Part A | 3.2
52 Part A Fundamentals
spectral density is given by
S.f / D 2 jF.f /j =n ;
(3.19)
where n is the number of points in the time series. Taking the square of the amplitude of the Fourier transform
removes the phase information from the record, but the
result is still a very irregular function of frequency.
A smooth version of the spectrum is found by filtering
S.f / over frequency or averaging spectra from several
ensembles. Glover et al. [3.15] give a good, practical
guide to the details of calculating power spectra.
Once the spectrum is known, the significant wave
height H S is defined as
H S D 4
Z
S.f /df D 4 ;
(3.20)
where is the variance of the wave record. The peak
wave frequency is the frequency at the highest point in
the power spectrum. The mean frequency f m and zerocrossing frequency f z are given by
f m D m 1 =m 0 ;
f z D m 2 =m 0 :
(3.21)
Where the spectral moments are calculated as
m n D
Z
S.f /f
n df :
(3.22)
Wave spectra for design are generally specified in an analytic form. The most popular of these is the Joint North
Sea Wave Observation Project (JONSWAP) spectral
form. It is given by
S.f / D ˇf
5 exp
"
5
4
 f
f p
à 4
#
exp
"
.ffp/
2
2 2 f 2
p
#
;
(3.23)
where
D
(
a D 0:07 if f Ä f p ;
b D 0:09 if f > f p :
(3.24)
The JONSWAP spectrum was originally proposed to
describe fetch-limited waves, but by adjusting its parameters, it can give a reasonable fit to most singlepeaked spectra. Given the significant wave height, peak
period, and peak enhancement factor , Goda [3.16]
showed that the scale factor is approximated by
ˇ D
5
16
H
2
s f
2
p
Â
1:15 C 0:168
0:925
1:909 C
à 1
:
(3.25)
The JONSWAP spectrum can be used to describe most
spectra with single peaks. However, combinations of
sea and swell in storms can result in spectra with two or
more peaks. The Ochi–Hubble [3.17] spectrum is often
used to describe double-peaked spectra in areas subject
to tropical storms. It is the sum of two Gamma distributions
S.f / D
2
X
jD1
H Sj T Pj .. j C 0:25/
j
4 .. j /.T Pj f / .4jC1/ exp
j C 0:25
T Pj f
4
!
:
(3.26)
This spectrum has three parameters for each of the two
wave systems, a significant wave height, a peak period,
and a shape factor .
The Torsethaugen and Haver [3.18] double-peaked
spectrum is also the sum of two Gamma functions.
Their paper gives parameters which were fit to measurements made in the North and Norwegian seas.
The spectral representation of waves makes it natural to think of them as a Gaussian random process. The
envelope of a Gaussian process has a Rayleigh distribution, and to first order, so do wave and crest heights.
However, the trough preceding a large crest is likely to
be on a lower part of the envelope. Trough to crest wave
height differences are, therefore, slightly smaller than
given by the Rayleigh distribution
P.h/ D exp
"
2
 h
H S
à 2
#
;
(3.27)
where H S is four times the standard deviation of the
wave trace.
The empirical distribution suggested by Forristall [3.19] accounts for the observed reduction in
wave height and has been shown to agree with many
observations, including measurements in water depths
less than 30 m. It is given by
P.h/ D exp
"
2:263
 h
H S
à 2:126
#
:
(3.28)
Crest heights in steep waves are higher than those
predicted by Gaussian theory because the waves are
nonlinear. The distribution produced from simulations
of second-order waves by Forristall [3.20] accounts for
the most important nonlinearity. It is a Weibull distribution of the form
P.Á 2 / D exp
"
 Á
˛H s
à ˇ
#
;
(3.29)
52 Part A Fundamentals
spectral density is given by
S.f / D 2 jF.f /j =n ;
(3.19)
where n is the number of points in the time series. Taking the square of the amplitude of the Fourier transform
removes the phase information from the record, but the
result is still a very irregular function of frequency.
A smooth version of the spectrum is found by filtering
S.f / over frequency or averaging spectra from several
ensembles. Glover et al. [3.15] give a good, practical
guide to the details of calculating power spectra.
Once the spectrum is known, the significant wave
height H S is defined as
H S D 4
Z
S.f /df D 4 ;
(3.20)
where is the variance of the wave record. The peak
wave frequency is the frequency at the highest point in
the power spectrum. The mean frequency f m and zerocrossing frequency f z are given by
f m D m 1 =m 0 ;
f z D m 2 =m 0 :
(3.21)
Where the spectral moments are calculated as
m n D
Z
S.f /f
n df :
(3.22)
Wave spectra for design are generally specified in an analytic form. The most popular of these is the Joint North
Sea Wave Observation Project (JONSWAP) spectral
form. It is given by
S.f / D ˇf
5 exp
"
5
4
 f
f p
à 4
#
exp
"
.ffp/
2
2 2 f 2
p
#
;
(3.23)
where
D
(
a D 0:07 if f Ä f p ;
b D 0:09 if f > f p :
(3.24)
The JONSWAP spectrum was originally proposed to
describe fetch-limited waves, but by adjusting its parameters, it can give a reasonable fit to most singlepeaked spectra. Given the significant wave height, peak
period, and peak enhancement factor , Goda [3.16]
showed that the scale factor is approximated by
ˇ D
5
16
H
2
s f
2
p
Â
1:15 C 0:168
0:925
1:909 C
à 1
:
(3.25)
The JONSWAP spectrum can be used to describe most
spectra with single peaks. However, combinations of
sea and swell in storms can result in spectra with two or
more peaks. The Ochi–Hubble [3.17] spectrum is often
used to describe double-peaked spectra in areas subject
to tropical storms. It is the sum of two Gamma distributions
S.f / D
2
X
jD1
H Sj T Pj .. j C 0:25/
j
4 .. j /.T Pj f / .4jC1/ exp
j C 0:25
T Pj f
4
!
:
(3.26)
This spectrum has three parameters for each of the two
wave systems, a significant wave height, a peak period,
and a shape factor .
The Torsethaugen and Haver [3.18] double-peaked
spectrum is also the sum of two Gamma functions.
Their paper gives parameters which were fit to measurements made in the North and Norwegian seas.
The spectral representation of waves makes it natural to think of them as a Gaussian random process. The
envelope of a Gaussian process has a Rayleigh distribution, and to first order, so do wave and crest heights.
However, the trough preceding a large crest is likely to
be on a lower part of the envelope. Trough to crest wave
height differences are, therefore, slightly smaller than
given by the Rayleigh distribution
P.h/ D exp
"
2
 h
H S
à 2
#
;
(3.27)
where H S is four times the standard deviation of the
wave trace.
The empirical distribution suggested by Forristall [3.19] accounts for the observed reduction in
wave height and has been shown to agree with many
observations, including measurements in water depths
less than 30 m. It is given by
P.h/ D exp
"
2:263
 h
H S
à 2:126
#
:
(3.28)
Crest heights in steep waves are higher than those
predicted by Gaussian theory because the waves are
nonlinear. The distribution produced from simulations
of second-order waves by Forristall [3.20] accounts for
the most important nonlinearity. It is a Weibull distribution of the form
P.Á 2 / D exp
"
 Á
˛H s
à ˇ
#
;
(3.29)