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 \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} where \vec u(t, x, y, z) is the fluid velocity, p is the
pressure, \rho the
density, \nu the
viscosity, and \vec{F_\text{ext}} 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 \phi(t,x,y,z): {\vec{\nabla}\times\vec{u}=\vec{0}}\Leftrightarrow{\vec{u}=\vec{\nabla}\phi}\text{,}which must satisfy the
Laplace equation, \nabla^2\phi=0\text{.}In an ideal flow, the viscosity is negligible and the only external force acting on the fluid is the earth gravity \vec{F_\text{ext}}=(0,0,-\rho g). In those circumstances, the
Navier–Stokes equations reduces to {\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), which integrates (spatially) to the
Bernoulli conservation law:{\partial\phi \over\partial t}+{1 \over2}\bigl(\vec\nabla\phi\bigr)^2 +{1 \over \rho} p + gz=(\text{const})\text{.}
Linear potential flow theory magnitude of fluid particles decreases exponentially 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. When considering small amplitude waves and motions, the quadratic term \left(\vec{\nabla}\phi\right)^2 can be neglected, giving the linear Bernoulli equation,{\partial\phi \over\partial t}+{1 \over \rho} p + gz=(\text{const})\text{.} and third Airy assumptions then imply\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} These constraints entirely determine
sinusoidal wave solutions of the form \phi=A(z)\sin{\!(kx-\omega t)}\text{,} where k determines the
wavenumber of the solution and A(z) and \omega are determined by the boundary constraints (and k ). Specifically,\begin{align} &A(z)={gH \over 2\omega}{\cosh(k(z+h)) \over \cosh(kh)} \\ &\omega=gk\tanh(kh)\text{.} \end{align} The surface elevation \eta can then be simply derived as \eta=-{1 \over g}{\partial \phi \over \partial t}={H \over 2}\cos(kx-\omega t)\text{:} a plane wave progressing along the x-axis direction.
Consequences 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. In deep water where the water depth is larger than half the
wavelength, the wave
energy flux is{{efn|The energy flux is P = \tfrac{1}{16} \rho g H_{m0}^2 c_g, with c_g the group velocity, 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.}} : 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. 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 : 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}}, 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
mean energy density per unit area of
gravity waves on the water surface is proportional to the wave height squared, according to linear wave theory: :E=\frac{1}{16}\rho g H_{m0}^2,{{efn|Here, the factor for random waves is , as opposed to for periodic waves – as explained hereafter. For a small-amplitude sinusoidal wave \eta = a \cos 2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right) with wave amplitude a, the wave energy density per unit horizontal area is E=\frac{1}{2}\rho g a^2, or E=\frac{1}{8}\rho g H^2 using the wave height H = 2a for sinusoidal waves. In terms of the variance of the surface elevation m_0 = \sigma_\eta^2 = \overline{(\eta-\bar\eta)^2} = \frac{1}{2}a^2, the energy density is E=\rho g m_0. Turning to random waves, the last formulation of the wave energy equation in terms of m_0 is also valid (Holthuijsen, 2007, p. 40), due to
Parseval's theorem. Further, the
significant wave height is
defined as H_{m0} = 4\sqrt{m_0}, leading to the factor in the wave energy density per unit horizontal area.}} 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, :P = E\, c_g, with
cg the group velocity (m/s). Due to the
dispersion relation for waves under gravity, the group velocity depends on the wavelength
λ, or equivalently, on the wave
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. == Wave energy converters ==