# Wave power

Wave power is the transport of energy by ocean surface waves, and the capture of that energy to do useful work – for example, electricity generation, water desalination, or the pumping of water (into reservoirs). Machinery able to exploit wave power is generally known as a wave energy converter (WEC).

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 to use it since at least 1890.1 In 2008, the first experimental wave farm was opened in Portugal, at the Aguçadoura Wave Park.2 The major competitor of wave power is offshore wind power.

## Physical concepts

When an object bobs up and down on a ripple in a pond, it experiences an elliptical trajectory.
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.
Photograph of the water particle orbits under a – progressive and periodic – surface gravity wave in a wave flume. The wave conditions are: mean water depth d = 2.50 ft (0.76 m), wave height H = 0.339 ft (0.103 m), wavelength λ = 6.42 ft (1.96 m), period T = 1.12 s.3

Waves are generated by wind passing over the surface of 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 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, making the water to go into the shear stress causes the growth of the waves.4

Wave height is determined by wind speed, the duration of time the wind has been blowing, 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. When this limit has been reached the sea is said to be "fully developed".

In general, larger waves are more powerful but wave power is also determined by wave speed, wavelength, and water density.

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 where the water depth is larger than half the wavelength, the wave energy flux isa

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

with P the wave energy flux per unit of wave-crest length, Hm0 the significant wave height, Te the wave energy period, ρ the water density and g the acceleration by gravity. The above formula states that wave power is proportional to the wave energy period and to the square of the wave height. When the significant wave height is given in metres, and the wave period in seconds, the result is the wave power in kilowatts (kW) per metre of wavefront length.5678

Example: Consider moderate ocean swells, in deep water, a few km off a coastline, with a wave height of 3 m and a wave energy 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 wave crest.

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 of power across each metre 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 average energy density per unit area of gravity waves on the water surface is proportional to the wave height squared, according to linear wave theory:49

$E=\frac{1}{16}\rho g H_{m0}^2,$ b10

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:114

$P = E\, c_g, \, \$

with cg 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:49

### Deep-water characteristics and opportunities

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 about twenty times the water depth, as found quite often near the coast, the group velocity is equal to the phase velocity.12

## History

The first known patent to use energy from ocean waves dates back to 1799 and was filed in Paris by Girard and his son.13 An early application of wave power was a device constructed around 1910 by Bochaux-Praceique to light and power his house at Royan, near Bordeaux in France.14 It appears that this was the first oscillating water-column type of wave-energy device.15 From 1855 to 1973 there were already 340 patents filed in the UK alone.13

Modern scientific pursuit of wave energy was pioneered by Yoshio Masuda's experiments in the 1940s.16 He has tested various concepts of wave-energy devices at sea, with several hundred units used to power navigation lights. Among these was the concept of extracting power from the angular motion at the joints of an articulated raft, which was proposed in the 1950s by Masuda.17

A renewed interest in wave energy was motivated by the oil crisis in 1973. A number of university researchers re-examined the potential to generate energy from ocean waves, among whom notably were Stephen Salter from the University of Edinburgh, Kjell Budal and Johannes Falnes from Norwegian Institute of Technology (now merged into Norwegian University of Science and Technology), Michael E. McCormick from U.S. Naval Academy, David Evans from Bristol University, Michael French from University of Lancaster, Nick Newman and C. C. Mei from MIT.

Stephen Salter's 1974 invention became known as Salter's duck or nodding 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 giving 81% efficiency.18

In the 1980s, as the oil price went down, wave-energy funding was drastically reduced. Nevertheless, a few first-generation prototypes were tested at sea. More recently, following the issue of climate change, there is again a growing interest worldwide for renewable energy, including wave energy.19

## Modern technology

Wave power devices are generally categorized by the method used to capture the energy of the waves, by location and by the power take-off system. Locations are shoreline, nearshore and offshore. Types of power take-off include: hydraulic ram, elastomeric hose pump, pump-to-shore, hydroelectric turbine, air turbine,20 and linear electrical generator. When evaluating wave energy as a technology type, it is important to distinguish between the four most common approaches: point absorber buoys, surface attenuators, oscillating water columns, and overtopping devices.

### Point Absorber Buoy

This device floats on the surface of the water, held in place by cables connected to the seabed. Buoys use the rise and fall of swells to drive hydraulic pumps and generate electricity. EMF generated by electrical transmission cables and acoustic of these devices may be a concern for marine organisms. The presence of the buoys may affect fish, marine mammals, and birds as potential minor collision risk and roosting sites. Potential also exists for entanglement in mooring lines. Energy removed from the waves may also affect the shoreline, resulting in a recommendation that sites remain a considerable distance from the shore.21

### Surface Attenuator

These devices act similarly to point absorber buoys, with multiple floating segments connected to one another and are oriented perpendicular to incoming waves. A flexing motion is created by swells that drive hydraulic pumps to generate electricity. Environmental effects are similar to those of point absorber buoys, with an additional concern that organisms could be pinched in the joints.21

### Oscillating Water Column

Oscillating water column devices can be located on shore or in deeper waters offshore. With an air chamber integrated into the device, swells compress air in the chambers forcing air through an air turbine to create electricity. Significant noise is produced as air is pushed through the turbines, potentially affecting birds and other marine organisms within the vicinity of the device. There is also concern about marine organisms getting trapped or entangled within the air chambers.21

### Overtopping Device

Overtopping devices are long structures that use wave velocity to fill a reservoir to a greater water level than the surrounding ocean. The potential energy in the reservoir height is then captured with low-head turbines. Devices can be either on shore or floating offshore. Floating devices will have environmental concerns about the mooring system affecting benthic organisms, organisms becoming entangled, or EMF effects produced from subsea cables. There is also some concern regarding low levels of turbine noise and wave energy removal affecting the nearfield habitat.21

### Oscillating Wave Surge Converter

These devices typically have one end fixed to a structure or the seabed while the other end is free to move. Energy is collected from the relative motion of the body compared to the fixed point. Oscillating wave surge converters often come in the form of floats, flaps, or membranes. Environmental concerns include minor risk of collision, artificial reefing near the fixed point, EMF effects from subsea cables, and energy removal effecting sediment transport.21 Some of these designs incorporate parabolic reflectors as a means of increasing the wave energy at the point of capture. These capture systems use the rise and fall motion of waves to capture energy.22 Once the wave energy is captured at a wave source, power must be carried to the point of use or to a connection to the electrical grid by transmission power cables.23

The table contains descriptions of some wave power systems:

Device Proponent Country of origin Capture method Location Power take off Year build Notes
Anaconda Wave Energy Converter Checkmate SeaEnergy.[25] UK Surface-following attenuator Offshore Hydroelectric turbine 2008 In the early stages of development, the device is a 200 metres (660 ft) long rubber tube which is tethered underwater. Passing waves will instigate a wave inside the tube, which will then propagates down its walls, driving a turbine at the far end.2425
AquaBuOY Finavera Wind Energy, later SSE Renewables Limited Ireland-Canada-Scotland Buoy Offshore Hydroelectric turbine 2003 In 2009 Finavera Renewables surrendered its wave energy permits from FERC.[27] In July 2010 Finavera announced that it had entered into a definitive agreement to sell all assets and intellectual property related to the AquaBuOY wave energy technology.26272829
AWS-iii AWS Ocean Energy UK (Scotland) Surface-following attenuator? Offshore Air turbine 2010 The AWS-III is a floating toroidal vessel. It has rubber membranes on the outer faces which deform as waves pass, moving air inside chambers which in turn drive air-turbines to generate electricity. AWS Ocean tested a 1/9 scale model in Loch Ness in 2010, and are now working on a full sized version which will be 60m across and should generate 2.5 MW. It is envisage these will be installed in offshore farms moored in around 100m depth of water.30313233
CETO Wave Power Carnegie Australia Buoy Offshore Pump-to-shore 1999 As of 2008, the device is being tested off Fremantle, Western Australia,[35] the device consists of a single piston pump attached to the sea floor with a float (buoy) 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 water desalination.3435
Cycloidal Wave Energy Converter Atargis Energy Corporation USA Fully Submerged Wave Termination Device Offshore Direct Drive Generator 2006 In the tank testing stage of development, the device is a 20 metres (66 ft) diameter fully submerged rotor with two hydrofoils. Numerical studies have shown greater than 99% wave power termination capabilities.36 These were confirmed by experiments in a small 2D wave flume37 as well as a large offshore wave basin.
FlanSea (Flanders Electricity from the Sea) FlanSea Belgium Buoy Offshore Hydroelectric turbine 2010 A point absorber buoy developed for use in the southern North Sea conditions.[31][32][33] It works by means of a cable that due to the bobbing effect of the buoy, generates electricity.383940
Islay LIMPET Islay LIMPET Scotland oscillating water column Onshore Air turbine 1991 500 kW shoreline device uses an oscillating water column to drive air in and out of a pressure chamber through a Wells turbine.414243
Lysekil Project Uppsala University Sweden Buoy Offshore Linear generator 2002 Direct driven linear generator placed on the seabed, connected to a buoy at the surface via a line. The movements of the buoy will drive the translator in the generator.4445
Oceanlinx Oceanlinx Australia OWC Nearshore & Offshore air turbine 1997 Wave energy is captured with an Oscillating Water Column and electricity is generated by air flowing through a turbine. The third medium scale demonstration unit near Port Kembla, NSW, Australia, a medium scale system that was grid connected in early 2010.46

In May 2010, the wave energy generator snapped from its mooring lines in extreme seas and sank on Port Kembla's eastern breakwater.47

A full scale commercial nearshore unit, greenWAVE, with a capacity of 1MW will be installed off Port MacDonnell in South Australia before the end of 2013.48

OE buoy Ocean Energy Ireland Buoy Offshore Air turbine 2006 In September 2009 completed a 2-year sea trial in one quarter scale form. The OE buoy has only one moving part.49
OWEL Ocean Wave Energy Ltd UK Wave Surge Converter Offshore Air turbine 2013 The surging motion of long period waves compresses air in a tapered duct which is then used to drive an air turbine mounted on top of the floating vessel.50 The design of a full scale demonstration project was completed in Spring 2013, ready for fabrication.51
Oyster wave energy converter Aquamarine Power UK (Scots-Irish) Oscillating wave surge converter Nearshore Pump-to-shore (hydro-electric turbine) 2005 A hinged mechanical flap attached to the seabed captures the energy of nearshore waves. It drives hydraulic pistons to deliver high pressure water to an onshore turbine which generates electricity. In November 2009, the first full-scale demonstrator Oyster began producing power on Orkney.52
Pelamis Wave Energy Converter Pelamis Wave Power UK (Scottish) Surface-following attenuator Offshore Hydraulic 1998 As waves pass along a series of semi-submerged cylindrical sections linked by hinged joints, the sections move relative to one another. This motion activates hydraulic cylinders which pump high pressure oil through hydraulic motors which drive electrical generators.53 The first working Pelamis machine in 2004 was at the European Marine Energy Center.54 The later P2, owned by E.ON, started grid connected tests off Orkney in 2010.55
PowerBuoy Ocean Power Technologies US Buoy Offshore Hydroelectric turbine 1997 The Pacific Northwest Generating Cooperative is funding construction of a commercial wave-power park at Reedsport, Oregon using buoys.56 The rise and fall of the waves moves a rack and pinion within the buoy and spins a generator.57 The electricity is transmitted by a submerged transmission line. The buoys are designed to be installed one to five miles (8 km) offshore in water 100 to 200 feet (60 m) deep.58
PB150 PowerBuoy with peak-rated power output of 150 kW.
R38/50 kW, R115/150 kW 40South Energy UK Underwater attenuator Offshore Electrical conversion 2010 These machines work by extracting energy from the relative motion between one Upper Member and one Lower Member, following an innovative method which earned the company one UKTI Research & Development Award in 2011.59 A first generation full scale prototype for this solution was tested offshore in 2010,606162 and a second generation full scale prototype was tested offshore during 2011.63 In 2012 the first units were sold to clients in various countries, for delivery within the year.6465 The first reduced scale prototypes were tested offshore during 2007, but the company decided to remain in a "stealth mode" until May 201066 and is now recognized as one of the technological innovators in the sector.67 The company initially considered installing at Wave Hub in 2012,68 but that project is on hold for now. The R38/50 kW is rated at 50 kW while the R115/150 kW is rated at 150 kW.
SDE Sea Waves Power Plant SDE Energy Ltd. Israel Buoy Nearshore Hydroelectric turbine 2010 A breakwater-based wave energy converter, this device is built close to the shore and utilizes the vertical motion of buoys for creating hydraulic pressure which in turn operates the system's generators. In 2010 it began construction of a new 250 kWh model in the port of Jaffa, Tel Aviv and preparing to construct its standing orders for a 100 MWh power plants in the islands of Zanzibar and Kosrae, Micronesia.
SeaRaser Alvin Smith (Dartmouth Wave Energy)\Ecotricity UK Buoy Nearshore Hydraulic ram 2008 Consisting of a piston pump(s) attached to the sea floor with a float (buoy) tethered to the piston. Waves cause the float to rise and fall, generating pressurized water, which is piped to resoviors onshore which then drive hydraulic generators.6970

It is currently "undergoing extensive modelling ahead of a sea trial" 71

Squid/ WaveNET AlbaTERN UK (Scotland) Multi-point absorber Nearshore Hydraulic? 2011 A 10 kW Squid prototype was tested at EMEC in 2011.72 The company have since secured funding through the WATERS2 project, to further develop the device including developing arrays.73
Unnamed Ocean Wave-Powered Generator SRI International US Buoy Offshore Electroactive polymer artificial muscle 2004 A type of wave buoys, built using special polymers, is being developed by SRI International.7475
Wavebob Wavebob Ireland Buoy Offshore Direct Drive Power Take off 1999 Wavebob have conducted some ocean trials, as well as extensive tank tests. It is an ocean-going heaving buoy, with a submerged tank which captures additional mass of seawater for added power and tunability, and as a safety feature (Tank "Venting")
Wavepiston Wavepiston ApS Denmark Oscillating wave surge converter Nearshore Pump-to-shore (hydro-electric turbine) 2013 The idea behind this concept is to reduce the mooring means for wave energy structures. Wavepiston systems use vertical plates to exploit the horizontal movement in ocean waves. By attaching several plates in parallel on a single structure the forces applied on the structure by the plates will tend to neutralize each other. This neutralization reduces the required mooring means. “Force cancellation” is the term used by the inventors of the technology to describe the neutralization of forces. A Wavepiston system will comprise a long (several wavelengths) floating structure, anchored by its ends, such that the predominant wave direction is along the structure.

The simplest and strongest floating structure imaginable is a steel cable fitted with floaters and anchored by slack moorings in both ends. An inherent feature of slack mooring is that the mooring system compensates for variations in water level due to tidal.

Wave Dragon Erik Friis-Madsen Denmark Overtopping device Offshore Hydroelectric turbine 2003 With the Wave Dragon wave energy converter large wing reflectors focus waves up a ramp into an offshore reservoir. The water returns to the ocean by the force of gravity via hydroelectric generators.
Wave Dragon seen from reflector, prototype 1:4½
WaveRoller AW-Energy Oy Finland Oscillating wave surge converter Nearshore Hydraulic 1994 The WaveRoller is a plate anchored on the sea bottom by its lower part. The back and forth movement of surge moves the plate. The kinetic energy transferred to this plate is collected by a piston pump. Full-scale demonstration project built off Portugal in 2009.7677
WaveRoller farm installation in Peniche, Portugal. August 2012
Wave Star Wave Star A/S Denmark Multi-point absorber Offshore Hydroelectric turbine 2000 The Wavestar machine draws energy from wave power with floats that rise and fall with the up and down motion of waves. The floats are attached by arms to a platform that stands on legs secured to the sea floor. The motion of the floats is transferred via hydraulics into the rotation of a generator, producing electricity. Wave Star has been testing a 1:10 machine since 2005 in Nissum Bredning, Denmark, it was taken out of duty in November 2011. A 1:2 Wave Star machine is in place in Hanstholm which has produced electricity to the grid since September 2009.78

### Environmental Effects

Common environmental concerns associated with marine energy developments include:

• The risk of marine mammals and fish being struck by tidal turbine blades;
• The effects of EMF and underwater noise emitted from operating marine energy devices;
• The physical presence of marine energy projects and their potential to alter the behavior of marine mammals, fish, and seabirds with attraction or avoidance;
• The potential effect on nearfield and farfield marine environment and processes such as sediment transport and water quality.

The Tethys database is an online knowledge management system that provides the marine energy community with access to information and scientific literature on environmental effects of marine energy developments.79

## Potential

The worldwide resource of wave energy has been estimated to be greater than 2 TW.80 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 north and south temperate zones have the best sites for capturing wave power. The prevailing westerlies in these zones blow strongest in winter.

World wave energy resource map

## Challenges

There is a potential impact on the marine environment. Noise pollution, for example, could have negative impact if not monitored, although the noise and visible impact of each design varies greatly.7 Other biophysical impacts (flora and fauna, sediment regimes and water column structure and flows) of scaling up the technology is being studied.81 In terms of socio-economic challenges, wave farms can result in the displacement of commercial and recreational fishermen from productive fishing grounds, can change the pattern of beach sand nourishment, and may represent hazards to safe navigation.82 Waves generate about 2,700 gigawatts of power. Of those 2,700 gigawatts, only about 500 gigawatts can be captured with the current technology.22

## Wave farms

### Portugal

• The Aguçadoura Wave Farm was the world's first wave farm. It was located 5 km (3 mi) offshore near Póvoa de Varzim, north of Porto, Portugal. The farm was designed to use three Pelamis wave energy converters to convert the motion of the ocean surface waves into electricity, totalling to 2.25 MW in total installed capacity. The farm first generated electricity in July 200883 and was officially opened on September 23, 2008, by the Portuguese Minister of Economy.8485 The wave farm was shut down two months after the official opening in November 2008 as a result of the financial collapse of Babcock & Brown due to the global economic crisis. The machines were off-site at this time due to technical problems, and although resolved have not returned to site and were subsequently scrapped in 2011 as the technology had moved on to the P2 variant as supplied to Eon and Scottish Power Renewables.86 A second phase of the project planned to increase the installed capacity to 21 MW using a further 25 Pelamis machines87 is in doubt following Babcock's financial collapse.

### United Kingdom

• Funding for a 3 MW 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 package for marine power in Scotland. The first of 66 machines was launched in May 2010.88
• A facility known as Wave hub has been constructed off the north coast of Cornwall, England, to facilitate wave energy development. 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 20 MW of capacity to be connected, with potential expansion to 40 MW. Four device manufacturers have so far expressed interest in connecting to the Wave hub.8990 The scientists have calculated that wave energy gathered at Wave Hub will be enough to power up to 7,500 households. The site has the potential to save greenhouse gas emissions of about 300,000 tons of carbon dioxide in the next 25 years.91

### Australia

• A CETO wave farm off the coast of Western Australia has been operating to prove commercial viability and, after preliminary environmental approval, is poised for further development.citation needed9293
• Ocean Power Technologies (OPT Australasia Pty Ltd) is developing a wave farm connected to the grid near Portland, Victoria through a 19 MW wave power station. The project has received an AU \$66.46 million grant from the Federal Government of Australia.94
• Oceanlinx will deploy a commercial scale demonstrator off the coast of South Australia at Port MacDonnell before the end of 2013. This device, the greenWAVE, has a rated electrical capacity of 1MW. This project has been supported by ARENA through the Emerging Renewables Program. The greenWAVE device is a bottom standing gravity structure, that does not require anchoring or seabed preparation and with no moving parts below the surface of the water.48

### United States

• Reedsport, Oregon – a commercial wave park on the west coast of the United States located 2.5 miles offshore near Reedsport, Oregon. The first phase of this project is for ten PB150 PowerBuoys, or 1.5 megawatts.9596 The Reedsport wave farm was scheduled for installation spring 2013.97 Project has ground to a halt because of legal and technical problems, August, 2013. See:-

http://www.oregonlive.com/environment/index.ssf/2013/08/oregon_wave_energy_stalls_off.html

## Notes

1. ^ The energy flux is $P = \tfrac{1}{16} \rho g H_{m0}^2 c_g,$ with $c_g$ the group velocity, see Herbich, John B. (2000). Handbook of coastal engineering. McGraw-Hill Professional. p. A.117, Eq. (12). ISBN 978-0-07-134402-9. The group velocity is $c_g=\tfrac{g}{4\pi}T$, see the collapsed table "Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to linear wave theory" in the section "Wave energy and wave energy flux" below.
2. ^ Here, the factor for random waves is 116, as opposed to 18 for periodic waves – as explained hereafter. 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 $\scriptstyle E=\frac{1}{2}\rho g a^2,$ or $\scriptstyle E=\frac{1}{8}\rho g H^2$ 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 $\scriptstyle E=\rho g m_0\,$. 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 116 in the wave energy density per unit horizontal area.
3. ^ 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.

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