Influid dynamics, aStokes wave is anonlinear andperiodicsurface wave on aninviscid fluid layer of constant mean depth.This type of modelling has its origins in the mid 19th century whenSir George Stokes – using aperturbation series approach, now known as theStokes 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 ofcoastal andoffshore structures, in order to determine the wavekinematics (free surface elevation andflow velocities). The wave kinematics are subsequently needed in thedesign process to determine thewave loads on a structure.^[2] For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of smallamplitude. In such shallow water, acnoidal 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 withstanding waves^[3] and even random waves.^[4][5]

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

According to Stokes's third-order theory, thefree surface elevationη, thevelocity potential Φ, thephase speed (or celerity)c and the wavephaseθ are, for aprogressivesurface gravity wave on deep water – i.e. the fluid layer has infinite depth:^[6]where

• x is the horizontal coordinate;
• z is the vertical coordinate, with the positivez-direction upward – opposing to the direction of theEarth's gravity – andz = 0 corresponding with themean surface elevation;
• t is time;
• a is the first-order waveamplitude;
• k is theangular wavenumber,k = 2π /λ withλ being thewavelength;
• ω is theangular frequency,ω = 2π /τ whereτ is theperiod, and
• g is thestrength of the Earth's gravity, aconstant in this approximation.

The expansion parameterka is known as the wave steepness. The phase speed increases with increasing nonlinearityka of the waves. Thewave heightH, being the difference between the surface elevationη at acrest and atrough, is:^[7]$${\displaystyle H=2a\left(1+\frac{1}{2}{k}^2{a}^2\right).}$$

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.^[6] Up to third order the orbital velocityfieldu = ∇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 approximationtrochoidal, as already noted byStokes (1847).^[8]

Stokes further observed, that although (in thisEulerian description) the third-order orbital velocity field consists of a circular motion at each point, theLagrangian paths offluid 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 theStokes drift.^[8]

The surface elevationη and the velocity potential Φ are, according to Stokes's second-order theory of surface gravity waves on a fluid layer ofmean depthh:^[6][9]

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

The ratioS of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:^[6]$${\displaystyle \mathcal{S}=ka\frac{3-{\tanh }^{2}kh}{4{\tanh }^{3}kh}.}$$

In deep water, for largekh the ratioS has theasymptote$${\displaystyle \underset{kh\to \mathrm{\infty }}{lim}\mathcal{S}=\frac{1}{2}ka.}$$

For long waves, i.e. smallkh, the ratioS behaves as$${\displaystyle \underset{kh\to 0}{lim}\mathcal{S}=\frac{3}{4}\frac{ka}{(kh{)}^3},}$$or, in terms of the wave heightH = 2a and wavelengthλ = 2π /k:$${\displaystyle \underset{kh\to 0}{lim}\mathcal{S}=\frac{3}{32{\pi }^2}\frac{H{\lambda }^2}{h^3}=\frac{3}{32{\pi }^2}\mathcal{U},}$$with$${\displaystyle \mathcal{U}\equiv \frac{H{\lambda }^2}{h^3}.}$$

HereU is theUrsell parameter (or Stokes parameter). For long waves (λ ≫h) of small heightH, i.e.U ≪ 32π²/3 ≈ 100, second-order Stokes theory is applicable. Otherwise, for fairly long waves (λ > 7h) of appreciable heightH acnoidal wave description is more appropriate.^[6] According to Hedges, fifth-order Stokes theory is applicable forU < 40, and otherwise fifth-ordercnoidal wave theory is preferable.^[10][11]

For Stokes waves under the action of gravity, the third-orderdispersion relation is – according toStokes's first definition of celerity:^[9]

This third-order dispersion relation is a direct consequence of avoidingsecular terms, 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):$${\displaystyle \underset{kh\to \mathrm{\infty }}{lim}{\omega }^2=gk\left\{1+{\left(ka\right)}^2\right\}+\mathcal{O}\left((ka{)}^4\right),}$$and in shallow water (long wavelengths compared to the depth):$${\displaystyle \underset{kh\to 0}{lim}{\omega }^2=k^{2}gh\left\{1+\frac{9}{8}\frac{{\left(ka\right)}^2}{{\left(kh\right)}^4}\right\}+\mathcal{O}\left((ka{)}^4\right).}$$

Asshown above, the long-wave Stokes expansion for the dispersion relation will only be valid for small enough values of the Ursell parameter:U ≪ 100.

A fundamental problem in finding solutions for surface gravity waves is thatboundary conditions have to be applied at the position of thefree 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 relevantpotential flow quantities in aTaylor series around the mean (or still) surface elevation.^[12] 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 theStokes expansion – in terms of a small parameter, most often the wave steepness. The unknown terms in the expansion can be solved sequentially.^[6][8] Often, only a small number of terms is needed to provide a solution of sufficient accuracy for engineering purposes.^[11] Typical applications are in the design ofcoastal andoffshore structures, and ofships.

Another property of nonlinear waves is that thephase speed of nonlinear waves depends on thewave height. In a perturbation-series approach, this easily gives rise to a spurioussecular variation of the solution, in contradiction with the periodic behaviour of the waves. Stokes solved this problem by also expanding thedispersion relationship into a perturbation series, by a method now known as theLindstedt–Poincaré method.^[6]

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 forwavelengths (λ) not large as compared with the mean depth (h). Inshallow water, the low-order Stokes expansion breaks down (gives unrealistic results) for appreciable wave amplitude (as compared to the depth). Then,Boussinesq approximations are more appropriate. Further approximations on Boussinesq-type (multi-directional) wave equations lead – for one-way wave propagation – to theKorteweg–de Vries equation or theBenjamin–Bona–Mahony equation. Like (near) exact Stokes-wave solutions,^[14] these two equations havesolitary wave (soliton) solutions, besides periodic-wave solutions known ascnoidal waves.^[11]

Already in 1914, Wilton extended the Stokes expansion for deep-water surface gravity waves to tenth order, although introducing errors at the eight order.^[15] A fifth-order theory for finite depth was derived by De in 1955.^[16] For engineering use, the fifth-order formulations of Fenton are convenient, applicable to both Stokesfirst andsecond definition of phase speed (celerity).^[17] The demarcation between when fifth-order Stokes theory is preferable over fifth-ordercnoidal wave theory is forUrsell parameters below about 40.^[10][11]

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 thevelocity potential andstream function as the independent variables, and the coordinates (x,z) as the dependent variables, withx andz being the horizontal and vertical coordinates respectively.^[18] 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 theradius of convergence of the rephrased series expansion reduces.^[19]

Another approach is by using theLagrangian frame of reference, following thefluid parcels. The Lagrangian formulations show enhanced convergence, as compared to the formulations in both theEulerian frame, and in the frame with the potential and streamfunction as independent variables.^[20][21]

An exact solution for nonlinear purecapillary waves of permanent form, and for infinite fluid depth, was obtained by Crapper in 1957. Note that these capillary waves – being short waves forced bysurface 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.^[22]

By use of computer models, the Stokes expansion for surface gravity waves has been continued, up to high (117th) order bySchwartz (1974). Schwartz has found that the amplitudea (ora₁) of the first-orderfundamental reaches a maximumbefore the maximumwave heightH is reached. Consequently, the wave steepnesska in terms of wave amplitude is not a monotone function up to the highest wave, and Schwartz utilizes insteadkH as the expansion parameter. To estimate the highest wave in deep water, Schwartz has usedPadé approximants andDomb–Sykes plots 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 Williams (1981,1985).

Several exact relationships exist between integral properties – such askinetic andpotential energy, horizontal wavemomentum andradiation stress – as found byLonguet-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).Cokelet (1978) harvtxt error: no target: CITEREFCokelet1978 (help), 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 theconservation laws for water waves, throughNoether's theorem.^[25]

In 2005, Hammack,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.^[26] 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.^[27]

Convergence of the Stokes expansion was first proved byLevi-Civita (1925) for the case of small-amplitude waves – on the free surface of a fluid of infinite depth. This was extended shortly afterwards byStruik (1926) for the case of finite depth and small-amplitude waves.^[28]

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 ofvelocity potential andstream function – does not converge for high-amplitude waves. While other formulations converge much more rapidly, e.g. in theEulerian frame of reference (with the velocity potential or stream function as a function of the spatial coordinates).^[19]

The maximum wave steepness, for periodic and propagating deep-water waves, isH /λ = 0.1410633 ± 4 · 10⁻⁷,^[29] so the wave height is about one-seventh (⁠1/7⁠) of the wavelength λ.^[24] And surface gravity waves of this maximum height have a sharpwave crest – with an angle of 120° (in the fluid domain) – also for finite depth, as shown by Stokes in 1880.^[18]

An accurate estimate of the highest wave steepness in deep water (H /λ ≈ 0.142) was already made in 1893, byJohn Henry Michell, using a numerical method.^[30] 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.^[31] The existence of the highest wave on deep water with a sharp-angled crest of 120° was proved byJohn Toland in 1978.^[32] 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.^[33][34]

The highest Stokes wave – under the action of gravity – can be approximated with the following simple and accurate representation of thefree surface elevationη(x,t):^[35]$${\displaystyle \frac{\eta }{\lambda }=A\left[\cosh \left(\frac{x-ct}{\lambda }\right)-1\right],}$$with$${\displaystyle A=\frac{1}{\sqrt{3}\sinh \left(\frac{1}{2}\right)}\approx 1.108,}$$for$${\displaystyle -\frac{1}{2}\lambda \le (x-ct)\le \frac{1}{2}\lambda ,}$$

and shifted horizontally over aninteger 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.^[35]

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 apendulum in agrandfather clock.^[36]

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 aPadé approximant can be found at StokesWave.org^[37]

In deeper water, Stokes waves are unstable.^[38] This was shown byT. Brooke Benjamin and Jim E. Feir in 1967.^[39][40] TheBenjamin–Feir instability is a side-band or modulational instability, with the side-band modulations propagating in the same direction as thecarrier wave; waves become unstable on deeper water for a relative depthkh > 1.363 (withk thewavenumber andh the mean water depth).^[41] The Benjamin–Feir instability can be described with thenonlinear Schrödinger equation, by inserting a Stokes wave with side bands.^[38] Subsequently, with a more refined analysis, it has been shown – theoretically and experimentally – that the Stokes wave and its side bands exhibitFermi–Pasta–Ulam–Tsingou recurrence: a cyclic alternation between modulation and demodulation.^[42]

In 1978Longuet-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${\displaystyle \lambda }$)^[43] and subharmonics (for perturbations at the spatial scales larger than${\displaystyle \lambda }$).^[44]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.^[45]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 withresonant wave interactions between five (or more) wave components.^[46][47][48]

In many instances, the oscillatory flow in the fluid interior of surface waves can be described accurately usingpotential flow theory, apart fromboundary layers near the free surface and bottom (wherevorticity is important, due toviscous effects, seeStokes boundary layer).^[49] Then, theflow velocityu can be described as thegradient of avelocity potential${\displaystyle \mathrm{\Phi }}$:

$${\displaystyle \mathbf{u}=\mathbf{\nabla }\mathrm{\Phi }.}$$

A

Consequently, assumingincompressible flow, the velocity fieldu isdivergence-free and the velocity potential${\displaystyle \mathrm{\Phi }}$ satisfiesLaplace's equation^[49]

$${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Phi }=0}$$

B

in the fluid interior.

The fluid region is described using three-dimensionalCartesian coordinates (x,y,z), withx andy the horizontal coordinates, andz the vertical coordinate – with the positivez-direction opposing the direction of thegravitational acceleration. Time is denoted witht. The free surface is located atz =η(x,y,t), and the bottom of the fluid region is atz = −h(x,y).

The free-surfaceboundary conditions forsurface gravity waves – using apotential flow description – consist of akinematic and adynamic boundary condition.^[50]Thekinematic boundary condition ensures that thenormal component of the fluid'sflow velocity,${\displaystyle \mathbf{u}=[\mathrm{\partial }\mathrm{\Phi }/\mathrm{\partial }x\mathrm{\partial }\mathrm{\Phi }/\mathrm{\partial }y\mathrm{\partial }\mathrm{\Phi }/\mathrm{\partial }z{]}^{\mathrm{T}}}$ in matrix notation, at the free surface equals the normal velocity component of the free-surface motionz =η(x,y,t):

$${\displaystyle \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}+\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }x}\frac{\mathrm{\partial }\eta }{\mathrm{\partial }x}+\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }y}\frac{\mathrm{\partial }\eta }{\mathrm{\partial }y}=\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}\text{ at }z=\eta (x,y,t).}$$

C

Thedynamic boundary condition states that, withoutsurface tension effects, the atmospheric pressure just above the free surface equals the fluidpressure just below the surface. For an unsteady potential flow this means that theBernoulli equation is to be applied at the free surface. In case of a constant atmospheric pressure, the dynamic boundary condition becomes:

$${\displaystyle \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}{\left|\mathbf{u}\right|}^2+g\eta =0\text{ at }z=\eta (x,y,t),}$$

D

where the constant atmospheric pressure has been taken equal to zero,without loss of generality.

Both boundary conditions contain the potential${\displaystyle \mathrm{\Phi }}$ as well as the surface elevationη. A (dynamic) boundary condition in terms of only the potential${\displaystyle \mathrm{\Phi }}$ can be constructed by taking thematerial derivative of the dynamic boundary condition, and using the kinematic boundary condition:^[49][50][51]$${\displaystyle (\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathbf{\nabla })\left(\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}|\mathbf{u}{|}^2+g\eta \right)=0}$$$${\displaystyle \Rightarrow \frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{t}^2}+g\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}+\mathbf{u}\cdot \mathbf{\nabla }\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }t}+\frac{1}{2}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(|\mathbf{u}{|}^2\right)+\frac{1}{2}\mathbf{u}\cdot \mathbf{\nabla }\left(|\mathbf{u}{|}^2\right)=0}$$

$${\displaystyle \Rightarrow \frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{t}^2}+g\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}+\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(|\mathbf{u}{|}^2\right)+\frac{1}{2}\mathbf{u}\cdot \mathbf{\nabla }\left(|\mathbf{u}{|}^2\right)=0\text{ at }z=\eta (x,y,t).}$$

E

At the bottom of the fluid layer,impermeability requires thenormal component of the flow velocity to vanish:^[49]

$${\displaystyle \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }n}=\frac{1}{\sqrt{1+{\left(\frac{\mathrm{\partial }h}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }h}{\mathrm{\partial }y}\right)}^2}}\left\{\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}+\frac{\mathrm{\partial }h}{\mathrm{\partial }x}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }x}+\frac{\mathrm{\partial }h}{\mathrm{\partial }y}\frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }y}\right\}=0,\text{ at }z=-h(x,y),}$$

F

whereh(x,y) is the depth of the bed below thedatumz = 0 andn is the coordinate component in the directionnormal to the bed.

For permanent waves above a horizontal bed, the mean depthh is a constant and the boundary condition at the bed becomes:$${\displaystyle \frac{\mathrm{\partial }\mathrm{\Phi }}{\mathrm{\partial }z}=0\text{ at }z=-h.}$$

The free-surface boundary conditions(D) and(E) apply at the yet unknown free-surface elevationz =η(x,y,t). They can be transformed into boundary conditions at a fixed elevationz = constant by use ofTaylor series expansions of the flow field around that elevation.^[49]Without loss of generality the mean surface elevation – around which the Taylor series are developed – can be taken atz = 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 byLevi-Civita (1925).

The following notation is used: the Taylor series of some fieldf(x,y,z,t) aroundz = 0 – and evaluated atz =η(x,y,t) – is:^[52]$${\displaystyle f(x,y,\eta ,t)={\left[f\right]}_0+\eta {\left[\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\right]}_0+\frac{1}{2}{\eta }^2{\left[\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}\right]}_0+\cdots }$$with subscript zero meaning evaluation atz = 0, e.g.:[f]₀ =f(x,y,0,t).

Applying the Taylor expansion to free-surface boundary conditionEq. (E) in terms of the potential Φ gives:^[49][52]

G

showing terms up to triple products ofη,Φ andu, as required for the construction of the Stokes expansion up to third-orderO((ka)³). Here,ka is the wave steepness, withk a characteristicwavenumber anda a characteristic waveamplitude for the problem under study. The fieldsη,Φ andu are assumed to beO(ka).

The dynamic free-surface boundary conditionEq. (D) can be evaluated in terms of quantities atz = 0 as:^[49][52]

H

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

Theperturbation series are in terms of a small ordering parameterε ≪ 1 – which subsequently turns out to be proportional to (and of the order of) the wave slopeka, see the series solution inthis section.^[53] So, takeε =ka:

When applied in the flow equations, they should be valid independent of the particular value ofε. By equating in powers ofε, each term proportional toε 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(G); it becomes:^[6]

The resulting boundary conditions atz = 0 for the first three orders are:

First order:

$${\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{\Phi }}_1}{\mathrm{\partial }{t}^2}+g\frac{\mathrm{\partial }{\mathrm{\Phi }}_1}{\mathrm{\partial }z}=0,}$$

J1

Second order:

$${\displaystyle \frac{{\mathrm{\partial }}^2{\mathrm{\Phi }}_2}{\mathrm{\partial }{t}^2}+g\frac{\mathrm{\partial }{\mathrm{\Phi }}_2}{\mathrm{\partial }z}=-{\eta }_1\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left(\frac{{\mathrm{\partial }}^2{\mathrm{\Phi }}_1}{\mathrm{\partial }{t}^2}+g\frac{\mathrm{\partial }{\mathrm{\Phi }}_1}{\mathrm{\partial }z}\right)-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(|{\mathbf{u}}_1{|}^2\right),}$$

J2

Third order:

J3

In a similar fashion – from the dynamic boundary condition(H) – the conditions atz = 0 at the orders 1, 2 and 3 become:

First order:

$${\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}_1}{\mathrm{\partial }t}+g{\eta }_1=0,}$$

K1

Second order:

$${\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}_2}{\mathrm{\partial }t}+g{\eta }_2=-{\eta }_1\frac{{\mathrm{\partial }}^2{\mathrm{\Phi }}_1}{\mathrm{\partial }t\mathrm{\partial }z}-\frac{1}{2}{\left|{\mathbf{u}}_1\right|}^2,}$$

K2

Third order:

K3

For the linear equations(A),(B) and(F) the perturbation technique results in a series of equations independent of the perturbation solutions at other orders:

L

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

The waves of permanent form propagate with a constantphase velocity (orcelerity), denoted asc. If the steady wave motion is in the horizontalx-direction, the flow quantitiesη andu are not separately dependent onx and timet, but are functions ofx −ct:^[55]$${\displaystyle \eta (x,t)=\eta (x-ct)\text{and}\mathbf{u}(x,z,t)=\mathbf{u}(x-ct,z).}$$

Further the waves are periodic – and because they are also of permanent form – both in horizontal spacex and in timet, withwavelengthλ andperiodτ respectively. Note thatΦ(x,z,t) itself is not necessary periodic due to the possibility of a constant (linear) drift inx and/ort:^[56]$${\displaystyle \mathrm{\Phi }(x,z,t)=\beta x-\gamma t+\phi (x-ct,z),}$$withφ(x,z,t) – as well as the derivatives ∂Φ/∂t and ∂Φ/∂x – being periodic. Hereβ is the mean flow velocity belowtrough level, andγ is related to thehydraulic head as observed in aframe of reference moving with the wave's phase velocityc (so the flow becomessteady in this reference frame).

In order to apply the Stokes expansion to progressive periodic waves, it is advantageous to describe them throughFourier series as a function of thewave phaseθ(x,t):^[48][56]$${\displaystyle \theta =kx-\omega t=k\left(x-ct\right),}$$

assuming waves propagating in thex–direction. Herek = 2π /λ is thewavenumber,ω = 2π /τ is theangular frequency andc =ω /k (=λ /τ) is thephase velocity.

Now, the free surface elevationη(x,t) of a periodic wave can be described as theFourier series:^[11][56]$${\displaystyle \eta =\sum _{n=1}^{\mathrm{\infty }}{A}_n\cos (n\theta ).}$$

Similarly, the corresponding expression for the velocity potentialΦ(x,z,t) is:^[56]$${\displaystyle \mathrm{\Phi }=\beta x-\gamma t+\sum _{n=1}^{\mathrm{\infty }}{B}_n[\cosh \left(nk(z+h)\right)]\sin (n\theta ),}$$

satisfying both theLaplace equation∇²Φ = 0 in the fluid interior, as well as the boundary condition∂Φ/∂z = 0 at the bedz = −h.

For a given value of the wavenumberk, the parameters:A_n,B_n (withn = 1, 2, 3, ...),c,β andγ have yet to be determined. They all can be expanded as perturbation series inε.Fenton (1990) provides these values for fifth-order Stokes's wave theory.

For progressive periodic waves, derivatives with respect tox andt of functionsf(θ,z) ofθ(x,t) can be expressed as derivatives with respect toθ:$${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}=+k\frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }\text{and}\frac{\mathrm{\partial }f}{\mathrm{\partial }t}=-\omega \frac{\mathrm{\partial }f}{\mathrm{\partial }\theta }.}$$

The important point for non-linear waves – in contrast to linearAiry wave theory – is that the phase velocityc also depends on thewave amplitudea, besides its dependence on wavelengthλ = 2π /k and mean depthh. Negligence of the dependence ofc on wave amplitude results in the appearance ofsecular terms, in the higher-order contributions to the perturbation-series solution.Stokes (1847) already applied the required non-linear correction to the phase speedc in order to prevent secular behaviour. A general approach to do so is now known as theLindstedt–Poincaré method. Since the wavenumberk is given and thus fixed, the non-linear behaviour of the phase velocityc =ω /k is brought into account by also expanding the angular frequencyω into a perturbation series:^[9]$${\displaystyle \omega ={\omega }_0+\epsilon {\omega }_1+{\epsilon }^2{\omega }_2+\cdots .}$$

Hereω₀ will turn out to be related to the wavenumberk through the lineardispersion relation. However time derivatives, through∂f/∂t = −ω ∂f/∂θ, now also give contributions – containingω₁,ω₂, etc. – to the governing equations at higher orders in the perturbation series. By tuningω₁,ω₂, etc., secular behaviour can be prevented. For surface gravity waves, it is found thatω₁ = 0 and the first non-zero contribution to the dispersion relation comes fromω₂ (see e.g. the sub-section "Third-order dispersion relation" above).^[9]

For non-linear surface waves there is, in general, ambiguity in splitting the total motion into a wave part and amean part. As a consequence, there is some freedom in choosing the phase speed (celerity) of the wave.Stokes (1847) identified two logical definitions of phase speed, known as Stokes's first and second definition of wave celerity:^[6][11][57]

1. Stokes's first definition of wave celerity has, for a pure wave motion, themean value of the horizontalEulerian flow-velocityŪ_E at any location belowtrough level equal to zero. Due to theirrotationality 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 aframe of reference moving with the mean horizontal velocityŪ_E. This is an advantageous approach when the mean Eulerian flow velocityŪ_E is known, e.g. from measurements.
2. Stokes's second definition of wave celerity is for a frame of reference where the mean horizontalmass transport of the wave motion equal to zero. This is different from the first definition due to the mass transport in thesplash zone, i.e. between the trough and crest level, in the wave propagation direction. This wave-induced mass transport is caused by the positivecorrelation between surface elevation and horizontal velocity. In the reference frame for Stokes's second definition, the wave-induced mass transport is compensated by an opposingundertow (soŪ_E < 0 for waves propagating in the positivex-direction). This is the logical definition for waves generated in awave flume in the laboratory, or waves moving perpendicular towards a beach.

As pointed out byMichael E. McIntyre, the mean horizontal mass transport will be (near) zero for awave 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).^[58] 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.

• Jun Zhang,Stokes waves applet, Texas A&M University, retrieved2012-08-09

• Fluid dynamics
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