Design of breakwater for conversion of wave energy into electrical energy

Ocean Engineering 51 (2012) 106–118

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Ocean Engineering

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Design of breakwater for conversion of wave energy into electrical energy

Paolo Boccotti n

Mediterranea University, Natural Ocean Engineering Laboratory (NOEL), Loc. Feo di Vito, 89122 Reggio Calabria, Italy

a r t i c l e i n f o

Article history:

Received 9 March 2011

Accepted 12 May 2012

Editor-in-Chief: A.I. Incecikthe top of a vertical duct instead of entering through a large opening in a vertical wall, as in some

conventional OWCs. Designing this and other wave energy converters is difficult for the simple reason
Available online 20 June 2012
Keywords:

Caisson breakwaters

Renewable energy sources

OWCs

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x.doi.org/10.1016/j.oceaneng.2012.05.011

þ39 0965875292; fax: þ390965875412.

ail address: [email protected]

a b s t r a c t

Some recent papers have dealt with the physics and expected performances of caisson breakwaters

embodying a U-OWC: seawater flows in and out of these plants through a relatively small opening at

that experience gained from similar plants built in the past cannot be exploited. Hence, there must be

attempts to foresee the main phenomena that may occur in the plant’s lifetime, which is the first aim of

this paper. The second aim is to suggest a holistic overview of the design of these converters. This

means looking at the plant as a whole and considering all of the following aspects together: safety,

performances (absorption of wave energy, production of electric power), extreme loads on the various

walls, overall stability.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Since the pioneering paper by Salter (1974), the possibility ofconverting wave energy into electric power has been an object ofstudy for over forty years. Cruz (2008) presents an updatedhistory of these forty years of research in his book.

A fundamental contribution was by Wells, who presented turbinesthat rotate in the same direction even if the current is reversed. Theseturbines and relevant performances were extensively illustrated byRaghunathan (1995) and Curran and Gato (1997); the latter pre-sented some plots to estimate the head losses and efficiency.

Many devices have been developed with the aim of convertingwave energy into electric power. In Japan and Europe, oscillatingwater columns (OWCs) appear to have gained the most attention.The pioneering work was a full-scale caisson breakwater makingup an OWC that was built at Sakata port (Takahashi et al., 1992).From the theoretical point of view, pioneering works includethose by Evans (1976, 1982) and Sarmento and Falc~ao (1985) onthe interaction between waves and a conventional OWC.

A conventional OWC is essentially a box with a large openingon the wave-beaten wall. This opening usually goes from theseabed to nearly the mean water level. An air pocket remainsbetween the roof and sea surface and is connected to the atmo-sphere by a tube with a Wells turbine. A typical defect ofconventional OWCs is that their eigenperiod is smaller than thewave period. To overcome this problem, some devices weredeveloped to create a sort of artificial resonance. These are knownas systems for latching (Korde, 1991).

ll rights reserved.

Recently, a new kind of OWC called U-OWC has been con-sidered (see Fig. 1). A U-OWC has an eigenperiod greater than thatof a conventional OWC; hence, it is expected to perform betterthan a conventional OWC with waves of large periods such asswells or sea storm waves (Boccotti, 2007a). However, U-OWCsperform well even with small wind waves because they commu-nicate with the sea through an opening close to the water level.This is because pressure fluctuations on the outer opening of aU-OWC are relatively large even with small-period waves, whichhave a large attenuation of wave pressure as the depth beneaththe mean water level increases.

Fig. 1 shows the scheme of a caisson breakwater embodying aU-OWC. There is a vertical duct along the entire wave-beaten wall.The upper opening of this vertical duct communicates with the sea.The vertical duct is connected to a chamber through an openingat the bottom. The chamber has a roof above the sea level andis connected to the atmosphere through tubes. Each tube isprovided with a Wells turbine. The chamber and duct are subdivided(by walls C) into a certain number of cells. These cells may beindependent of one another: each cell has one exhaust tube providedwith a Wells turbine. In this case, the turbines typically have adiameter of about 0.7 m. An alternative option is to connect a numberof cells with each other (air can flow from one cell to another throughholes in walls C) with a unique turbine for a set of cells. In this case,fewer turbines are employed, and these turbines are larger. Wedetermined the first option (independent cells) to be the best, bothbecause it enables us to employ some turbines with well-establishedcharacteristics (diameter, speed) and because the calculation of waterand air flow inside the plant is easier and more reliable.

The plant works under clearly evident principles: under waveaction, the pressure on the upper opening of the vertical ductfluctuates; as a consequence, water enters and exits (alternately)

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Nomenclature

List of symbols

a depth of the outer opening;A absorption coefficient (¼ratio between the average

power absorbed by the plant and the mean energyflux of the incident waves);

Ap conversion coefficient (¼ratio between the averageelectric power produced by the plant and the meanenergy flux of the incident waves);

cG group celerity;cR propagation speed of the reflected wave energy;C ratio between F and E;D diameter;E frequency spectrum;E wave energy per unit surface;E mean energy per unit surface of the standing

wave field;Ein mean energy per unit surface, of the incident waves;h given threshold of Hs;h0, h00 energy per unit weight;H wave height at point xo,yo;H height of the incident wave;Hs significant wave height;k wave number;L lifetime;Lp dominant wave length;p probability density function;pa air pressure;P probability of exceeding;P* weight in still water;Q water discharge of waves at section y¼0 (breakwater);Qp water discharge inside the plant;R resonance coefficient (R¼0: resonance);Rv vertical reaction of the soil;s width of the vertical duct;S directional spectrum;

Sw uplift force;t time;T wave period, or time lag;Te eigenperiod;T time lag between wave pressure on the outer opening

and water discharge in the plant;T* lag of the absolute minimum of the autocovariance at

point xo, yo;u current velocity in the vertical duct;x horizontal axis parallel to the breakwater;X,Y local coordinates axis with origin at xo, yo;xo, yo coordinate of a point struck home by a wave group at

the apex of its development stage;y horizontal axis orthogonal to the breakwater;z vertical axis;b ratio between the wave height at a breakwater

embodying a U-OWC and the wave height at aconventional breakwater;

Dx interaxis of adjacent cells;Dp wave pressure on the outer opening;Dp wave pressure on the outer opening, with the QD

theory;DRv vertical force due to the water flow inside the plant;ga specific weight of water;d random phase angle;e phase angle;Z free surface displacement;Z free surface displacement with the QD theory;y angle between the wave direction and y-axis;t time lag;C convariance of the random free surface displacement;F actual wave energy flux at the breakwater;F mean wave energy flux at the breakwater;Fp actual power absorbed by the plant;Fp average power absorbed by the plant;o angular frequency.w value of the ratio cR/cG;

P. Boccotti / Ocean Engineering 51 (2012) 106–118 107

through the opening and lets the air pocket compress and expand.The pressure fluctuations in the air pocket yield some currentsthat drive the Wells turbines. The eigenperiod of a U-OWC growswith the ratio between the width of the chamber and the width ofthe vertical duct, and with the length of the vertical duct. There-fore, it is easy for the designer to let the eigenperiod be equal tothe period of the waves conveying the largest amount of waveenergy in a year.

The solution for the interaction between linear waves andU-OWCs is given in a previous paper (Boccotti, 2007b). A novelconcept of this solution was to consider the transient when thestanding wave field gradually expands towards the open sea aftera wave train has hit the breakwater. In Sections 2 and 3, the logicpath of the aforementioned solution and operation sequence forestimating the wave energy absorption are summarized. The newcontributions of these two sections are on the effect of the wavesteepness, which emerges from a comparison between theoreticalpredictions and evidence from a small-scale field experiment weconducted earlier (Boccotti et al., 2007).

When verifying safety and estimating the largest loads on thevarious walls, it is convenient to resort to the quasi-determinism(QD) theory. Section 4 gives the reasons why the QD theory issuited to this. Later sections represent a first step towards anoverview of the design, where performances, safety, loads, and

overall stability should be considered together. To the best of ourknowledge, there is no study in the scientific literature that hasdealt with the overall design of a wave energy converter, whilethere have been many studies on specific topics: the efficiency ofcertain Wells turbines; how waves change due to the interactionwith an OWC or U-OWC; good devices for latching and so on.

Guidelines for the design of U-OWCs will only be possible aftera full-scale prototype has been built and tested. The present paperaims to suggest some ideas that should be helpful for the designof such a prototype.

2. Summary of the solution (2007) for interaction betweenwaves and U-OWC

Given the wave pressure Dp on the upper opening of thevertical duct, the water flow inside the plant may be calculated bya finite difference approach; the energy equation is applied toboth the water mass inside the chamber and duct and the air inthe exhaust tube, and the continuity equation is applied to boththe water mass inside the chamber and duct and the air in thechamber and exhaust tube. However, we do not know Dp becausethe waves at the breakwater are modified by the current enteringor exiting the vertical duct. Hence, the reasoning goes as follows.

Fig. 2. Control volume before a breakwater.

Fig. 1. Caisson breakwater embodying a U-OWC: (a) cross-section and (b) horizontal

section.

P. Boccotti / Ocean Engineering 51 (2012) 106–118108

Let us suppose a periodic wave train with a crest-to-troughheight H,

Zðy,tÞ ¼H

2cosðky�otÞ ð1Þ

which hits a caisson breakwater embodying a U-OWC. We do notknow how the waves are modified. Hence, we write theirexpression in the most general form, which proves to be

Zðy,tÞ ¼ ½bcosðkyÞcosðo tÞþasinðkyÞcosðo tþeÞ�H, ð2Þ

where a, b, and e are currently unknown. (Note that the waveelevation and wave pressure at the breakwater (y¼0) dependonly on b and not on a nor e because the second standing wave onthe RHS of Eq. (2) has a node at y¼0.)

There are N1 triplets a, b, e that have been proven to satisfy

the two equations of wave-breakwater interaction at every timeinstant: that is,

QpðtÞ ¼Q ðtÞ, ð3aÞ

FpðtÞ ¼FðtÞ: ð3bÞ

Each of these triplets a, b, e corresponds to a new value of theratio cR/cG. The equation of cR is

cR ¼Fin�FE�E in

ð4Þ

and is obtained by applying the energy equation to a controlvolume before the breakwater (Fig. 2) during the transient whenthe incident wave field (Eq. (1)) is still at the seaward end of thecontrol volume and the standing wave field (2) gradually expandstowards the open sea. Hence, there are infinite solutions to theproblem of wave-plant interaction, which are characterized bythe value of the ratio w¼cR/cG.

The actual solution is one of the infinite solutions with0rwr1. If we assume that the propagation speed of the waveenergy is always equal to cG, the solution is that yielding w¼1, andthe problem of wave-plant interaction has arrived at closure: thesolution is represented by the triplet a, b, e for which cR/cG isequal to 1.

Otherwise, the reasoning can be as follows. Power per unitwidth flows from the open sea to the plant, and this requires thatan amount of wave energy per unit area of the surface bedistributed along the path towards the plant. In other words,energy travels over a ‘‘road’’ consisting of a quantity of energydistributed per unit surface. In this light, the most efficientsolution is to maximize the ratio between the amount of powerper unit width flowing towards the plant and the amount ofenergy per unit area of the surface that must be distributed alongthe path towards the plant. Hence, if we assume that the actualsolution is the one that is the most efficient, we must find thesolution that gives the largest value of the ratio

C¼FE¼�2absin ea2þb2

cG: ð5Þ

Factor C depends on the resonance coefficient

R¼4T

T, ð6Þ

which is zero under resonance conditions, approaches 1 when theeigenperiod of the plant is much greater than the wave period,and approaches �1 when the eigenperiod is much smaller thanthe wave period.

The following three concepts are useful for comparison of thetheory with the results of a small-scale field experiment.

First, the amplification factor b (ratio between the crest-to-trough wave height at the breakwater embodying the U-OWC andthe crest-to-trough height 2H at a conventional vertical break-water) is always smaller than 1 for the solution w¼1 and may begreater than 1 only for solutions with w smaller than 1.

Second, for a given 9R9, the factor C is usually almost the samefor every value of w, with only one important exception: if 9R9 isvery close to 1, C greatly increases when passing from solutionw¼1 to w¼0.

Third, if 9R9 is very close to 1, the amplification factor b takeson the largest values, which are much greater than 1.

The conclusion is that we may expect to find b to be smaller orgreater than 1 for a great majority of the domain (�1,1) of R; onthe other hand, we will only find very larger b if 9R9 is close to 1.


The small-scale field experiment described in an earlier paper(Boccotti et al., 2007) confirmed this overall view for negative R

only. This conclusion can be verified by examining the series offigures in that paper that compared the spectrum at the break-water to the spectrum of the incident waves. These figures enableus to directly evaluate the amplification factor b. For multimodalspectra, the figures indicate the value of b for various parts of thespectrum of different modes. In particular, the figures easily verifythe theory that there are some systematically very large ampli-fications (b up to 2.5) when R is close to �1. In contrast, there isno b greater than 1 when R is positive. This strong differencebetween positive and negative R with regard to the amplificationfactor b is not an effect of the response characteristics, which arefrequency-dependent. In fact, two distinct frequencies with twoopposite values of R should have nearly the same responsecharacteristics (see Fig. 4 in Section 3). Hence, the reason forthe experimental evidence of asymmetry between positive andnegative R must be found elsewhere.

We assumed that the actual solution is the most efficient one,and we assumed that the most efficient solution is the oneyielding the largest value of ratio C between F and E. However,with solution w¼0, the waves may become too steep and break(or partially break) because w¼0 implies the largest values of theamplification factor b. Thus, if the incident waves are rather steep,w¼0 may be not the most efficient solution even if it yields thelargest value of C. This is because of energy dissipation due tototal or partial wave breaking.

In the above-cited field experiment, the wind waves had asteepness on an order of magnitude greater than that of the swelland had a positive R whereas the swell had a negative R (refer totable 1 in Boccotti et al., 2007). The great difference in wavesteepness between the negative and positive domains of R shouldbe the cause of the asymmetry between positive and negative R

with regard to the amplification factor b.

3. Main software: first program for designing the plant

Once Dp(t) (wave pressure on the outer opening of the verticalduct) is known, it is possible to calculate the water and air flowinside the plant. The plots of Curran and Gato (1997) are veryuseful for this purpose since they permit the calculation of boththe head losses in the exhaust tube and the electric power that isproduced.

For our design, two main programs are needed to calculate thewater and air flow inside the plant. The first starts from anumerical simulation of the random pressure fluctuation Dp(t)of a sea state with a given spectrum. The program estimates the

Fig. 3. Cross-correlation of pressure fluctuation at

wave energy that is absorbed and converted from a sea state withgiven characteristics. Since the power absorbed from a sea statemay be a very large share of the total energy flux of incidentwaves, the standing waves before the breakwater embodying aU-OWC may exhibit a huge difference from the standing wavesbefore a conventional caisson breakwater. Hence, this differencecannot be neglected when numerically simulating Dp(t) of thesesea states.

In order to estimate the absorption and conversion of waveenergy from a given sea state, we perform the following pre-liminary operations:

a)

the

take the frequency spectrum of the incident waves;
b) fix the number N of elementary waves; c) fix the set of N small wave heights (H1,H2,y,HN ) and set of N
angular frequencies (o1,o2,y,oN ) to fit the given spectrum;
d) choose N phase angles (d1,d2,y,dN) that are stochastically
independent from one another and uniformly distributed over(0, 2p);

e)
calculate the mean energy flux and mean energy per unitsurface of the incident waves:
Fin ¼ rgXN

1

1

8H2

i cGi, ð7aÞ

E in ¼ rgXN

1

1

8H2

i : ð7bÞ

We then

(i)

out

fix a tentative value of b;
(ii) calculate Dp on the outer opening of the vertical duct by
means of

DpðtÞ ¼ brgXN

i ¼ 1

Hicosh½kiðd�aÞ�

coshðkidÞcosðoitþdiÞ; ð8Þ

(iii)
calculate the water flow inside the plant from Dp(t) toobtain Qp(t) and Fp(t);
(iv)
obtain the average Fp and F ¼Fp from Fp(t); (v) obtain the cross-correlation
cðtÞ ¼ oDpðtÞQpðtþtÞ4 ; ð9Þ

(vi)
obtain T and T from c(t) (see Fig. 3); (vii) obtain the group celerity cG for the wave period T and the
water depth at the breakwater;

er opening and discharge in the plant.


(viii)
obtain the phase angle
e¼�p2

1þ4T

T

!; ð10Þ

(ix)
compute
a¼ F4bsin9e9Fin

; ð11Þ

(x)
calculate the mean energy per unit surface of the wave field(2) before the breakwater
E ¼ 2ða2þb2ÞE in; ð12Þ

(xi)
calculate cR by means of Eq. (4); (xii)
Fig. 4. Example of amplification factor b and absorption coefficient A as functions

of resonance coefficient R.

check that the following inequalities are fulfilled:

FrFin, ð13aÞ

EZE in, ð13bÞ

cRrcG: ð13cÞ

If even one of these inequalities is not satisfied go back to(i) and fix a new value of b; otherwise, store the values ofthe triplet a, b, e and the value of the ratio cR/cG;

(xiii)
obtain the function b vs. cR/cG.
This procedure is based on the theoretical solution summar-ized in the previous section, which is exact for monochromaticwaves. However, this procedure is extended to a sea state with avalue of b unique for all of the harmonic components because thisapproach has proved to be effective for the special scope ofestimating the performances of the plant. Boccotti et al. (2007)used Hi and oi from the actual spectra of incident waves and thevalue of b relevant to cR/cG¼1 to find values of

A¼Fp=Fin, ð14Þ

that closely agreed with the actual A of sea states consistingof wind waves. Moreover, using the value of b relevant tocR/cG¼0.25, they found values of A that closely agreed with theactual A of sea states consisting of swells.

We define ‘‘wind waves’’ as waves in the generating area and‘‘swells’’ as waves from outside the generating area. The differ-ence is in the wave steepness. In the aforesaid field experiment,wind waves had an average Hs/Lp0 of 3.7�10�2, whereas theswell had an average Hs/Lp0 of 3�10�3. Naturally, a new experi-ment should be performed to examine a wider range of wavesteepness. For preliminary design purposes, we resort to solutionw¼1 if waves have an Hs/Lp0 of the order of 10�2 and solutionw¼0.25 if waves have an Hs/Lp0 of the order of 10�3.

Fig. 4 shows typical plots of b and A as functions of R accordingto two solutions: one for w¼cR/cG¼1 and the other for w¼0.25. Asshown, for a given value of w, b takes its minimum and A takes itsmaximum at R¼0 (resonance). For a given R, both b and A grow asw is reduced. Note that the extreme solution w¼0 yields A¼100%for every R. Note that if A was a linear function of w, for a given R,the difference between A of solution w¼0.25 and A of solutionw¼1 should be considerably greater than the actual difference.

On some ocean coasts where the largest share of the waveenergy that arrives in one year comes from swells, a plant isexpected to have excellent performances since swells allow forsolution w¼0.25. However, even in seas where the largest share ofthe annual wave energy comes from wind waves, the U-OWC

should perform much better than a conventional OWC for thereasons given by Boccotti, 2007a.

4. Main software: second program for designing the plant

We now consider the second program, which estimates theeffects of the maximum expected wave over the plant’s lifetime.If we run the first program for the design sea state, we typicallyfind a very small percentage of absorbed wave energy and anamplification factor b close to 1. This means that in the designsea state, the wave amplitude at a breakwater embodying aU-OWC is only slightly smaller than the wave amplitude at aconventional wall breakwater. For this reason, when verifying theplant’s performance under actions due to extreme sea conditions,we may assume that the wave amplitude at a caisson breakwaterembodying a U-OWC is equal to the wave amplitude at aconventional caisson breakwater.

The maximum expected wave in the structure’s lifetime willnot only be an exceptionally large wave for the given location butalso will probably be an exceptionally large wave even for thevery strong sea state in which it will occur (cf. Boccotti, 2000,Section 7.8). According to the quasi-determinism theory (QD) ofsea waves, a wave with a given height that is exceptionally largewith respect to its sea state will be the central wave of a wavegroup that will strike at the apex of its development stage (i.e., astage at which the height of the central wave of a group grows toits maximum thanks to a spontaneous shrinking of the three-dimensional envelope). A consequence of the two previoussentences is that the maximum expected wave in the lifetime ofa structure will very probably be the central wave of a group thatwill strike the structure at the apex of its development stage. Inour case, the structure is a caisson of a breakwater embodying aU-OWC. The quasi-determinism theory lets us predict the space-time configuration of the free surface of the wave group, particlevelocities and accelerations, and pressure fluctuations beneaththe water surface. The free surface displacement at the section ofthe breakwater struck by the wave group is (see the Appendix)

ZðtoþTÞ ¼H

R10 EðoÞ½cosðoTÞ�cosðoT�oTn

Þ�doR10 EðoÞ½1�cosðoTn

Þ�do, ð15Þ


and the pressure fluctuation at depth z¼�a of the upper openingof the vertical duct is

DpðtoþTÞ ¼ gaH

R10 EðoÞ cosh½kðd�aÞ�=coshðkdÞ

� �½cosðoTÞ�cosðoT�oTn

Þ�doR10 EðoÞ½1�cosðoTn

Þ�do,

ð16Þ

where E is the frequency spectrum of the design sea state, H is themaximum expected wave height in the lifetime (i.e., the height ofthe incident wave), T* is the lag of the absolute minimum of theautocovariance, and to is the instant when the crest of themaximum expected wave occurs.

The program calculates the water and air flows inside thecaisson under the known DpðtÞ. The water flow inside the plantdepends upon the difference between the energy per unit weighth0 at the upper opening of the vertical duct and the energy perunit weight h00on the water surface inside the chamber. If Z isgreater than the elevation �a, h0 is the energy per unit weight ofthe wave at depth z¼�a. If Z is smaller than the elevation �a,then

h0 ¼ z0 þu2=2g, ð17Þ

where z0(r�a) is the actual elevation of the water surface in thevertical duct during the time intervals in which the upper openingof the vertical duct remains above the wave surface. Fig. 5 showsZ at the breakwater, the water discharge in the vertical duct(positive if the water flows downwards), and elevation z0 of thefree surface in the vertical duct (this is shown for the time intervalt1, t4 when the upper opening of the vertical duct remains abovethe wave surface). In interval t1, t2, the flow in the vertical duct isupwards; hence, z0 remains equal to �a. In interval t2, t3, the flowin the vertical duct is downwards; hence, the water level in theduct falls. In the short interval t3�t4, the water level in the ductcontinues to gradually rise because the flow in the duct has beenreversed. At time instant t4 the water level in the vertical ductsuddenly rises because the wave surface rises high enough forwater to flow through the upper opening of the vertical duct.

The program has to calculate

1)

Figver

the highest elevation reached by the water in the chamberduring the collision of the wave group of the maximumexpected wave;

2)
the largest positive and negative pressures in the air pocket; 3) the lowest level of water in the vertical duct; 4) the largest positive and negative loads on walls A, B, D, and E
and on the roof of the chamber (a load is positive if it acts fromthe plant outwards);

. 5. Interval (t1, t4): the wave surface is beneath the outer opening of the vertical d

tical duct decreases. (t3, t4): the water level in the vertical duct rises.

5)

uct

the largest loads on wall C when the pressure in the air pocket(pa) is greater or less than the atmospheric pressure (patm) (forthis job we must run the second program several times, asshown in Section 7);

6)
the vertical force DRv due to the water flow inside a caisson atthe instant of the crest of the maximum expected wave in thelifetime, which is the instant of both the largest horizontalforce and largest overturning moment (DRv depends on: (i) thevariation in water mass in the chamber, (ii) the inertia of thewater mass in the chamber and vertical duct, (iii) the flux ofmomentum through the upper opening of the vertical duct,and (iv) the wave pressure on the upper opening of the verticalduct and on the top of wall E).
The first calculation (highest water level reached in thechamber) is crucial. As shown in Section 6, if the water level inthe chamber exceeds the elevation of the exhaust tube, thepressure of the air trapped between the water surface and roofmay grow rapidly to very high peaks of short durations. In otherwords, an excessive rise in the water level in the chamber yields awater hammer that is dangerous for both the roof and walls of thechamber. Thus, these calculations must be performed conserva-tively. We suggest the following precautions. First, assume thatthe turbine has stopped: a stoppage of the turbine implies smallerhead losses in the exhaust tube, so the rise of water in thechamber is greater. Second, assume that the maximum expectedwave height occurs with the high tide. Third, repeat the calcula-tion twice: first assume that the equation of state is an isotherm,then a second time assuming that the equation of state isadiabatic. Take the larger value of the two calculations for therise of water in the chamber (there is usually only a smalldifference between the results of the two runs of the program).

5. Designing the chamberþduct

We begin with the crucial relationship between the plant’sdesign and performance. Specifically, we consider how A andAp—the ratio between the electric power produced by the break-water and the wave energy flux arriving at the breakwater—varyif we let the width s of the vertical duct vary for given values of d,a, bx

0

, by0

, h1, and h2 (see Fig. 1) and for a given configuration of theexhaust tube and turbine. The procedure can then be repeatedwith new values of the above listed parameters, particularly newvalues of by

0

. This task can be done with the first program.

. (t1,t2): water flows out from the vertical duct. (t2, t3): the water level in the


As an example, Fig. 6(a) shows a hypothetical breakwater forthe Mediterranean Sea. Fig. 6(b) shows A, Ap, and R as functions ofthe width s of the outer opening for a typical sea state of windwaves (Hs¼2.0 m, Tp¼6.0 s). Since Hs/Lp0¼3.6�10�2, calcula-tions are performed with solution w¼1. The plots inFig. 6(b) shows the effects on efficiency due to variations in thewidth s. When s¼1 m, Ap is 12%, which is rather small for tworeasons: the opening of the duct is rather small, so the intake ofwave energy is also small; and R is far from zero and so is far fromresonance (bearing in mind that the domain of R is (�1,1)). Ap hasa large rate of growth up to s¼2 m because from s¼0 to s¼2 m,the opening of the intake of wave energy grows and 9R9 getscloser and closer to zero. For s from 2 m to 4 m, Ap remains nearlyconstant thanks to two opposite effects that counterbalance eachother: the widening opening for increasing intake of wave energyand 9R9 moving further away from zero. When s is greater than4 m, the effect of 9R9 moving further away from zero becomesmore dominant than the effect of the growing opening for intakeof the wave energy, so the efficiency is gradually reduced.

We now consider Fig. 7, which shows a hypothetical break-water for the Atlantic coast of Morocco. We consider the effects ofa sea state consisting of a swell with Hs of 1.5 m and Tp of 12 s;this is a typical wave size for the Atlantic coast of north Africa.Since the wave steepness Hs/Lp0 is 6.7�10�3, we use solutionw¼0.25. With this swell, the breakwater of Fig. 6(a) yields thegreatest efficiency with a rather small value of width s: Ap¼28.0%when s¼1.25 m. This is because the wave period is so large thatwe can only get resonance with a very small duct width. In orderto improve efficiency—that is, develop resonance with a largeropening of the energy intake—the length of the duct is increased.

Fig. 7. (a) Hypothetical plant on the Atlantic coast of Morocco. The exhaust tube, turbi

(b) See the caption of Fig. 6.

Fig. 6. (a) Hypothetical plant in the Mediterranean Sea. The configuration of the ex

monoplane Wells turbine with a maximum angular speed of 4000 rpm. Each cell is ind

(b) Absorption coefficient A, conversion coefficient Ap, and resonance coefficient R vs. s (

the same shape of multimodal spectrum used by Boccotti, 2007a.

As shown in Fig. 7(b), if s grows, the efficiency of the oceanbreakwater grows rapidly to a maximum and then graduallydecreases. The same observations for the Mediterranean break-water hold here too. With the given swell of 1.5 m Hs and 12.0 sTp, the breakwater of Fig. 7 reaches its maximum estimatedproduction of 3.7 MW/km when s¼1.50 m, whereas the break-water of Fig. 6 reaches its maximum estimated production of3.3 MW/km when s¼1.25 m. Note that for this comparisonbetween the U-OWCs of Fig. 6(a) and Fig. 7(a), we have assumedthat the height of the roof was the same (6 m above sea level) forthe two breakwaters. For the actual height (8 m above sea level)of the second breakwater, Ap remains nearly unchanged, and A

slightly increases (the largest A increases from 71.5% to 76.5%).Once the size of the chamber and duct has been defined to

obtain a good efficiency, the safety and loads on the various wallsmust be checked. We first consider safety; we examine the loadsin Section 7. Safety is essentially connected to the highestelevation reached by the water in the chamber. According to theQD theory, when the central wave of a group at the apex of itsdevelopment stage hits the breakwater, the pressure of the airpocket reaches its maximum at nearly the same instant, so the airdischarge through the exhaust tube also reaches its maximum. Atthis instant, the water is still rising in the chamber. Afterwards,there are two possible scenarios.

In the first scenario (Fig. 8), the water level in the chamberrises up to a maximum that remains beneath the elevation of theexhaust tube. At time instant t1, which is when the water level inthe chamber reaches its maximum, the pressure in the air pocketis zero (atmospheric pressure). After t1, the air pressure decreasesto below the atmospheric pressure, so air is sucked from the

ne, and dimensions (bx0 , by

0) are the same as those for the plant shown in Fig. 6(a).

haust tube is that of Fig. 11a with d1¼0.6 m, D1¼0.45 m, D2¼0.75 m. It has a

ependent from the others and has a chamber with dimensions bx0 ¼4 m, by

0 ¼4 m.

the width of the outer opening). Calculations: the sequence given in Section 3 with

Fig. 9. QD theory: expected result when the largest wave in the plant’s lifetime hits a breakwater embodying a U-OWC. The case when the water level in the chamber

exceeds the exhaust tube.

Fig. 8. QD theory: expected result when the largest wave in the plant’s lifetime hits a breakwater embodying a U-OWC. The case when the water level in the chamber

remains beneath the exhaust tube.


atmosphere. In the second scenario (Fig. 9), the water level in thechamber exceeds the elevation of the exhaust tube, so the airmass between the water surface and roof can no longer flowtowards the atmosphere. In this case, the air pressure in thechamber grows rapidly (the growth is quasi-impulsive) up to amaximum at instant t2 when the water level in the chamber

reaches its maximum elevation. After t2, the water level in thechamber decreases; the air pocket is stretched with the conse-quence of decreasing the air pressure.

In the second scenario, two peaks of pressure occur in the airpocket. The second of these two peaks may be very high; thus, itmay be dangerous even though it has a very short duration. There


are two ways to limit the height of the second peak (or eliminateit completely). The first way is to partially close or reduce thediameter of the exhaust tube, which somewhat increases the firstpeak. This possibility is examined in the next section. The secondway is to increase the height h1 of the roof of the chamber.

6. Designing the exhaust tube

We now present two alternative designs for the exhaust tube(see Figs.10a and b).

Alternative (a): the exhaust tube is a circular cylinder withdiameter D2 for the turbine. If we keep the valve fully open, thereshould almost certainly be a very large increase in pressure underthe design wave due to the low resistance to the air flow.A scenario such as that given in Fig. 9 with an exceptionally largeheight of the second pressure peak should occur. Alternative (a)minimizes the head loss in the exhaust tube to increase theenergy absorbed by the turbine. The valve is kept fully open withthe sea states conveying nearly all of the total wave energy in ayear. The valve is partially closed for very strong sea states; thispartial closure does not restrict the production of electric powersince in those strong seas, the production of electric power isalready limited by the power of the generator coupled with theturbine. With exceptionally strong seas such as the design seastate, the valve is partially closed to yield some very large headlosses. The valve (typically a butterfly valve) must be motorizedand controlled by software that uses measurements of theincident waves or water level in the chamber as input data.

Alternative (b): The exhaust tube consists of a piece with adiameter D1 that is smaller than diameter D2 of the turbine, andthere is a gradual expansion from the former to the latter. Thevalve is in the piece with the smaller diameter. We can choose D1

Fig. 10. Two alternative configurations of the exhaust tube.

so that the plant is safe even when the valve is fully open duringthe design sea state. With alternative (b), the valve may bemanually operated and closed only for maintenance work. Thevalve may be fully open during seasons when extreme stormscannot occur (e.g., from April to November in the MediterraneanSea). The valve is partially closed at the beginning of the seasonfor the strongest seas.

With alternative (b), the estimated annual production ofelectric power is only slightly smaller than the production withalternative (a). Alternative (b) has an annual production of electricenergy that is typically at least the 95% of that with alternative(a). If we accept this small reduction in annual electric energyproduction, we gain two benefits, the first of which is particularlyimportant: 1) safety does not depend on control software but on aphysical phenomenon (the head losses in a tube) and 2) the valveis less expensive because its diameter is smaller than thediameter of the valve of alternative (a) and it does not need tobe motorized.

The re-entrant opening of alternative (b) (Fig. 10b) leads to agreater load on the flange. However, the re-entrant opening letsus reduce the length of the tube out of the chamber. Indeed, evenif the valve is put very close to the wall, it does not interact withthe vena contracta.

The designer may choose to make a hole through the rein-forced concrete wall with a diameter D3 greater than the diameterD1 of the exhaust tube (e.g., D1¼0.45 m, D3¼0.70 m). This is fortwo reasons. First, this allows for the possibility of changing D1 inthe future. Second, this gives an opening suitable for entering thechamber (i.e., if the exhaust tube is removed, a person can enterthrough the hole of diameter D3).

At the instant of highest pressure of the air pocket (pressurepamax), a uniform outward load pamax acts on the flange betweenthe circumferences of diameters D1 and D3. Moreover, on theinner circumference of the flange (that of diameter D1), there is aoutward load q per unit length that can be conservativelyestimated to be

q¼pamaxD1

4, ð18Þ

neglecting the flux of linear momentum from the piece of tube ofdiameter D2 towards the atmosphere.

Usually, at the instant when the pressure of the air pocketreaches its maximum, distance d2 between the tube and the waterlevel in the chamber (see Fig.11) is greater than distance d1

between the roof and the tube. The ratio d2/d1 typically rangesbetween 1 and 5. This implies that air entering the exhaust tubein the chamber mainly comes from below. Thus, we have aupward linear momentum that is transformed into a horizontallinear momentum inside the tube, similar to what happens at anelbow. This implies that the tube near the left opening is subjectto an upward force. The linear momentum entering the exhausttube also has an x-component that does not yield a force Fx on thetube if the tube is at the center of a cell.

7. Extreme loads on the various walls

The largest loads on walls A, B and D (see Fig. 1) under theaction of the maximum expected wave are those with the airpocket under pressure. These loads act from the inside towardsthe outside of the chamber. If the largest pressure peak in the airpocket is the first peak and the second peak is markedly smallerthan the first peak (or the second peak does not exist), wall Bproves to be less loaded than walls A and D. Indeed, at the instantof the first pressure peak in the air pocket, wall B experiences thewave crest, which means that the maximum load on the internal

Fig. 11. Loads on the flange of the exhaust tube.


face of wall B occurs together with the maximum load on theexternal face; thus, the load on the external face partiallycounterbalances the load on the internal face. If the secondpressure peak is not markedly smaller than the first peak, theload on wall B may reach its maximum at the time instant of thesecond peak. This is because at the instant of the second pressurepeak, the wave elevation at the breakwater has fallen, so the loadon the external face of wall B is smaller than that at the timeinstant of the first pressure peak.

The largest load acts on the roof and on the upper part of wallD. The pressure of the air pocket acts on the internal face, and theatmospheric pressure acts on the external face. We can assumethat the load on wall A varies linearly from the load on wall B tothe load on wall D. Usually, the largest outward load on wall Eoccurs at the base of this wall at the time instant when thepressure of the air pocket reaches its maximum.

The largest inward loads may occur either at the top of walls A,B, and D or beneath the mean water level. On the top of walls A, Band D, the largest inward load coincides with the lowest (nega-tive) pressure of the air pocket. The inward load on wall D at agiven elevation beneath the mean water level is equal to thedifference between the pressure of the fillet (sand) of the cellsbehind the chamber and the pressure of the chamber at the givenelevation. For the inward load on wall E, it typically occurs as acaisson is being towed.

We now consider the loads on wall C. If waves are long-crested, the wave attack is orthogonal to the breakwater, and theconfiguration of the hydraulic plant is the same for all cells, wall Cwill not be loaded; in other words, it has the same pressure on theleft and right faces. The shorter the wave crest, the greater theangle between the wave crest and breakwater; the greater thedifference in the configuration of the hydraulic plant from one cellto the next, the larger the load on wall C. The length of the wavecrest of a wave group decreases as the directional spread of thesea state grows. For the angle between the wave crest andbreakwater, the QD theory shows that the direction of a wavegroup advance in the open sea has a high probability of coincidingwith the dominant direction of the directional spectrum. How-ever, the quasi-determinism theory shows that there are some

anomalous wave groups whose direction of advance differs fromthe dominant direction of the spectrum. In particular, this theorypredicts that an exceptionally high wave in part of the shelteredarea behind a breakwater is probably due to an anomalous wavegroup with a direction of advance slightly different from thedominant direction of the spectrum of the incident waves(cf. Boccotti, 2000, Section 10.7). Bearing in mind this fact, wesuggest cautiously assuming increments of 101 for the anglebetween the most probable direction of the advance of a wavegroup and the orthogonal to the breakwater. For the configurationof the hydraulic plant, we suggest assuming that the turbine in acell is running at maximum speed while the turbine in theneighboring cell has stopped. We denote the cells before andafter wall C under examination as cells 1 and 2, respectively. Wemust cover the following situations:

(i)
the center of the wave crest strikes the center of cell 1;the turbine of cell 1 is stopped;the turbine of cell 2 is running at maximum speed;
(ii)
the center of the wave crest strikes the center of cell 1;the turbine of cell 1 is running at maximum speed;the turbine of cell 2 is stopped;
(iii)
the center of the wave crest strikes the center of cell 2;the turbine of cell 1 is stopped;the turbine of cell 2 is running at maximum speed;
(iv)
the center of the wave crest strikes the center of cell 2;the turbine of cell 1 is running at maximum speed;the turbine of cell 2 is stopped.
According to the QD theory, the largest wave height during theevolution of a wave group occurs at X¼0, Y¼0. Hence, X¼0, Y¼0coincides with the center of either cell 1 or cell 2 according towhether situations (i), (ii) or (iii) and (iv) are applicable. In orderto manage situation (i), we have to run the second program twotimes: the first time, the input pressure Dp is the pressure of thewave group at point X¼0, Y¼0; and the second time, the inputpressure Dp is the pressure of the wave group at point X¼Dx,Y¼0 (where Dx is the width of a cell). Each run creates a file witha sequence of pressure values at various elevations in thechamber. A new program is then used; it is the third programof software needed to design our breakwater. This program readsthe time series of the two files created with the two runs of thesecond program and calculates the pressure differences on thetwo faces of wall C separating cell 2 from cell 1. Typically, thelargest loads on wall C prove to be smaller than the largest loadson walls A, B and D.

To calculate the pressure exerted by the wave group on theouter opening of the vertical duct at point X¼0, Y¼0, we canresort to Eq. (16). To calculate the pressure exerted by the wavegroup at point X¼Dx, Y¼0, we must resort to the generalequation of the QD theory, which is in terms of the directionalspectrum of the incident waves (see the Appendix). We obtain

DpðxoþDx,toþTÞ ¼ gaHR1

0

R 2p0 Sðo,yÞ cosh½kðd-aÞ�=coshðkdÞ

� ��½cosðkDxsin y�oTÞ�cosðkDxsin y

�oðT�TnÞÞ�dydo

�Z 10

Z 2p

0Sðo,yÞ½1-cosðoTn

Þ�dydo: ð19Þ

8. Designing the ballasted cells and the superstructure

Elevation h1 is usually prescribed. The total width by must bedesigned so that the overall stability with the prescribed safetyfactors is ensured. The calculation of the characteristics of the design


sea state in deep water at the location of the breakwater may bedone as described in Sections 7.1–5 of the book of Boccotti (2000).The calculations of Hs, the maximum expected wave height at thedepth of the breakwater, and the wave pressures on the wall andbase of the breakwater may be done with Goda’s algorithm (Goda,2000). The overall stability can be verified in a similar manner as thatof a conventional caisson breakwater (chapter 13 of Boccotti, 2000).

So far, all of the above calculations are the same as those for aconventional caisson breakwater. One novelty is the vertical forcedue to the water flow inside the breakwater. This force (DRv) iscalculated with the second program (see Section 4). The verticalforce due to the water flow in the breakwater varies during thecollision of a wave group. At the instant the horizontal force onthe breakwater reaches its maximum (instant of the crest of thelargest wave of the group), the aforesaid vertical force actsdownwards to increase the vertical reaction of the soil and hencethe resistance against sliding. Moreover, the new downward forceincreases the resistance against overturning. The downward forceDRv is typically about 10% of the weight in still water P*

For analysis of the overall stability, a fourth program isnecessary. This program starts from the characteristics of thedesign sea state in deep water, calculates the pressure distribu-tion on both the wall and the base of the breakwater with theGoda algorithm, reads the value of DRv as an input, computes theweight in still water P*, moment Ms of P* and DRv with respect tothe axis of rotation O1, overturning moment MR of the horizontalwave force Fo on the wall, and the uplift force Sw, and finally

Fig. 12. Section connecting the U-

computes the vertical reaction of the soil

Rv ¼ PnþDRv�Sw ð20Þ

and the eccentricity eR of Rv with respect to the axis of rotation O1

eR ¼ ðMs�MRÞ=Rv: ð21Þ

At this point, all the information needed for calculating thevarious safety factors (cf. Boccotti, 2000, Section 13.3) has beengathered.

Walls A0 and C0 (see Fig. 1) may be designed in a similarmanner to the walls of a conventional caisson breakwater.However, the connection between these walls and the chamberrepresents a novelty. Here, a critical situation could occur a prioriunder the action of wave troughs given that under a wave trough,the chamber and duct is subject to a seaward force, and resistanceto this force is only given by the section shown in Fig. 12c. Noforce is given by the superstructure, which is simply in contactwith wall D of the chamber. The chamber exerts a tensile force,shear force, and bending moment on the section shown inFig. 12c. To compute these forces and moment, we need a fifthprogram that estimates the forces exerted by the largest wavetrough during the plant’s lifetime. The horizontal force F anduplift force Sw exerted by the largest wave trough in the plant’slifetime are examined in Sections 13.1 and 13.2 of the book ofBoccotti (2000). The pressure distribution exerted by the soilunder the largest wave trough is obtained by applying theequilibrium equations to the whole caisson.

OWC with the ballasted cells.


Once all the forces acting on the chamber and duct are known(including the force exerted on wall D by the ballast in the cellsbehind this wall), the equilibrium equations applied to thechamber and duct yield the forces and bending moments actingon the vertical section shown in Fig. 12c. Note that the verticalshear force exerted on said vertical section usually proves to be aupward force since

R0v4Pn0

þS0wþDR0

v ð22Þ

where R0v,Pn0 ,S0w represent the vertical soil reaction, weight in stillwater, and vertical wave force on the chamberþduct, and DRv

0

isthe vertical force (positive downwards) due to the water flow inthe breakwater, under the largest wave trough. The bendingmoment is counterclockwise, and the horizontal force is seaward.The superstructure cannot exert tensile stresses on the chamber,so resistance to the aforesaid forces and moment is only given bythe section of reinforced concrete shown in Fig. 12c. The largesttension occurs at the top of this vertical section. Usually, the samewidths as that of a conventional caisson breakwater prove to beenough for walls A0 and C0.

9. Conclusions

A crucial question when designing a U-OWC is how theefficiency varies with the width s of the duct. By letting s growfrom zero, the conversion coefficient Ap grows rapidly at firstbecause of the increased size of the energy intake and because theresonance coefficient becomes closer to zero. The efficiency thenplateaus because the increased opening of the energy intake isnearly counterbalanced by the fact that the resonance coefficientbecomes far from zero. Finally, at some point, letting s growfurther reduces the efficiency because the effect of the resonancecoefficient becoming further from zero dominates the effect offurther growth in the opening of the energy intake.

A second crucial question concerns safety: what happensinside the system chamber and duct when the breakwater is hitby the wave group of the maximum expected wave height duringthe plant’s lifetime? If the highest elevation reached by the waterlevel exceeds the elevation of the exhaust tube, the air pressure inthe chamber experiences a high peak of short duration. In order toincrease safety, the height of this sharp peak should be reduced,or the peak should be completely eliminated. One way to do so isincreasing the elevation h1 of the roof. Alternatively, the valve ofthe exhaust tube may be partially closed during strong storms.One solution for greatly increasing safety with only a small loss ofefficiency is using an exhaust tube consisting of two pieces—onewith the diameter of the turbine and one with a smallerdiameter—with a gradual expansion between the two pieces.

Because of the air flow and air pressure, the flange of the exhausttube is subject to orthogonal loads; if the tube is a re-entrant,vertical shear stress and bending moment due to a vertical force areexerted by the air sucked from the chamber.

The same calculation done to evaluate the highest elevationreached by the water level in the chamber also yields the largestloads on the various walls. At the time instant when a wave groupexerts the maximum force on the breakwater, the pressure in theair pocket of the chamber is nearly at its maximum, so wall B issubject to two opposite forces. This is why the largest load on wallB is smaller than the largest loads on walls A and D. Wall C isloaded only if the dominant wave direction makes an angle withthe orthogonal to the breakwater and/or there is some differencein the configuration of two adjacent cells—in particular, if one cellhas a turbine running and the adjacent cell has a turbine stopped.

The same calculation done to estimate the highest elevationreached by the water level in the chamber also yields the vertical

force DRv that is exerted on the breakwater because of the waterflow inside the system chamber and duct. The overall stability isverified with the same approach applied for a conventionalcaisson breakwater: one novelty is the presence of the new force(DRv) that enhances stability. Indeed, DRv acts downwards at thetime instant when the wave exerts the maximum horizontal forceon the breakwater.

For walls A0 and C0, which are also present in conventionalcaisson breakwaters, a new verification is needed: the cross-sectionat the connection between these walls and the chamber must beverified when subject to the largest wave trough in the plant’slifetime. The size of these walls could depend on this verification.

Appendix: On the Eqs. (15), (16) and (19)

The equation of the expected free surface displacement,according to the linear quasi-determinism (QD) theory is (seeBoccotti, 2000, Section 10.1)

ZðxoþX,yoþY ,toþTÞ ¼CðX,Y ,T; xo,yoÞ�CðX,Y ,T�Tn; xo,yoÞ

Cð0,0,0; xo,yoÞ�Cð0,0,Tn; xo,yoÞ

H

2,

ðA1Þ

where C is the covariance, with both space and time lags, of therandom free surface displacement of stationary wind-generatedwaves; T* is the lag of the absolute minimum of autocovarianceC(0,0,T;xo,yo); xo,yo are the coordinates of the point where anexceptionally large wave occurs; H is the given height of this wave,which is assumed to be exceptionally large with respect to the rmsfree surface displacement at point xo,yo; to is the time instantwherein the crest of the exceptionally large wave occurs at xo,yo.Eq.(A1), typically, represents a three-dimensional wave group thatstrikes home point xo,yo, at the apex of its development stage.

The definition of C is

CðX,Y ,T; xo,yoÞ ¼ oZðxo,yo,tÞZðxoþX,yoþY ,tþTÞ4 , ðA2Þ

where the angle brackets represent an average with respect totime t, and Z is the random free surface displacement of thestationary wind-generated waves.

Eq. (A1) holds for every configuration of the solid boundary,provided that the water flow is frictionless. What changes with theconfiguration of the solid boundary is C. If the solid boundaryconsists in a long vertical breakwater along the line y¼0, C isgiven by

CðX,Y ,T; xo,yoÞ ¼

Z 10

Z 2p

0Sðo,yÞcosðkyocos yÞcos½kðyoþYÞ

�cos y�cosðkXsin y�oTÞdydo, ðA3Þ

where S(o,y) is the directional spectrum of the incident waves (seeBoccotti,2000, Section 8.9). From (A1) and (A3) it follows that:

ZðxoþX,yoþY ,toþTÞ ¼H

2

Z 10

Z 2p

0Sðo,yÞcosðkyocos yÞcos½kðyoþYÞ

�cos y�fcosðkXsin y�oTÞ�cos½kXsin y�oðT�TnÞ�g

�dydoZ 1

0

Z 2p

0Sðo,yÞcos2ðkyocos yÞ½1�cosðoTn

Þ�dydo:,

ðA4Þ

As to the wave height, here H denotes the actual wave heightat point xo,yo, that is the height of the standing wave; whereas,the height of the incident wave is denoted by H. H will be themaximum expected wave height in the lifetime if the breakwaterwas not there; H will be the maximum expected wave height atpoint xo,yo, with the presence of the breakwater. The relationship


between H and H is

H¼ 2HðCdðyoÞÞ=ðCdð0ÞÞ, ðA5Þ

where Cd is the diffraction coefficient (see Boccotti, 2000,Section 8.8).

Because of the general relationship between directional spec-trum and frequency spectrum

EðoÞ ¼Z 2p

0Sðo,yÞdy, ðA6Þ

in the case that yo¼0, X¼0, Y¼0, Eq. (A4) is reduced to Eq. (15).The wave pressures in the water, to the first order in a Stokes

expansion, proceed directly from the deterministic wavefunction (A4). If yo¼0 and Y¼0, the wave pressure at the depthz¼�a of the outer opening, is given by Eq. (16) or Eq. (19),respectively, if X¼0 or X¼Dx.

The probability PðHmaxðLÞ4HÞ that the largest wave height in agiven lifetime L at a given location exceeds a fixed threshold H

depends on functions

(i)
P(H;Hs¼h)¼probability that an individual wave height in asea state of given Hs exceeds a given threshold H;
(ii)
P(Hs4h)¼probability that the significant wave heightexceeds a given threshold h at the given location;
(iii)
bðhmÞ¼regression of the duration of the equivalent triangularstorms (hm being the largest value of Hs in a ETS);
(iv)
T ðhÞ¼average wave period in a sea state wherein Hs has agiven value h;
(see Boccotti, 2000, Section 7.8, for more details). The H employedin Eqs. (15) and (16) is obtained from the inverse functionHmaxðL,PÞ, which gives the threshold for a given probability P tobe exceeded by the largest wave height in lifetimeL. Specifically, H is equal to HmaxðL,PÞ with the values of L and P

being prescribed by various guidelines for the design of maritimestructures.

In order to apply Eqs. (15) and (16), it is necessary todetermine the spectrum of the sea state wherein HmaxðL,PÞwilloccur. The probability density function pðHs ¼ h;Hmax ¼HÞ can beused for this. Here, the random variable is Hs of the sea statewhere an individual wave with a given height H and being thelargest wave of its storm will occur at the given location. Thesolution for this PDF is expressed in terms of the above-listedfunctions (i)–(iv). As it is the largest wave height in the plant’slifetime, H¼HmaxðL,PÞ must also be the largest wave height of itsown sea storm. Hence, pðHs ¼ h;Hmax ¼HÞ may represent the PDFof the Hs of the sea state where the wave of given heightH¼HmaxðL,PÞ occurs. This PDF proves to be very narrow, so themode represents the Hs of the sea state where the wave of givenheight H¼HmaxðL,PÞ most probably occurs. Said mode is denotedby Hs(L,P). The frequency spectrum E(o) is a characteristicspectrum of wind waves with the given significant wave heightHs(L,P).

It is possible that the most critical condition does not occur apriori when the largest wave height in the plant’s lifetime occursat the breakwater. As an example, we consider the case where awave group reaches the apex of its development stage one half-wavelength before the breakwater (yo¼�Lp/2). According to the

QD theory, the breakwater is hit by four consecutive waves withnearly the same crest-to-trough height of about 1.5H, whereas ifthe incident wave group reaches the apex of its developmentstage at the breakwater (yo¼0), the breakwater is hit by a wave ofcrest-to-trough height 2H that is preceded and followed by awave of crest-to-trough height 1.2H. (This example assumes thatthe spectrum is the mean JONSWAP and the waves are in deepwater.). Of course, a priori we cannot say whether is it moredangerous that the breakwater is hit by a wave of height 2H

preceded and followed by a wave of height 1.2H, or by fourconsecutive waves with the same height 1.5H. After manycalculations, it has been concluded that the most severe condi-tions should occur if the wave group reaches the apex of itsdevelopment stage just at the breakwater.

Depending on the frequency spectrum, Eqs. (15) and (16) holdwhen a wave group reaches the apex of the development stage atthe breakwater (yo¼0). Otherwise, the configuration of the wavegroup at the breakwater depends on the directional spectrum ofthe incident waves (Eq. (A4)). The directional spectrum in shallowwater may be obtained from knowing the directional spectrum indeep water (e.g., Boccotti, 2000, Section 8.4).

Finally, the QD theory, like the Stokes theory, can be expandedto higher orders of approximation (see Arena et al., 2008; Arenaand Guedes Soares, 2009) to consider non-linear effects. However,non-linearity only locally alters the individual waves withoutmodifying the characteristic (complex) mechanics of wave groupsas disclosed by the linear QD theory.

References

Arena, F., Guedes Soares, C., 2009. Nonlinear crest, trough and wave heightdistributions in sea states with double-peaked spectra. ASME J. Offshore Mech.Ocean Eng. 131, 1–8. paper 041105.

Arena, F., Ascanelli, A., Nava, V., Pavone, D., Romolo, A., 2008. Non-linear three-dimensional wave groups in finite water depth. Coastal Eng. 55 (12),1052–1061.

Boccotti, P., 2000. Wave mechanics for Ocean Engineering. Elsevier, Amsterdam1–495.

Boccotti, P., 2007a. Comparison between a U-OWC and a conventional OWC.Ocean Eng. 34, 799.

Boccotti, P., 2007b. Caisson breakwaters embodying an OWC with a small opening.Part I: Theory. Ocean Eng. 34, 806.

Boccotti, P., Filianoti, P., Fiamma, V., Arena, F., 2007. Caisson breakwaters embody-ing an OWC with a small opening. Part II: a small scale field experiment. OceanEng. 34, 820.

Cruz, J., 2008. Ocean Wave Energy. Springer 1–431.Curran, R., Gato, L.M.C., 1997. The energy conversion performance of several types

of Wells turbine designs. Proc. Inst. Mech. Eng. 211, 133.Evans, D.V., 1976. A theory for wave-power absorption by oscillating bodies.

J. Fluid Mech. 77, 1.Evans, D.V., 1982. Wave-power absorption by system of oscillating surface

pressure distributions. J. Fluid Mech. 114, 481.Goda, Y., 2000. Random seas and design of maritime structures. World Scientific

Publications.Korde, U.A., 1991. On the control of wave energy devices in multi-frequency

waves. Appl. Ocean Res. 13, 132.Raghunathan, S., 1995. The Wells air turbine for wave energy conversion. Prog.

Aerosp. Sci. 31, 335.Salter, S., 1974. Wave power. Nature 249, 720.Sarmento, A.J., Falc~ao, A.F. de O., 1985. Wave generation by an oscillating surface-

pressure and its application in wave-energy extraction. J. Fluid Mech. 150, 467.Takahashi, S., Nakada, H., Ohneda, H., Shikamori, M., 1992. Wave power conver-

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