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Wave power
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== Physical concepts == {{Main|Airy wave theory}} Like most fluid motion, the interaction between ocean waves and energy converters is a high-order nonlinear phenomenon. It is described using the [[incompressible Navier–Stokes equations]] <math display="block">\begin{align} \frac{\partial\vec{u}}{\partial t}+(\vec{u}\cdot\vec{\nabla})\vec{u}&=\nu\Delta\vec{u}+\frac{\vec{F_\text{ext}}-\vec{\nabla}p}{\rho} \\ \vec{\nabla}\cdot\vec{u}&=0 \end{align} </math>where <math display="inline">\vec u(t, x, y, z)</math> is the fluid velocity, <math display="inline">p </math> is the [[pressure]], <math display="inline">\rho </math> the [[density]], <math display="inline">\nu </math> the [[viscosity]], and <math display="inline">\vec{F_\text{ext}} </math> the net external force on each fluid particle (typically [[gravity]]). Under typical conditions, however, the movement of waves is described by [[Airy wave theory]], which posits that * fluid motion is roughly [[irrotational]], * pressure is approximately constant at the water surface, and * the [[seabed]] depth is approximately constant. In situations relevant for energy harvesting from ocean waves these assumptions are usually valid. === Airy equations === The first condition implies that the motion can be described by a [[velocity potential]] <math display="inline"> \phi(t,x,y,z)</math>:<ref>{{Cite book |title=Numerical modelling of wave energy converters : state-of-the-art techniques for single devices and arrays |date=2016 |first=Matt |last=Folley |isbn=978-0-12-803211-4 |publisher=Academic Press |location=London, UK |oclc=952708484}}</ref><math display="block"> {\vec{\nabla}\times\vec{u}=\vec{0}}\Leftrightarrow{\vec{u}=\vec{\nabla}\phi}\text{,}</math>which must satisfy the [[Laplace's equation|Laplace equation]],<math display="block"> \nabla^2\phi=0\text{.}</math>In an ideal flow, the viscosity is negligible and the only external force acting on the fluid is the earth gravity <math> \vec{F_\text{ext}}=(0,0,-\rho g)</math>. In those circumstances, the [[Navier–Stokes equations]] reduces to <math display="block">{\partial\vec\nabla\phi \over\partial t}+{1 \over2}\vec \nabla\bigl(\vec\nabla\phi\bigr)^2= -{1 \over \rho}\cdot\vec\nabla p +{1 \over \rho}\vec\nabla\bigl(\rho gz\bigr), </math>which integrates (spatially) to the [[Bernoulli Equation|Bernoulli conservation law]]:<math display="block">{\partial\phi \over\partial t}+{1 \over2}\bigl(\vec\nabla\phi\bigr)^2 +{1 \over \rho} p + gz=(\text{const})\text{.} </math> === Linear potential flow theory === [[File:Wave motion-i18n-mod.svg|thumb|Motion of a particle in an ocean wave.<br /> '''A''' = At deep water. The [[circular motion]] magnitude of fluid particles decreases exponentially with increasing depth below the surface.<br /> '''B''' = At shallow water (ocean floor is now at B). The elliptical movement of a fluid particle flattens with decreasing depth.<br /> '''1''' = Propagation direction. <br /> '''2''' = Wave crest.<br /> '''3''' = Wave trough.]] When considering small amplitude waves and motions, the quadratic term <math display="inline">\left(\vec{\nabla}\phi\right)^2 </math> can be neglected, giving the linear Bernoulli equation,<math display="block">{\partial\phi \over\partial t}+{1 \over \rho} p + gz=(\text{const})\text{.} </math>and third Airy assumptions then imply<math display="block">\begin{align} &{\partial^2\phi \over\partial t^2} + g{\partial\phi \over\partial z}=0\quad\quad\quad(\text{surface}) \\ &{\partial\phi \over\partial z}=0\phantom{{\partial^2\phi \over\partial t^2}+{}}\,\,\quad\quad\quad(\text{seabed}) \end{align} </math>These constraints entirely determine [[sinusoidal]] wave solutions of the form <math display="block">\phi=A(z)\sin{\!(kx-\omega t)}\text{,} </math>where <math>k </math> determines the [[wavenumber]] of the solution and <math>A(z) </math> and <math>\omega </math> are determined by the boundary constraints (and <math>k </math>). Specifically,<math display="block">\begin{align} &A(z)={gH \over 2\omega}{\cosh(k(z+h)) \over \cosh(kh)} \\ &\omega=gk\tanh(kh)\text{.} \end{align} </math>The surface elevation <math>\eta </math> can then be simply derived as <math display="block">\eta=-{1 \over g}{\partial \phi \over \partial t}={H \over 2}\cos(kx-\omega t)\text{:} </math>a plane wave progressing along the x-axis direction. ==== Consequences ==== [[Ocean surface wave#Science of waves|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 [[microseism]]s.<ref name="Phillips" /> Pressure fluctuations at greater depth are too small to be interesting for wave power conversion. The behavior of Airy waves offers two interesting regimes: water deeper than half the wavelength, as is common in the sea and ocean, and shallow water, with wavelengths larger than about twenty times the water depth. Deep waves are [[Dispersion (water waves)|dispersionful]]: Waves of long wavelengths propagate faster and tend to outpace those with shorter wavelengths. Deep-water group velocity is half the [[phase velocity]]. Shallow water waves are dispersionless: group velocity is equal to phase velocity, and [[wavetrain]]s propagate undisturbed.<ref name="Phillips" /><ref name="Dean_Dalrymple">{{cite book |author1=R. G. Dean |title=Water wave mechanics for engineers and scientists |author2=R. A. Dalrymple |publisher=World Scientific, Singapore |year=1991 |isbn=978-981-02-0420-4 |series=Advanced Series on Ocean Engineering |volume=2 |name-list-style=amp}} See page 64–65.</ref><ref name="Goda" /> The following table summarizes the behavior of waves in the various regimes: {| class="wikitable collapsible collapsed" style="width:65%; text-align:center;" |+ Airy gravity waves on the surface of deep water, shallow water, or intermediate depth<!-- Data is not present --> |- ! style="width:10%;" | quantity ! style="width:7%;" | symbol ! style="width:7%;" | units ! style="width:15%;" | deep water<br>(''h'' > {{1/2}} ''λ'') ! style="width:25%;" | shallow water<br>(''h'' < 0.05 ''λ'') ! style="width:25%;" | intermediate depth<br>(all ''λ'' and ''h'') |- style="height:120px" ! [[phase velocity]] | <math> c_p=\frac{\lambda}{T}=\frac{\omega}{k}</math> || m / s || <math>\frac{g}{2\pi} T</math> || <math>\sqrt{g h}</math> || <math>\sqrt{\frac{g\lambda}{2\pi}\tanh\left(\frac{2\pi h}{\lambda}\right)}</math> |- style="height:120px" ! [[group velocity]]{{efn|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''.}} | <math> c_g= c_p^2 \frac{\partial\left(\lambda/c_p\right)}{\partial\lambda}=\frac{\partial\omega}{\partial k}</math> || m / s || <math>\frac{g}{4\pi} T</math> || <math>\sqrt{g h}</math> || <math>\frac{1}{2} c_p \left( 1 + \frac{4\pi h}{\lambda}\frac{1}{\sinh\left( \frac{4\pi h}{\lambda}\right)} \right)</math> |- style="height:120px" ! ratio | <math> \frac{c_g}{c_p}</math> || – || <math>\frac{1}{2}</math> || <math> 1</math> || <math>\frac{1}{2} \left( 1 + \frac{4\pi h}{\lambda}\frac{1}{\sinh\left( \frac{4\pi h}{\lambda}\right)} \right)</math> |- style="height:120px" ! wavelength | <math>\lambda</math> || m || <math>\frac{g}{2\pi} T^2</math> || <math>T \sqrt{g h}</math> || for given period ''T'', the solution of:<br> <br><math> \left(\frac{2\pi}{T}\right)^2=\frac{2\pi g}{\lambda}\tanh\left(\frac{2\pi h}{\lambda}\right)</math> |- ! colspan="6" | general |- style="height:80px" ! wave energy density | <math> E</math> | J / m<sup>2</sup> | colspan="3" | <math>\frac{1}{16} \rho g H_{m0}^2</math> |- style="height:80px" ! wave energy [[flux]] | <math> P</math> | W / m | colspan="3" | <math> E\;c_g</math> |- style="height:80px" ! angular [[frequency]] | <math> \omega</math> | [[radian|rad]] / s | colspan="3" | <math>\frac{2\pi}{T}</math> |- style="height:80px" ! [[wavenumber]] | <math> k</math> | rad / m | colspan="3" | <math>\frac{2\pi}{\lambda}</math> |} === Wave power formula === [[File:Orbital wave motion-Wiegel Johnson ICCE 1950 Fig 6.png|thumb|Photograph of the elliptical trajectories of water particles under a – progressive and periodic – [[surface gravity wave]] in a [[wave flume]]. The wave conditions are: mean water depth ''d'' = {{convert|2.50|ft|m|abbr=on}}, [[wave height]] ''H'' = {{convert|0.339|ft|m|abbr=on}}, wavelength λ = {{convert|6.42|ft|m|abbr=on}}, [[period (physics)|period]] ''T'' = 1.12 s.<ref>Figure 6 from: {{cite book |last1=Wiegel |first1=R.L. |title=Proceedings 1st International Conference on Coastal Engineering |url=https://repository.tudelft.nl/record/uuid:5c12e11b-a0fc-4245-a23f-4bbc75571c33 |pages=5–21 |date=October 1950 |chapter=Elements of wave theory |location=Long Beach, California |publisher=[[American Society of Civil Engineers|ASCE]] |last2=Johnson |first2=J.W. |editor-last=Johnson |editor-first=J.W. |series=Coastal Engineering Proceedings |volume=1 |doi=10.9753/icce.v1.2 |doi-access=free }}</ref>]] In deep water where the water depth is larger than half the [[wavelength]], the wave [[energy flux]] is{{efn|The energy flux is <math>P = \tfrac{1}{16} \rho g H_{m0}^2 c_g,</math> with <math>c_g</math> the group velocity,<ref>{{Cite book | publisher = McGraw-Hill Professional | isbn = 978-0-07-134402-9 | last = Herbich | first = John B. | title = Handbook of coastal engineering | year = 2000 | no-pp = yes |page=A.117, Eq. (12) }}</ref> The group velocity is <math>c_g=\tfrac{g}{4\pi}T</math>, 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 "''[[#Energy and energy flux|Wave energy and wave energy flux]]''" below.}} :<math> 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, </math> with ''P'' the wave energy flux per unit of wave-crest length, ''H''<sub>''m0''</sub> the [[significant wave height]], ''T''<sub>''e''</sub> the wave energy [[period (physics)|period]], ''ρ'' the water [[density]] and ''g'' the [[Earth's gravity|acceleration by gravity]]. The above formula states that wave power is proportional to the wave energy period and to the [[Square (algebra)|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.<ref>{{cite book |title=Waves in ocean engineering |year=2001 |publisher=Elsevier |location=Oxford |isbn=978-0080435664 |pages=35–36 |author=Tucker, M.J. |edition=1st |author2=Pitt, E.G. |editor=Bhattacharyya, R. |editor2=McCormick, M.E. |chapter=2}}</ref><ref>{{cite web |title=Wave Power |publisher=[[University of Strathclyde]] |url=http://www.esru.strath.ac.uk/EandE/Web_sites/01-02/RE_info/wave%20power.htm |access-date=November 2, 2008 |archive-url=https://web.archive.org/web/20081226032455/http://www.esru.strath.ac.uk/EandE/Web_sites/01-02/RE_info/wave%20power.htm |archive-date=December 26, 2008 |url-status=live}}</ref><ref name="ocs">{{cite web |url=http://www.ocsenergy.anl.gov/documents/docs/OCS_EIS_WhitePaper_Wave.pdf|title=Wave Energy Potential on the U.S. Outer Continental Shelf |publisher=[[United States Department of the Interior]] |access-date=October 17, 2008 |archive-url=https://web.archive.org/web/20090711052514/http://ocsenergy.anl.gov/documents/docs/OCS_EIS_WhitePaper_Wave.pdf |archive-date=July 11, 2009}}</ref><ref>[http://www.scotland.gov.uk/Publications/2006/04/24110728/10 Academic Study: Matching Renewable Electricity Generation with Demand: Full Report] {{Webarchive|url=https://web.archive.org/web/20111114015028/http://www.scotland.gov.uk/Publications/2006/04/24110728/10 |date=November 14, 2011 }}. Scotland.gov.uk.</ref> For 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 s. Solving for power produces :<math> 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}}, </math> or 36 kilowatts of power potential per meter of wave crest. In major storms, the largest offshore sea states have significant wave height of about 15 meters and energy period of about 15 seconds. According to the above formula, such waves carry about 1.7 MW of power across each meter of wavefront. An effective wave power device captures a significant portion of the wave energy flux. As a result, wave heights diminish in the region behind the device. === Energy and energy flux === In a [[sea state]], the [[arithmetic mean|mean]] [[energy density]] per unit area of [[gravity wave]]s on the water surface is proportional to the wave height squared, according to linear wave theory:<ref name="Phillips" /><ref name="Goda">{{cite book | first=Y. | last=Goda | title=Random Seas and Design of Maritime Structures | year=2000 | publisher=World Scientific | isbn=978-981-02-3256-6 }}</ref> :<math>E=\frac{1}{16}\rho g H_{m0}^2,</math>{{efn|Here, the factor for random waves is {{frac|1|16}}, as opposed to {{frac|1|8}} for periodic waves – as explained hereafter. For a small-amplitude sinusoidal wave <math display="inline"> \eta = a \cos 2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)</math> with wave amplitude <math> a,</math> the wave energy density per unit horizontal area is <math display="inline"> E=\frac{1}{2}\rho g a^2,</math> or <math display="inline"> E=\frac{1}{8}\rho g H^2</math> using the wave height <math display="inline"> H = 2a</math> for sinusoidal waves. In terms of the variance of the surface elevation <math display="inline"> m_0 = \sigma_\eta^2 = \overline{(\eta-\bar\eta)^2} = \frac{1}{2}a^2,</math> the energy density is <math display="inline"> E=\rho g m_0</math>. Turning to random waves, the last formulation of the wave energy equation in terms of <math display="inline"> m_0</math> is also valid (Holthuijsen, 2007, p. 40), due to [[Parseval's theorem]]. Further, the [[significant wave height]] is ''defined'' as <math display="inline"> H_{m0} = 4\sqrt{m_0}</math>, leading to the factor {{frac|1|16}} in the wave energy density per unit horizontal area.}}<ref>{{Cite book | last = Holthuijsen | first = Leo H. | year = 2007 | title = Waves in oceanic and coastal waters | publisher = Cambridge University Press | isbn = 978-0-521-86028-4 | location = Cambridge }}</ref> where ''E'' is the mean wave energy density per unit horizontal area (J/m<sup>2</sup>), the sum of [[kinetic energy|kinetic]] and [[potential energy]] density per unit horizontal area. The potential energy density is equal to the kinetic energy,<ref name="Phillips" /> both contributing half to the wave energy density ''E'', as can be expected from the [[Equipartition theorem#Potential energy and harmonic oscillators|equipartition theorem]]. The waves propagate on the surface, where crests travel with the phase velocity while the energy is transported horizontally with the [[group velocity]]. The mean transport rate of the wave energy through a vertical [[plane (mathematics)|plane]] of unit width, parallel to a wave crest, is the energy [[flux]] (or wave power, not to be confused with the output produced by a device), and is equal to:<ref>{{cite journal | last=Reynolds |first=O. | author-link=Osborne Reynolds | year=1877 |title=On the rate of progression of groups of waves and the rate at which energy is transmitted by waves | journal=Nature | volume=16 |issue=408 | pages=343–44 | doi = 10.1038/016341c0 |bibcode = 1877Natur..16R.341. | doi-access=free }}<br>{{cite journal | title=On progressive waves | author=Lord Rayleigh (J. W. Strutt) | author-link=Lord Rayleigh | year=1877 | journal=Proceedings of the London Mathematical Society | volume=9 | issue=1 | pages=21–26 | doi=10.1112/plms/s1-9.1.21 | url=https://zenodo.org/record/1447762 }} Reprinted as Appendix in: ''Theory of Sound'' '''1''', MacMillan, 2nd revised edition, 1894.</ref><ref name="Phillips" /> :<math>P = E\, c_g, </math> with ''c<sub>g</sub>'' the group velocity (m/s). Due to the [[dispersion (water waves)|dispersion relation]] for waves under gravity, the group velocity depends on the wavelength ''λ'', or equivalently, on the wave [[period (physics)|period]] ''T''. [[Wave height]] is determined by wind speed, the length of time the wind has been blowing, fetch (the distance over which the wind excites the waves) and by the [[bathymetry]] (which can focus or disperse the energy of the waves). A given wind speed has a matching practical limit over which time or distance do not increase wave size. At this limit the waves are said to be "fully developed". In general, larger waves are more powerful but wave power is also determined by [[wavelength]], water [[density]], water depth and acceleration of gravity.
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