Stokes wave

It is used in the design of coastal and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities).

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:[6]

As a result, the surface elevation of deep-water waves is to a good approximation trochoidal, as already noted by Stokes (1847).

[8] 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.

Observe that for finite depth the velocity potential Φ contains a linear drift in time, independent of position (x and z).

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

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.

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.

In shallow water, the low-order Stokes expansion breaks down (gives unrealistic results) for appreciable wave amplitude (as compared to the depth).

[16] For engineering use, the fifth-order formulations of Fenton are convenient, applicable to both Stokes first and second definition of phase speed (celerity).

[10][11] Different choices for the frame of reference and expansion parameters are possible in Stokes-like approaches to the nonlinear wave problem.

[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.

[20][21] An exact solution for nonlinear pure capillary waves of permanent form, and for infinite fluid depth, was obtained by Crapper in 1957.

[22] By use of computer models, the Stokes expansion for surface gravity waves has been continued, up to high (117th) order by Schwartz (1974).

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 Williams (1981, 1985).

Several exact relationships exist between integral properties – such as kinetic and potential energy, horizontal wave momentum and radiation stress – as found by Longuet-Higgins (1975).

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 the conservation laws for water waves, through Noether's theorem.

[27] Convergence of the Stokes expansion was first proved by Levi-Civita (1925) for the case of small-amplitude waves – on the free surface of a fluid of infinite depth.

[18] An accurate estimate of the highest wave steepness in deep water (H / λ ≈ 0.142) was already made in 1893, by John Henry Michell, using a numerical method.

[31] The existence of the highest wave on deep water with a sharp-angled crest of 120° was proved by John Toland in 1978.

[33][34] 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):[35]

[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 a Padé approximant can be found at StokesWave.org[37] In deeper water, Stokes waves are unstable.

[38] 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.

[42] In 1978 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

[46][47][48] In many instances, the oscillatory flow in the fluid interior of surface waves can be described accurately using potential flow theory, apart from boundary layers near the free surface and bottom (where vorticity is important, due to viscous effects, see Stokes boundary layer).

(D) can be evaluated in terms of quantities at z = 0 as:[49][52] The advantages of these Taylor-series expansions fully emerge in combination with a perturbation-series approach, for weakly non-linear waves (ka ≪ 1).

Stokes (1847) already applied the required non-linear correction to the phase speed c in order to prevent secular behaviour.

[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.

Surface elevation of a deep water wave according to Stokes ' third-order theory. The wave steepness is: ka = 0.3, with k the wavenumber and a the wave amplitude . Typical for these surface gravity waves are the sharp crests and flat troughs .
Model testing with periodic waves in the wave–tow tank of the Jere A. Chase Ocean Engineering Laboratory, University of New Hampshire .
Undular bore and whelps near the mouth of Araguari River in north-eastern Brazil. View is oblique toward mouth from airplane at approximately 100 ft (30 m) altitude. [ 1 ] The undulations following behind the bore front appear as slowly modulated Stokes waves.
Third-order Stokes wave in deep water under the action of gravity. The wave steepness is: ka = 0.3.
The three harmonics contributing to the surface elevation of a deep water wave, according to Stokes's third-order theory. The wave steepness is: ka = 0.3. For visibility, the vertical scale is distorted by a factor of four, compared to the horizontal scale.
Description: * the dark blue line is the surface elevation of the 3rd-order Stokes wave, * the black line is the fundamental wave component, with wavenumber k ( wavelength λ, k = 2 π / λ ), * the light blue line is the harmonic at 2 k (wavelength 1 ⁄ 2 λ), and * the red line is the harmonic at 3 k (wavelength 1 ⁄ 3 λ).
The ratio S = a 2 / a of the amplitude a 2 of the harmonic with twice the wavenumber (2 k ), to the amplitude a of the fundamental , according to Stokes's second-order theory for surface gravity waves. On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the wavelength , while the vertical axis is the Stokes parameter S divided by the wave steepness ka (with k = 2 π / λ ).
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.
Nonlinear enhancement of the phase speed c = ω / k – according to Stokes's third-order theory for surface gravity waves , and using Stokes's first definition of celerity – as compared to the linear-theory phase speed c 0 . On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the wavelength , while the vertical axis is the nonlinear phase-speed enhancement ( c − c 0 ) / c 0 divided by the wave steepness ka squared.
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.
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.
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 crests and flat troughs as found in nonlinear surface gravity waves.
Validity of several theories for periodic water waves, according to Le Méhauté (1976). [ 13 ] 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 theory, for high waves ( H > 1 ⁄ 4 H breaking ).
Several integral properties of Stokes waves on deep water as a function of wave steepness. [ 23 ] The wave steepness is defined as the ratio of wave height H to the wavelength λ. The wave properties are made dimensionless using the wavenumber k = 2π / λ , gravitational acceleration g and the fluid density ρ .
Shown are the kinetic energy density T , the potential energy density V , the total energy density 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 ( kH ) 2 and 1/8 ( kH ) 2 , being the values of the integral properties as derived from (linear) Airy wave theory . The maximum wave height occurs for a wave steepness H / λ ≈ 0.1412 , above which no periodic surface gravity waves exist. [ 24 ]
Note that the shown wave properties have a maximum for a wave height less than the maximum wave height (see e.g. Longuet-Higgins 1975 ; Cokelet 1977 ).
Stokes waves of maximum wave height on deep water, under the action of gravity.
Animation of steep Stokes waves in deep water, with a wavelength of about twice the water depth, for three successive wave periods . The wave height is about 9.2% of the wavelength .
Description of the animation : The white dots are fluid particles, followed in time. In the case shown here, the mean Eulerian horizontal velocity below the wave trough is zero. [ 54 ]