Lecture Note
Wave Spectra and Short/Long Term Statistics
-
University:
Massachusetts Institute of Technology -
Course:
2.22 | Design Principles for Ocean Vehicles (13.42) Academic year:
2021
-
Views:
388
Pages:
21
Author:
Nathan Ferguson
13.42 Design Principles for Ocean Vehicles
Spring 2005
1. Ocean Wave Spectra
1. Wave energy spectra. Red text indicates wave generation mechanisms and blue text indicates
damping/restoring forces.
The majority of ocean waves are wind generated. Other wave generating mechanisms
include earthquakes and planetary forces. Planetary forces drive tides and cause long
1 period waves on the order of 12 to 24 hours. Earthquakes are the major cause of tsunamis
which, while rare, can be catastrophic if the earthquake occurs near or on the coast.
Waves also encounter forces that tend to restore them to a flat surface. For small
wavelength (high frequency) waves surface tension plays a large role in damping out these
waves. The majority of waves are restored by gravity and longer period waves are damped
by the Coriolis force.
As wind begins to blow (between 0.5 - 2 knots) on a calm surface small ripples, capillary
waves or “cat-paws”, tend to form. These small waves are on the order of less than 2 cm.
As the wind becomes stronger wave amplitude increases and the waves become longer in
order to satisfy the dispersion relationship. This growth is driven by the Bernoulli effect,
frictional drag, and separation drag on the wave crests.
Wind must blow over long periods of time and large distances to reach a fully developed
sea state. When the phase speed of the wave crest matches the wind speed non-linear
interactions stop (except friction) and the phase speed is maximized. The limiting
frequency of the waves can be determined by the equation for phase speed and the
dispersion relationship:
C p » U w = w/k = g /w
wc »
g
Uw
(1)
(2)
where U w is the wind speed and wc is the limting frequency. Once wind stops viscosity
erodes the waves slowly. The smallest wavelengths decay the fastest. Sample spectrum
shapes are shown in figure 2.
2 For a storm with wind speed, U w , the effects of the storm can be felt at a distance from the
storm, R . The number of wave cycles between the storm and the observation location is
N = R/l . The amplitude of the waves decays as e -g t where g = 2n k 2 = 2nw 4 /g 2 (from
Landau and Lifshitz).
The development of storms can be tabulated. Fetch is the length over which the wind must
blow to have fully developed seas (given in standard miles), and the storm duration, given
in hours, is the time the storm must last to result in a fully developed sea.
Wind
warnings
small craft
gale
hurricane
Beaufort
scale
Wind
speed (mph)
Fetch
(miles)
Storm
duration (hr)
3-4
12
15
3
5-6
25
100
12
7
35
400
28
9
50
1050
50
70+
3 2. Typical Wave Spectra
Researchers have studying ocean waves have proposed several formulation for wave
spectra dependent on a a number of parameters (such as wind speed, fetch, or modal
frequency). These formulations are very useful in the absence of measured data, but they
can be subject to geographical and seasonal limitations.
Most ocean wave spectra take a standard form following the mathematical formulation:
S + (w ) =
A - B/w 4
e
w5
(3)
The frequency peak is called the modal frequency. The area under the spectrum is the
zeroth moment, M o , which may be defined in terms of the significant wave height. For a
narrow-banded spectrum the significant wave height is approximately four times the
square root of the zeroth moment. Since the significant wave height depends on the wind
speed, the spectrum could be formulated in terms of the wind speed instead of the
significant wave height. While certain spectra can have more than one peak, it is assumed
that a single storm produces a single-peaked spectrum and any second peak is due to a
distant storm that sends waves to the considered location.
Several more specific theoretical representations of wave spectra have been developed
using data collected by observation platforms and satellite data in various regions. These
spectra are discussed below. When considering which spectrum formulation it is important
to take into account the specific criteria that were used in developing the spectrum.
Typically parameters that influence the spectrum are:
•
Fetch limitations, i.e. whether the location we are considering has some physical
boundaries that do not permit the waves to fully develop.
•
Whether the seas are developing or decaying
•
Seafloor topography: Deep water wave spectra are invalid in shallow waters, and
vice versa. It may also be necessary to account for wave diffraction.
•
Local currents: Strong currents may significantly impact the wave spectrum
4 •
Presence of swells: Swells are waves that result from distant storms that travel a
significant distance and arrive often at an angle that differs from the wind direction.
If we use a spreading function to correct a unidirectional spectrum it will not
account for the presence of swell. It is also important when measuring waves that
the component that results from swell be accounted for separately.
The Pierson-Moskowitz spectrum (equation 11) was developed for fully developed seas in
the Northern Atlantic Ocean generated by local winds.
2 2
8.1 g 2 -0032(
S (w ) = 3 5 e . g/zw )
10 w
+
(4)
where z is the significant wave height,
z ” H 1/ 3 = 4 M o ,
(5)
wm = 0.4 g /z .
(6)
and wm is the modal frequency,
This spectrum is developed under the following conditions: unidirectional seas, North
Atlantic Ocean, fully developed local wind generation with unlimited fetch. The most
critical of these assumptions is the fully developed assumption. For it is possible to achieve
a larger heave response for a platform from a developing sea, even though the significant
wave height may be smaller that that of a fully developed sea, since the modal frequency is
higher and heave motions tend to have higher natural frequencies. In the case of a rolling
ship the decaying sea might excite a larger roll motion since the natural frequency of roll
tends to be relatively low.
In order to overcome the limitation of fully developed seas, a two parameter spectrum was
developed. This spectrum is the Bretschneider spectrum (equation 14). The B-S spectrum
5 replaced the Pierson-Moskowitz spectrum as the ITTC standard.
S + (w ) =
1.25 wm4
. (wm /w )4
z e-125
5
4 w
(7)
where again, z is the significant wave height,
z ” H 1/ 3 = 4 M o ,
(8)
If wm satisfies equation 13 then equation 14 reduces to equation 11. By allowing the user
to specify the modal frequency and significant wave height, this spectrum can be used for
sea states of varying severity from developing to decaying.
The Ochi Spectrum (equation 16) is a three parameter spectrum that allows the user to
specify the significant wave height, the modal frequency, and the steepness of the spectrum
peak.
l
� 4l +1
4�
�� � 4l +1 �� wm � 4 ��
1 � 4 wm �ł z 2
S (w ) = Ł
exp
�- �
��
��
4 G(l ) w 4l +1
� Ł 4 łŁ w ł �
+
(9)
where G ( l ) is the gamma function, and l is the parameter that controls the spectrum
steepness. For l = 1 , equation 16 reduces to equation 14. The Ochi spectrum is limited in
that it also considers only unidirectional seas and unlimited fetch, but the designed can now
specify the spectrums severity (z ), the state of development (peak frequency wm ) and
isolate the important frequency range by dictating the spectrum width ( l ). The ability to
dictate l allows the designer to account for swell from a distant storm.
The JONSWAP spectrum (equation 17) was developed by the Joint North Sea Wave
Project for the limited fetch North Sea and is used extensively by the offshore industry.
This spectrum is significant because it was developed taking into consideration the growth
of waves over a limited fetch and wave attenuation in shallow water. Over 2,000 spectra
were measured and a least squares method was used to obtain the spectral formulation
6 assuming conditions like near uniform winds.
ag 2 - 54 ( wwm ) d
g
e
w5
(10)
(w - wm )2
2s 2wm2
(11)
4
S + (w ) =
where
d =-
a = 0.076 x (-0 .2 2 )
(12)
gx
U2
(13)
0.07; w £ wm (21) ��
�
�
0.09; w > wm ��
�
(14)
x=
�
�
s =�
The wind speed in knots is U , x is the fetch in nautical miles, and the modal frequency
can be found as
.
wm = 2p * 3.5 * ( g /U ) x -033
.
(15)
To recap, in general for a narrow banded spectrum:
�z �
�-¥ S (w ) dw = M o = �Ł 4 �ł
¥
+
2
(16)
We can account for the effects of two separate storms by adding the respective spectrums:
S + (w ) = S1+ (w ) + S 2+ (w )
(17)
We can also correct for directionality multiplying the spectrum by a spreading function,
7 M (m) ,
+
(w ) M (m )
S + (w, m ) = S BS
(18)
where M ( m ) spreads the energy over a certain angle contained within the interval [ -p , p ]
from the wind direction. The integral of M ( m ) over this interval is one.
�
p
-p
M (m) d m = 1
(19)
For example we can choose a spreading function such that
M (m) =
2
cos2 m
p
(20)
on the interval
-
m
m
a1/N )}
This is the expected value given am is greater than a1/N as can be represented as
11
(29) a1/N = � 1 /N a p ( (am = a) | (am > a1/N ) ) da
¥
a
(30)
where the probability, p ��Ł am = a | am > a1/N ��ł , is simply
�
�
Ł
p am = a | am > a
1/N �
�
ł
=
P ( (am = a) ˙ (am > a1/N ) )
P (am > a1/N )
(31)
Keeping the amplitude in non-dimensional form we can calculate the Nth highest average
wave height using the approximate pdf given in the last reading.
h 1/N »
2 N 1 -e 2
1+ 1- e 2
�
¥
h
1 /N
h o2 e -ho / 2dh o
2
(32)
For a value of N = 3 , a1/N is considered the significant wave amplitude where
a1/N » 2s = 2 M o for e < 0 .5 . The significant wave height is defined as twice the
significant wave amplitude,
H 1/ 3 = 2 a1/ 3 .
(33)
This value is very close to that which a casual observer would estimate as the wave height
when watching the sea. This makes the significant wave height a very useful statistic. Maps
of the significant wave height over the entire earth can be seen on satellite images. There
are several links on the course webpage that illustrate this quantity over the earth surface.
From such a satellite composite image we can see that the southern ocean is the most
tumultuous ocean and has the highest significant wave height.
12 5. Long Term Statistics
Structural design analysis of offshore structures over a long time, specifically the total life
span of the structure or T , requires knowledge of the short term statistics of waves at the
system installation location. Over the life of a platform or structure there are a total of i
storms, one of these can also include a “storm” which is described completely calm
conditions, similar to the null set in probability. The probability of each storm event is, Pi ,
which can also be looked at as the fraction of the total life of the structure over which a
certain storm, i , exists. Thus the life of each storm, Ti is dictated by equation 1.
Ti = T Pi
(34)
From short term statistics we have the frequency of waves, upcrossings, exceeding a
certain amplitude, ao .
n(ao ) =
1
2p
M 2 -ao2 / ( 2 M o ) 1 - ao2/ ( 2 M o ) 1
=
e
=
e
m
Mo
T
T
(35)
Where T is the average period of waves in a storm. So it is easy to find the total number
of times the level ao is exceeded in a storm i , given by Ni as
N i = n(ao ) Ti =
Ti
mi
Ti
(36)
Where Ti is the life of the storm i , Ti is average period of the waves within that storm.
The total number of times level ao is exceeded during the life of the structure then
becomes:
N ao = � N i = �
all i
i
T
- a2/ ( 2M )
Pi e o o i
T
(37)
where M o i is the zeroth moment of the spectrum of storm i . It is necessary to make the
assumption that the spectrum is narrow banded so that we can write the zeroth moment in
13 terms of the significant wave height. This simplifies the problem since the significant wave
height is the data most often available that describe sea conditions over a certain time
period.
Next we can adjust the equation for total number of upcrossings to reflect a number of
upcrossings exceeding the height, ho which is simply defined as twice the amplitude, ao .
N ho = �
i
� 2h2 �
-� 2o �
T
Pi e Ł z i ł
T
(38)
Equation 5 represents the total number of upcrossings past a level ho over a time
interval T . The total number of upcrossings past the mean water level, ho = 0 , is given by
equation 6.
No = �
i
T
Pi
T
(39)
Having determined the equation for the total number of waves exceeding a level ho and
also the total number of upcrossings during the life of the structure, we can find the
probability of a wave height exceeding a design height ho in equation (40)
P(h > ho ) =
Nho
No
�
=
i
Pi
�
e -2ho /z i
2
T
Ti
i
Pi
2
(40)
T
Ti
This equation represents the fraction of total waves above level ho . Realizing the life span
of the structure, T , is constant we can cancel it out of equation (40) and rewrite it in the
form
P (h > ho ) =
N ho
No
=
�
i
�
�
�
�
Ł
1
Ti
2
� ( )P
i
14
2 �
e -2 ho /z i ��� Pi
1
Ti
i
ł
(41) The numerator and denominator of equation (41) each have the form of the expected value
formula for a random process with random variables T and z . So we can rewrite the
probability as
�
P(h > ho ) =
E ��� T1 e -2ho /z
2
�
E
{ }
2 �
�
�
�
�
(42)
1
T
Note the subscript i is dropped since T is the random variable and Ti is a single random
event. Thus the expected value of the random variable is the sum over all the ith random
events. This can also be extended to a random process.
We now have and equation for the probability of waves above a level that is not implicitly
dependent on the life span of the structure. In practical application the average wave period
can be neglected (cancelled) and the probability approximated by
�
2 �
P (h > ho ) » E ��� e-2 ho /z ��� .
�
2
�
(43)
One valuable calculation is related to the hundred year wave. This is the wave that exceeds
some value, h1 00 , on the average only once in every one hundred years.
P(h > h1 00) =
N (# of waves over h100 )
T
1
=
=
N o (# of total upcrossings) 100 years /T 3.16 *109 seconds
(44)
Note: This equation takes into consideration 100 years with the appropriate number of leap
years. So now we have an equation for the probability of the hundred year wave.
�
E ��� e -2h100 /z
�
2
2 �
�
�
�
�
=
T
3.16e + 9
(45)
Where T is the average period of waves over all storms. In order to solve for h100 we must
have statistics for very rough storms with high significant wave heights. Smaller, more
15 frequent, values of wave height do not play a role in this calculation and it is only the most
ferocious storms that contribute to this value. Thus you can imagine in order to collect this
data observations during major storms, hurricanes, typhoons, etc., must be performed over
time. Under these conditions it is too risky for human observation and often equipment is
lost at sea making obtaining this challenging.
From probability theory it is possible to approximate the probability of extreme events by
using a Weibull distribution in the form
(
P ( x > xo ) = e
-
xo -x1
x 2 -x1
)
g
(46)
where x1 , x2 , and g are estimated from historical data and x1 is a threshold level below
which the statistics will not effect the extremes.
6. Encounter Frequency
The principle motions of a ship or structure in seas are defined as surge, sway, heave, pitch,
roll and yaw. A diagram of the coordinate system is shown in figure 1.
16 4. Motion of a ship defined with a right handed coordinate system with forward motion as the positive
x-direction and positive heave upwards.
Waves incident on the structure or ship can be described as head seas, following seas, beam
seas, or quartering seas depending on the incident direction. Figure 2 illustrates the first
three cases and quartering seas are defined as those that approach the ship from the either
the port (left) or starboard (right) stern quarter of the ship (between 90o and 180 o or 180 o
and 270o ).
5. Incident sea description. Figure at left shows head, beam and following seas. The incident angle, m ,
is measured from the bow (x-axis is
m = 0o ) counterclockwise.
The motion of a ship, forward or otherwise, affects the way incident waves are viewed by
someone aboard the vessel. For example if the ship is making way in head seas with a
constant velocity, U , then the waves will appear to meet the ship at a faster rate than the
actual frequency of the waves. This new, or observed, frequency is termed the encounter
frequency, we . If the waves are incident on the ship at some angle, m , then the component
of the speed of the ship in the direction of wave propagation is U a = U cos m . The wave
crests move at the phase speed, C p = w/k and the relative speed between the ship and the
17 waves is
U r = Ua + Cp = U cos m + w/k
(47)
Thus it appears that the waves have a phase speed U r such that
Ur =
we
w
= U cos m +
k
k
(48)
Using the dispersion relationship for waves in deep water we can rewrite the equation for
encounter frequency as
we =
Uw 2
cos m + w
g
(49)
for w > 0 . In practice we usually have the encounter frequency since it is observed and we
would like to calculate the actual wave frequency, so taking equation (49) we can solve for
frequency, w .
�
g
U cos m
��
w=
� -1– 1 + 4we
�
2U cos m �
g
�
��
�
�
�
(50)
Looking at the equation 17 we see that there are several possible solutions for w . These
are dependent on the incident angle m . For 4we
U cos m
g
> -1 (real values of w ) we can look
at different incident angles:
(1) HEAD SEAS: incident angle between -p/2 < m < p/2 and the cosine of the angle is
positive ( cos m > 0 ). The encounter frequency is always positive for head seas.
(2) FOLLOWING SEAS: incident angle between p/2 < m < 3p /2 and cos m < 0 . Here
there are three possible scenarios:
18 | U cos m |> C p ,
we < 0 ,
ship overtakes waves
| U cos m |= C p ,
we = 0 ,
ship “surfs” waves
| U cos m |< C p ,
we > 0 ,
waves overtake ship
The final case, where the waves move faster than the ship, can tend to cause problems in
controlling a vessel, especially when the seas are not directly from behind the ship. This
effect is greatest in roll and yaw. When a ship is moving with the waves but overtaking
them it can also appear as if the seas are approaching from the bow, thus the actual incident
angle is ambiguous and could be m or m + p .
The effect of encounter frequency also changes the observed spectrum of the seas. Energy
must be preserved under the spectrum thus we can look at the following
S (we )| dwe |= S (w )| dw |
S (we ) =
S (w )
dwe
dw
(51)
(52)
We can find the derivative of encounter frequency by the actual frequency by using
equation (49).
dwe
2w
= 1+
U cos m
dw
g
(53)
So the spectrum of the encounter frequency becomes
S (w e ) =
S (w )
1+ 2wg U cos m
19
(54) This spectrum has an integrable singularity at
-g
.
2U cos m
w=
(55)
At this value of w the encounter frequency is
we =
g
.
4U | cos m |
(56)
Given U , m , we can look the two cases with encounter frequency
we = w +
w |w |
U cos m
g
1. if m is in ( -p /2 ,p /2 ) then we > 0 for w > 0 and we > w for w > 0 .
g
2. if m is in ( p/2, 3p/ 2 ) then we < 0 for w < 0 and for w > U |cos
m| .
6. Encounter frequency versus actual frequency for following seas ( cos m < 0 ) and w > 0 .
20
(57) 7. Encounter frequency versus actual frequency for the three conditions: head, beam and following
seas.
7. Useful References
Read section four of the supplemental notes: Triantafyllou and Chryssostomidis, (1980)
"Environment Description, Force Prediction and Statistics for Design Applications in
Ocean Engineering"
21
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