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Wave power refers to the energy of ocean surface waves and the capture of that energy to do useful work — including electricity generation, desalination, and the pumping of water (into reservoirs). Wave power is a form of renewable energy. Though often co-mingled, wave power is distinct from the diurnal flux of tidal power and the steady gyre of ocean currents. Wave power generation is not currently a widely employed commercial technology although there have been attempts at using it since at least 1890^[1].

The world's first commercial wave farm is based in Portugal,^[2] at the Aguçadora Wave Park, which consists of three 750 kilowatt Pelamis devices.^[3]

The north and south temperate zones have the best sites for capturing wave power. The prevailing westerlies in these zones blow strongest in winter.

Physical concepts

See Energy, Power and Work for more information on these important physical concepts.

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.^[4] The wave height increases with increases in (see Ocean surface wave):

wind speed,
time duration of the wind blowing,
fetch — the distance of open water that the wind has blown over, and
water depth (in case of shallow water effects, for water depths less than half the wavelength).^[5]

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.^[4] 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).

Wave power formula

In deep water, if the water depth is larger than half the wavelength, the wave energy flux is

P={\frac {\rho g^{2}}{64\pi }}H_{m0}^{2}T\approx \left(0.5{\frac {\text{kW}}{{\text{m}}^{3}\cdot {\text{s}}}}\right)H_{m0}^{2}\;T,

where

P the wave energy flux per unit wave crest length (kW/m);
H_m0 is the significant wave height (meter), as measured by wave buoys and predicted by wave forecast models. By definition,^[6] H_m0 is four times the standard deviation of the water surface elevation;
T is the wave period (second);
ρ is the mass density of the water (kg/m³), and
g is the acceleration by gravity (m/s²).

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 meters, and the wave period in seconds, the result is the wave power in kilowatts (kW) per meter wavefront length.^[7]^[8]

Example: Consider moderate ocean swells, in deep water, a few kilometers off a coastline, with a wave height of 3 meters and a wave period of 8 seconds. Using the formula to solve for power, we get

P\approx 0.5{\frac {\text{kW}}{{\text{m}}^{3}\cdot {\text{s}}}}(3\cdot {\text{m}})^{2}(8\cdot {\text{s}})\approx 36{\frac {\text{kW}}{\text{m}}},

meaning there are 36 kilowatts of power potential per meter of coastline.

In major storms, the largest waves offshore are about 15 meters high and have a period of about 15 seconds. According to the above formula, such waves carry about 1.7 MW/m of power across each meter of wavefront.

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 density per unit area of gravity waves on the water surface is proportional to the wave height squared, according to linear wave theory:^[4]^[6]

E={\frac {1}{16}}\rho gH_{m0}^{2},

^[9]

where E is the mean wave energy density per unit horizontal area (J/m²), the sum of kinetic and potential energy density per unit horizontal area. The potential energy density is equal to the kinetic energy,^[4] both contributing half to the wave energy density E, as can be expected from the equipartition theorem. In ocean waves, surface tension effects are negligible for wavelengths above a few decimetres.

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:^[10]^[4]

P=E\,c_{g},\,\

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:^[4]^[6]

Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to linear wave theory
quantity	symbol	units	deep water ( h > ½ λ )	shallow water ( h < 0.05 λ )	intermediate depth ( all λ and h )
phase velocity	$\displaystyle c_{p}={\frac {\lambda }{T}}={\frac {\omega }{k}}$	m / s	${\frac {g}{2\pi }}T$	${\sqrt {gh}}$	${\sqrt {{\frac {g\lambda }{2\pi }}\tanh \left({\frac {2\pi h}{\lambda }}\right)}}$
group velocity^[11]	$\displaystyle c_{g}=c_{p}^{2}{\frac {\partial \left(\lambda /c_{p}\right)}{\partial \lambda }}={\frac {\partial \omega }{\partial k}}$	m / s	${\frac {g}{4\pi }}T$	${\sqrt {gh}}$	${\frac {1}{2}}c_{p}\left(1+{\frac {4\pi h}{\lambda }}{\frac {1}{\sinh \left(\displaystyle {\frac {4\pi h}{\lambda }}\right)}}\right)$
ratio	$\displaystyle {\frac {c_{g}}{c_{p}}}$	-	$\displaystyle {\frac {1}{2}}$	$\displaystyle 1$	${\frac {1}{2}}\left(1+{\frac {4\pi h}{\lambda }}{\frac {1}{\sinh \left(\displaystyle {\frac {4\pi h}{\lambda }}\right)}}\right)$
wavelength	$\displaystyle \lambda$	m	${\frac {g}{2\pi }}T^{2}$	$T{\sqrt {gh}}$	for given period T, the solution of: $\displaystyle \left({\frac {2\pi }{T}}\right)^{2}={\frac {2\pi g}{\lambda }}\tanh \left({\frac {2\pi h}{\lambda }}\right)$
general
wave energy density	$\displaystyle E$	J / m²	${\frac {1}{16}}\rho gH_{m0}^{2}$
wave energy flux	$\displaystyle P$	W / m	$\displaystyle E\;c_{g}$
angular frequency	$\displaystyle \omega$	rad / s	${\frac {2\pi }{T}}$
wavenumber	$\displaystyle k$	rad / m	${\frac {2\pi }{\lambda }}$

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.^[5]

Modern technology

Wave power devices are generally categorized by the method used to capture the energy of the waves. They can also be categorized by location and power take-off system. Method types are point absorber or buoy; surfacing following or attenuator; terminator, lining perpendicular to wave propagation; oscillating water column; and overtopping. Locations are shoreline, nearshore and offshore. Types of power take-off include: hydraulic ram, elastomeric hose pump, pump-to-shore, hydroelectric turbine, air turbine,^[12] and linear electrical generator. Some of these designs incorporate parabolic reflectors as a means of increasing the wave energy at the point of capture.

These are descriptions of some wave power systems:

In the United States, the Pacific Northwest Generating Cooperative^[13] is funding the building of a commercial wave-power park at Reedsport, Oregon.^[14] The project will utilize the PowerBuoy^[15] technology which consists of modular, ocean-going buoys. The rising and falling of the waves moves the buoy-like structure creating mechanical energy which is converted into electricity and transmitted to shore over a submerged transmission line. A 40 kW buoy has a diameter of 12 feet (4 m) and is 52 feet (16 m) long, with approximately 13 feet of the unit rising above the ocean surface. Using the three-point mooring system, they are designed to be installed one to five miles (8 km) offshore in water 100 to 200 feet (60 m) deep.
An example of a surface following device is the Pelamis Wave Energy Converter. The sections of the device articulate with the movement of the waves, each resisting motion between it and the next section, creating pressurized oil to drive a hydraulic ram which drives a hydraulic motor.^[16]
With the Wave Dragon wave energy converter large "arms" focus waves up a ramp into an offshore reservoir. The water returns to the ocean by the force of gravity via hydroelectric generators.
The AquaBuOY, made by Finavera Renewables Inc., wave energy device: Energy transfer takes place by converting the vertical component of wave kinetic energy into pressurized seawater by means of two-stroke hose pumps. Pressurized seawater is directed into a conversion system consisting of a turbine driving an electrical generator. The power is transmitted to shore by means of a secure, undersea transmission line. A commercial wave power production facility utilizing the AquaBuOY technology is beginning initial construction in Portugal.^[17] The company has 250 MW of projects planned or under development on the west coast of North America.^[18]
A device called CETO, currently being tested off Fremantle, Western Australia,^[19] consists of a single piston pump attached to the sea floor, with a float tethered to the piston. Waves cause the float to rise and fall, generating pressurized water, which is piped to an onshore facility to drive hydraulic generators or run reverse osmosis desalination.^[20]

Challenges

These are some of the challenges to deploying wave power devices:

Efficiently converting wave motion into electricity; generally speaking, wave power is available in low-speed, high forces, and the motion of forces is not in a single direction. Most readily-available electric generators operate at higher speeds, and most readily-available turbines require a constant, steady flow.
Constructing devices that can survive storm damage and saltwater corrosion; likely sources of failure include seized bearings, broken welds, and snapped mooring lines. Knowing this, designers may create prototypes that are so overbuilt that materials costs prohibit affordable production.
High total cost of electricity; wave power will only be competitive when the total cost of generation is reduced. The total cost includes the primary converter, the power takeoff system, the mooring system, installation & maintenance cost, and electricity delivery costs.

Wave farms

The world's first commercial wave farm opened in 2008 at the Aguçadora Wave Park near Póvoa de Varzim in Portugal. It uses three Pelamis P-750 machines with a total installed capacity of 2.25MW.^[21]^[3] A second phase of the project is now planned to increase the installed capacity to 21MW using a further 25 Pelamis machines.^[22]

Funding for a 3MW wave farm in Scotland was announced on February 20, 2007 by the Scottish Executive, at a cost of over 4 million pounds, as part of a £13 million funding packages for marine power in Scotland. The farm will be the world's largest with a capacity of 3MW generated by four Pelamis machines.^[23]

Funding has also been announced for the development of a Wave hub off the north coast of Cornwall, England. The Wave hub will act as giant extension cable, allowing arrays of wave energy generating devices to be connected to the electricity grid. The Wave hub will initially allow 20MW of capacity to be connected with potential expansion to 40MW. Four device manufacturers have so far expressed interest in connecting to the Wave hub.

The scientists have calculated that wave energy gathered at Wave Hub will be enough to power up to 7,500 households. Savings that the Cornwall wave power generator will bring are significant: about 300,000 tons of carbon dioxide in the next 25 years.^[24]

A CETO wave farm of the coast of Western Australia has been operating to prove commercial viability and after preliminary environmental approval is poised for further development. One benefit of CETO is that the buoys that capture the wave motion are submersed and therefore, are not a visual pollutant. Furthermore, the underwater deployment makes them less prone to storm damage.

Potential

It is estimated that world-wide the economically recoverable power from waves is 2 Terawatt annual average ^[25]. Locations with the most potential for wave power include; the western seaboard of Europe, the northern coast of the UK and the Pacific coastlines of North and South America, Southern Africa, Australia and New Zealand. The UK has an estimated recoverable resource of between 50-90TWh of electricity a year, this is roughly 15-25% of the current UK electricity demand ^[26]

Discussion of Salter's Duck

While historic references to the power of waves do exist, the modern scientific pursuit of wave energy was begun in the 1970s by Professor Stephen Salter of the University of Edinburgh, Scotland in response to the Oil Crisis.

His invention became known as Salter's Duck, although it was officially referred to as the Edinburgh Duck. In small scale controlled tests, the Duck's curved cam-like body can stop 90% of wave motion and can convert 90% of that to electricity.^[27] The machine has never gone to sea, primarily because its complex hydraulic system is not well suited to incremental implementation, and the costs and risks of a full-scale test would be high. Most of the designs being tested currently absorb far less of the available wave power, and as a result their Mass to Power Ratios remain far away from the theoretical maximum.

According to sworn testimony before the House of Parliament, The UK Wave Energy program was shut down on March 19, 1982, in a closed meeting,^[28] the details of which remain secret. The members of the meeting were recruited largely from the nuclear and fossil fuels industries, and the wave programme manager, Clive Grove-Palmer, was excluded.

An analysis^[29] of Salter's Duck resulted in a miscalculation of the estimated cost of energy production by a factor of 10, an error which was only recently identified. Some wave power advocates believe that this error, combined with a general lack of enthusiasm for renewable energy in the 1980s (after oil prices fell), hindered the advancement of wave power technology.^[30]

Patents

U.S. patent 3,928,967 — Apparatus and method of extracting wave energy - The original "Salter's Duck" patent
U.S. patent 4,134,023 — Apparatus for use in the extraction of energy from waves on water - Salter's method for improving "duck" efficiency
U.S. patent 6,194,815 — Piezoelectric rotary electrical energy generator - Assignee: Ocean Power Technologies
US application 20,040,217,597 — Wave energy converters utilizing pressure differences - PowerBuoy Video

References

Twidell, John; Weir, Anthony D.; Weir, Tony (2006), Renewable Energy Resources, Taylor & Francis, ISBN 0419253300, 601 pp.

Notes

^ Christine Miller (Aug 2004). "Wave and Tidal Energy Experiments in San Francisco and Santa Cruz". Retrieved 2008-08-16.
^ Wave power scientist enthused by green energy
^ ^a ^b Alok Jha (25 September 2008). "Making waves: UK firm harnesses power of the sea ... in Portugal". The Guardian. Retrieved 2008-10-09.
^ ^a ^b ^c ^d ^e ^f Phillips, O.M. (1977). The dynamics of the upper ocean (2nd edition ed.). Cambridge University Press. ISBN 0 521 29801 6. {{cite book}}: |edition= has extra text (help)
^ ^a ^b R. G. Dean and R. A. Dalrymple (1991). Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering. Vol. 2. World Scientific, Singapore. ISBN 978-9810204204. See page 64–65.
^ ^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
^ For a small-amplitude sinusoidal wave $\scriptstyle \eta =a\,\cos \,2\pi \left({\frac {x}{\lambda }}-{\frac {t}{T}}\right)$ with wave amplitude $\scriptstyle a,\,$ the wave energy density per unit horizontal area is $\scriptstyle E={\frac {1}{2}}\rho ga^{2},$ or $\scriptstyle E={\frac {1}{8}}\rho gH^{2}$ using the wave height $\scriptstyle H\,=\,2\,a\,$ for sinusoidal waves. In terms of the variance of the surface elevation $\scriptstyle m_{0}=\sigma _{\eta }^{2}={\overline {(\eta -{\bar {\eta }})^{2}}}={\frac {1}{2}}a^{2},$ the energy density is $\scriptstyle E=\rho gm_{0}\,$ . Turning to random waves, the last formulation of the wave energy equation in terms of $\scriptstyle m_{0}\,$ is also valid (Holthuijsen, 2007, p. 40), due to Parseval's theorem. Further, the significant wave height is defined as $\scriptstyle H_{m0}=4{\sqrt {m_{0}}}$ , leading to the factor 1⁄16 in the wave energy density per unit horizontal area.
Reference: Holthuijsen, Leo H. (2007). Waves in oceanic and coastal waters. Cambridge: Cambridge University Press. ISBN 0521860288.
^ 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.
^ Embedded Shoreline Devices and Uses as Power Generation Sources Kimball, Kelly, November 2003
^ PNGC Power
^ Agreement to Develop Wave Power Park in Oregon from www.renewableeneregyaccess.com February 2007
^ OPT | Ocean Power Technologies
^ Jenny Haworth (24 September 2008). "If Portugal can rule the waves, why not Scotland?". The Scotsman. Retrieved 2008-10-09.
^ Wave Energy: Figueira da Foz, Portugal Finavera Renewables
^ Wave Energy Device Deployed
^ Stephen Cauchi (October 5, 2008). "New wave of power in renewable energy market". The Age. Retrieved 2008-10-10.
^ CETO Overview Carnegie Corporation
^ "Portugal Goverenment". Retrieved 2008-09-24.
^ Joao Lima. "Babcock, EDP and Efacec to Collaborate on Wave Energy Projects". Bloomberg Television. Retrieved 2008-09-24. {{cite web}}: Cite has empty unknown parameter: |1= (help)
^ Orkney to get 'biggest' wave farm BBC February 2007
^ Go-ahead for £28m Cornish wave farm
^ "Wave-Energy" (PDF). Retrieved 2008-10-13.
^ "Pelamis Wave Power". pelamiswave.com. Retrieved 2008-10-13. {{cite web}}: Italic or bold markup not allowed in: |publisher= (help)
^ Edinburgh Wave Energy Project
^ Memorandum submitted by Professor S H Salter, Department of Mechanical Engineering, University of Edinburgh House of Commons, UK Parliament
^ Water Power Devices
^ The untimely death of Salter's Duck from GreenLeftOnline July 1992

External links

Institutional links

News articles and compilations

Kate Galbraith (September 22, 2008). "Power From the Restless Sea Stirs the Imagination". New York Times. Retrieved 2008-10-09.
"Wave Power: The Coming Wave" from the Economist, June 5, 2008
"Is wave power commercially viable?"
"The untimely death of Salter's Duck"
"Ocean Power Fights Current Thinking"
"Wave energy in New Zealand"
"How it works: Wave power station"
"Oceanography: Waves. Theory and principles of waves, how they work and what causes them", at Seafriends, New Zealand
"Waves"
Waves power future from the Corvallis Gazette Times, February 5, 2005
EU Wavetrain project — A series of full-text, on-line scientific publications on physical concepts.

Wave climate and forecasts

Information on the long-term statistics of ocean waves can be found at the KNMI wave atlas.
The US National Oceanic and Atmospheric Administration provides imagery and forecasts of wave height on a global scale. On-line NOAA animation of a user-specified region-of-interest can be set to either wave height or wave period forecasts.

[1] Christine Miller (Aug 2004). "Wave and Tidal Energy Experiments in San Francisco and Santa Cruz". Retrieved 2008-08-16.

[2] Wave power scientist enthused by green energy

[TheGuardian_2008_09_25-3] Alok Jha (25 September 2008). "Making waves: UK firm harnesses power of the sea ... in Portugal". The Guardian. Retrieved 2008-10-09.

[Phillips-4] ^ ^a ^b ^c ^d ^e ^f Phillips, O.M. (1977). The dynamics of the upper ocean (2nd edition ed.). Cambridge University Press. ISBN 0 521 29801 6. {{cite book}}: |edition= has extra text (help)

[Dean_Dalrymple-5] R. G. Dean and R. A. Dalrymple (1991). Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering. Vol. 2. World Scientific, Singapore. ISBN 978-9810204204. See page 64–65.

[Goda-6] Goda, Y. (2000). Random Seas and Design of Maritime Structures. World Scientific. ISBN 978 981 02 3256 6.

[7] Wave Power

[8] Technology White Paper on Wave Energy Potential on the U.S. Outer Continental Shelf

[9] For a small-amplitude sinusoidal wave $\scriptstyle \eta =a\,\cos \,2\pi \left({\frac {x}{\lambda }}-{\frac {t}{T}}\right)$ with wave amplitude $\scriptstyle a,\,$ the wave energy density per unit horizontal area is $\scriptstyle E={\frac {1}{2}}\rho ga^{2},$ or $\scriptstyle E={\frac {1}{8}}\rho gH^{2}$ using the wave height $\scriptstyle H\,=\,2\,a\,$ for sinusoidal waves. In terms of the variance of the surface elevation $\scriptstyle m_{0}=\sigma _{\eta }^{2}={\overline {(\eta -{\bar {\eta }})^{2}}}={\frac {1}{2}}a^{2},$ the energy density is $\scriptstyle E=\rho gm_{0}\,$ . Turning to random waves, the last formulation of the wave energy equation in terms of $\scriptstyle m_{0}\,$ is also valid (Holthuijsen, 2007, p. 40), due to Parseval's theorem. Further, the significant wave height is defined as $\scriptstyle H_{m0}=4{\sqrt {m_{0}}}$ , leading to the factor 1⁄16 in the wave energy density per unit horizontal area.
Reference: Holthuijsen, Leo H. (2007). Waves in oceanic and coastal waters. Cambridge: Cambridge University Press. ISBN 0521860288.

[10] 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.

[11] 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.

[12] Embedded Shoreline Devices and Uses as Power Generation Sources Kimball, Kelly, November 2003

[13] PNGC Power

[14] Agreement to Develop Wave Power Park in Oregon from www.renewableeneregyaccess.com February 2007

[15] OPT | Ocean Power Technologies

[16] Jenny Haworth (24 September 2008). "If Portugal can rule the waves, why not Scotland?". The Scotsman. Retrieved 2008-10-09.

[17] Wave Energy: Figueira da Foz, Portugal Finavera Renewables

[18] Wave Energy Device Deployed

[19] Stephen Cauchi (October 5, 2008). "New wave of power in renewable energy market". The Age. Retrieved 2008-10-10.

[20] CETO Overview Carnegie Corporation

[21] "Portugal Goverenment". Retrieved 2008-09-24.

[22] Joao Lima. "Babcock, EDP and Efacec to Collaborate on Wave Energy Projects". Bloomberg Television. Retrieved 2008-09-24. {{cite web}}: Cite has empty unknown parameter: |1= (help)

[autogenerated1-23] Orkney to get 'biggest' wave farm BBC February 2007

[24] Go-ahead for £28m Cornish wave farm

[25] "Wave-Energy" (PDF). Retrieved 2008-10-13.

[26] "Pelamis Wave Power". pelamiswave.com. Retrieved 2008-10-13. {{cite web}}: Italic or bold markup not allowed in: |publisher= (help)

[27] Edinburgh Wave Energy Project

[28] Memorandum submitted by Professor S H Salter, Department of Mechanical Engineering, University of Edinburgh House of Commons, UK Parliament

[29] Water Power Devices

[30] The untimely death of Salter's Duck from GreenLeftOnline July 1992

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