Page 125 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

116 Fundamentals of Ocean Renewable Energy

N

∞

E tot = ρg S(σ)dσ = ρg S(σ i )δ(σ) → E(σ) = ρgS(σ) (5.25)
0
1
Considering Eqs (5.23), (5.25), we can see that the variance density spectrum
represents the square of the amplitude of each individual sine wave:
1 2
a = S(σ i )δσ (5.26)
i
2
The moments of the variance density spectrum are frequently used to
compute statistical wave properties. The rth moment (m r ) is defined as
∞

r
m r = σ S(σ)dσ (5.27)
0
For instance, the significant wave height H mo , which is the mean of the highest
one-third of waves in a record, is given by
√
H mo = 4 m 0 (5.28)

Wave Power for Irregular Waves
Let us start with the total wave energy of a wave spectrum. By definition, the
wave energy (in Joules) can be computed as
∞ 1

2
E = ρg S(σ)dσ = ρgm 0 → E = ρgH mo (5.29)
0 16
in which we used the following relation
√ 1 2
H mo = 4 m 0 → m 0 = H mo (5.30)
16

Because irregular waves can be represented by a wave energy spectrum, we
can assume that the total wave power is calculated by adding the wave power
of individual waves, which construct the wave spectrum. The wave power of an
individual wave with a frequency of σ is given by
σ 2kd
1
ΔP = ρgC g (σ)ΔS(σ) = ρgC g S(σ)Δσ = ρg 1 + S(σ)Δσ
k 2 sinh 2kd
(5.31)
Consequently, the total power becomes
∞

P = ρg C g (σ)S(σ)dσ
0
∞
σ 1 2kd 2

= 1 + S(σ)dσ and σ = gk tanh (kd) (5.32)
0 k 2 sinh 2kd