Environmental Engineering Reference
In-Depth Information
where
m
H
is the principal direction of propagation of the sea state and S
z
(
v
,
m
) the direction
spectrum.
The following applies for the variance of the short-crested sea state:
Z
1
m
H
þp=
Z
2
2
z
mðÞ¼
m
0
mðÞ¼
s
S
z
v; m
ð
Þ
d
m
d
v
0
m
H
p=
2
Statements regarding the distribution and frequency of certain values during a steady-state
sea condition, for example maxima or zero crossings of a given level, have a certain
practical significance. The nature of such distributions depends on the magnitude of the
dimensionless width parameter
, which is a measure of the width of the sea state spectrum:
e
0
:
5
m
2
m
0
m
4
e
¼
1
where m
0
,m
2
and m
4
are zero-, second- and fourth-order moments respectively for the area
beneath the spectrum. Taking the limit values
e
¼
0 (very
narrow spectrum), we get a normal or a Rayleigh distribution respectively for the maximum
values of the long-crested sea state. The following applies for the distribution densities:
e
¼
1 (very wide spectrum) and
2
M
2
m
0
1
2
p
m
0
z
f
zðÞ¼
p
exp
for
e
¼
1
2
M
2
m
0
zðÞ¼
z
M
z
f
m
0
exp
for
e
¼
0
These days we work almost exclusively approximately using a Rayleigh distribution for the
maxima although the sea state spectra are not narrow. Assuming a Rayleigh distribution for
the maxima results in the height of the wave being overestimated. We get the following
distribution density from f (
z
M
):
H
2
8
m
0
H
4
m
0
exp
f
ðÞ¼
We can use these equations to calculate the probabilities with which the maximum value of
the sea state function
z
M
or the wave height H exceeds or does not exceed certain values.
The following applies for the distribution function:
¼
1
exp
z
2
M
2
m
0
¼
exp
2
M
2
m
0
z
M
M
F
zðÞ¼
P
z
M
z
and P
z
M
> z
or
H
2
8
m
0
H
2
8
m
0
FH
ðÞ¼
PH
H
H
ð
Þ ¼
1
exp
and P H
ð
>
Þ ¼
exp
The significant wave height was introduced to characterise the irregular sea state for
practical engineering applications. By presuming a Rayleigh distribution for the sea state it
