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36.1 Description of the Ocean Environment

Back

In modeling offshore structures, one needs to account for the forces exerted by the surrounding fluid.

In-depth studies are given in Kinsman (1965), Sarpkaya and Isaacson (1981), Wilson (1984), Chakrabarti

(1987), and Faltinsen (1993). The vibration characteristics of a structure can be significantly altered when

it is surrounded by water. For example, damping by the fluid (or the added mass) lowers the natural

36-1

frequency of vibration. When considering the dynamics of an offshore structure, one must also consider

the forces due to the surrounding fluid. The two important sources of fluid motion are ocean waves and

ocean currents.

Most steady large currents are generated by the drag of the wind passing over the surface of the water,

and they are confined to a region near the ocean surface. Tidal currents are generated by the gravitational

attraction of the sun and the moon, and they are most significant near coasts. The ultimate source of the

ocean circulation is the uneven radiation heating of the Earth by the Sun.

Isaacson (1988) suggested an empirical formula for the current velocity in the horizontal direction as a

function of depth:

UcðxÞ ¼ ðUtide ðdÞ þ Ucirculation ðdÞÞ

􀀏 􀀐1=7

þ Udrift ðdÞ

x 2 d þ d0

􀀏 􀀐

ð36:1Þ

where Udrift is the wind-induced drift current, Utide is the tidal current, Ucirculation is the low-frequency

long-term circulation, x is the vertical distance measured from the ocean bottom, d is the depth of the

water, and d0 is the smaller of the depth of the thermocline and 50 m. The value of Utide is obtained from

tide tables, and Udrift is about 3% of the 10 min mean wind velocity at 10 m above the sea level.

It should be noted that these currents evolve slowly compared with the time scales of engineering

interests. Therefore, they can be treated as a quasisteady phenomenon. Waves, on the other hand, cannot

be treated as a steady phenomenon. The underlying physics that govern wave dynamics are too complex

and, therefore, waves must be modeled stochastically. The subsequent section discusses the concept of the

spectral density, available ocean wave spectral densities, a method to obtain the spectral density from

wave time histories, methods to obtain a sample time history from a spectral density, the short-term and

long-term statistics, and a method to obtain fluid velocities and accelerations from wave elevation using

linear wave theory.

36.1.1 Spectral Density

Here, we will consider only surface gravity waves.

Let us first consider a regular wave in order to

familiarize ourselves with the terms that are used

to describe a wave. The wave surface elevation is

denoted as hðx; tÞ and can be written as hðx; tÞ ¼

A cosðkx 2 vtÞ; where k is the wave number, and v

is the angular frequency. Figure 36.1 shows the

surface elevation at two time instances (t ¼ 0 and

t ¼ t) and the surface elevation at a fixed location

ðx ¼ 0Þ: A is the amplitude, H is the wave height or

the distance between the maximum and minimum

wave elevation or twice the amplitude, and T is the

period given by T ¼ 2p=v:

In practice, waves are not regular. Figure 36.2

shows a schematic time history of an irregular

wave surface elevation. The wave height and

frequency are not easy to find. Therefore, we rely

on a statistical description for the wave elevation

such as the wave spectral density. The spectral

density tells us how the energy of the system is distributed among frequencies. The random surface

elevation hðtÞ can be thought of as a summation of regular waves with different frequencies. The surface

elevation hðtÞ is related to its Fourier transform XðvÞ by

hðtÞ ¼

ð1

XðvÞ expð2ivtÞdv

t = 0

x = 0

t = t

Time

Distance

Wave Elevation h(t) Wave Elevation h(x)

FIGURE 36.1 Regular wave.

36-2 Vibration and Shock Handbook

Suppose that the energy of the system is proportional

to h2ðtÞ so that we can write the energy as

E ¼

Ch2ðtÞ

where C is the proportionality constant.

Let us assume that the expected value of the

energy is given by

E{E } ¼

CE{h2ðtÞ}

where E{h2ðtÞ} is the mean square of hðtÞ: If hðtÞ is an ergodic process (see Chapter 5 and Chapter 30),

then the mean square of hðtÞ can be approximated by the time average over a long period of time:

E{h2ðtÞ} ¼ lim

Ts !þ1

ðTs =2

2Ts =2

h2ðtÞdt ¼ lim

Ts !þ1

ð1

lXðvÞl2 dv ð36:2Þ

where we have used Parseval’s theorem

ð1

h2ðtÞdt ¼

ð1

lXðvÞl2 dv ð36:3Þ

where

lXðvÞl2 ¼ XðvÞXpðvÞ; XðvÞ ¼

ð1

hðtÞ expð2ivtÞdt; XpðvÞ ¼

ð1

hðtÞ expðivtÞdt

We define the power spectral density (or simply the spectrum) as (see Chapter 5 and Chapter 30)

Shh ðvÞ ; 1

2pTs

lXðvÞl2 ð36:4Þ

so that E{h2ðtÞ} is given by

E{h2ðtÞ} ¼

ð1

Shh ðvÞdv ð36:5Þ

For a zero-mean process, E{h2ðtÞ} is also the variance s2h

: The spectral density has units of h2t: Where h

is the wave elevation, the spectral density has a unit of m2 sec.

It can also be shown that Shh ðvÞ is related to the autocorrelation function, RðtÞ; by the Wiener –

Khinchine relations (Wiener, 1930; Khinchine, 1934):

Shh ðvÞ ¼

ð1

Rhh ðtÞ expð2ivtÞdt; Rhh ðtÞ ¼

ð1

Shh ðvÞ expðivtÞdv ð36:6Þ

It should be noted that, in some textbooks, the factor 1=2p appears in the second equation instead of the

first. Figure 36.3 shows some important pairs of Shh ðvÞ and Rhh ðtÞ:

There are a few properties of the spectral density that readers should become familiar with. The first

property is that the spectral density function of a real-valued stationary process is both real and

symmetric. That is, Shh ðvÞ ¼ Shh ð2vÞ (Equation 36.4). Secondly, the area under the spectral density is

equal to E{h2ðtÞ} (Equation 36.5) and is also equal to Rhh ð0Þ ¼ s2h

2 m2h

; where s2h

is the variance and m2h

is the mean of hðtÞ: In most cases, we only consider a zero-mean process so that the area under the

spectral density is just s2h

: If the process does not have a zero mean, the mean can be subtracted from it so

that the process has a zero mean.

For ocean applications, a one-sided spectrum in terms of cycles per second (cps) or hertz is often used.

We will denote the one-sided spectrum with a superscript “o”. The one-sided spectrum can be obtained

from the two-sided spectrum by

hh ðvÞ ¼ 2Shh ðvÞ; v $ 0

Time

Wave Elevation h(t)

FIGURE 36.2 Time history of random wave.

Fluid-Induced Vibration 36-3

The two-sided spectrum in terms of v can be transformed to the spectrum in terms of f (where v ¼ 2pf )

Shh ðf Þ ¼ 2pShh ðvÞ; f ; v $ 0

Then, the two-sided spectrum in terms of v can be transformed to the one-sided spectrum in terms of

cps (or hertz) by

hh ðf Þ ¼ 4pShh ðvÞ; f ; v $ 0

R(t) = S(w)eiw tdw S(w) =

d(w)

2pd(w)

d(w +w0)

cos w0t

2p/T

1/a

2pe-aΩtΩ

2pe-aΩtΩcos w0t

sin w0t

•

-•

R(t)e-iwtdw

•

-•

a2+w2

4 sin2 (wT/2)

Tw2

-T T

d(w-w0)

-w0 w0

FIGURE 36.3 Relationship between the autocorrelation function and the power spectral density.

36-4 Vibration and Shock Handbook

It should be noted that the spectral density that we have defined here is the amplitude half-spectrum. The

amplitude, height, and height double spectra are related to the amplitude half-spectrum by

SAðvÞ ¼ 2SðvÞ; SH ðvÞ ¼ 8SðvÞ; S2H ðvÞ ¼ 16SðvÞ

36.1.2 Ocean Wave Spectral Densities

In this section, we will discuss spectral density models to describe a random sea. An excellent review of

existing spectral density models is given in Chapter 4 of Chakrabarti (1987).

The ocean wave spectrum models are semiempirical formulas. That is, they are derived mathematically

but the formulation requires one or more experimentally determined parameters. The accuracy of the

spectrum depends significantly on the choice of these parameters.

In formulating spectral densities, the parameters that influence the spectrum are fetch limitations,

decaying vs. developing seas, water depth, current, and swell. The fetch is the distance over which a wind

blows in a wave-generating phase. Fetch limitation refers to the limitation on the distance due to some

physical boundaries so that full wave development is prohibited. In a developing sea, the sea has not yet

reached its stationary state under a stationary wind. In contrast, a wind has blown for a sufficient time in

a fully developed sea, and the sea has reached its stationary state. In a decaying sea, the wind has dropped

off from its stationary value. Swell is the wave motion caused by a distant storm and persists even after the

storm has died down or moved away.

The Pierson –Moskowitz (P– M) spectrum (Pierson and Moskowitz, 1964) is the most extensively used

spectrum for representing a fully developed sea. It is a one-parameter model in which the sea severity can

be specified in terms of the wind velocity. The P–M spectrum is given by

hh ðf Þ ¼

8:1 £ 1023g2

v5 exp 20:74

Uw;19:5 m

􀁻 !4

v24

􀁻 !

where g is the gravitational constant and Uw;19:5 m is the wind speed at a height of 19.5 m above the still

water. The P–M spectrum is also called the wind-speed spectrum because it requires wind data. It can

also be written in terms of the modal frequency vm as

hh ðf Þ ¼

8:1 £ 1023g2

v5 exp 21:25

􀀏 􀀐4 􀀏 􀀐

ð36:7Þ

Note that the modal frequency is the frequency at which the spectrum is the maximum.

In some cases, it may be more convenient to express the spectrum in terms of significant wave height

rather than the wind speed or modal frequency. For a narrowband Gaussian process1, the significant wave

height is related to the standard deviation by Hs ¼ 4sh : The standard deviation is the square root of the

area under the spectral density,

1 Shh ðvÞdv ¼ s2h

: Then, the spectrum can be written as

hh ðf Þ ¼

8:1 £ 1023g2

v5 exp 2

0:0324g2

v24

􀁻 !

ð36:8Þ

and the peak frequency and the significant wave height are related by

vm ¼ 0:4

ffiffiffiffiffiffi

g=Hs

ð36:9Þ

The P–M spectrum is applicable for deep water, unidirectional seas, fully developed and local-windgenerated

seas with unlimited fetch, and was developed for the North Atlantic. The effect of swell is not

accounted for. Although it was developed for the North Atlantic, the spectrum is valid for other locations.

However, the limitation that the sea is fully developed may be too restrictive because it cannot model the

1See Section 36.1.5 for details.

Fluid-Induced Vibration 36-5

effect of waves generated at a distance. Therefore,

we consider a two-parameter spectrum, such as the

Bretschneider spectrum, in order to model a sea

that is not fully developed as well as a fully

developed sea.

The Bretschneider spectrum (Bretschneider,

1959, 1969) is a two-parameter spectrum in

which both the sea severity and the state of

development can be specified. The Bretschneider

spectrum is given by

hh ðf Þ ¼ 0:169

v4s

v5 H2

s exp 20:675

􀀏 􀀐4 􀀏 􀀐

where vs ¼ 2p=Ts and Ts is the significant

period. The sea severity can be specified by Hs

and the state of development can be specified by

vs: It can be shown that the relationship vs ¼ 1:167vm (equivalent to vs ¼ 1:46=

ffiffiffiffi

Hs p ) renders the

Bretschneider spectrum and the P–M spectrum equivalent. Figure 36.4 shows the Bretschneider

spectra for Hs ¼ 4 m. When vs ¼ 0:731 rad/sec, the P–M and the Bretschneider spectra are identical.

It should be noted that the developing sea will have a slightly higher modal frequency than the fully

developed sea, and can be described by vs greater than 1:46=

ffiffiffiffi

Hs p :

Other two-parameter spectral densities that are often used are the International Ship Structures

Congress (ISSC) and the International Towing Tank Conference (ITTC) spectra. The ISSC spectrum is

written in terms of the significant wave height and the mean frequency, where the mean frequency is

given by

v􀀊 ¼

ffiðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

vSðvÞdv

ð1

SðvÞdv

vuuuuut

¼ 1:30vm

Thus, the ISSC spectrum is given by

hh ðf Þ ¼ 0:111

v􀀊4

v5 H2

s exp 20:444

v􀀊

􀀏 􀀐4 􀀏 􀀐

The ITTC spectrum is based on the significant wave height and the zero crossing frequency and is

given by

hh ðf Þ ¼ 0:0795

v4z

v5 H2

s exp 20:318

􀀏 􀀐4 􀀏 􀀐

where the zero crossing frequency, vz, is given by

vz ¼

ffiðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

v2SðvÞdv

ð1

SðvÞdv

vuuuuut

¼ 1:41vm

The Bretschneider, ITTC, and ISSC spectra are called two-parameter spectra, and they can be written as

hh ðf Þ ¼

v~4

v5 H2

s exp 2A

􀀏 􀀐4 􀀏 􀀐

with A and v given in Table 36.1.

0.5

1.5

2.5

0.5 1 1.5

Frequency w (rad/s)

Spectral Density (m2 s)

2 2.5 3

ws = 1.315

ws = 1.169

ws = 1.023

ws = 0.877

Pierson-Moskowitz spectrum

ws = 0.731 Hs = 4m {

FIGURE 36.4 Bretschneider spectrum with various

values of vs :

36-6 Vibration and Shock Handbook

The spectra that we have discussed so far do not allow us to generate spectra with two peaks to

represent local or distant storms or to specify the sharpness of the peaks. The Ochi – Hubble (O – H)

spectrum (Ochi and Hubble, 1976) is a six-parameter spectrum with the form:

hh ðvÞ ¼

i¼1

ðð4li þ 1Þv4

mi=4Þli

GðliÞ

v4li þ1 exp 2

4li þ 1

􀀏 􀀐

vmi

􀀏 􀀐4 􀀏 􀀐

where GðliÞ is the Gamma function, Hs1; vm1; and l1 are the significant wave height, modal frequency,

and shape factor for the lower frequency components, respectively, and Hs2; vm2; and l2 are those for the

higher frequency component. Assuming that the entire spectrum is that of a narrow band, the equivalent

significant wave height is given by

Hs ¼

ffiffiffiffiffiffiffiffiffiffiffiffi

s1 þ H2

For l1 ¼ 1 and l2 ¼ 0; the spectrum reduces to the P–M spectrum. With the assumption that the entire

spectrum is narrowband, the value of l1 is much higher than l2: The O –H spectrum represents

unidirectional seas with unlimited fetch. The sea severity and the state of development can be specified by

Hsi and vmi; respectively. In addition, li can be selected to control the frequency width of the spectrum. For

example, a small li (wider frequency range) describes a developing sea, and a large li (narrower frequency

range) describes a swell condition. Figure 36.5 shows the O – H spectrum with l1 ¼ 2:72; vm1 ¼ 0:626 rad/

sec, Hs1 ¼ 3:35 m, l2 ¼ 2:72; vm2 ¼ 1:25 rad/sec, and Hs2 ¼ 2:19 m.

Finally, another spectrum that is commonly used is the Joint North Sea Wave Project (JONSWAP)

spectrum developed by Hasselmann et al. (1973). It is a fetch-limited spectrum because the growth over a

limited fetch is taken into account. The attenuation in shallow water is also taken into account. The

JONSWAP spectrum is written as

hh ðvÞ ¼

ag2

v5 exp 21:25

􀀏 􀀐4 􀀏 􀀐

gexpð2ðv2vm Þ=2t 2v2m

where g is the peakedness parameter and t is the shape parameter. The peakedness parameter g is the

ratio of the maximum spectral energy to the maximum spectral energy of the corresponding P– M

spectrum. That is, when g ¼ 7; the peak spectral energy is seven times that of the P–M spectrum.

g ¼

7:0 for very peaked data

3:3 for mean of selected JONSWAP data

1:0 for P – M spectrum

8>><

>>:

t ¼

0:07 for v # vm

0:09 for v . vm

(

a ¼ 0:076ðX􀀊 Þ20:22 or 0:0081 if fetch independent

X􀀊 ¼ gX=U2

X ¼ fetch length ðnautical milesÞ

Uw ¼ wind speed ðknotsÞ

vm ¼ 2p £ 3:5ðg=Uw Þ X􀀊 20:33

TABLE 36.1 Two-Parameter Spectrum Models

hh ðvÞ ¼ ðA=4ÞH2

sv~4=v5 expð2Aðv=v~Þ24 Þ

Model A v~

Bretschneider 0.675 vs

ITTC 0.318 vz

ISSC 0.4427 v􀀊

0 0.5 1

Spectrum Density (m2 s)

1.5

Frequency w (rad/s)

Hs = 4 m

higher frequency spectrum

lower frequency

spectrum

2.5

FIGURE 36.5 Ochi – Hubble spectrum.

Fluid-Induced Vibration 36-7

Figure 36.6 shows the JONSWAP spectrum

when a ¼ 0:0081 and vm ¼ 0:626 rad/sec for

three peakedness parameters.

36.1.3 Approximation of Spectral

Density from Time Series

From the time history of the wave elevation, the

spectral density function can be obtained by two

methods.

The first method is to use the autocorrelation

function Rhh ðtÞ; which is related to the spectral

density function Shh ðvÞ by the Wiener– Khinchine

relations (Equation 36.6).

The autocorrelation Rhh ðtÞ is the expected value

of hðtÞhðt þ tÞ or Rhh ðtÞ ¼ E{hðtÞhðt þ tÞ};

where t is an arbitrary time and t is the time lag.

For a weakly stationary process, the autocorrelation

is a function of the time lag only.

Assuming that the process is ergodic, the autocorrelation function for a given time history of length Ts

can be approximated as

R^ hh ðtÞ ¼ lim

Ts!1

Ts 2 t

ðTs 2t

hðtÞhðt þ tÞdt for 0 , t , Ts

Note that the superscript ‘ is used to emphasize that the variable is an approximation based on a sample

time history of length Ts: The spectral density is then obtained by taking the Fourier cosine transform of

R^ hh ðtÞ;

S^hh ðvÞ ¼

ðTs

R^ hh ðtÞ cos vt dt ð36:10Þ

The second method for obtaining the spectral density function is to use the relationship between the

spectral density and the Fourier transform of the time series. They are related by

S^hh ðvÞ ¼ lim

Ts!1

2pTs

lX^ ðvÞ X^ pðvÞl ð36:11Þ

where X^ ðvÞ is given by

X^ ðvÞ ¼

ðTs

hðtÞ expð2ivtÞdt

and X^ pðvÞ is the complex conjugate given by

X^ p ðvÞ ¼

ðTs

hðtÞ expðivtÞdt

In order to obtain the Fourier transforms of the time series (see Chapter 2, Chapter 10, Chapter 21,

and Appendix 2A), the discrete Fourier transform (DFT) or the fast Fourier transform (FFT) procedure

can be used. For detailed descriptions of how this is done, see Appendix 1 in Tucker (1991). Nowadays,

spectral analysis is almost always carried out via FFTs because it is easier to use and faster than the formal

method via correlation function.

It should be noted that the length of the sample time history only needs to be long enough so that the

limits converge. Taking a longer sample will not improve the accuracy of the estimate. Instead, one

should take many samples or break one long sample into many parts. For n samples, the spectral densities

g = 7.0

g = 3.3

g = 1

a= 0.0081

wm = 0.626

Frequency w (rad/s)

Spectral Density (m2s)

0.5 1 1.5 2 2.5 3

FIGURE 36.6 JONSWAP spectrum for g ¼ 1.0, 3.3,

and 7.0.

36-8 Vibration and Shock Handbook

are obtained for each sample time history using either Equation 36.10 or Equation 36.11, and they are

averaged to give the estimate.

The determination of the spectral density from wave records depends on the details of the procedure

such as the length of the record, sampling interval, degree and type of filtering and smoothing, and time

discretization.

36.1.4 Generation of Time Series from a Spectral Density

In a nonlinear analysis, the structural response is found by a numerical integration in time. Therefore,

one needs to convert the wave elevation spectrum into an equivalent time history. The wave elevation can

be represented as a sum of many sinusoidal functions with different angular frequencies and random

phase angles. That is, we write hðtÞ as

hðtÞ ¼

i¼1

cosðvit 2 wiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Shh ðviÞDvi

ð36:12Þ

where wi is a uniform random number between 0 and 2p, vi are discrete sampling frequencies,

Dvi ¼ vi 2 vi21; and N is the number of partitions. Recall that the area under the spectrum is equal to

the variance, sh

2: The incremental area under the spectrum, Shh ðviÞDvi; can be denoted as si

2 such that

the sum of all the incremental area equals the variance of the wave elevation or sh

2 ¼

PNi

¼1 si

2: The time

history can be written as

hðtÞ ¼

i¼1

cosðvit 2 wiÞ

ffiffi

2 p si

The sampling frequencies, vi; can be chosen at equal intervals such that vi ¼ iv1: However, the time

history will then have the lowest frequency of v1 and will have a period of T ¼ 2p=v1: In order to avoid this

unwanted periodicity, Borgman (1969) suggested that the frequencies are chosen so that the area under the

spectrum curve for each interval is equal or si

2 ¼ s 2 ¼ sh

2=N: The time history is written as

hðtÞ ¼

ffiffiffiffi

i¼1

cosðv􀀊it 2wiÞ ð36:13Þ

where v􀀊i ¼ ðvi þvi21Þ=2: The discrete frequencies, vi, are chosen such that the area between the interval

0 , v , vi is equal to i=N of the total area under the curve between the interval 0 , v , vN or

ðvi

Shh ðvÞdv ¼

ðvN

Shh ðvÞdv for i ¼ 1; …; N

where it is assumed that the area under the spectrum beyond vN is negligible. If hðtÞ is a narrowband

Gaussian process, the standard deviation can be replaced by sh ¼ Hs=4; and the time history can be

written as

hðtÞ ¼

ffiffiffiffi

r XN

i¼1

cosðv􀀊it 2wiÞ

Shinozuka (1972) proposed that the sampling frequencies, v􀀊i, in Equation 36.13 should be randomly

chosen according to the density function, f ðvÞ ; So

hh ðvÞ=s2h

: This is equivalent to performing an

integration using the Monte Carlo method. The random frequencies v distributed according to f ðvÞ can be

obtained from uniformly distributed random numbers, x, by v ¼ F21ðxÞ; where FðvÞ is the cumulative

distribution of f ðvÞ:

The random frequencies obtained this way are used in Equation 36.13 to generate a sample time series.

It should be noted that many sample time histories should be obtained and averaged to synthesize a time

history for use in numerical simulations.

Fluid-Induced Vibration 36-9

36.1.5 Short-Term Statistics

In discussing wave statistics, we often use the

term significant wave to describe an irregular sea

surface. The significant wave is not a physical

wave that can be seen but rather a statistical

description of random waves. The concept of

significant wave height was first introduced by

Sverdrup and Munk (1947) as the average height

of the highest one third of all waves. Usually,

ships co-operate in programs to find sea statistics

by reporting a rough estimate of the storm

severity in terms of an observed wave height.

This observed wave height is consistently very close to the significant wave height.

Stationarity and ergodicity are two assumptions that are made in describing short-term waves

statistics. These assumptions are valid only for “short” time intervals — approximately two hours or the

duration of a storm — but not for weeks or years. The wave elevation is assumed to be weakly stationary

so that its autocorrelation is a function of time lag only. As a result, the mean and the variance are

constant, and the spectral density is invariant with time. Therefore, the significant wave height and the

significant wave period are constant when we consider short-term statistics. In this case, the individual

wave height and wave period are the stochastic variables. We then need to determine certain statistics for

the analysis and design of offshore structures when we consider short time intervals.

Consider a sample time history of a zero-mean random process, as shown in Figure 36.7. The questions

that we ask are how often is a certain level (e.g., z in the figure) exceeded, and how are the maxima

distributed? Likewise, we can ask when we can expect to see that a certain level is exceeded for the first time,

and what are the values of the peaks of a random process? The first question is important when a structure

may fail due to a one-time excessive load, and the second question is important when a structure may fail

due to cyclic loads.

It is found that the rate at which a random process XðtÞ crosses Z with a positive slope (zero upcrossing)

may be calculated from

nzþ ¼

ð1

vfXX_ ðz; vÞdv

where fXX_ ðx; x_Þ is the joint probability density function of X and X_ ðtÞ: The expected time of the first upcrossing

is then the inverse of the crossing rate or

E{T} ¼ 1=nzþ

The probability density function of the maxima, A, can be calculated from

fAðaÞ ¼

ð0

2vfXX_X€ ða; 0;vÞdv

ð0

2vfX_X€ ð0;vÞdv

where fXX_X€ ðx; x_; x€Þ is the joint probability density function of X, X_ ; and X€ :

If XðtÞ is a Gaussian process, then we can write the joint probability density functions as

fXX_ ðx; x_Þ ¼

2psXsX_

exp 2

􀀏 􀀐2

sX_

􀁻 !2 " #

; 2 1 , x , 1; 2 1 , x_ , 1

and

fXX_X€ ðx; x_; x€Þ ¼

ð2pÞ3=2lMl1=2 exp 2

2 ð{x} 2 {mX }ÞT½M􀀉21ð{x} 2 {mX }Þ

􀀒 􀀓

Random process

positive maxima

negative maxima

Time

FIGURE 36.7 A sample time history.

36-10 Vibration and Shock Handbook

where

½M􀀉 ¼

X 0 s2

0 s2

X_ 0

X_ 0 s2

X€

6664

7775

and {x} 2 {mX } ¼

x 2 mX

x_ 2mX_

x€ 2mX€

664

775

Then, for a stationary Gaussian process, the up-crossing rate is given by

nþz

ð1

fX_X€ ðZ; x_Þx_ dx_ ¼

2psXsX_

exp 2

􀀏 􀀐2 􀀒 􀀓ð1

exp 2

sX_

􀁻 !2 " #

x_ dx_

sX_

2psX

exp 2

􀀏 􀀐2 􀀒 􀀓

ð36:14Þ

and the probability density function of maxima is given by the Rice density function (Rice, 1954)

fAðaÞ ¼

ffiffiffiffiffiffiffiffiffi

p12a2

ffiffiffiffi

2p p sh

exp 2

2ð1 2 a2Þ

" #

þ a

2 F

ffiffiffiffiffiffiffiffiffi

a2 2 1 p

􀁻 !

exp 2

􀁻 !

for 2 1 , a , 1

where FðxÞ is the cumulative distribution function of standard normal random variable

FðxÞ ¼

1ffiffiffiffi

p2p

ðx

expð2z2=2Þdz

and a is the irregularity factor equivalent to the ratio of the number of zero up-crossings (number of

times that h½t􀀉 crosses zero with a positive slope) to the number of peaks. a ranges from 0 to 1, and it is

also equal to

a ¼

sh_

2s € h

If XðtÞ is a broadband process, a ¼ 0 and the Rice distribution is reduced to the Gaussian probability

density function given by

fAðaÞ ¼

1 ffiffiffiffi

p2psh

exp 2

s2h

" #

for 2 1 , a , 1

If XðtÞ is a narrowband process, it is guaranteed that it will have a peak whenever hðtÞ crosses its mean.

In this case, the irregularity factor is close to unity, and the Rice distribution is reduced to the Rayleigh

probability density function given by

fAðaÞ ¼

s2h

exp 2

s2h

" #

for 0 , a , 1

In other words, the amplitudes of a narrowband

stationary Gaussian process are distributed

according to the Rayleigh distribution.

Figure 36.8 shows the Rice distribution for

various values of a. Note that the Rice distribution

includes both positive and negative maxima except

when a ¼ 1; in which case all the maxima are

positive. The positive maxima are the local

maxima that occur above the mean of XðtÞ; and

the negative maxima are the local maxima that

occur below the mean, as shown in Figure 36.7.

In some cases, the negative maxima may not mean

-4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-3 -2 -1 0

Maxima, A

fA (a)

1 2 3 4

a = 0.8

a = 1 (Rayleigh)

a = 0 (Gaussian) a = 0.6

a = 0.4

a = 0.2

FIGURE 36.8 Rice distribution for maxima.

Fluid-Induced Vibration 36-11

much physically. In those cases, we can use the

truncated Rice distribution, where only the

positive portion of fAðaÞ is used. fAðaÞ is normalized

by the area under the probability density for

positive maxima (Longuet-Higgins, 1952; Ochi,

1973):

f trunc

A ðaÞ ¼

fAða ð Þ 1

fAðaÞda

; a $ 0

The truncated Rice distribution is shown in

Figure 36.9.

If XðtÞ is the wave elevation, its maxima, A, are

the amplitudes of the wave elevation. The wave

height, H ¼ 2A; is then distributed according to

fH ðhÞ ¼ fAðH=2Þ

dH ¼

4s2h

exp 2

4s2h

" #

for 0 , h , 1

For any given wave, the probability that the height is less than h (the cumulative distribution) is

FH ðhÞ ¼ 1 2 exp 2

4s2h

" #

for 0 , h , 1

If hðtÞ is a stationary narrowband process so that the peaks are distributed according to the Rayleigh

distribution, we find that the root-mean-square wave height,

ffiffiffiffiffiffiffiffi

E{H2}

; is given by

ffiffiffiffiffiffiffiffi

E{H2}

ð1

h2fH ðhÞdh ¼ 2

ffiffi

2 p sh

In addition, it can be shown that the average and the significant wave heights are given by

H0 ; E{H} ¼

ffiffiffiffi

2p p sh ; Hs ; E{H1=3} ¼ 4sh ð36:15Þ

where E{H1=3} means that it is the expectation of the highest one third of the waves.

36.1.6 Long-Term Statistics

Because offshore structures are designed for long life spans, we must also consider long-term wave

statistics. Previously, when we considered the short-term statistics, the significant wave height and

spectrum were assumed to be invariant with time. This assumption is valid only over time periods of

days at most. For longer time periods, the significant wave height has its own statistics and is a

random variable.

When one uses short-term statistics to describe long-term events, improbable events seem

unjustifiably probable. For example, let us consider the probability that the wave height, distributed

according to the Rayleigh distribution, exceeds a certain extreme value. Let us assume that the mean

period of this wave is 10 sec and the probability that the height of any given wave is greater than 300 ft is

10210. The value is small and the occurrence of a 300 ft wave seems improbable. However, the probability

that the height will exceed 300 ft at least once in 10 years (3 £ 108 sec) is given by

1 2 ð1 2 10210Þ3£108 =10 ¼ 0:997

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.5 1 1.5 2

Maxima, A

trunc (a)

2.5 3 3.5 4

a = 1 (Rayleigh)

a = 0.8

a = 0.6

a = 0.4

a = 0.2

a = 0

FIGURE 36.9 Truncated Rice distribution.

36-12 Vibration and Shock Handbook

Thus, the statistical description states that it is almost certain that the wave height will exceed 300 ft at

least once in 10 years. This prediction is a shortcoming of the short-term statistics since waves of this

magnitude do not arrive at this probability.

In order to compute the probability that a wave height will exceed a certain extreme value, we require

statistics for these extreme events. The actual maximum amplitude in a sequence of random amplitudes

is a random variable itself. It has a probability distribution with mean value, standard deviation and other

statistical properties. In fact, the distributions of these maximum values are called the extreme value

distributions (EVDs). Gumbel (1958) obtained three methods of extrapolation known as three

asymptotes. They are the Gumbel, Fretchet, and Weibull distributions. We will discuss the Gumbel and

Weibull distributions in the next section. For the moment, we will discuss the concept of the N-year

storm.

In long-term statistics, we often speak of an N-year storm. It means that, for any given year, the

probability that we will have an N-year storm is

p ¼

It follows that the probability that we will have m storms in n years is given by

Pr{mN-year storms in n years} ¼n Pm

􀀏 􀀐m

1 2

􀀏 􀀐n2m

where n Pm is the permutation given by

n Pm ¼

ðn 2 mÞ!

The probability that we will have at least one N-year storm in n years is

Pr{at least one N-year storms in n years} ¼ 1 2 1 2

􀀏 􀀐n

For a large N, the probability can be approximated as 1 2 expðn=NÞ: It should be noted that the

probability that we will have exactly one N-year storm in N years is not one, but

Pr{one N-year storm in N years} ¼ 1 2

􀀏 􀀐M21

As N ! 1; we find that

P ¼ 1=e < 0:3679

The probability that we will have at least one N-year storm in N years is

Pr{at least one N-year storms in n years} ¼ 1 2 1 2

􀀏 􀀐N

As N ! 1; we find that

P ¼ 1 2 1=e < 0:6321 ð36:16Þ

36.1.6.1 Weibull Distribution

The Weibull distribution fits probabilities of extremes quite satisfactorily. In long-term statistics, the

significant wave height follows the Weibull distribution closely. The probability density and the

cumulative distribution are given by

f ðhÞ ¼

h 2 g

􀀏 􀀐m21

exp 2

h 2 g

􀀏 􀀐m 􀀏 􀀐

; FðhÞ ¼ 1 2 exp 2

h 2 g

􀀏 􀀐m 􀀏 􀀐

for g , h ð36:17Þ

Fluid-Induced Vibration 36-13

where m is called the shape parameter. Manipulating the cumulative distribution, we can write

lnð2ln{1 2 FðhÞ}Þ ¼ m{lnðh 2 gÞ2 lnðbÞ}

where the left-hand side is known from data. If we let y ¼ lnð2ln{1 2 FðhÞ}Þ and x ¼ lnðh 2 gÞ; y is a

straight line with slope of m and a y-intercept of 2 m ln b:

y ¼ mx 2 m ln b

Suppose we have significant wave height data over a long period of time, and our goal is to find the

Weibull parameters, g, b, and m that best fit the distribution of the significant wave heights. These

parameters can be determined by the least-squares method or using the Weibull paper. Using the latter

method, we first guess g so that the discrete points ðx; yÞ or ðlnðh 2 gÞ; lnð2ln{1 2 FðhÞ}ÞÞ form a straight

line. The slope of this line is m, and the value of y when the line intersects the y axis is 2 m ln b. This

method will be illustrated in Section 36.3.3.

The Gumbel distribution is given by

f ðhÞ ¼ a expð2aðh 2 bÞÞexp{exp½2aðh 2 bÞ􀀉}

FðhÞ ¼ exp{ 2 exp½2aðh 2 bÞ􀀉} for 2 1 , h , 1 ð36:18Þ

When ln½2ln FðhÞ􀀉 is plotted against h, the result is a line with a slope of 2a and y intercept of ab.

Another distribution that may be used is the lognormal distribution given by (Jasper, 1954)

f ðhÞ ¼

1 ffiffiffiffi

p2psh

exp 2

ðln h 2 mÞ2

" #

; FðhÞ ¼ F

ln h 2 m

􀀏 􀀐

for 0 # h ð36:19Þ

36.1.6.2 Wave Velocities via Linear Wave Theory

The wave velocities that correspond to the wave

elevation given in Equation 36.12 can be obtained

by linear wave theory. Linear wave theory, also

called airy wave theory, sinusoidal wave theory,

and small-amplitude theory, is the simplest wave

theory. It is also the most important wave theory

because it forms the basis for the probabilistic

spectral description of waves.

Linear wave theory assumes that the wave height

is small compared with the wavelength and wave

depth. In addition, fluid particles are assumed to

follow a circular orbit. The readers should refer to

Kinsman (1965) and LeMehaute (1976) for

detailed descriptions.

In linear wave theory, the surface elevation is given by

hðy; tÞ ¼ A cosðvt 2 kyÞ ð36:20Þ

which is a plane wave traveling to the right in Figure 36.10. Linear wave theory relates this sinusoidal

surface elevation to the wave velocities given by

wy ðx; y; tÞ ¼ Av

cosh kx

sinh kd

cosðvt 2 kyÞ; wy ðx; y; tÞ ¼ Av

sinh kx

sinh kd

sinðvt 2 kyÞ ð36:21Þ

where k, v, and A are wave number, angular frequency, and amplitude of a surface wave, respectively.

The velocities vary with time, horizontal coordinate y, and depth x measured from the ocean floor. The

wave velocities are sinusoidal in y and t, but exponentially decrease with the distance from the surface.

A,H/2

FIGURE 36.10 A schematic of a simple sinusoidal

wave shown at two different times.

36-14 Vibration and Shock Handbook

The frequency v is related to the wave number k by the dispersion relation given by

v2 ¼ gk tanh kd

where d is the water depth. For deep water, tanh kd approaches unity and the frequency is given by

lim

d!1

v2 ¼ gk

For the surface elevation given in Equation 36.12, the surface elevation and the wave velocities are

given by

hðy; tÞ ¼

i¼1

cosðvit 2 kiy 2 wiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Shh ðviÞDvi

wy ðx; y; tÞ ¼

i¼1

cosh kix

sinh kid

cosðvit 2 kiy 2 wiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Shh ðviÞDvi

wy ðx; y; tÞ ¼

i¼1

sinh kix

sinh kid

sinðvit 2 kiy 2 wiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Shh ðviÞDvi

ð36:22Þ

The wave accelerations can be obtained by differentiating the wave velocities with respect to time. Sample

time histories of the wave velocity and acceleration can be obtained using either Borgman’s or

Shinozuka’s method.

36.1.7 Summary

In this section, the concept of spectral density is introduced. It is then shown how the concept is used to

describe the ocean wave heights. The spectral density or spectrum is related to the autocorrelation

function by the Wiener– Khinchine relations:

Shh ðvÞ ¼

ð1

Rhh ðtÞexpð2ivtÞdt

Rhh ðtÞ ¼

ð1

Shh ðvÞexpð2ivtÞdv

In addition, the spectral density function of a real-valued stationary process is also real and symmetric, or

Shh ðvÞ ¼ Shh ð2vÞ ð36:23Þ

and the area under the spectral density is given by

ð1

Shh ðvÞdv ¼ Rhh ð0Þ ¼ s2h

2 m2h

ð36:24Þ

If the spectral density is given only for v $ 0; then this one-sided spectrum is related to the two-sided

spectrum by

hh ðvÞ ¼ 2Shh ðvÞ; v $ 0

When the frequency is given in Hertz instead of in rad/sec, the spectra are related by

Shh ðf Þ ¼ 2pShh ðvÞ; v $ 0

The spectra that are often used to describe wave heights are the P– M, Bretschneider, ITTC, ISSC,

O – H, and JONSWAP spectra. The most widely used spectrum is the P–M spectrum, which is a

single parameter spectrum. The P–M spectrum is applicable for deep water, unidirectional seas, fully

developed and local-wind-generated sea with unlimited fetch, and was originally developed for the

North Atlantic. The single parameter for this spectrum can be expressed as the wind velocity at

19.5 m above sea level or the significant wave height that specifies the sea severity. When it is written

Fluid-Induced Vibration 36-15

in terms of the significant wave height, it is given by

hh ðf Þ ¼

8:1 £ 1023g2

v5 exp 2

0:0324g2

v24

􀁻 !

Bretschneider, ITTC, and ISSC spectra are two-parameter spectra in which the state of development

as well as the sea severity can be specified. The O –H spectrum is a six-parameter spectrum that

allows us to represent local and distant storm effects and to specify sharpness of the peaks as well as

to specify the sea severity and the state of development. The JONSWAP spectrum allows us to

account for growth over a limited fetch.

For a given ocean spectrum, a sample time history can be obtained by

hðtÞ ¼

ffiffiffiffi

i¼1

cosðv􀀊it 2wiÞ

In Borgman’s method, the sampling frequencies v􀀊i ¼ ðvi þvi21Þ=2 are chosen so that the area between

vi21 and vi are equal. In Shinozuka’s method, the sampling frequencies are chosen randomly. The

traditional method of choosing the sampling frequency at even intervals is not recommended.

When a relatively short interval of time is considered, for example, about two hours or the

duration of a storm, it can be assumed that the spectrum and its statistics are invariant with time. In

this case, the distribution of local maxima or peaks of a stationary Gaussian process is described by

the Rice distribution. The Rice distribution can be reduced to the Rayleigh distribution when the

process is narrowband and to the Gaussian distribution when the process is broadband. In the longterm

statistics, the spectrum and its statistics may vary with time. In this case, the term “N-year

storm” is often used to indicate the sea severity, and the significant wave heights closely follow the

Weibull distribution.

Finally, the wave velocities and accelerations are related to the wave velocities using linear wave

theory.

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