Specifications

44
The power spectral density is obtained from the Fourier coefficients
2
00
)( HfPSD = (5.6.5a)
12/1)(
22
−=+=
−
NlHHfPSD
lNll
K (5.6.5b)
2
2/2/
)(
NN
HfPSD =
(5.6.5c)
where frequencies range from 0.0 Hz to 0.64 Hz in steps of 0.005 Hz. Actually, there is one
more step, all coefficients are smoothed according to
11
4
1
2
1
4
1
+−
++=
llll
PSDPSDPSDPSD (5.6.6)
To limit the number of frequencies low frequency coefficients (f
l
≤ 0.1 Hz) are left as they are,
while only every other smoothed coefficient on the high frequency side(f
l
> 0.1 Hz) is kept in
the spectrum file. Finally, 8 consecutive spectra covering 1600 s are averaged and used to
compute the half-hourly wave spectrum. Each half-integral hour (1800 s) a new cycle starts.
5.6.2 Wave direction spectrum
So far only the vertical displacements have been processed to give the wave power spectral
density. When north and west displacements are included into the processing, much more wave
information can be obtained. Starting from the time-series of north, west and vertical (n, w, v)
displacements, the three associated Fourier series may be calculated. Each Fourier series
consists of a number of Fourier coefficients, which in turn consist of a real and imaginary part.
Thus six Fourier components per frequency f are obtained α
nf
, β
nf
, α
wf
, β
wf
, α
vf
and β
vf
or in
vector notation:
nfnfnf
iA
β
α
+= (5.6.7a)
wfwfwf
iA
β
α
+
= (5.6.7b)
vfvfvf
iA
β
α
+= (5.6.7c)
Building on this, co- (C) and quadrature-spectra or quad-spectra (Q) may be formed, e.g. (we
shall omit the frequency subscript hereafter)
wfnfwfnfwfnfnw
AAC ββαα +=⋅= (5.6.8)
nfvfnfvfnfvfvn
AAQ αββα −=×= (5.6.9)
In total 9 components arranged in a 3x3 matrix will be obtained for both co- and quad-spectra.
However, not all components need to be calculated. By definition we have
0===
vvwwnn
QQQ
(5.6.10)
Furthermore, Q represents rotation. To give an example, a wave rolling eastward will have a
rotation component directed to the north (right-handed screw) and hence Q
vw
≠ 0 and Q
wv
≠ 0.
The rotation in the waves is particularly clear for breaking waves in the surf zone. A rotation
component directed vertically would represent eddy currents which are not part of the physics
of waves, therefore we also have
0==
nwwn
QQ (5.6.11)