PETRONAS TECHNICAL STANDARDS

DESIGN AND ENGINEERING PRACTICE

REPORT (SM)

WAVES AND MARINE STRUCTURES

PTS 20.088

MARCH 1972

PREFACE

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CONTENTS

Summary

Main symbols

1. Introduction

2. Regular waves

3. A phenomenological description of waves

4. A statistical description of waves

5. The wave spectrum

6. The response of marine structures to waves

7. References

Figures 1 - 8

One Table

SUMMARY

From among the various environmental conditions to which marine structures are subjected, the waveaction is isolated and discussed in some detail. After a short introduction on a few basic concepts ofregular waves, the irregular wave pattern as actually observed at sea is discussed.

First of all a description of irregular waves is attempted from their appearance in a wave record. Thisphenomenological description cannot be very extensive, but it logically results in the introduction of anumber of suitable parameters. Statistically, however, a comprehensive description is possible usingonly a few parameters. The statistical relations are illustrated by an example.

Next a mathematical model to represent irregular waves is given. For this purpose the wave spectrumis introduced. This also permits a discussion of the behaviour of structures due to wave action. Thisaspect is discussed briefly.

KEYWORDS

Offshore structure, environmental condition, wave load, regular wave, random wave, statisticaldescription, wave spectrum, transfer function.

MAIN SYMBOLS

c Wave celerity (regular wave)

d Water depth

g Acceleration due to gravity

k = lp2 Wave number (regular wave)

mo Mean square of V (t); area under the wave spectrum

z Vertical movement of floating unit

H = sVa Wave height (regular wave)

Sx (w) Spectral density function of quantity x

T Wave period (regular wave)

e Parameter for irregularity of wave record = width of spectrum; phaseangle

V Elevation of water surface

Va Wave amplitude (regular wave)

l Wave length (regular wave)

r Specific mass of water

s Standard deviation

w T2p=

Circular wave frequency

For notations with irregular waves see Fig. 4.

WAVES AND MARINE STRUCTURES

1. INTRODUCTION

Offshore activities for the exploration and production of oil, gas and minerals are increasingcontinuously and rapidly. The structures used for these activities are exposed to a greatvariety of environmental conditions, such as wind, current, waves, tides, fog, icing, soilconditions and earthquakes. Each of these conditions may form an operational and/orconstructional hazard, which must be investigated in greater or less detail. The forces onmarine structures also vary greatly: own weights, operational loads such as hook loads,buoyancy, wind forces, current forces, wave forces, anchor line forces or soil reactions andpossibly other loads as well.

In the following only the phenomena associated with wave action will be discussed. There arethree reasons for this restriction:

While some of the environmental conditions need to be considered only in special cases or incertain areas, any marine structure is subject to wave action.

The forces associated with wave action are very large and often dominant in the total patternof loading.

Wave action is essentially of a dynamic nature, as opposed to practically all otherenvironmental conditions, which can reasonably well be approximated statically.

The water surface at sea varies irregularly, even chaotically, in time. Its exact time history isunpredictable; the surface profile can only be classified in statistical terms. The magnitude of theforces involved in wave action and their dynamic nature become increasingly important whenentering more exposed areas further away from shore in deeper water, as clearly the present trendis. It is therefore of great importance to have a good understanding of what waves are and whatwaves do to marine structures.

It should be noted that the discussion on waves is even somewhat more restricted here in thesense that we shall be discussing the properties and consequences of the continuous variation ofthe water surface in the open sea, as is typical for many offshore installations. Limitations such asthe presence of a coast or water depths of, says less than about 80 m may have importantinfluences which are not discussed here. Thus special effects such as ground swells and impactloads by waves breaking against the structure are not included. The latter phenomenon, forexample, is of great importance for coastal structures, where the sloping sea bottom causes thewaves to approach the coast perpendicularly and to become steeper and steeper. In this way themechanism is created for a more or less frontal confrontation and fairly regular breaking of wavesat the location of the structure occurs. In the open sea such mechanisms generally do not existand consequently impact loading can be disregarded in most cases. If it is expected to occur,however, it should be investigated independently.

Finally it is emphasized that in a short article such as this the entire subject cannot be coveredexhaustively. Any suggestion of completeness or of a rigid physical and mathematical treatment isunintentional and would clearly be incorrect. This article is simply an attempt to give a correctunderstanding of the physical occurrences in sea waves and to formulate them in a logical andcoherent description.

2. REGULAR WAVES

A few basic concepts can best be shown for regular waves. An artificial, regular andprogressive wave to be of sinusoidal form. The surface profile is a function of location andtime. If the wave moves in the positive x direction this can be expressed in this equation:

( )tkxsina w-V=V (2.1)

At a fixed moment, for which t = 0 can be taken without any loss of generality, the function oflocation is given by:

kxsinaV=V (2.2)

This is illustrated in Fig. 1a. The wave length of one cycle being l, it is obvious that if x = lthen k l= 2 p, from which it follows that k = 2p/l. This parameter is called the wave numberand is expressed in m -1

At a fixed location, say x = 0, the general equation is reduced to a function of time:

V = V a sin (- wt) = - V a sin wt (2.3)

The minus sign is not important for the principle of the discussion. It indicates, however, thatfor wave moving in the positive x direction the water surface in the origin will initially fall belowthe mean level. This is also shown in Fig. 1a. If the direction of wave propagation is reserved( - wt) has to replaced by (+ wt) and the water surface will initially rise above the mean level.In a consistent mathematical formulation this should be taken into account. If the period ofone complete cycle is denoted by T as shown in Fig. 1b, clearly for t = T the argument of thesine-function is 2 p. Thus: wT = 2 p, or w = 2 p / T. The parameter w is the (circular) wavefrequency in rad/sec. It can theoretically be derived that there is a fixed relation between thewave length and the wave period. The relation is:

lp=l

pd2tanh

2

gT2(2.4)

where d is the water depth.

During one period T a wave crest progresses over one wave length l, so that the wavevelocity is c = l/T. However, a wave motion is not a continuous transportation of mass ofwater as might appear from the propagation of the wave crests. The water particles move incircular orbits, the centre of which remains in the same position, at least in the theoretical andartificial concept of a regular wave. This can be easily be verified by watching the movementsof a cork or small raft in waves. It will move both to a fro and up and down in a circularmanner, but on average stay in the same place. The orbital motion of the water particles is foreach point along the x axis delayed by a time interval kx/w = Tx/l = x/c. This causes thewaves crests and troughs to progress as is clearly illustrated in Fig. 2.

The orbital motion decreases exponentially with depth by the factor e-kz. At the surface z = 0the radius is equal to the amplitude of the wave V a, while a depth of z = l the radius is V ae-p 0.043 V a. This is hardly perceptible and the water is therefore considered to be deep aslong as d > l. In deep water the wave is not affected by the bottom, does not "feel" thebottom. For shallower water the influence is considerable, however. The circular orbits deforminto ellipse with the longer axis horizontal. The short axis is the long axis horizontal. The shortaxis is somewhat smaller than the radius of the corresponding circle, the long axis isconsiderable larger. The orbital motions at various depths are shown in Figure 3.

One more important thing on regular waves should be mentioned: the energy. A wavecontains potential energy owing to the variable profile and kinetic energy owing to the orbitalmotion of the wave particles. The average energy over a wave of the wave particles. Theaverage energy over a wave length and per unit of horizontal area of the water surface is:

potential Ep = r g Va2 = 1/16 r g H2 )

) )

kinetic Ek = r g Va2 = 1/16 r g H2 )

) )

total Et = r g Va2 = 1/8 r g H2 )

) ) (2.5)

in which r is the mass per unit of volume of the water and g the gravitational acceleration; theproduct rg is the specific weight g. The equations (2.5) are easily derived from the definitionsof potential and kinetic energy, integration over the wave length and the water depth and thenaveraging over the former. Reference is made to the textbooks.

3. A PHENOMENOLOGICAL DESCRIPTION OF WAVES

At sea the surface profile will always be more or less irregular. A swell originating from adistant wave field and existing without a corresponding local wind field usually has a fairlyregular appearance. A locally generated wind sea, however, has no resemblance whatsoeverto a regular wave. But even for swell it would be an over-simplification to represent it by aperfectly regular wave form as discussed in Chapter 2.

In Figure 4 a recording is shown of the instantaneous water elevations at a certain point as afunction of time. It represents a surface profile at the point under consideration over, forexample, a time interval of 30 minutes. A meaningful description is in the first place given bytwo averages: the main value, V and the root-mean-square value, r.m.s. (V). The mean valueis defined by:

+

-

V=V

2T

2T

dt)t(lim T1

T (3.1)

where T is the length of the record. It can also be approximated by taking the average of alarge number of discrete value V i these being the instantaneous water elevations above thezero level:

=

V=VN

1iiN

1 (3.2)

It will further be assumed that V =0, analogous with the mean value of the regular wave formin Fig. 1. This amounts to a shift of the zero level to the water level without the waves beingpresent.

The root-mean-square value is the average magnitude of the water surface elevation V in anabsolute sense. To avoid cancellation of positive and negative values the signal is squaredand average, while finally the root is taken. The definition is fully analogous with the meanvalue:

r.m.s (V) = { }

21

2T

2T

dt)t(limm 2T1

To

V= +

-

(3.3)

and an approximation from the discrete value V i is clearly:

r.m.s (V) = 21

N

1i

2io N

1m

V=

=

(3.4)

The parameter mo introduced above stands for the mean square of the instantaneous water

elevations. For a regular wave r.m.s. (V ) = ..2 aV

Obviously in Fig. 4 one cannot indicate "the amplitude" and "the period" as in Fig. 1b. Alldistance between the mean or zero level and a stationary point in the record with a horizontaltangent are therefore called apparent wave amplitudes and indicated by a

~V . All crests aremeasured in an upward direction, so that both positive (above zero level) and negative crests(below zero level) can occur. The same applies to troughs, which are measured positively in adownward direction. For example, in a record of 30 minutes approximately 200 crests and200 troughs, i.e. 400 values of a

~V . will be observed. It is clearly desirable to have anindication of the orders of magnitude involved in one or two parameters instead of 400. This isindeed usual. The highest crests or deepest trough in the record is the maximum apparentamplitude, indicated by maxa

~V ; there is, of course, only one maximum. The value is also

given in terms of the maximum wave height of the record maxH~

, measured from crests to

preceding or following trough. The other very important parameter is the so-called significantwave amplitude

31a

~V . It is obtained by taking the top one-third of all the apparent

amplitudes and averaging these. In the example given: sum the 133 highest out of 400 crest

and trough amplitudes and divide by 133. The significant wave height 31H

~ is derived in a

similar way by taking apparent wave heights, crest trough, instead of amplitudes from thezero level. Obviously there are only roughly 200 wave heights as opposed to 400 amplitudes.

The 67 highest are therefore averaged. The 31H

~ thus found is very nearly equal to twice the

31a

~V .

For a regular wave from the period as measured from zero-crossings, crests, troughs or anyother corresponding set of points is identical. For an irregular signal this is not generally thecase. The reference for the determination of a period must be therefore be clearly defined.Obvious references are the zero-crossing and the crests of troughs, the apparent period

denoted by zT~ and cT

~ respectively. Since between any two adjacent zero-crossings there

must be at least one crests or trough, the total number of crests and troughs together , Nc,cannot be smaller than the total number of zero crossing Nz , up ward and downwardtogether. The more negative crests and trough occur, the larger Nc is than Nz and the moreirregular and appearance of the recording.

The average zero-crossing period is called the mean period and is found from:

zzmean N

recordoflength2T

~T == (3.5)

while the average crest period is defined by:

cc N

recordoflength2T

~ = (3.6)

Owing to the facts Nc > Nz, the average crests period is always smaller than the mean period.An indication of the degree of irregularity in the record is given by the ratio of both periods, orrather by the parameter e defined as:

2

z

c2

T~T~

1

-=e (3.7)

For the regular sine wave form .0andT~

T~

zc =e= For an extremely irregular wave recording

cT~

is considerably smaller than zT~

, so that 2

zc T~/T

~

tends to zero and e tends to

one. So the parameter e has a value between 0 and 1; the larger e the more irregular thecharacter of the wave record. To state a few values: in a swell e may be of the order of 0.30,in a typical wind sea of the order of 0.50 to 0.80.

One other period which can be used to characterize a wave recording to some extent shouldalso be mentioned, namely the significant period. This is defined in two ways. Firstly in a

similar manner as for the significant wave height. The significant period 31T~

is then the

average of the top one-third of all zT~. Secondly by considering only the top one-third of the

waves, i.e. in the aforementioned example only the 67 individual wave from which 31H

~ was

obtained. Now only the zT~ of these waves are averaged. The significant period thus defined

is denoted by Ts to distinguish it clearly from 31T~

, for these are by no means equal to one

another. If a wave record is characterized by a significant wave height and a significant period

usually 31H

~ and its associated Ts are meant.

In Figure 4 an example of a wave is shown, while various quantities are illustrated. It shouldbe noted that no uniform nomenclature or notation is used, so that in different works differentindications may be found. It is believed, however, that the maximum and significantamplitudes (or heights), the average zero-crossing and crests periods the parameter e areuniversally accepted, although the notation may differ.

4. A STATISTICAL DESCRIPTION OF WAVES

Clearly the description of sea waves given so far only provide a rough indication of what isobserved visually or actually recorded by instruments. It is insufficient for numerical evaluationin engineering applications. Therefore a more detail representation of waves themselves anda mathematical model of the physical occurrences are needed in order to calculate the effectof waves on structures. By analysing a great number of wave records it will be found that thestatistical distributions of various quantities are unvarying, so that even a comprehensiverepresentation of wave properties is possible through the use of only a few parameters.

Thus mathematical formulation is sought which gives an adequate description of the waveproperties observed. Similar to a Fourier analysis the following formulation is suggested:

e+w-q+qV=V

=nnnnnnan

1ntsinykcosxksin (4.1)

This represent the sum of an infinite number of regular wave components as given inequation (2.1), each with its own direction q, frequency w and amplitude V a. Moreover thecomponents are not necessarily in phase at any time, so that a phase difference e to somecommon references introduced for each component. When looking at the water surface at afixed point, for which x = y = 0 can be chosen, the formulation is simplified to:

( )

=e+wV=V

1nnnan tsin (4.2)

Omitting the minus in wn t simply means that each en is in fast replaced by (en + 180), whichagain reduced to en by adjusting the common reference. Thus (4.2) is supposed to describethe wave profile at a fixed location as shown in Fig. 5a. Starting from (4.2) various theoreticalrelation have been derived for the instantaneous wave elevations V i, for the apparentamplitudes VV of~a and for the probability that a certain maxa

~V occurs. The question now is

whether these relations correspond with the physical properties actually observed for seawaves. If so, (4.2) can serve as a mathematical model for the further discussion of wave andthe interaction between waves and structures. It should be realized, however, that anymodelling can only give partial description of reality. A mathematical model is extremelyuseful, even indispensable, but always has a limited validity. This should constantly be bornein mind.

Assume that a wave record has shown in Fig. 5a is available and that the parameters mo ande of the previous chapter have been determined. If the actual instantaneous wave elevation V iabove the still water surface are arranged by magnitudes it will be found that there are anequal number of positive and negative values and that the distribution is a normal orGaussian distribution, as shown in Fig. 5b. The standard deviation s of the normal distributionwill be equal to the r.m.s. value:

om=s (4.3)

Thus by parameter mo the whole distribution function is fixed. It is theoretically known that thenormal distribution is obtained for quantities whish are subject to a large number of mutuallyindependent causes. In the case under consideration this means that all wave component areindependent, in other words the phase relations en in (4.2) are completely arbitrary or random.

The theoretical distribution of the apparent amplitudes a~V in the model (4.2) is found to

depend only on the two parameters mo and e.It is shown graphically in Fig. 5c. if all a~V of thewave recording in Fig. 5a are arranged by magnitudes the distribution will indeed closelyfollow the curve of Fig. 5c for the appropriate e. Only a few negative amplitudes will occur,while a peak is found around a

~V = om . The less irregular the character of wave record,

the smaller is e, as discussed in the previous chapter. The distribution function in Fig. 5c thenshow that less and less negative crests and troughs will occur. For the limiting case 0e ofa fairly regular wave profile (although not a real regular wave) the distribution starts at theorigin and only covers positive values of a

~V . This curve of the family is the Rayleighdistribution . It is fully determined by mo alone and is often a sufficiently accuraterepresentation of the apparent amplitudes for practical applications, especially since one isusually mostly interested in the larger value of a

~V .

The one wave record shown in fig. 5a has a certain maximum apparent amplitude maxa

~V . If a

large number R of wave records were available for identical environmental conditions (suchas wind direction and velocity, duration that the wind is blowing, water depth, exposure etc.)i.e for the same wave conditions, it would notice that mo and e of each record were the same,but that the

maxa~V values were different . The explanation is simple and will be given in

Chapter 5. Thus it appears that maxa

~V is not a unique feature and that again only a

distribution of the maxa

~V value of the R records can be given. The theoretical distribution of

maxa~V has been obtained from equation (4.2) and it represented in Fig. 5d. Obviously the

maximum occurring depends on the number N of complete cycle investigated (N= Nz). Iftwo records taken at the same location, starting at the same time and one lasting for 30 min.ant the other for 60 min. , the

maxa~V in the longer record cannot be smaller than that in the

shorter one. But clearly there is a possibility that it is larger. Thus the form and the position ofthe distribution function for

maxa~V depend on the three parameters mo, e and N. N

determines in the first place the position of the distribution; which increasing N the wholecurve moves upwards. The influence of e is small and it is usually sufficiently accurate to usethe distribution for the limiting case e = 0. The form of the distribution is jointly determined bymo and N. For the scarce experimental evidence available in this connection, the theoreticaland actually observed distribution in wave measurements also agree.

Through the distribution discussed above and shown in Fig, 5a comprehensive statisticaldescription of irregular waves at sea is possible my means of only three parameters : mo, eand N. It will now be discussed with the aid of an example what all this really means. It will beassumed that we are considering a storm of approximately 3 hours duration containing 1000wave cycles (N=1000). It will further be assumed that the relation for e= 0 are sufficientlyaccurate, so that they are completely determined by the mean square wave elevation mo. Letmo be 4 m

2. The results are summarized in the accompanying Table.

The normal distribution for the instantaneous water surface elevations V i reveals that 68.26%

of all V i are between m2mandm2m oo +=+-=- . In other words for 68.26% ofthe time, or 122.9 min. of the 180 min. , the water surface is 2 m from the mean level. As readfrom the table, for only 4.46% of the time or 8.2 min. the water surface is more than 4 m andin only 0.26% of the time or 0.5 min. more than 6 m from the mean level. An indication of howfar these values are exceeded is given by the apparent wave amplitudes, which follow theRayleigh distribution. Select the top 1/n-th part of all amplitudes and average these.

This value is incated by .~n/1aV The significant wave amplitude discussed in the previous

chapter is one of this family, namely .~3/1aV The mean of all wave amplitudes is clearly

.~1/1aV For the convenience of the discussion we will assume roughly that the number of

crests, which is equal to the number of troughs, is also a thousand since N = 1000. As shown

in table, m10.5m55.2~ oa 10/1 ==V . Thus the average of the 100 (10%) highest

amplitudes is 5.10 m. If the Rayleigh distribution in this range were linear, 95 % of all a~V

would be smaller. But the distribution is exponential and actually 96.2% of all amplitudes aresmaller.

Thus the table shows that 135 apparent wave amplitudes will exceed 4 m, 38 will exceed5.10m, 19 will exceed 5.62m and only 4 will be higher than 6.68m. How high this 4 waves are,and especially what the largest amplitudes, cannot be derived from the Rayleigh distribution.This question will be answered by the distribution of

maxa~V . The peak of this distribution is at

A om in Fig. 5d; for N = 1000 and e= 0 then A = 3.72. This value is called the mostprobability maximum, in this case present in 1000 wave crests or 1000 wave troughs.However, since the distribution is far from symmetric, the area below it is 37% and above it63% of the total: the most probable maximum has a probability of 63% of being exceeded .Going more into the tail of the distribution a maximum with, for example, a 10% or a 5%probability of being exceeded can be determined. The correct values are indicated in thetable. It is seen that a significant wave amplitude of 4.00m, the most probable maximumwave amplitudes is 7.44 m, but that there is a chance of 10% that is larger than 8.58m and achance of 5% it exceeding even 8.88m. This, of course, could be extended to any probabilitylevel. But it is debatable to which point the tails of the theoretical statistical distributions will bea valid representation of the phenomena in nature. Maximum amplitudes of a magnitude nearto the 5% probability level have, however, actually been measured.

A few further comments should be made. If wave heights rather than wave amplitudes arestudied all values are simply doubled. The theoretical distributions have been derived foramplitudes and not for heights. However, the theory does not distinguish between crests andtroughs and indications are that heights are very reasonable approximated by twice theamplitudes. Furthermore, all data given in the table are statistical data. Thus is any particularcase of finite duration studied, certain deviations may surely occur. The multiplication factorsgiven in column 6 for the maximum apparent amplitude depend on N. They are larger N andsmaller N. The factor for the average of the top 1/nth part of all amplitudes depends on e , butif e is neglected they are constants. Finally, the percentages of time for the water surfaceelevation to be between certain limits are universally valid, regardless of the degree ofirregularity of the sea (e) and the duration considered (N).

TABLE : EXAMPLE OF WAVE STATISTICS FOR A DURATION OF APPROXIMATELY 3 HRS (N = 1000);2

o m4mand0 ==e

5. THE WAVE SPECTRUM

From the previous chapter it can been seen that an irregular sea can be thought of as thesum of a very large number of regular waves. This is shown schematically in Fig. 6. Adescription of the elevation of the water surface at a fixed location was found to be:

( )

=e+wV=V

1nnnan tsin (5.1)

This formulations is equivalent to the assumption that all wave components in equation (4.1)progress in the same direction. In this case the x axis is taken along the direction ofpropagation and the term kn x is included in the phase difference en, which is possible sincethey are completely arbitrary. The wave direction is an important parameter for the descriptionof a wave field over a larger area instead of at one point and for the interaction betweenwaves and a structure. For reasons of simplifications, however, only the unidirectional casewill be considered here.

Suppose now that the waves during a particular sea state in a particular area are measured ata number of locations, or that at one fixed location the waves are measured during a numberof periods of, say, 30 minutes, one after the other, and assuming, of course, that theenvironmental conditions do not change in the meantime. For sea waves both methods areequivalent. Suppose next that each record is anlaysed to break it down into its regular wavecomponents. There are indeed methods for doing this. It will be found that the componentsare the same in all cases; only their mutual phase relation are different. Thus for each w n thesame V an but a different e n is found from each record. In other words, the wave componentsare typical for a particular sea state due to particular environmental conditions, but theirposition, one with respect to the other, and thus the appearance of the sum, is not at alltypical. This explains the statement made in Chapter 4 that statistical averages of a recordand quantities derived therefrom (such as mo and e ) are fully determined by theenvironmental conditions, but that the particular

maxa~V observed is purely a chance result.

Since the phase angles e n turn out to be completely arbitrary or random a particular waverecording is in fact an absolutely unique representation of the existing sea state. At no othertime or location will the record ever occur in precisely the same form. Consequently, seawaves essentially stochastic in nature; they can only be described adequately in statisticalterms. The random phase model (5.1) exhibits these characteristics correctly.

In the mathematical model (5.1) the frequencies wn cover in principle the entire frequencyrange from w= 0 to infinity and are densely distributed within it. These phases en are random,which means that they have a uniform distribution between e = 0 and 2p. As stated in theprevious paragraph, for each wn a fixed anV will be found for a particular sea state. The value

anV are not only fixed, but even form a fairly smooth distribution over wn. It is usual to givethe distribution in the quadratic form of anV

2 as a function of wn. The reason is that thisquantity is a direct measure of the total energy contained in each wave component, seeequation (2.5). This is illustrated in Fig. 7a for a restricted number of wave components. Thegraph is called a line spectrum. If the number of components is increased, each linesegments the total length of which is equal to the original segment. Thus the line segmentsbecome smaller and smaller. To prevent this it is preferable to use the density of the quantity anV

2 instead of the quantity itself. The density s is defined as haly times the amplitude

squared divided by the frequency interval ( ) :aroundn1n1n21 ww-w=wD -+

wD

V=

2an2

1

nS (5.2)

The density Sn remains constant with an increasing number of components. In an actual seaan infinite number of wave components is presents and Sn becomes a continuous function ofw see Fig. 7b, defined by:

( ) 2ad

d21

21

21

dS V=ww w+w

w-wV (5.3)

The function is called the wave ( )wVS spectral density function, or more shortly the wavespectrum, energy spectrum or power spectrum. It specifies the distribution of the energycontained in each regular wave component of the mathematical model (5.1) over the wavefrequency, or what is equivalent: it gives the relation between wave amplitudes and waveperiods of the regular wave components making up an irregular sea. Apparently thedistribution of the total wave energy over the frequencies of the regular wave components ischaracteristic for a particular sea state.

The mean square of the wave elevation, mo, was introduced in equation (3.3). By substituting(5.1) in the definition it can easily be shown that:

{ } 2an21

1n

2T1

To dt)t(limm

2T

2T

V=V=

=-

+

(5.4)

Thus mo is measure of the total amount of energy contained in an irregular sea. Fromequations (5.3) and (5.4) it is obvious that mo is equal to the area under the wave spectrum:

ww= V

d)(Sm oo (5.5)

The parameter e indicating the irregularity of the wave record and introduced by equation(3.7) is also related to the wave spectrum. It can be shown that:

4o

224o2

mm

mmm -=e (5.6)

where mn is the nth moment of the spectrum defined by:

www= V

d)(Sm non (5.7)

e is usually called the width of the spectrum.

It was assumed in the above that the environmental conditions were constant and alreadyexisted for some time. Consequently the sea is fully developed and stationary underprevailing meteorological conditions. There is equilibrium between the energy input by thewind and the energy lost in the wave motion itself and by radiating energy in the form ofwaves propagating outside the wind field area. So as long as the sea is stationary the wavespectrum and the statistical relations discussed form a valid and comprehensive description

of the sea. The total amount of energy (mo) and its distribution ( ))(S wV are constant. Howlong this situation will last is difficult to say, for it depends entirely on circumstances. From thehuge scale of the physical occurrences it will be obvious that mo will not change rapidly. For atypical wind-driven sea it is usually assumed that it is constant over a period of from hr to3 hr; for a swell it may be constant for many hours or even days.

As stated in the beginning of this chapter, the formulation (5.1) is valid for a unidirectional sea.The wave crests then all run parallel and are infinitely long, as opposed to what is observed atsea. Fully analogously, however, a directional mathematical model and a directional wavespectrum can be defined, which meet the requirement of short-crestedness. For manytechnical applications this is an unnecessary sophistication.

6. THE RESPONSE OF MARINE STRUCTURES TO WAVES

Using the mathematical model of the waves discussed it is also possible to give a gooddescription of the behaviour of marine structures placed in an irregular sea, at least as long asthe response of the structure can reasonable well be approximated linearly. Take as examplethe vertical motion of a floating drilling unit. Each regular, sinusoidal wave component forcesthe drilling unit into an equally regular, sinusoidal vertical movement:

)t(sinzz nznnan Ve+e+w= (6.1)

where zan is the amplitude of the movement and nzVe the phase difference between the

movement and the wave motion. By summing all motion components one obtains:

=

=V

e+w=

e+e+w=

1nn*

nan1n

nznnan tsinztsinzz (6.2)

which is similar to the wave ,model (5.1). Everything said in Chapter 3, 4 and 5 can berepeated if "wave" is replaced by the appropriate quantity, in this case "vertical motion".

In accordance with equation (5.3) a vertical motion spectrum can be defined by:

2a2

1d

dz zd)(S

21

21

w+w

w-w=ww (6.3)

Obviously the following relation exists between the motion spectrum and the wave spectrum:

)(S)(z

)(S

2

a

az wV

wV

=w (6.4)

This relation is actually used in carrying out calculations for the behaviour in irregular waves.The ratio z/a V a is the amplitude ratio or transfer function for the vertical movement. Thus ananalysis of the behaviour of a marine structure in irregular waves is reduced to thedetermination of the transfer function for the quantity under considerayion, either by

calculations or by experiments in a series of regular waves of various frequencies. It will beclear that the transfer function is influenced by a number of circumstances, such as waterdepth, dimensions of the structure and wave direction.

Equation (6.4) can also be reserved:

)(S

)(S)(

z z

a

aww

=wV V

(6.5)

This allows for the analysis of experiments in irregular waves or full-scale measurements atsea. If the vertical motion spectrum and the wave spectrum are measured simultaneously, thetransfer function of the vertical motion can be obtained from the quotient. It can next becompared with calculations to check theoretical approach to the problem. In Fig. 8 anexample for the semi-submersible drilling unit STAFLO of Shell U.K. Ltd. Is shown. Themeasurements were performed during operations in the North Sea. Of course, the proceduredescribed can be applied to any quantity other than vertical motion, such as angular motions,wave loads, dynamic stresses produced by waves, etc.

In the foregoing wave action and the result of the wave action on structures was discussed.This discussion covers only the variable part of motions, forces, stresses, etc., as a result ofwave action. The mean level of the quantities or variations due to other circumstances mustbe separated from the wave action and dealt with independently; see also the introduction.This, of course, presumes that a linear treatment of the problem studied is a reasonableapproximation of reality. In the opinion of the author this is true for the great majority ofengineering problems in offshore operations.

7. REFERENCES

There is a great deal of literature available on the subject. Usually, however, it is of a ratherspecialized nature and not very accessible. For a good and clear discussion of the resolutionof signals into their components and spectral analysis of stochastic signals, Dutch readersmay be referred to Chapters 15 and 16 of "Regeltechniek" by J.C. Cool, F.J. Schijff and T.J.Viersma, published by Agon Elsevier in 1969.

A mathematical and more detailed treatment of wave action is found in the following papers:

Cartwright, D.E. and M.S. Longuet-Higgins, "The statistical distribution of the maxima of arandom function", Proc. Royal Society of London, A, Vol. 237, 1956.

Longuet-Higgins, M.S., "On the statistical distribution of the heights of sea waves", J. ofMarine Research, Vol. XI, No. 3. 1952.

Cartwright, D.E., "On estimating the mean energy of sea waves from the highest waves in arecord", Proc. Royal Soc. of London, A, Vol. 247, 1958.

The analysis of marine structures in waves is discussed more elaborately in two other papersby the author:

"The role of model tests and their correlation with full-scale observations", Proc. Symposiumon Offshore Hydrodynamics, Wageningen, 25th-26th August 1971.

"De analyse van maritieme constructies", lecture delivered at the Vreedenburgh Day of theKoninklijk Instituut van Ingenieurs, Delft, November 1971 and to be published in "DeIngenieur".

FIG. 1 - REGULAR WAVES

FIG.2 - ORBITAL MOTION AND WAVE PROPAGATION

FIG.3 - ORBITAL MOTION AND WATER DEPTH

FIG.4 - EXAMPLE OF A WAVE RECORDING WITH NOMENCLATURE

FIG.5 - STATISTICAL DISTRIBUTION OF VARIOUS WAVE QUANTITIES

FIG.6 - SUMMATION OF REGULAR WAVE COMPONENTS

FIG.7 - THE DISTRIBUTION OF WAVE ENERGY OVER THE WAVE COMPONENTS

FIG.8 - COMPARISON BETWEEN CALCULATIONS AND FULL SCALE MEASUREMENTS FORTHE VERTICAL MOTION OF THE ROTARY TABLE OF THE SEMI-SUBMERSIBLE

DRILLING UNIT STAFLO

TITLEPREFACETABLE OF CONTENTSSUMMARYMAIN SYMBOLS1. INTRODUCTION2. REGULAR WAVES3. A PHENOMENOLOGICAL DESCRIPTION OF WAVES4. A STATISTICAL DESCRIPTION OF WAVES5. THE WAVE SPECTRUM6. THE RESPONSE OF MARINE STRUCTURES TO WAVES7. REFERENCESFIGURES 1 - 8 FIG. 1 - REGULAR WAVESFIG.2 - ORBITAL MOTION AND WAVE PROPAGATIONFIG.3 - ORBITAL MOTION AND WATER DEPTHFIG.4 - EXAMPLE OF A WAVE RECORDING WITH NOMENCLATUREFIG.5 - STATISTICAL DISTRIBUTION OF VARIOUS WAVE QUANTITIESFIG.6 - SUMMATION OF REGULAR WAVE COMPONENTSFIG.7 - THE DISTRIBUTION OF WAVE ENERGY OVER THE WAVE COMPONENTSFIG.8 - COMPARISON BETWEEN CALCULATIONS AND FULL SCALE MEASUREMENTS FOR

TABLEEXAMPLE OF WAVE STATISTICS FOR A DURATION OF APPROXIMATELY 3 HRS (N = 1000);