Wave Power Physical Concepts

Motion of a particle in an ocean wave.
A = At deep water. The orbital motion of fluid particles decreases rapidly with increasing depth below the surface.
B = At shallow water (ocean floor is now at B). The elliptical movement of a fluid particle flattens with decreasing depth.
1 = Propagation direction.
2 = Wave crest.
3 = Wave trough.

Waves are generated by wind passing over the sea: as long as the waves propagate slower than the wind speed just above the waves, there is an energy transfer from the wind to the most energetic waves. Both air pressure differences between the upwind and the lee side of a wave crest, as well as friction on the water surface by the wind shear stress cause the growth of the waves.^[3] The wave height increases with increasing wind speed, duration since the wind started to blow, and of the fetch (the distance of open water that the wind has blown over), see Ocean surface wave.

In general, large waves are more powerful. Specifically, wave power is determined by wave height, wave speed, wavelength, and water density.

Wave size is determined by wind speed and fetch (the distance over which the wind excites the waves) and by the depth and topography of the seafloor (which can focus or disperse the energy of the waves). A given wind speed has a matching practical limit over which time or distance will not produce larger waves. This limit is called a "fully developed sea."

Oscillatory motion is highest at the surface and diminishes exponentially with depth. However, for standing waves (clapotis) near a reflecting coast, wave energy is also present as pressure oscillations at great depth, producing microseisms.^[3] These pressure fluctuations at greater depth are too small to be interesting from the point of view of wave power.

The waves propagate on the ocean surface, and the wave energy is also transported horizontally with the group velocity. The mean transport rate of the wave energy through a vertical plane of unit width, parallel to a wave crest, is called the wave energy flux (or wave power, which must not be confused with the actual power generated by a wave power device). In deep water, if the water depth is larger than half the wavelength, the wave energy flux is:

The above formula states that wave power is proportional to the wave period and to the square of the wave height. When the significant wave height is given in meter, and the wave period in second, the result is the wave power in kW (kilo watt) per meter wavefront length.^[5]^[6]

In major storms, the largest waves offshore are about 15 meters high and have a period of about 15 seconds. According to the above deep-water formula, such waves carry about 3.2 MW/m of power across each meter of wavefront. At moderate conditions, for a wave height of about 3 m and a wave period of 8 seconds, a wave power location will have an average flux much less than this: about 70 kW/m.

An effective wave power device captures as much as possible of the wave energy flux. As a result the waves will be of lower height in the region behind the wave power device.

Wave energy and wave energy flux

In a sea state, the energy per unit area of gravity waves on the water surface is proportional to the wave height squared, according to linear wave theory:^[3]^[4]

where E is the mean wave energy per unit horizontal area (J/m²), the sum of kinetic energy and potential energy. The potential energy is equal to the kinetic energy,^[3] both contributing half to the wave energy E, as can be expected from the equipartition theorem.

As the waves propagate, their energy is transported. The energy transport velocity is the group velocity. As a result, the wave energy flux, through a vertical plane of unit width perpendicular to the wave propagation direction, is equal to:^[7]^[3]

with c_g the group velocity (m/s). Due to the dispersion relation for water waves under the action of gravity, the group velocity depends on the wavelength λ, or equivalently, on the wave period T. Further, the dispersion relation is a function of the water depth h. As a result, the group velocity behaves differently in the limits of deep and shallow water, and at intermediate depths:^[3]^[4]

Deep water corresponds with a water depth larger than half the wavelength, which is the common situation in the sea and ocean. In deep water, longer period waves propagate faster and transport their energy faster. The deep-water group velocity is half the phase velocity. In shallow water, for wavelengths larger than twenty times the water depth, as found quite often near the coast, the group velocity is equal to the phase velocity.^[9]

References

^ Nauman, Matt. "PG&E to invest in wave energy", San Jose Mercury News, 2007-12-18. Retrieved on 2007-12-18.
^ Wave power scientist enthused by green energy
^ ^a ^b ^c ^d ^e ^f Phillips, O.M. (1977). The dynamics of the upper ocean, 2nd edition, Cambridge University Press. ISBN 0 521 29801 6.
^ ^a ^b ^c Goda, Y. (2000). Random Seas and Design of Maritime Structures. World Scientific. ISBN 978 981 02 3256 6.
^ Wave Power
^ Technology White Paper on Wave Energy Potential on the U.S. Outer Continental Shelf
^ Reynolds, O. (1877). "On the rate of progression of groups of waves and the rate at which energy is transmitted by waves". Nature 16: 343–44.
Lord Rayleigh (J. W. Strutt) (1877). "On progressive waves". Proceedings of the London Mathematical Society 9: 21–26. doi:10.1112/plms/s1-9.1.21. Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.
^ For determining the group velocity the angular frequency ω is considered as a function of the wavenumber k, or equivalently, the period T as a function of the wavelength λ.
^ R. G. Dean and R. A. Dalrymple (1991). Water wave mechanics for engineers and scientists, Advanced Series on Ocean Engineering 2. World Scientific, Singapore. ISBN 978-9810204204. See page 64–65.

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