Page 301 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

286 Fundamentals of Ocean Renewable Energy

summary of the physical mechanisms of wave-tide interaction, and tools that
can be used to investigate them. More details can be found in other resources
(e.g. [24–26]).
In general, tides are long waves with periods in the range of hours and
days, whereas wind-generated waves are short waves with periods in the order
of seconds; therefore, wave-tide interaction can be regarded as the interaction
between long and short waves.

Effect of Tidal Currents on Wave Energy
Considering a wave that propagates in the presence of currents, the frequency
of the wave changes by an ambient current as follows (i.e. Doppler shift):
ω = σ + ku (10.1)
where σ is the relative wave frequency, ω is the absolute wave frequency, u
is the ambient current velocity, and k is the wave number. The absolute wave
frequency can be measured by a stationary observer (i.e. the wave frequency
that would be experienced by a wave energy device), but the relative frequency
can be measured by an observer moving at the same speed as the currents.
Equations derived from linear wave theory are valid for an observer moving
with the currents [27].
As mentioned before (Section 5.1.3), wave energy propagates at the group
velocity, C g . Due to the Doppler shift, we can write
dσ d(ω − ku) dω *
C g = = = − u → C g = C − u (10.2)
g
dk dk dk
*
where C is the absolute group velocity (stationary observer). Referring back to
g
Chapter 5, the wave power for monochromatic waves is given by
1 2

P = ρgH C g = EC g (10.3)
8
where P is the wave power, H is the wave height, and E is wave energy averaged
over the wave period. Referring to Eq. (10.2), we can conclude that
*
*
C = C g + u → EC = EC g + uE (10.4)
g
g
*
EC is the wave power in the presence of currents; it is a combination of the
g
wave power originating from waves (EC g ), and the wave energy transport by
tidal currents (uE). The interaction of tidal currents and waves leads to further
complications for wave energy resource assessment. When waves propagate in
the presence of currents, the principle of wave energy conservation is no longer
valid, due to the energy exchange between wave and current fields. This is the
reason why spectral wave models such as SWAN do not use conservation of
wave energy in their formulation. Instead of wave energy, wave action (i.e. E/σ)
is conserved in the presence of ambient currents (Section 8.4.1).