Specifications

45
Thus, one obtains:
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
vvvnvw
nvnnnw
wvwnww
CCC
CCC
CCC
(5.6.12)
and
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
0
00
00
vnvw
nv
wv
QQ
Q
Q
(5.6.13)
Given these components a whole set of informative wave parameters such as: wave direction,
direction spread, wave ellipticity can be obtained. Before discussing their meaning in more
detail, first, all formulas will be given.
()
vvwwnn
nv
CCC
Q
a
+
=
1
(5.6.14)
()
vvwwnn
wv
CCC
Q
b
+
−
=
1
(5.6.15)
wwnn
wwnn
CC
CC
a
+
−
=
2
(5.6.16)
wwnn
nw
CC
C
b
+
−
=
2
2
(5.6.17)
These are the first four Fourier coefficients of the normalized directional distribution ),( fG
θ
⎭
⎬
⎫
⎩
⎨
⎧
+++++=
...2sin2cossincos
2
11
),(
2211
θθθθ
π
θ babafG (5.6.18)
alternatively cast as
⎭
⎬
⎫
⎩
⎨
⎧
+−+−+−+=
...)(2sin)(2cos)cos(
2
11
),(
020201
θθθθθθ
π
θ nmmfG
(5.6.19)
where
),arctan(
110
ab=
θ
(5.6.20)
2
1
2
11
bam += (5.6.21)
02022
2sin2cos
θ
θ
bam += (5.6.22)
02022
2cos2sin
θ
θ
ban +−= (5.6.23)
The m- and n- coefficients are known as the centred Fourier coefficients [Kuik88] or the second
harmonic of the directional energy distribution recalculated to the mean wave direction.