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Fifth-Order Stokes Wave

This article highlights developments to be released in FLOW-3D version 9.4.

Generating Non-Linear Waves

The capability to generate linear waves at a mesh boundary has existed in FLOW-3D since version 9.2. It is based on Airy's linear wave theory for waves of infinitesimally small amplitude (Airy, 1845). The surface elevation of a linear wave simply has a sinusoidal shape. Although the linear theory can handle many engineering problems with ease and reasonable accuracy, its prediction of wave motion becomes unacceptable when wave amplitude is no longer negligible compared to wavelength. For example, it cannot account for the phenomena that the wave crest is higher above the mean water surface than the trough is below the mean water surface.

In version 9.4 of FLOW-3D, a non-linear wave boundary condition is implemented so that users can simulate more realistic wave propagation into the computational domain. It uses the fifth-order Stokes wave theory developed by Fenton (1985) for limited-amplitude waves. Compared to the linear wave theory, the fifth-order Stokes wave theory has a higher order of accuracy and can describe larger waves. In mathematics, a linear wave is simply the first-order approximation of the fifth-order Stokes wave.

Propagation of a 5th-Order Stokes Wave

Stokes wave propagation.

Figure 1 shows the surface profiles of a linear wave and a fifth-order Stokes wave. While the linear wave has equally high and equally long crests and troughs, the Stokes wave has shorter and higher crests and longer and shallower troughs. The shape of the fifth-order Stokes wave agrees with observation of real wave motion. The animation above shows the propagation of a fifth-order Stokes wave calculated using FLOW-3D.

Velocity Profile of a 5th-Order Stokes Wave

Comparison of profiles of the linear wave and the 5th order Stokes wave.
Figure 1: Comparison of profiles of the linear wave and the 5th-order Stokes wave. d: mean water depth; h: wave height; L: wave length.

Figure 2 compares the horizontal velocity profiles underneath the wave crest measured by Le Mehaute, et. al. (1968) and calculated using the fifth-order Stokes wave theory. The horizontal and vertical axes represent the normalized horizontal velocity and the vertical distance from domain bottom, respectively. The figure shows a reasonable agreement between the theory and the experiment. There is a small shift between the two results. Fenton claimed that this shift is caused by experimental error (Rienecker and Fenton, 1981, Fenton 1985). In fact, Fenton's fifth-order theory makes better kinetics and pressure predictions than other Stokes wave theories (Fenton, 1985, USACE, 2008).

Figure 2: Comparison of horizontal velocity profiles measured by Le Méhauté et al (1968)
Figure 2: Comparison of horizontal velocity profiles measured by Le Méhauté et al (1968) and calculated by Fenton's fifth-order Stokes wave theory. d: water depth; z: vertical distance from bottom; u: wave-induced velocity; g: gravitational acceleration.

Application Range

Applicability ranges of various wave theories (after Le Méhauté 1976 and USACE 2008).
Figure 3: Applicability ranges of various wave theories (after Le Méhauté 1976 and USACE 2008). d: mean water depth; H: wave height; T: wave period; g: gravitational acceleration.

Figure 3 shows the application ranges of various wave theories. It is seen that the linear wave theory is valid only for waves with small amplitude. With increasing wave amplitude or decreasing water depth, Stokes wave theories at higher-order accuracy should be used. Since the linear wave and the low-order Stokes wave (second to fourth orders) are approximations of the fifth-order Stokes wave, the fifth-order Stokes wave theory can also be applied, with better accuracy, to all cases where the linear or low-order Stokes wave theory is valid.

More About the Model

In FLOW-3D, the Stokes wave is generated at mesh boundaries. It also allows a horizontal water current with uniform and constant velocity at a wave boundary. Although the Stokes wave theory cannot describe breaking waves, after entering the computational domain, a Stokes wave can evolve into any type of wave with or without breaking. Water flow inside the computational domain can be inviscid, laminar or turbulent. The fifth-order Stokes wave generator works with most other physical models in FLOW-3D and supports many applications in ocean, coastal and maritime engineering.


  1. Airy, G. B., 1845, Tides and Waves, Encyc. Metrop. Article 102.
  2. Fenton, J. D., 1985, A Fifth-Order Stokes Theory for Steady Waves, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 111, No. 2.
  3. Le Méhauté, B., 1976, An Introduction to Hydrodynamics and Water Waves, Springer-Verlag.
  4. Le Méhauté, B., Divoky, D., and Lin, A., 1968, Shallow Water Waves: A Comparison of Theories and Experiments, Proceedings of 11th Conference of Costal Engineering, Vol. 1.
  5. Rienecker, M., M., and Fenton, J., D., 1981, A Fourier Approximation Method for Steady Water Waves, Journal of Fluid Mechanics, Vol. 104.
  6. USACE (U.S. Army Corps of Engineers), 2008, Coastal Engineering Manual EM 1110-2-1100, Part II, Washington, DC.