Part A | 3.7
70 Part A Fundamentals
distribution with the short-term distributions discussed
in Sect. 3.2.2. Borgman [3.134] showed how the maximum wave and crest height during a storm can be
estimated by integrating the short-term wave and crest
height distributions over the storm’s sea state history.
Tucker and Pitt [3.135] give a thorough description
of how the Borgman integral has been applied to the
calculation of extreme wave heights. Forristall [3.136]
validated these methods using long-time series of individual waves.
If the probability that the wave or crest height exceeds Á is given by P.Á/, then the probability that the
height will not exceed Á in N waves is given by
P.Á max < Á/ D Œ1 P.Á/
N :
(3.43)
For a sequence of records i D 1 : : : k during a storm, the
probability of non-exceedance becomes
P.Á max < Á/ D
k
Y
iD1
Œ1 P i .Á/
Ni :
(3.44)
The calculations can be performed more accurately by
taking the logarithm of (3.44) to give
log ŒP.Á max < Á/ D
k
X
iD1
N i log Œ1 P.Á/ : (3.45)
Equation (3.45) is applied to calculate the expected
maximum wave and crest height in each storm. The sets
of maximum heights are then fit to extreme value distributions to determine the return period individual wave
and crest heights.
Fitting an extreme value distribution to the most
probable maxima does not account for the fact that values higher than the most probable value can occur in
any storm. Tromans and Vanderschuren [3.137] proposed a method for taking account of this short-term
variability. If the probability distribution of the most
probable maxima in a storm is P.H mp / and the distribution of the maximum given H mp is P.HjH mp /, then the
distribution of the maximum in a single random storm
is
P.Hjsrs/ D
Z
P.HjH mp /p.H mp /dH mp :
(3.46)
Tromans and Vanderschuren found that P.HjH mp / was
very similar from one storm to another, and that its
mean could be described by the function
P.HjH mp / D
exp exp
(
log N
" Â H
H mp
à 2
1
#)!
; (3.47)
where N is an equivalent number of waves in a storm.
The value of ln N can be estimated from the short-term
distributions in the historical storms. Values between ln
N D 8 and ln N D 10 are typical, and the results are
not very sensitive to the exact value. The result is to
increase the estimates of extreme maximum wave and
crest heights to about 5% more than the most probable
values.
3.7.9 Rogue Waves
It is generally agreed that a rogue wave is one with
a height greater than 2.2 times the significant wave
height or a crest greater than 1.25 times the significant wave height. There have been many reports of such
waves in the literature in the last few years. The best
known is the Draupner wave, recorded in the North
Sea on January 1, 1995 [3.138]. The crest height of this
wave was 1.55 times the significant wave height. Unfortunately, very little is known about the instrumentation
used for this measurement. The Andrea wave [3.139]
is a much better documented case. It was also recorded
in the North Sea on November 9, 2007. Essentially, the
same wave was recorded by four laser altimeters. Analysis of the intensity of the return signals indicated that
there was no sea spray at the wave crest. The height
of the Andrea wave was 2.49 times the significant wave
height, and the crest was 1.63 times the significant wave
height.
The central question in the study of rogue waves is
whether they can be explained as a statistical anomaly
or whether they require a physical explanation different
than second-order theory [3.140]. The statistical explanation for something like the Andrea wave is certainly
a stretch. According to second-order statistics, its crest
had a probability of 6 10
8 . However, Christou and
Ewans [3.141] did a careful study of over 10
8 measured waves and found that the sample crest distribution
was only slightly higher than predicted by second-order
statistics.
Some processes that produce very high waves are
understood reasonably well. Waves traveling into an opposing current can steepen and become much higher.
Many ships have been damaged when they encountered
such waves in the Agulhas Current south of Africa.
Bottom features can refract waves, making them much
larger in localized areas. Surfers are well aware of this
phenomenon. The more difficult cases to explain are unusually high waves in deep water far from shore.
Theoretical attempts to explain rogue waves involve
the integration of nonlinear equations that approximate
the development of steep random waves [3.142]. All of
these show a modulation of the wave envelope similar to the Benjamin–Feir instability observed in regular
70 Part A Fundamentals
distribution with the short-term distributions discussed
in Sect. 3.2.2. Borgman [3.134] showed how the maximum wave and crest height during a storm can be
estimated by integrating the short-term wave and crest
height distributions over the storm’s sea state history.
Tucker and Pitt [3.135] give a thorough description
of how the Borgman integral has been applied to the
calculation of extreme wave heights. Forristall [3.136]
validated these methods using long-time series of individual waves.
If the probability that the wave or crest height exceeds Á is given by P.Á/, then the probability that the
height will not exceed Á in N waves is given by
P.Á max < Á/ D Œ1 P.Á/
N :
(3.43)
For a sequence of records i D 1 : : : k during a storm, the
probability of non-exceedance becomes
P.Á max < Á/ D
k
Y
iD1
Œ1 P i .Á/
Ni :
(3.44)
The calculations can be performed more accurately by
taking the logarithm of (3.44) to give
log ŒP.Á max < Á/ D
k
X
iD1
N i log Œ1 P.Á/ : (3.45)
Equation (3.45) is applied to calculate the expected
maximum wave and crest height in each storm. The sets
of maximum heights are then fit to extreme value distributions to determine the return period individual wave
and crest heights.
Fitting an extreme value distribution to the most
probable maxima does not account for the fact that values higher than the most probable value can occur in
any storm. Tromans and Vanderschuren [3.137] proposed a method for taking account of this short-term
variability. If the probability distribution of the most
probable maxima in a storm is P.H mp / and the distribution of the maximum given H mp is P.HjH mp /, then the
distribution of the maximum in a single random storm
is
P.Hjsrs/ D
Z
P.HjH mp /p.H mp /dH mp :
(3.46)
Tromans and Vanderschuren found that P.HjH mp / was
very similar from one storm to another, and that its
mean could be described by the function
P.HjH mp / D
exp exp
(
log N
" Â H
H mp
à 2
1
#)!
; (3.47)
where N is an equivalent number of waves in a storm.
The value of ln N can be estimated from the short-term
distributions in the historical storms. Values between ln
N D 8 and ln N D 10 are typical, and the results are
not very sensitive to the exact value. The result is to
increase the estimates of extreme maximum wave and
crest heights to about 5% more than the most probable
values.
3.7.9 Rogue Waves
It is generally agreed that a rogue wave is one with
a height greater than 2.2 times the significant wave
height or a crest greater than 1.25 times the significant wave height. There have been many reports of such
waves in the literature in the last few years. The best
known is the Draupner wave, recorded in the North
Sea on January 1, 1995 [3.138]. The crest height of this
wave was 1.55 times the significant wave height. Unfortunately, very little is known about the instrumentation
used for this measurement. The Andrea wave [3.139]
is a much better documented case. It was also recorded
in the North Sea on November 9, 2007. Essentially, the
same wave was recorded by four laser altimeters. Analysis of the intensity of the return signals indicated that
there was no sea spray at the wave crest. The height
of the Andrea wave was 2.49 times the significant wave
height, and the crest was 1.63 times the significant wave
height.
The central question in the study of rogue waves is
whether they can be explained as a statistical anomaly
or whether they require a physical explanation different
than second-order theory [3.140]. The statistical explanation for something like the Andrea wave is certainly
a stretch. According to second-order statistics, its crest
had a probability of 6 10
8 . However, Christou and
Ewans [3.141] did a careful study of over 10
8 measured waves and found that the sample crest distribution
was only slightly higher than predicted by second-order
statistics.
Some processes that produce very high waves are
understood reasonably well. Waves traveling into an opposing current can steepen and become much higher.
Many ships have been damaged when they encountered
such waves in the Agulhas Current south of Africa.
Bottom features can refract waves, making them much
larger in localized areas. Surfers are well aware of this
phenomenon. The more difficult cases to explain are unusually high waves in deep water far from shore.
Theoretical attempts to explain rogue waves involve
the integration of nonlinear equations that approximate
the development of steep random waves [3.142]. All of
these show a modulation of the wave envelope similar to the Benjamin–Feir instability observed in regular