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CHAPTER

METHODFNALYSISOFRANDOMAVEEXPERIMENTSWITHEFLECTINGCOASTALTRUCTURES

byPierreGaillard,MichelGauthierndForrestHollySogreah,Grenoble,France

ABSTRACTAmethodofevaluationoftheincidentandreflectedwavespectrainlabo-ratoryexperimentsorfieldinvestigations,basedontheanalysisofwaverecordsobtainedwithathree-gaugearray,ispresented.Resultsofalaboratoryinvestigationwitharubble-moundbreakwaterandwithaspecialtypeofsea-wall(ARCsystem)aregiventoillustratetheapplicationsofthemethod.

INTRODUCTIONRandom waveexperimentsperformedwithreflectingcoastalstructuresrequirecalibrationofthewavegeneratormovementinordertoobtainincidentwavesconformingtospecifiedcharacteristics.Thiscalibra-tionraisesaproblembecauseofmultiplewavereflectionsfromthestructureandthewavepaddle.Atechniqueoftenusedinsuchcases,aswellasinfieldinvestigations,consistsin measuringthewaterlevelvariationinfrontofthestructureandinapplyinga methodofanalysiswhichestimatestheincidentandreflectedwaves.Severalmethodsofanalysisbasedon measurementswithtwowave-gaugeshaveformerlybeenpublishedbyKajima(3],ThorntonandCalhoun[5],GodaandSuzuki[2),Morden,RicheyandChristensen[4],Inthese methods,however,theincidentwavespectraldensitycannotbeestimatedcorrectlyinthevicinityofadiscretenumberofcriticalfrequencies,forwhichthedistancebetweenthetwowave-gaugesisamultipleofhalfthecorres-ponding wave-length.Inthispaperadifferentmethod,basedontheanalysisofwaverecordsobtainedwithathreegaugearrayispresented.Withthismethod,itispossibletoestimatetheincidentandreflectedwavespectraforthewholerangeoffrequenciesofinterest.

2 0 4

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RANDOMWAVEEXPERIMENTS05BASIC ASSUMPTIONSTheincident wavesareconsidereda sastationary,ergodicrandom processwithgaussian properties.Thewatersurfacelevelvariationsrelatedt owavespropagating alongt h e positivex axis are accordingly represented b y :

n.(x,t)= Re {2/0 e 2* " + }/s^Cf)df}1 )with:

224 "fmgtanh( m h ) 2 )fwavefrequencymwave numberhWaterdeptha tresti | >randomphasewith constantprobability densityStwo-sided powerspectraldensity o ftheincident wavesWeshallconsider acoastalstructurei nthewavet a n k ,withi t slongitu-dinalaxisparallelt ot h e wavefronts.Thestructuresectioni sassumeduniform s ot h a tdiffraction effectsare negligible.Thereflectedwavescanthus b erepresentedb y :

n(x,t)= Re 2 /'T(x.f)ei(2rft" mx + >/Toldf }3 )roThetransferfunction(x,f) relatedt ot h e wavereflection,i safunc-tiono ft h elocation o fthe pointo fobservation,and o ffrequency.I nthefollowing,weconsider a sreferencet h eabscissa o ft h eseawardfaceo ft h estructure,wherethetransferfunction takesthe value:-(x.f) -(f) st)T s =(xs, f ) -R( f )e^svi' 4 )

C( f ) reflection coefficiento ft h estructureKj ( f ) phasel a gduet o wavereflection.Providedt h a ttherei s n o wavedampingo r breakingalongt h e wavef l u m e ,thetransferfunction associated withany abscissai srelatedt o b y :sT -( x) - i29Kk 5)Theobjectiveo ft h eanalysisi st oestimatefrom recordso fwaterlevelvariationsthree unknownfunctionso ffrequency:theincident wavepowerspectraldensityT, t h ereflection coefficient( ) andt h ephaseJ -lagj >( f ) duet othisreflection.

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206OASTALENGINEERING-1980Let u s nowconsiderthe relationshipbetweentheincidentwavesandt h epartiallyo rtotallystanding wavesobservedi n fronto ft h estructure.Sincet h eobservedwaterlevelvariationsaresimply alinearsuperposi-tiono ft h eincidentandreflected wavesgiven by( 1 )and( 3 ) ,t h erelationshipbetweenincident wavesandobserved wavesi sdefinedbyt h etransferfunction:

T r y i . =+ T( xk, t ) 6 )andt h erelationshipbetweent h espectrum o ft h eobserved waves S ( f )andt h espectrum o ft h eincidentwavesT(f) i s :

SR( f ) =I(f).TrlKni.TT Kr11 7)Where*tandsfort h ecomplex conjugateo f.Fromequations( 4 )t o( 7 ) ,t h efollowingexpressionsresult:

S( f )=S(f).tl+2Re(T c o s2 92Im(T sin2 9+I T2 ( 8 )I xx\ SSR(f)=S^fMl+2 CR(f).cos( 2 9R+( ( O+ CR2(f))9 )ANALYSIS WITHTWOWAVEGAUGES

Frommeasurementso f waterlevelvariations. , n , att w olocations. ,J kx , ,i ti s possiblet oestimatet h espectra. ( f ) and( f ) , which areKrelated to thenknownunctions S , C and j byquationssimilar to(8) an d (9) Itislsoossible tobtain thecross-spectrum:

S,,f) - C ,,f) - i,,f) f R .,t) e i2lTftdt -. ,wx.,(f) =/* R . , ( t )- _i2lTftjjkv - i j k ' sjkv > < > jkvR j k (T)=E t^j'^' v*t) (10)

Thefollowingrelationshipholdsbetweenthecross-spectrum..(f) andtheincidentwavespectrum(f):

s .fc I(f)Tnj ,1r*kJ1Sjk x(f).(1+TJ (1+T*)'SJk ^f).(1+Tse129j) MV2\) 11)

B yseparatingtherealandimaginarypartsof(11),weobtaintheco-spectrumandquad-spectrum:

CJk(f)=Sxf){(l+|Tg).cos(e.-efc)+2Re(Tg)cos(9.+9)-2Im(T)sin(9.+9, ) }s j

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RANDOMWAVEEXPERIMENTS 207

Q .k( f ) = Sl(f ) ( 1-|TJ ). sin( 9 .- 9kForsimplicity,weshallwritehereafter:

(13)

j k m(Xj-x^ (14)Withtheexpressionsof.(f), , ( f )ivenby( 8 )andtheexpressionsofthecoandquad-spectraJgivenBy(12)(13),wehaveasetoffourlinearequationswiththreeunknownfunctionsoffrequencye(T) ,

2Im(T) and I T A necessaryandsufficientconditionforhavinganonzerosolutionofthissetofequationsis:

=0 (15)

2cos 29.J -2 sin9 YV1 2os 29,k -2in9.k W2cos (9.+9J k -2in (9.+9,J k) cos9Jk

-sin 0jkjkVsrslnV

Thisconditiongivesthefollowingexpressionoftheincidentwavespec-trum:

sijkfs . + s .J k 2(C,c o s 9 . ,jk jk Q . ,in9 . , )Jkk

4s in. . Jk (16)Th esubscriptksaddedtoindicatethatthisisanestimateof( f )derivedfromthewaverecordsatabscissas. and.Thespectraoftheincidentandreflectedwavesarerelatedby:

SR(f) CR( f ) .sx(f) | T s | s:(f) (17)From(16)and(17)wegetthefollowingexpressionofthereflectedwavespectrum:

SRjkf) S.+S,1 k 2cikosVQjkln4sinjk (18)Expressions(16)and(18)aresimilartothoseusedbyKajima( 3 ) .Differ-encescomeonlyfromtheorientationofthepositive xisinoppositedirections.Fromtheprecedingsetofequationswecanalsoderivethefollowingexpressionforthetransferfunction:

Tsjk-Se-219k-Se-219J+C

4ST(f).sin.I Jk(19)

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208OASTALENGINEERING-1980whichcanalsobewrittenas:

Se 2i9kSe 219J e WTSjkf) h20)sjV2

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RANDOMWAVEEXPERIMENTS09ANALYSISWITHTHREEWAVEGAUGESFromsimultaneousmeasurementsofwaterlevelvariationswithathree-gaugearray,ninefunctionsoffrequencyarederivedi.e.thethreespec-tra.,S_ ,Sorrespondingtotheobservedpartiallystanding wavesatthegaugelocations,thethreeco-spectra,,C1,C ndquad-spectra .,Qo,Qonssociatedwiththedifferentcouplesofgauges.Witheachofthesefunctions,isassociatedanequationsuchas(8),(12)or(13).Wethushaveasetofnineequationsforthreeunknownfunctionsonly.Sincetherearemoredataavailablethanunknownparametersmanydif-ferentformulaecanbederivedforevaluatingtheincidentwavespectrumandthetransferfunctionrepresentingthe wavereflectionfromthestruc-ture.Manyoftheseformulaeareinappropriatebecausetheycannotbeappliedforcriticalfrequenciesassociatedwithsomeorallofthegaugeintervals.Atheoreticalanalysisdevelopedin[ 1 ]ledustothefollowingformulaeforevaluatingtheincidentandreflectedwavespectra:

V) " 2 1 S 2 1 (f) + 3 1 S I 3 1 (f) + 3 2 )S R (f) 2 1 ) + " 3 1 ) + 3 2 S R 3 2 (f)

(24)

2 1+ 3 1+a3 2 jk 25)S , and S , areestimatesoftheincidentandreflectedwavespectraIjk Rjkobtainedfrom(16)and(18)forthedifferentpossibleassociationsofgaugesbypairs.The weightingfactors. , arepositivefunctionsoffrequency,definedJkinsuchawaythat(24)bevalidforallfrequencies,ofinterest,includ-ingforthecriticalvaluesassociatedwiththedifferentgaugeintervals.F orthistobetrue,thegaugelocationsmustbechosensoastoavoidanycoincidenceofthecriticalfrequenciesassociatedwiththedifferentgaugespacings;otherwise,effectssimilartothoseobservedwithatwo-gaugearraywouldbeencounteredforthesefrequencies.Forcriticalfrequencies19, S cannotbecomputedfrom(16)asshownintheprecedingsection,so01 shouldvanishforthisfrequency.Byananalysisofequations(8),(12),(13),itcanbeshownthatsolutionsgiven by(16)for,_, S andby(18)forS. . aretheoreticallyequivalentforthesecriticalfrequencies.However,sinceslightdifferencesmayresultfrombackgroundnoiseST(f) andS( f ) ar eestimatedby:1

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210OASTALENGINEERING-1980Sl fcl2) -0.5SI31+SI32) (26)SR(fcl2) -0 .5SR31+SR32)

(29)

Similarformulaear eusedforthecriticalfrequencies f,.,rr9sso-ciatedwiththeothergaugeintervals.The weightingfunctionsar ethussubjecttotheconditions:

ajk(f)-0a ( f )-1/ 2 for-f... 27)a (f)-1/ 2JlC

Weusethefollowingfunctions,whichsatisfyconditions(25)and(27):nQajk(f ) =1 jk=1 ,2 ,3...S^n) 28)

(n) 2n 2nr t 2nr tS^ =in +sin3,+sin32Insertionofthesefunctionsin(24),for=1and2 ,leadsto:

Sl(f) Sn)-Cn)Qn)

1 x+ 2+ 3C1) - C21cos21+31cos31+32os3230)Q1) - Q21sin21+31sin31+32in32 (2)S.sCSj+SjJsin21+0.5(S1+S3)sin31+o.5 S2+S3)sin32(2)CC2lCOs921sin62 1+C3 1COSe31Sln93 1+C3 2COSe3 2Sln93 2 (31)(2) 33 QQ21sin92 1+Q3 1Sln63 1+Q3 2sln93 2Thetransferfunctionassociatedwiththewavereflectiononthecoastalstructureisestimatedby meansof:T( f ).21sln292 1+N31sln2e31+N3 2sln293 2 32)

D21sin2921+D31sin2e31+D32sin2932with:

D., =S. +, - 2 C. cos,, -., sin.,jk jkk xjkk'

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RANDOM WAV EEXPERIMENTSN . , and. , arethetermsinvolvedin(20)whenestimatingthetransferjk jkfunctionfromeachpairofgauges.Thereflectioncoefficientofthestructurecanbeestimatedbytwopos-siblemethods:

. thefirstisbytheuseof(17)withtheexpressionsof STndgivenby(24),i.e.by(29)(30)(31)forthetwolowestordersofthe weightingfunctions.thesecondisbytheuseof(4 )withthetransferfunctiongivenby(32)and(33).

Inthefollowing werefertothesetwoestimatesasR ( f ) an d1( f ) respectively.

APPLICATIONTOLABORATORYEXPERIMENTSExamplesofapplicationofthepresentmethodtorandom-waveexperimentsconductedatSogreahfslaboratorywillnow begiven.

EXPERIMENTALSET-UPTheexperimentswereperformedinawavetankof39mlength,1.40 mdepthand0. 6 0mwidth,withtheset-upindicatedinfigure1 .Thebottomprofileconformstoanactualsituationfor whichbreakwaterstabilitytestswererequired.Twokindsofstructureswereconsideredatascaleof1/50:a )rubblemoundbreakwaterwithatwolayertetrapodcoveringona

seawardslopeof4/3.b)verticalsea-wallwithananti-reflectionchamber(ARCsystem),

asdescribedin( 6 )Thoughathree-gaugearrayisgenerallysufficientforapplyingthepresentmethod,afour-gaugearraywasusedintheseparticularexperi-ments,inordertocomparetheresultsprovidedbythefourdifferentcombinationsofgaugesbytriplets.Waterlevelvariationsweremeasuredwithcapacitance-typewavegauges.Figure showsoneofthewave-gaugearrangementsused.Inthiscase,gauges1and4arelocatedrespectivelyatthethirdanti-nodeandthirdnodeofthepartiallystandingwavescorrespondingtothespectralpeakfrequency.Dataretrievalanddigitizationofwaterlevelmeasurementswereperfor-medonaGeneralAutomationGA220 minicomputerof1 6Kwordscapacity.AFortrancomputercodebasedonthepreviouslydescribedmethod,isimplementedonthiscomputerforcurrentexperiments.

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212 COASTALENGINEERING-1980

00Q u'D

Fig.1XPERIMENTALET-UP

Model BREAKWATERITHETRAPODS Modti l SEA-WALLWITHAR CYSTEM

S,If)SoO)

INCIDENTWAVESREFLECTEDWAVES

0.05.10.15 .20Fig.2Incidentandreflectedwavespectraobtainedwithmodel

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RANDOM WAVEEXPERIMENTS13Forprocessingthedataofthefour-gaugearrayandforcomparingtheresultsofthefourdifferenttriplets,aspecialcodewasimplementedonanITELAS6computer.Waverecordswereanalysedoveraperiodof16 0s ,withatimestepof0 .1s.Autoandcross-covariancefunctionswerecomputedwithamaximumlagof _ +10s .Thisanalysisprovidedvaluesofthe16followingfunc-tionswithafrequencyintervalof0. 0 2Hzatmodelscale,i.e.of0 . 0 0 2 8Hzatprototypescale:

. thespectraofwavesobservedatthegaugelocations S.,S S S

. thesixco-spectra,C^,C^,C^,C^,C

. thesixquad-spectra Q21>Q32Q ,Q32,Q^, Thenumberofdegreesoffreedomassociatedwiththeprecedingparametersisapproximately40,withaHammingspectralwindow.FromtheChi-squarelawitisconcludedthatthe8 0 %confidenceintervalliesbetween73%and130%oftheestimatedpowerspectraldensity.

INCIDENTANDREF L ECT EDWAVESPECTRAFigure2showstheincidentandreflectedwavespectraderivedfromthewaverecordsofgauges2 ,3 ,4offigure1 ,withthefirsttypeofstruc-ture,byapplicationofformulae(29)and(31).Figure3givesasimultaneousplotoftheincidentwavespectraobtainedbyanalysisofthefourdifferentcombinationsofwavegauges( 0standsforgauges1 ,2 ,3 ;vforgauges1 ,2 ,4 ;xforgauges1 ,3 ,4andXforgauges2 ,3 ,4).Asthisgraphresultsfromaprinteroutput,coincidentvaluesofthepowerspectraldensitiesareplottedbyasinglesymbol.Thefourestimatesoftheincidentwavespectrumareinexcellentagree-ment.Figure4givesasimultaneousploto ftheincidentwavespectraobtainedbyanalysisofthefourgaugeswiththetwolowestordersofweigthingfunctions(28)( *standsfor=lnd or=2).Nosignificantdifferenceappearsherebetweenthetwoproceduresofestimationoftheincidentwavespectrum.Thesameconclusionsweredrawnfromtestswiththesecondtypeofstruc-ture.Withthistechnique,hegaugelocationsshouldbechosensoastoavoidtooproximatevaluesfthecriticalfrequenciesassociatedwiththedifferentintervals.igure5referstoaparticularwavegaugearrange-ment,1 ,2 ,4 ,whereriticalfrequenciesassociatedwiththegaugeintervalsareverynearlycoincident,sothat definedby(28),adropstoaverylow valueforfrequenciesnear>1.55Hztmodelscaleand f=0.2 2Hztprototypescale.Thissituationleadstoundesirableeffectsshowninfigures6and7 .

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214OASTALENGINEERING-1980GAUGES12-3 1-3-42-3-4

0-05.10.15.20.2E Fig.3Incidentwavespectrumresultingf rom 4different3-augerrayswithn =2WEIGHTINGFUNCTIONSNo N

.-rrsEs:.:::::::::;:::Fig.4Incidentwavespectrum obtainedwithweightingfunct ionso fordern= 2

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RANDOM WAV EEXPERIMENTS5si n o21sin2 t4 1sin2 042SI1 Iba,.12)

0.05.10.15.20Fig.5Weightingfunct ionswithnearlycoincidentcriticalfrequencies 1231-2-4134

6.10.16.20Fig.6Compar isonofresultsinthe si tuat ionofFig.5,forg a u g e s1-2-4

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216 COASTALENGINEERING-1980

WEIGHTINGFUNCTIONSN

0 N

0.20 Fig.7Effectofweightingparameterninthesituationoffig.5

#s

0.05.10.15 Fig.8Reflectioncoefficientsa ndphasela gformodel

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RANDOMWAV EEXPERIMENTS17Infigure6 ,itisseenthattheincidentwavespectrumestimatedfromgauges1 ,2 ,4exhibitsasmallhumpinthevicinityofthepreviouslymentionedfrequency,whilethisisnotobservedwiththespectraestimat-edfromthethreeothergaugeassociations.Figure7isagraphsimilartofigure4.Itshowsthattheweightingfunctionsoforder givesahumpofgreatermagnitudethanthoseoforder2 ,inthefrequencyintervalwherecriticalfrequenciesar enearlycoincident,whileresultsarealmostthesameforotherfrequencies.Becauseofthis,preferenceshouldbegivento(31)forevaluatingtheincidentandreflectedwavespectra.Anotherpointshouldbestressedaboutthepresentationofwavespectraderivedthroughthisanalysis.Thefrequency=0sacriticalfrequencywhateverthechoiceofthegaugeintervals,sincein9 . , vanishesforthisvalue.F orthisreason,asharpriseintheabsolutevalueofT(f)andCf) isobservedatverylowfrequencies,withathree-gaugearrayRaswellaswithatwo-gaugearray.Thiseffectwasillustratedin( 1 )withnumericallysimulatedrandom waves,forwhichitwasclearthatnoenergywasactuallypresentinthelowfrequencyrange.Thisspuriouseffectaffectsafrequencyintervalequaltothewidthofthespecificspectralwindowusedforsmoothing wavespectra.Acut-offfrequencyequaltothisbandwidthisintroducedinthecomputations.

REFLECTIONCOEFFICIENTSFigures8and9showthevariation withfrequencyofthereflectioncoef-ficientsCf), C '( f ) forthetwostructurespreviously mentioned,

Rtogetherwiththephaselag j >(f).C( f ) isderivedherebymeansoftheweightingfunctionsoforder2 .Itwasshownin(lj,withnumericallysimulatedwaterwaves,thatagoodestimateofthereflectioncoefficientscannotbeobtainedforfrequencieswherethepowerspectraldensityisverylow.Athresholdvaluedependentonthepeakspectraldensity,wasaccordinglychosenforselectingthefrequencyrangeoverwhichtheseparametersarecomputed.Resultsarepresentedhereforathresholdvalueof10%.Thetwoproceduresdescribedforevaluatingthereflectioncoefficientofthestructureversusfrequency,thoughtheoreticallyequivalent,giveslightlydifferentestimatesofthisparameter.Bothproceduresgivethesamegeneraltrendwhichistypicalofthestructuresconsidered.Therubble-moundbreakwaterexhibitsaprogressivedecreaseinreflectionasfrequencyincreases,duetohigherdissipationaswavesimpingeontheartificialblocks.TheARCsystemontheotherhandexhibitsaU-shapedcurve,withminimumreflectionforafrequencydependentonthedimensionsoftheanti-reflectionchamber,whichactsasaresonator.

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218 COASTALENGINEERING-1980

=RIx CRf)o ^ s

0.05.10.15Fig.9Reflectioncoefficientsa ndphasela gformode l2| C Rfl WEIGHTINGFUNCTIONS

No N

s

f0.05.10.15Fig.10ReflectioncoefficientsCp(f)esultingfromdifferent gaugea r r a yswithn= &2

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RANDOMWAVEEXPERIMENTS19Asshownonfigures8and9( f ) hasgenerallyamoreregularvariationasafunctionoffrequencythanC(f). Thereasonforthisbehaviourhasnotyetbeenfullyunderstood.Fromourexperiments,thedifferencesbetween( f ) and C '( f ) , didnotseemtobeconnectedwithnon-lineareffects,astheydidnotincreasewiththesignificantwaveheightofthewavesconsidered.Figure10comparesthereflectioncoefficients_(f), associatedwithKthefirstandsecondorderweightingfunctions(28):*standsfor=l,0 for= 2. Asthisgraphresultsfrom aprinteroutput,coincidentvaluesofR(f) bybothmethodsareplottedbyasinglesymbol.Resultsshownherecomefromtheanalysisofthefourcombinationsofgauges.Agenerallygoodagreementisobservedbetweenresultsofdifferenttriplets,andbetweenthetwomethodsofestimationsof cR(f) frfrequencieswherewaveenergyishigh.Thescatterofexperimentalpointsincreasesontheboundariesofthewavespectrum.

CONCLUSIONInconclusion,themethodofanalysisjustdescribedhasproventobeavaluabletoolforestimatingincidentandreflectedwavespectrainins-tanceswherewavereflectiononstructureshastobeaccountedfor.Examplesofapplicationofthemethodtoflumeexperimentshavebeengivenhere.Themethodshouldalsobeapplicabletofieldinvestigations,pro-videdthattheincomingwavesaredirectednormallytothereflectingstructure.Thenumericalandexperimentaltestshaveshownthattheincidentwavespectrumisestimatedaccurately,providedthatthegaugespacingsareproperlyselected,soastoavoidtoosmallafrequencyintervalbetweencriticalfrequencies.Theincidentwavespectrumispracticallyinsen-sitivetotheorderoftheweightingfunctionsused,exceptwhencriticalfrequenciesarenearlycoincident.Themethodalsoprovestobeaninterestingmeansofstudyingthereflec-tioncoefficient(andtheassociatedphaselag)ofcoastalstructuresasafunctionoffrequency.I tenablesabetterassessmentoftheefficiencyofwaveabsorbingdevices,suchastheARCsystem.Twoproceduresofevaluationofthereflectioncoefficienthavebeeninvestigated,onedirectlyconnectedwiththeincidentandreflectedwavespectraldensitiesbyequations(17)and(29),theotherderivedfromthetheoreticaltransferfunction(32).Thefirstoneisconsideredpreferablesinceitshoweda moreregularvariationofthereflectioncoefficientversusfrequency.

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220OASTALENGINEERING-1980REFERENCES

( 1 ) P.GaillardMethoded'analysedelahouleirreguliereproduiteencanalenpresenced'unouvragereflechissant-SOGREAH,NT1834,Decembre1976

( 2 ) Y.Goda&Y.SuzukiEstimationofincidentandreflectedwavesinrandom waveexperiments.Proc.15thCoastalEng.Conf.Honolulu,Hawaii,July1976,p.8 2 8 - 8 4 5

( 3 ) R .KajimaEstimationofanincidentwavespectrumundertheinfluenceofreflection-Proc.13thCongressofIAHR,KyotoJapan1969,Vol.5 .1(Seminars),p .2 8 5 - 2 8 8

( 4 ) D.B.Morden,E.P.Richey& D.R.ChristensenDecompositionofcoexistingrandomwaveenergy.Proc.15thCoastalEng.Conf.Honolulu,Hawaii,July1976,p.8 4 6 - 8 6 5

( 5 ) E.B.Thornton&R.J.CalhounSpectralresolutionofbreakwaterreflectedwaves.J.Waterways,harborsandcoastalEng.Proc.ASCEWW4November1972,p.4 4 3 -4 6 0

( 6 ) L.Tourmen,J.P.Montaz,L.DoubletOnnouveaudispositifamortissantlareflexiondep vaguessurlesparoisverticales lachambreanti-reflexion(ARC),17 eCongresAIRHBaden-Baden,RFA,aout1977,SujetC6 ,p.41-48.