In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for 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 and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities). The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. 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 waves and even random waves.
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.
Third-order Stokes wave on deep water
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 θ are, for a progressive surface gravity wave on deep water – i.e. the fluid layer has infinite depth:
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 = 0 corresponding with the mean surface elevation; t is time; a is the first-order wave amplitude; k is the angular wavenumber, with λ being the wavelength; ω is the angular frequency, where τ is the period, and g is the strength of the Earth's gravity, a 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 and a trough, is:
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. Up to third order the orbital velocity field u = ∇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, as already noted by .
Stokes further observed, that although (in this Eulerian description) the third-order orbital velocity field consists of a circular motion at each point, the Lagrangian paths of fluid 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 the Stokes drift.
Second-order Stokes wave on arbitrary depth
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 mean depth h:
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 2θ) in Φ vanish for deep-water waves.
Stokes and Ursell parameters
The ratio of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:
In deep water, for large kh the ratio has the asymptote
For long waves, i.e. small kh, the ratio behaves as
or, in terms of the wave height and wavelength :
with
Here is the Ursell parameter (or Stokes parameter). For long waves () of small height H, i.e. , second-order Stokes theory is applicable. Otherwise, for fairly long waves () of appreciable height H a cnoidal wave description is more appropriate. According to Hedges, fifth-order Stokes theory is applicable for , and otherwise fifth-order cnoidal wave theory is preferable.
Third-order dispersion relation
For Stokes waves under the action of gravity, the third-order dispersion relation is – according to Stokes's first definition of celerity:
This third-order dispersion relation is a direct consequence of avoiding secular 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):
and in shallow water (long wavelengths compared to the depth):
As shown above, the long-wave Stokes expansion for the dispersion relation will only be valid for small enough values of the Ursell parameter: .
Overview
Stokes's approach to the nonlinear wave problem
A fundamental problem in finding solutions for surface gravity waves is that boundary conditions have to be applied at the position of the free surface, which is not known beforehand and is thus a part of the solution to be found. Sir George Stokes solved this nonlinear wave problem in 1847 by expanding the relevant potential flow quantities in a Taylor series around the mean (or still) surface elevation. As a result, the boundary conditions can be expressed in terms of quantities at the mean (or still) surface elevation (which is fixed and known).
Next, a solution for the nonlinear wave problem (including the Taylor series expansion around the mean or still surface elevation) is sought by means of a perturbation series – known as the Stokes expansion – in terms of a small parameter, most often the wave steepness. The unknown terms in the expansion can be solved sequentially. Often, only a small number of terms is needed to provide a solution of sufficient accuracy for engineering purposes. Typical applications are in the design of coastal and offshore structures, and of ships.
Another property of nonlinear waves is that the phase speed of nonlinear waves depends on the wave height. In a perturbation-series approach, this easily gives rise to a spurious secular variation of the solution, in contradiction with the periodic behaviour of the waves. Stokes solved this problem by also expanding the dispersion relationship into a perturbation series, by a method now known as the Lindstedt–Poincaré method.
Applicability
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 wavelengths (λ) not large as compared with the mean depth (h). In shallow 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 the Korteweg–de Vries equation or the Benjamin–Bona–Mahony equation. Like (near) exact Stokes-wave solutions, these two equations have solitary wave (soliton) solutions, besides periodic-wave solutions known as cnoidal waves.
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. A fifth-order theory for finite depth was derived by De in 1955. For engineering use, the fifth-order formulations of Fenton are convenient, applicable to both Stokes first and second definition of phase speed (celerity)., also (including corrections) in The demarcation between when fifth-order Stokes theory is preferable over fifth-order cnoidal wave theory is for Ursell parameters below about 40.
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. 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.
Another approach is by using the Lagrangian frame of reference, following the fluid parcels. The Lagrangian formulations show enhanced convergence, as compared to the formulations in both the Eulerian frame, and in the frame with the potential and streamfunction as independent variables.
An exact solution for nonlinear pure capillary waves 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.
By use of computer models, the Stokes expansion for surface gravity waves has been continued, up to high (117th) order by . Schwartz has found that the amplitude a (or a1) of the first-order 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 approximants and Domb–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 .
Several exact relationships exist between integral properties – such as kinetic and potential energy, horizontal wave momentum and radiation stress – as found by . 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). , 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 laws for water waves, through Noether's theorem.
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. 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.
Convergence and instability
Convergence Convergence of the Stokes expansion was first proved by for the case of small-amplitude waves – on the free surface of a fluid of infinite depth. This was extended shortly afterwards by for the case of finite depth and small-amplitude waves.