Page 122 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Wave Energy Chapter | 5 113

where p D is the dynamic pressure and u is the horizontal velocity given in
Table 5.1. Note that wave power is conserved in linear wave theory (i.e. the
RHS of Eq. (5.16) is constant).
1 2
In the above formulation, E = ρgH is the total mechanical energy (i.e.
8
potential and kinetic) of waves per unit surface area averaged over a wave
period. In other words, wave power is simply the product of wave energy
and wave group velocity. The group velocity C g (which is dependent on the
wave celerity C, wave number k, and water depth) is the speed of wave
energy propagation. Referring to the dispersion equation, it is also defined
as [1]

∂σ ∂
C g = = gk tanh(kh) (5.18)
∂k ∂k
In realistic ocean environments, ocean currents, which are generated by
tides, wind, or temperature, affect the propagation of wave energy. Tidal waves
and wind waves can be regarded as long and short waves, which interact in
various ways. The Doppler shift (which is explained in Chapter 7) is an example
where the frequency of waves is influenced by currents:

∂ω ∂σ
ω = σ + ku → = + u (5.19)
∂k ∂k
where σ is the relative wave frequency (observed in a coordinate system moving
with the same velocity as the ambient current), ω is the absolute wave frequency
(observed in a fixed frame), and u is the ambient current velocity. The wave
energy flux when we have ambient currents becomes

*
*
C = C g + u → EC = EC g + uE = cst (5.20)
g g
*
where C is the group velocity in the presence of ambient currents, EC g is
g
the wave energy transport by the group velocity, and uE is the wave energy
transport by tidal currents. When waves propagate in the presence of currents,
the wave energy flux is no longer conserved. This happens due to the exchange
of energy between wave and current fields. To avoid this issue, wave models
apply the conservation of the wave action (E/σ), rather than conservation of
wave energy in their formulations (e.g. [2]). Nevertheless, the total energy flux
due to waves and currents is conserved. Therefore, the conservation of energy,
in a more general way, can be expressed as follows [3]

1 C g 1

3
EC g + Eu + ρghu + u 2 − E = cst (5.21)
2 C 2
It is clear that when ambient currents are zero, the earlier equations reduce to
P = EC g = cst.