Environmental Engineering Reference
In-Depth Information
2.5.6 Short-term statistics for the sea state
By assuming that individual short-term sea states are of a steady-state nature over
periods of approx. 3 h, it has been possible to develop so-called sea state spectra for the
stochastic description of the sea state [21]. The sea state function
(t), that is the
corresponding Gaussian process, arising according to the model of the superposition of
primary waves can be presented as follows for a long-crested sea state (see also [11]):
z
Z
1
q
2
S
z
ðÞ
d
z
ðÞ¼
v
cos
½
v
t
e
ðÞ
0
where S
z
(
) a phase angle that, as a randomnumber with the
same probability, takes on all the values in the range from 0 to 2
p
v
)isthesea state spectrumand
(
v
e
. This equation is not a
Riemann integral, but rather themathematical interpretation of the fact that an infinite number
p
of primary waves with amplitudes
) are superposed.
It is expedient and in no way a constraint to agree that the function
z
(t) be measured
from the mean value. The normal distribution density is then
2
S
z
ðÞ
d
v
and random phases
(
v
e
!
2
2
2
m
0
1
2
p
z
1
2
p
m
0
z
p
p
f
ðÞ¼
exp
¼
exp
2
z
2
s
s
z
where
Z
1
2
z
¼
m
0
¼
s
S
z
ðÞ
d
v
0
and m
0
is the variance of the distribution, which is equal to the area beneath the spectrum
(moment of zero order). All the derivations of the time function
z
(t) likewise exhibit
normal distributions:
z
¼
2
1
2
p
m
2
z
f
p
exp
2
m
2
z
2
z
¼
1
2 p m
4
p
f
exp
2 m
4
where
Z
1
v
n
S
z
ðÞ
d
m
n
¼
v
0
and m
n
is the moment of
n
th order for the area beneath the spectrum.
When representing the short-crested sea state it is assumed that the direction of
propagation of the primary waves is scattered over a range of
p
/2 about the mean
principal direction of propagation of the sea state (roughly the wind direction). In a
fixed system of coordinates the short-crested sea state can be represented as follows:
Z
1
m
H
þp=
Z
2
q
2
S
z
v; m
z
ð
x
;
y
;
t
Þ¼
ð
Þ
d
m
d
v
cos k
½
ð
x
cos
m þ
y
sin
m
Þ v
t
e
v; m
ð
Þ
0
m
H
p=
2
