One of the most commonly used models for the directional spreading function is
the cosine-power function defined as:
Γ(s +1)
D(θ) =
cos2s (θ - θp )
for | θ - θp | < π / 2
(47)
π Γ(s +1/ 2)
where θp is the principal direction of wave propagation and Γ is the gamma func-
tion. The parameter s is an index describing the degree of directional spreading
with s → ∞ representing a unidirectional wave field. Figure 4 shows a plot of the
of the cosine power spreading function for different values of the spreading
index, s.
0.035
s = 10
0.03
s= 5
0.025
0.02
s= 3
0.015
s= 1
0.01
0.005
0
-90
-60
-30
0
30
60
90
θ (deg)
Figure 4.
Cosine-power spreading function for different values of the spreading
index s
A number of other models have been proposed for the directional spreading
function including the normal, circular normal, and wrapped-normal distribution
(Borgman 1969). A more intuitive and universal parameter to describe the degree
of directional spreading in a multidirectional sea state is the standard deviation of
the directional spreading function, σθ, which is defined as:
θp +π/ 2
∫
σθ
=
D(θ) (θ - θ p )2 dθ
2
(48)
θp -π / 2
20
Chapter 3 Numerical Solution
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