Page 126 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Wave Energy Chapter | 5 117

The earlier equation is hard to implement, unless we use spectral wave
models or measured data that include the full wave spectrum. Let us formulate
the wave power equation for deep waters, in which the dispersion equation
is much simpler. As mentioned in Section 5.1.2, in deep waters, tanh kd
approaches 1. Therefore, the dispersion equation and the group velocity are
given by

tanh kd ≈ 1 → σ = gk (5.33)
dσ g 1 g gT
→ C g = = √ = = (5.34)
dk 2 gk 2σ 4π
Replacing the group velocity in Eq. (5.32), results in

∞ ∞ g 1 ∞

P = ρg C g (σ)S(σ)dσ = ρg S(σ)dσ = ρg 2 σ −1 S(σ)dσ
0 0 2σ 2 0
(5.35)
or
1 2
P = ρg m −1 (5.36)
2

For the case of monochromatic waves in deep waters, the wave power
becomes
gT 1 2 ρg 2 2
P = C g E = ρgH = H T (5.37)
4π 8 32π

If we wanted to use simple wave statistical parameters to compute the wave
power (something similar to the previous equation, which only depends on the
wave height and the wave period), we can also use the deep water approximation
for irregular waves.
Assuming the deep water approximation, the group velocity (which is
gT E
the speed of propagation of the wave energy) is C g = ; we called the
4π
period T E the energy wave period, as it determines the average speed of wave
energy propagation. Further, the total wave energy of a spectrum is given
by Eq. (5.29) in terms of m o or the significant wave height. Therefore, we
can write

gT E gT E 1 2
P = C g E = (ρgm 0 ) = ρgH mo (5.38)
4π 4π 16
ρg 2 2 ρg 2 √ 2
T
P = H mo E = H mo / 2 T E (5.39)
64π 32π
which is very similar to Eq. (5.37). In fact, we can use the monochromatic wave
equation, if we replace the wave period by the energy wave period, and wave
√
height by H mo / 2, which is called H rms or the root-mean-square wave height.