block">\nabla^2 \Phi = 0</math>|{{EquationRef|B}}}}
in the fluid interior.

The fluid region is described using three-dimensional [[Cartesian coordinate]]s (''x'',''y'',''z''), with ''x'' and ''y'' the horizontal coordinates, and ''z'' the vertical coordinate – with the positive ''z''-direction opposing the direction of the [[gravity of Earth|gravitational acceleration]]. Time is denoted with ''t''. The free surface is located at {{math|1=''z'' = ''η''(''x'',''y'',''t'')}}, and the bottom of the fluid region is at {{math|1=''z'' = −''h''(''x'',''y'')}}.

The free-surface [[boundary condition]]s for [[surface gravity wave]]s – using a [[potential flow]] description – consist of a ''kinematic'' and a ''dynamic'' boundary condition.<ref name="Mei_4_6">{{harvtxt|Mei|1989|pages=4–6}}</ref>
The ''kinematic'' boundary condition ensures that the [[normal component]] of the fluid's [[flow velocity]], <math>\mathbf u = [\partial\Phi/\partial x ~~~ \partial\Phi/\partial y ~~~ \partial\Phi/\partial z]^{\mathrm T}</math> in matrix notation, at the free surface equals the normal velocity component of the free-surface motion {{math|1=''z'' = ''η''(''x'',''y'',''t'')}}:
{{NumBlk||<math display="block">
  \frac{\partial \eta}{\partial t}
+ \frac{\partial \Phi}{\partial x}\, \frac{\partial \eta}{\partial x}
+ \frac{\partial \Phi}{\partial y}\, \frac{\partial \eta}{\partial y}
= \frac{\partial \Phi}{\partial z}
  \qquad \text{ at } z=\eta(x,y,t).
</math>|{{EquationRef|C}}
}}

The ''dynamic'' boundary condition states that, without [[surface tension]] effects, the atmospheric pressure just above the free surface equals the fluid [[pressure]] just below the surface. For an unsteady potential flow this means that the [[Bernoulli's principle|Bernoulli equation]] is to be applied at the free surface. In case of a constant atmospheric pressure, the dynamic boundary condition becomes:
{{NumBlk||<math display="block"> \frac{\partial \Phi}{\partial t} + \tfrac 1 2\, \left| \mathbf{u} \right|^2 + g\, \eta = 0 \qquad \text{ at } z=\eta(x,y,t),</math>|{{EquationRef|D}}}}
where the constant atmospheric pressure has been taken equal to zero, [[without loss of generality]].

Both boundary conditions contain the potential <math>\Phi</math> as well as the surface elevation ''η''. A (dynamic) boundary condition in terms of only the potential <math>\Phi</math> can be constructed by taking the [[material derivative]] of the dynamic boundary condition, and using the kinematic boundary condition:<ref name="Phillips">{{Citation
| title = Dynamics of the upper ocean
| edition = 2nd
| publisher = Cambridge University Press
| first = O.M.
| last = Phillips
| author-link=Owen Martin Phillips 
| year = 1980
| isbn = 978-0-521-29801-8
| pages = 33–37
}}</ref><ref name="Mei_4_6"/><ref>{{Citation
| doi = 10.1017/S0022112062000233
| volume = 12
| issue = 3
| pages = 321–332
| last = Longuet-Higgins
| first = M.S.
| author-link = Michael Longuet-Higgins
| title = Resonant interactions between two trains of gravity waves
| journal = Journal of Fluid Mechanics
| year = 1962
|bibcode = 1962JFM....12..321L | s2cid = 122810532
}}</ref>
<math display="block">
  {\color{Gray}{
  \Bigl( \frac{\partial}{\partial t}
  + \mathbf{u} \cdot \boldsymbol{\nabla} \Bigr)\,
    \left( \frac{\partial\Phi}{\partial t} + \tfrac12\, |\mathbf{u}|^2 + g\, \eta \right) 
    = 0}}
</math>
<math display="block">
  {\color{Gray}{
  \Rightarrow \quad
  \frac{\partial^2 \Phi}{\partial t^2}
  + g\, \frac{\partial \Phi}{\partial z}
  + \mathbf{u} \cdot \boldsymbol{\nabla} \frac{\partial \Phi}{\partial t}
  + \tfrac12\, \frac{\partial}{\partial t} \left( |\mathbf{u}|^2 \right)
  + \tfrac12\, \mathbf{u} \cdot \boldsymbol{\nabla} \left( |\mathbf{u}|^2 \right)
  = 0}}
</math>
{{NumBlk||<math display="block">
  {\color{Gray}{
  \Rightarrow \quad
  }}
  \frac{\partial^2 \Phi}{\partial t^2} 
  + g\, \frac{\partial \Phi}{\partial z}
  + \frac{\partial}{\partial t} \left( |\mathbf{u}|^2 \right)
  + \tfrac12\, \mathbf{u} \cdot \boldsymbol{\nabla} \left( |\mathbf{u}|^2 \right)
  = 0
  \qquad \text{ at } z=\eta(x,y,t).
</math>|{{EquationRef|E}}
}}

At the bottom of the fluid layer, [[Permeability (earth sciences)|impermeability]] requires the [[normal component]] of the flow velocity to vanish:<ref name="Phillips"/>
{{NumBlk||<math display="block">
  \frac{\partial\Phi}{\partial n}
    = \frac{1}{\sqrt{ 1 
             + \left( \frac{\partial h}{\partial x} \right)^2 
             + \left( \frac{\partial h}{\partial y} \right)^2
    }}\,
    \left\{
        \frac{\partial \Phi}{\partial z} 
      + \frac{\partial h}{\partial x}\, \frac{\partial \Phi}{\partial x}  
      + \frac{\partial h}{\partial y}\, \frac{\partial \Phi}{\partial y}
    \right\}
    = 0,
    \qquad \text{ at } z=-h(x,y),
</math>|{{EquationRef|F}}
}}

where ''h''(''x'',''y'') is the depth of the bed below the [[datum (geodesy)|datum]] {{math|1=''z'' = 0}} and ''n'' is the coordinate component in the direction [[surface normal|normal to the bed]].

For permanent waves above a horizontal bed, the mean depth ''h'' is a constant and the boundary condition at the bed becomes:
<math display="block">\frac{\partial\Phi}{\partial z} = 0 \qquad \text{ at } z = -h.</math>

===Taylor series in the free-surface boundary conditions===

The free-surface boundary conditions {{EquationNote|D|(D)}} and {{EquationNote|E|(E)}} apply at the yet unknown free-surface elevation {{math|1=''z'' = ''η''(''x'',''y'',''t'')}}. They can be transformed into boundary conditions at a fixed elevation {{math|1=''z'' = constant}} by use of [[Taylor series]] expansions of the flow field around that elevation.<ref name="Phillips"/>
Without loss of generality the mean surface elevation – around which the Taylor series are developed – can be taken at {{math|1=''z'' = 0}}. 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 {{harvtxt|Levi-Civita|1925}}.

The following notation is used: the Taylor series of some field {{math|''f''(''x'',''y'',''z'',''t'')}} around {{math|1=''z'' = 0}} – and evaluated at {{math|1=''z'' = ''η''(''x'',''y'',''t'')}} – is:<ref name="Mei_607_608">{{harvtxt|Mei|1989|pages=607–608}}</ref>
<math display="block">
  f(x,y,\eta,t) = 
    \left[ f \right]_0
    + \eta\, \left[ \frac{\partial f}{\partial z} \right]_0
    + \frac12\, \eta^2\, \left[ \frac{\partial^2 f}{\partial z^2} \right]_0
    + \cdots
</math>
with subscript zero meaning evaluation at {{math|1=''z'' = 0}}, e.g.: {{math|1=[''f'']<sub>0</sub> = ''f''(''x'',''y'',0,''t'')}}.

Applying the Taylor expansion to free-surface boundary condition {{EquationNote|E|Eq. (E)}} in terms of the potential Φ gives:<ref name="Phillips"/><ref name="Mei_607_608"/>
{{NumBlk||<math display="block">
\begin{align}
  & \left[
    \frac{\partial^2 \Phi}{\partial t^2} 
    + g\, \frac{\partial \Phi}{\partial z}
  \right]_0
  + \eta \left[ \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi}{\partial t^2} 
              + g\, \frac{\partial \Phi}{\partial z}
            \right) 
          \right]_0
  + \left[ \frac{\partial}{\partial t} \left( |\mathbf{u}|^2 \right) \right]_0
  \\ & \quad
  + \tfrac12\, \eta^2 
           \left[ \frac{\partial^2}{\partial z^2} 
             \left( \frac{\partial^2 \Phi}{\partial t^2}  + g\, \frac{\partial \Phi}{\partial z} \right)
            \right]_0
  + \eta \left[ \frac{\partial^2}{\partial t\, \partial z} \left( |\mathbf{u}|^2 \right) \right]_0
  + \left[ \tfrac12\, \mathbf{u} \cdot \boldsymbol{\nabla} \left( |\mathbf{u}|^2 \right) \right]_0
  \\ & \quad
  + \cdots
  = 0,
\end{align}</math>|{{EquationRef|G}}
}}

showing terms up to triple products of ''η'', ''Φ'' and '''''u''''', as required for the construction of the Stokes expansion up to third-order {{mathcal|O}}((''ka'')<sup>3</sup>). 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 ''{{mathcal|O}}''(''ka'').

The dynamic free-surface boundary condition {{EquationNote|D|Eq. (D)}} can be evaluated in terms of quantities at {{math|1=''z'' = 0}} as:<ref name="Phillips"/><ref name="Mei_607_608"/>
{{NumBlk||<math display="block">\begin{align}
  & \left[ 
    \frac{\partial \Phi}{\partial t} + g\, \eta
  \right]_0
  + \eta \left[ \frac{\partial^2 \Phi}{\partial t\, \partial z} \right]_0
  + \biggl[ \tfrac12\, \left| \mathbf{u} \right|^2 \biggr]_0
  \\ & \quad
  + \tfrac12\, \eta^2 \left[ \frac{\partial^3 \Phi}{\partial t\, \partial z^2} \right]_0
  + \eta \left[ 
           \frac{\partial}{\partial z}
           \left(
             \tfrac12\, \left| \mathbf{u} \right|^2
           \right) 
         \right]_0
  + \cdots
  = 0.
\end{align}
</math>|{{EquationRef|H}}
}}

The advantages of these Taylor-series expansions fully emerge in combination with a perturbation-series approach, for weakly non-linear waves {{math|(''ka'' ≪ 1)}}.

===Perturbation-series approach===

The [[perturbation series]] are in terms of a small ordering parameter {{math|''ε'' ≪ 1}} – which subsequently turns out to be proportional to (and of the order of) the wave slope ''ka'', see the series solution in [[#Third-order Stokes wave on deep water|this section]].<ref>By [[non-dimensionalization]] of the flow equations and boundary conditions, different regimes may be identified, depending on the scaling of the coordinates and flow quantities. In deep(er) water, the characteristic [[wavelength]] is the only length scale available. So, the horizontal and vertical coordinates are all non-dimensionalized with the wavelength. This leads to Stokes wave theory. However, in shallow water, the water depth is the appropriate characteristic scale to make the vertical coordinate non-dimensional, while the horizontal coordinates are scaled with the wavelength – resulting in the [[Boussinesq approximation (water waves)|Boussinesq approximation]]. For a discussion, see:
* {{Citation| doi = 10.1016/0378-3839(94)00031-R | volume = 25| issue = 1–2 | pages = 81–85 | last = Beji | first = S. | title = Note on a nonlinearity parameter of surface waves | journal = Coastal Engineering | year = 1995 | bibcode = 1995CoasE..25...81B}};
* {{Citation| doi = 10.1016/S0378-3839(98)00024-6| volume = 34| issue = 1–2| pages = 163–168| last = Kirby| first = J.T. | title = Discussion of 'Note on a nonlinearity parameter of surface waves' by S. Beji | journal = Coastal Engineering| year = 1998 | bibcode = 1998CoasE..34..163K}} and
* {{Citation | doi = 10.1016/S0378-3839(98)00018-0 | volume = 34 | issue = 1–2 | pages = 169–171 | last = Beji | first = S. | title = Author's closure to J.T. Kirby's discussion 'Note on a nonlinearity parameter of surface waves' | journal = Coastal Engineering | year = 1998 | bibcode = 1998CoasE..34..169B }}</ref> So, take {{math|1=''ε'' = ''ka''}}:
<math display="block">\begin{align}
  \eta &= \varepsilon\, \eta_1 + \varepsilon^2\, \eta_2 + \varepsilon^3\, \eta_3 + \cdots ,
  \\
  \Phi &= \varepsilon\, \Phi_1 + \varepsilon^2\, \Phi_2 + \varepsilon^3\, \Phi_3 + \cdots 
  \quad \text{and}
  \\
  \mathbf{u} &= \varepsilon\, \mathbf{u}_1 + \varepsilon^2\, \mathbf{u}_2 + \varepsilon^3\, \mathbf{u}_3 + \cdots .
\end{align}</math>

When applied in the flow equations, they should be valid independent of the particular value of ''ε''. By equating in powers of ''ε'', each term proportional to ''ε'' to a certain power has to equal to zero. As an example of how the perturbation-series approach works, consider the non-linear boundary condition {{EquationNote|G|(G)}}; it becomes:<ref name="Dingemans"/>
<math display="block">\begin{align}
  & \varepsilon\, 
  \left\{
    \frac{\partial^2 \Phi_1}{\partial t^2} 
    + g\, \frac{\partial \Phi_1}{\partial z}
  \right\}
  \\
  & + \varepsilon^2\, 
  \left\{
    \frac{\partial^2 \Phi_2}{\partial t^2} 
    + g\, \frac{\partial \Phi_2}{\partial z}
    + \eta_1\, \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi_1}{\partial t^2} 
              + g\, \frac{\partial \Phi_1}{\partial z}
            \right) 
    + \frac{\partial}{\partial t} \left( |\mathbf{u}_1|^2 \right)
  \right\}
  \\
  & + \varepsilon^3\, 
  \left\{
    \frac{\partial^2 \Phi_3}{\partial t^2} 
    + g\, \frac{\partial \Phi_3}{\partial z}
    + \eta_1\, \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi_2}{\partial t^2} 
              + g\, \frac{\partial \Phi_2}{\partial z}
            \right) 
  \right.
  \\ & \qquad \quad \left.
    + \eta_2\, \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi_1}{\partial t^2} 
              + g\, \frac{\partial \Phi_1}{\partial z}
            \right) 
    + 2\, \frac{\partial}{\partial t} \left( \mathbf{u}_1 \cdot \mathbf{u}_2 \right)
  \right.
  \\ & \qquad \quad \left.
  + \tfrac12\, \eta_1^2\, 
           \frac{\partial^2}{\partial z^2} 
             \left(
               \frac{\partial^2 \Phi_1}{\partial t^2} 
                + g\, \frac{\partial \Phi_1}{\partial z}
              \right)
  + \eta_1\, \frac{\partial^2}{\partial t\, \partial z} \left( |\mathbf{u}_1|^2 \right)
  + \tfrac12\, \mathbf{u}_1 \cdot \boldsymbol{\nabla} \left( |\mathbf{u}_1|^2 \right)
  \right\}
  \\ &
  + \mathcal{O}\left( \varepsilon^4 \right)
  = 0, 
  \qquad \text{at } z=0.
\end{align}</math>

The resulting boundary conditions at {{math|1=''z'' = 0}} for the first three orders are:

;'''First order:'''
{{NumBlk||<math display="block">
    \frac{\partial^2 \Phi_1}{\partial t^2} 
    + g\, \frac{\partial \Phi_1}{\partial z}
    = 0,
</math>|{{EquationRef|J1}} }}

;'''Second order:'''
{{NumBlk||<math display="block">
    \frac{\partial^2 \Phi_2}{\partial t^2} 
    + g\, \frac{\partial \Phi_2}{\partial z}
    = - \eta_1\, \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi_1}{\partial t^2} 
              + g\, \frac{\partial \Phi_1}{\partial z}
            \right) 
    - \frac{\partial}{\partial t} \left( |\mathbf{u}_1|^2 \right),
</math>|{{EquationRef|J2}} }}

;'''Third order:'''
{{NumBlk||<math display="block">
\begin{align}
  \frac{\partial^2 \Phi_3}{\partial t^2} 
    + g\, \frac{\partial \Phi_3}{\partial z}
    = & - \eta_1\, \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi_2}{\partial t^2} 
              + g\, \frac{\partial \Phi_2}{\partial z}
            \right) 
    - \eta_2\, \frac{\partial}{\partial z} 
           \left(
             \frac{\partial^2 \Phi_1}{\partial t^2} 
              + g\, \frac{\partial \Phi_1}{\partial z}
            \right) 
  \\ &
    - 2\, \frac{\partial}{\partial t} \left( \mathbf{u}_1 \cdot \mathbf{u}_2 \right)
    - \tfrac12\, \eta_1^2\, 
           \frac{\partial^2}{\partial z^2} 
             \left(
               \frac{\partial^2 \Phi_1}{\partial t^2} 
                + g\, \frac{\partial \Phi_1}{\partial z}
              \right)
  \\ &
  - \eta_1\, \frac{\partial^2}{\partial t\, \partial z} \left( |\mathbf{u}_1|^2 \right)
  - \tfrac12\, \mathbf{u}_1 \cdot \boldsymbol{\nabla} \left( |\mathbf{u}_1|^2 \right).
\end{align}
</math>|{{EquationRef|J3}} }}

In a similar fashion – from the dynamic boundary condition {{EquationNote|H|(H)}} – the conditions at {{math|1=''z'' = 0}} at the orders 1, 2 and 3 become:

;'''First order:'''
{{NumBlk||<math display="block">
  \frac{\partial \Phi_1}{\partial t} + g\, \eta_1 = 0,
</math>|{{EquationRef|K1}} }}

;'''Second order:'''
{{NumBlk||<math display="block">
  \frac{\partial \Phi_2}{\partial t} + g\, \eta_2 
  = - \eta_1\, \frac{\partial^2 \Phi_1}{\partial t\, \partial z} 
    - \tfrac12\, \left| \mathbf{u}_1 \right|^2,
</math>|{{EquationRef|K2}} }}

;'''Third order:'''
{{NumBlk||<math display="block">
\begin{align}
  \frac{\partial \Phi_3}{\partial t}  + g\, \eta_3 = & 
    - \eta_1\, \frac{\partial^2 \Phi_2}{\partial t\, \partial z} 
    - \eta_2\, \frac{\partial^2 \Phi_1}{\partial t\, \partial z} 
    - \mathbf{u}_1 \cdot \mathbf{u}_2
  \\ &
    - \tfrac12\, \eta_1^2\, \frac{\partial^3 \Phi_1}{\partial t\, \partial z^2}
    - \eta_1\, \frac{\partial}{\partial z} \left( \tfrac12\, \left| \mathbf{u}_1 \right|^2 \right).
\end{align} 
</math>|{{EquationRef|K3}} }}

For the linear equations {{EquationNote|A|(A)}},  {{EquationNote|B|(B)}} and {{EquationNote|F|(F)}} the perturbation technique results in a series of equations independent of the perturbation solutions at other orders:
{{NumBlk||<math display="block">
\left.
\begin{array}{rcl}
  \mathbf{u}_j &=& \boldsymbol{\nabla} \Phi_j,
  \\[1ex]
  \nabla^2 \Phi_j &=& 0,
  \\[1ex]
  \displaystyle \frac{\partial \Phi_j}{\partial n} &=& 0 \quad \text{ at } z=-h,
\end{array}
\right\} \qquad \text{for all orders } j \in \mathbb{N}^+.
</math>|{{EquationRef|L}} }}

The above perturbation equations can be solved sequentially, i.e. starting with first order, thereafter continuing with the second order, third order, etc.

===Application to progressive periodic waves of permanent form===
[[File:Deep water wave.gif|thumb|right|300px|Animation of steep Stokes waves in deep water, with a [[wavelength]] of about twice the water depth, for three successive wave [[period (physics)|period]]s. The [[wave height]] is about 9.2% of the [[wavelength]].<br>''Description of the animation'':
The white dots are fluid particles, followed in time. In the case shown here, the [[average|mean]] [[Lagrangian and Eulerian specification of the flow field|Eulerian]] horizontal [[flow velocity|velocity]] below the wave [[trough (physics)|trough]] is zero.<ref>The wave physics are computed with the Rienecker & Fenton (R&F) [[streamfunction]] theory. For a computer code to compute these see: {{citation| first=J.D. |last=Fenton |year=1988 |title=The numerical solution of steady water wave problems |journal=Computers & Geosciences |volume=14 |issue=3 |pages=357–368 |postscript=.| bibcode=1988CG.....14..357F| doi=10.1016/0098-3004(88)90066-0}} The animations are made from the R&F results with a series of [[MATLAB|Matlab]] scripts and [[shell script]]s.</ref>]]

The waves of permanent form propagate with a constant [[phase velocity]] (or [[:wikt:celerity|celerity]]), denoted as ''c''. If the steady wave motion is in the horizontal ''x''-direction, the flow quantities ''η'' and '''u''' are not separately dependent on ''x'' and time ''t'', but are functions of {{math| ''x'' − ''ct''}}:<ref name="Wehausen_Laitone">{{harvtxt|Wehausen|Laitone|1960|pages=653–667, §27}}</ref>
<math display="block"> \eta(x,t) = \eta(x-ct)  \quad \text{and} \quad  \mathbf{u}(x,z,t) = \mathbf{u}(x-ct,z). </math>

Further the waves are periodic – and because they are also of permanent form – both in horizontal space ''x'' and in time ''t'', with [[wavelength]] ''λ'' and [[period (physics)|period]] ''τ'' respectively. Note that ''Φ''(''x'',''z'',''t'') itself is not necessary periodic due to the possibility of a constant (linear) drift in ''x'' and/or ''t'':<ref name="Whitham_16.6"/>
<math display="block">\Phi(x,z,t) = \beta x - \gamma t + \varphi(x-ct,z),</math>
with ''φ''(''x'',''z'',''t'') – as well as the derivatives ∂''Φ''/∂''t'' and ∂''Φ''/∂''x'' – being periodic. Here ''β'' is the mean flow velocity below [[trough (physics)|trough]] level, and ''γ'' is related to the [[hydraulic head]] as observed in a [[frame of reference]] moving with the wave's phase velocity ''c'' (so the flow becomes [[steady flow|steady]] in this reference frame).

In order to apply the Stokes expansion to progressive periodic waves, it is advantageous to describe them through [[Fourier series]] as a function of the [[wave phase]] ''θ''(''x'',''t''):<ref name="Dias_Kharif"/><ref name="Whitham_16.6"/>
<math display="block"> \theta = k x - \omega t = k \left( x - c t \right), </math>

assuming waves propagating in the ''x''–direction. Here {{math|1=''k'' = 2''π'' / ''λ''}} is the [[wavenumber]], {{math|1=''ω'' = 2''π'' / ''τ''}} is the [[angular frequency]] and {{math|1=''c'' = ''ω'' / ''k'' (= ''λ'' / ''τ'')}} is the [[phase velocity]].

Now, the free surface elevation ''η''(''x'',''t'') of a periodic wave can be described as the [[Fourier series]]:<ref name="Fenton"/><ref name="Whitham_16.6">{{harvtxt|Whitham|1974|pages=553–556, §16.6}}</ref>
<math display="block">\eta = \sum_{n=1}^{\infty} A_n\, \cos\, (n\theta).</math>

Similarly, the corresponding expression for the velocity potential ''Φ''(''x'',''z'',''t'') is:<ref name="Whitham_16.6"/>
<math display="block"> \Phi = \beta x - \gamma t + \sum_{n=1}^\infty B_n\, \biggl[ \cosh\, \left( nk\, (z+h) \right) \biggr]\, \sin\, (n\theta),</math>

satisfying both the [[Laplace equation]] {{math|1=∇<sup>2</sup>''Φ'' = 0}} in the fluid interior, as well as the boundary condition {{math|1=∂''Φ''/∂''z'' = 0}} at the bed {{math|1=''z'' = −''h''}}.

For a given value of the wavenumber ''k'', the parameters: ''A''<sub>n</sub>, ''B''<sub>n</sub> (with {{math|1=''n'' = 1, 2, 3, ...}}), ''c'', ''β'' and ''γ'' have yet to be determined. They all can be expanded as perturbation series in ''ε''. {{harvtxt|Fenton|1990}} provides these values for fifth-order Stokes's wave theory.

For progressive periodic waves, derivatives with respect to ''x'' and ''t'' of functions ''f''(''θ'',''z'') of ''θ''(''x'',''t'') can be expressed as derivatives with respect to ''θ'':
<math display="block">
  \frac{\partial f}{\partial x} = +k\, \frac{\partial f}{\partial \theta} 
  \qquad \text{and} \qquad 
  \frac{\partial f}{\partial t} = -\omega\, \frac{\partial f}{\partial \theta}.
</math>

The important point for non-linear waves – in contrast to linear [[Airy wave theory]] – is that the phase velocity ''c'' also depends on the [[wave amplitude]] ''a'', besides its dependence on wavelength {{math|1=''λ'' = 2π / ''k''}} and mean depth ''h''. Negligence of the dependence of ''c'' on wave amplitude results in the appearance of [[secular term]]s, in the higher-order contributions to the perturbation-series solution. {{harvtxt|Stokes|1847}} already applied the required non-linear correction to the phase speed ''c'' in order to prevent secular behaviour. A general approach to do so is now known as the [[Lindstedt–Poincaré method]]. Since the wavenumber ''k'' is given and thus fixed, the non-linear behaviour of the phase velocity {{math|1=''c'' = ''ω'' / ''k''}} is brought into account by also expanding the angular frequency ''ω'' into a perturbation series:<ref name="Whitham_13.13"/>
<math display="block"> \omega = \omega_0 + \varepsilon\, \omega_1 + \varepsilon^2\, \omega_2 + \cdots.</math>

Here ''ω''<sub>0</sub> will turn out to be related to the wavenumber ''k'' through the linear [[dispersion (water waves)|dispersion relation]]. However time derivatives, through {{math|1=∂''f''/∂''t'' = −''ω'' ∂''f''/∂''θ''}}, now also give contributions – containing ''ω''<sub>1</sub>, ''ω''<sub>2</sub>, etc. – to the governing equations at higher orders in the perturbation series. By tuning ''ω''<sub>1</sub>, ''ω''<sub>2</sub>, etc., secular behaviour can be prevented. For surface gravity waves, it is found that {{math|1=''ω''<sub>1</sub> = 0}} and the first non-zero contribution to the dispersion relation comes from ''ω''<sub>2</sub> (see e.g. the sub-section "[[#Third-order dispersion relation|Third-order dispersion relation]]" above).<ref name="Whitham_13.13"/>

==Stokes's two definitions of wave celerity {{anchor|Stokes' first definition of wave celerity|Stokes' second definition of wave celerity|Stokes first definition|Stokes second definition|Stokes' two definitions of wave celerity}}==

For non-linear surface waves there is, in general, ambiguity in splitting the total motion into a wave part and a [[average|mean]] part. As a consequence, there is some freedom in choosing the phase speed (celerity) of the wave. {{harvtxt|Stokes|1847}} identified two logical definitions of phase speed, known as Stokes's first and second definition of wave celerity:<ref name="Dingemans"/><ref name="Fenton"/><ref>{{Citation | publisher = Van Nostrand Reinhold | isbn = 9780442254025 | last1 = Sarpkaya | first1 = Turgut | last2 = Isaacson | first2 = Michael | title = Mechanics of wave forces on offshore structures | year = 1981 | page=183 }}</ref>

#'''Stokes's first definition of wave celerity''' has, for a pure wave motion, the [[average|mean value]] of the horizontal [[Lagrangian and Eulerian specification of the flow field|Eulerian]] flow-velocity ''Ū''<sub>E</sub> at any location below [[trough (physics)|trough]] level equal to zero. Due to the [[irrotational flow|irrotationality]] of potential flow, together with the horizontal sea bed and periodicity the mean horizontal velocity, the mean horizontal velocity is a constant between bed and trough level. So in Stokes first definition the wave is considered from a [[frame of reference]] moving with the mean horizontal velocity ''Ū''<sub>E</sub>. This is an advantageous approach when the mean Eulerian flow velocity ''Ū''<sub>E</sub> is known, e.g. from measurements.
#'''Stokes's second definition of wave celerity''' is for a frame of reference where the mean horizontal [[mass transfer|mass transport]] of the wave motion equal to zero. This is different from the first definition due to the mass transport in the [[splash zone]], i.e. between the trough and crest level, in the wave propagation direction. This wave-induced mass transport is caused by the positive [[correlation (statistics)|correlation]] between surface elevation and horizontal velocity. In the reference frame for Stokes's second definition, the wave-induced mass transport is compensated by an opposing [[undertow (wave action)|undertow]] (so ''Ū''<sub>E</sub>&nbsp;<&nbsp;0 for waves propagating in the positive ''x''-direction). This is the logical definition for waves generated in a [[wave flume]] in the laboratory, or waves moving perpendicular towards a beach.

As pointed out by [[Michael E. McIntyre]], the mean horizontal mass transport will be (near) zero for a [[wave group]] approaching into still water, with also in deep water the mass transport caused by the waves balanced by an opposite mass transport in a return flow (undertow).<ref>{{Citation | doi = 10.1017/S0022112081001626 | volume = 106 | pages = 331–347 | last = McIntyre | first = M.E. | title = On the 'wave momentum' myth | journal = Journal of Fluid Mechanics | year = 1981 | doi-broken-date = 29 November 2024 |bibcode = 1981JFM...106..331M | s2cid = 18232994 }}</ref> This is due to the fact that otherwise a large mean force will be needed to accelerate the body of water into which the wave group is propagating.

==Notes==
{{reflist|2}}

==References==
{{refbegin}}

===By Sir George Gabriel Stokes===

*{{citation
| first=G.G.
| last=Stokes
| author-link=George Gabriel Stokes
| year= 1847
| title= On the theory of oscillatory waves
| journal= Transactions of the Cambridge Philosophical Society
| volume= 8
| pages= 441–455
| postscript=.
}}<br>
:Reprinted in: {{citation
| first= G.G.
| last= Stokes
| year= 1880a
| contribution= On the theory of oscillatory waves
| title= Mathematical and Physical Papers, Volume I
| publisher= Cambridge University Press
| pages= 197–229
| isbn= 9781001435534
| url=https://archive.org/details/mathphyspapers01stokrich
| oclc = 314316422
}}
*{{citation
| first= G.G.
| last= Stokes
| year= 1880b
| contribution= Supplement to a paper on the theory of oscillatory waves
| title= Mathematical and Physical Papers, Volume I
| publisher= Cambridge University Press
| pages= 314–326
| isbn= 9781001435534
| url=https://archive.org/details/mathphyspapers01stokrich
| oclc = 314316422
}}

===Other historical references===

* {{citation
| doi = 10.1017/S0022112057000348
| last = Crapper
| first = G.D.
| volume = 2
| issue = 6
| pages = 532–540
| title = An exact solution for progressive capillary waves of arbitrary amplitude
| journal = Journal of Fluid Mechanics
| year = 1957
| doi-broken-date = 2 December 2024
|bibcode = 1957JFM.....2..532C | s2cid = 120377950
}}
* {{citation
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| pages = 713–736
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| first = S.C.
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| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| year = 1955
|bibcode = 1955PCPS...51..713D | s2cid = 122721263
}}
* {{Citation
| title = Détermination rigoureuse des ondes permanentes d'ampleur finie
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* {{Citation
| title = Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie
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| pages = 595–634
| doi = 10.1007/BF01206629
| s2cid = 122656179
}}
* {{citation
| author = Lord Rayleigh
| author-link = John Strutt, 3rd Baron Rayleigh
| title = On periodic irrotational waves at the surface of deep water
| journal = Philosophical Magazine |series=Series 6
| volume = 33
| issue = 197
| pages = 381–389
| year = 1917
| doi = 10.1080/14786440508635653
| url = https://zenodo.org/record/1430750
| postscript=.
}} <br>
:Reprinted in: {{citation
| publisher = Cambridge University Press
| url = https://archive.org/details/scientificpapers06rayliala
| last = Strutt
| first = John William (Lord Rayleigh)
| volume = 6
| issue = 1911–1919
| title = Scientific Papers
| year = 1920
| pages = 478–485, §419
| oclc = 2316730
}}
*{{citation
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| year = 1914
| url = https://zenodo.org/record/1430642
}}

===More recent (since 1960)===

*{{citation
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| issue = 1335
| pages = 183–230
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| first = E.D.
| title = Steep gravity waves in water of arbitrary uniform depth
| journal = Philosophical Transactions of the Royal Society
| year = 1977
|bibcode = 1977RSPTA.286..183C | s2cid = 119957640
}}
*{{citation
| doi = 10.1146/annurev.fluid.37.061903.175836
| volume = 37
| issue = 1
| pages = 23–42
| last = Craik
| first = A.D.D.
| title = George Gabriel Stokes on water wave theory
| journal = Annual Review of Fluid Mechanics
| year = 2005
|bibcode = 2005AnRFM..37...23C }}
*{{citation
| doi = 10.1146/annurev.fluid.31.1.301
| volume = 31
| pages = 301–346
| first1 = F. | last1 = Dias
| first2 = C. | last2 = Kharif
| title = Nonlinear gravity and capillary–gravity waves
| journal = Annual Review of Fluid Mechanics
| year = 1999
|bibcode = 1999AnRFM..31..301D }}
* {{citation
| title = A fifth-order Stokes theory for steady waves
| first = J.D. | last = Fenton
| year = 1985
| journal = Journal of Waterway, Port, Coastal, and Ocean Engineering
| volume = 111
| issue = 2
| pages = 216–234
| doi = 10.1061/(ASCE)0733-950X(1985)111:2(216)
| citeseerx = 10.1.1.461.6157 }}<br>
: And in (including corrections):
: {{citation
| contribution = Nonlinear wave theories
| first = J.D. | last = Fenton
| year = 1990
| title = Ocean Engineering Science
| series = The Sea
| editor1-first = B. | editor1-last = LeMéhauté
| editor2-first = D.M. | editor2-last = Hanes
| publisher = Wiley Interscience
| volume = 9A
| pages = 3–25
| isbn = 9780674017399
| url = http://www.johndfenton.com/Papers/Fenton90b-Nonlinear-wave-theories.pdf
}}
* {{citation
| volume = 342
| issue = 1629
| pages = 157–174
| last = Longuet-Higgins
| first = M.S.
| author-link = Michael Longuet-Higgins
| title = Integral properties of periodic gravity waves of finite amplitude
| journal = [[Proceedings of the Royal Society A]]
| year = 1975
| doi = 10.1098/rspa.1975.0018
|bibcode = 1975RSPSA.342..157L | s2cid = 123723040
}}
*{{Citation
| publisher = World Scientific
| isbn = 9789971507893
| last = Mei
| first = C.C.
| author-link = Chiang C. Mei
| title = The Applied Dynamics of Ocean Surface Waves
| year = 1989
}}
* {{citation
| doi = 10.1017/S0022112074000802
| volume = 62
| issue = 3
| pages = 553–578
| last = Schwartz
| first = L.W.
| title = Computer extension and analytic continuation of Stokes's expansion for gravity waves
| journal = Journal of Fluid Mechanics
| year = 1974
| doi-broken-date = 2 December 2024
|bibcode = 1974JFM....62..553S | s2cid = 120140832
}}
* {{citation
| doi = 10.1146/annurev.fl.14.010182.000351
| volume = 14
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* {{citation
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{{refend}}

==External links==
*{{citation | url=http://ceprofs.civil.tamu.edu/jzhang/oe671class/nonLinear.html | title=Stokes waves applet | publisher=Texas A&M University | author=Jun Zhang | access-date=2012-08-09 }}

{{physical oceanography}}

[[Category:Fluid dynamics]]
[[Category:Water waves]]
[[Category:Articles containing video clips]]