Transformation of statistical and spectral wave periods crossing a smooth low-crested structure

Transcription

1 ransformation of statistical and spectral wave periods crossing a smooth low-crested structure doi: /oc OCEANOLOGIA, 54(1), pp C Copyright by Polish Academy of Sciences, Institute of Oceanology, KEYWORDS Smooth submerged breakwater Wave period transformation Statistical wave parameters Spectral wave parameters Smooth emerged breakwater DaliborCarevic Goran Loncar Marko Prsic Faculty of Civil Engineering, University of Zagreb, Kaciceva 26, Zagreb 10000, Croatia; car@grad.hr correspondingauthor Received 6 June 2011, revised 28 November 2011, accepted 12 December Abstract We carried out experimental studies of a smooth submerged breakwater in a wave channel in order to study such a structure impacts on the changes of statistically and spectrally defined representative wave periods as waves cross it. We discuss the impact of relative submersion, i.e. the relationship between the breakwater crown submersionandtheincomingsignificantwavelength R c /L s i,ontherepresentative wave periods. he mean periods, estimated using statistical and spectral methods, were compared in front of and behind the breakwater: the two periods turned out to be identical. Based on the measurements of the spectral mean wave periods in front of and behind the breakwater, an empirical model is derived for estimating the reduction in mean spectral period for submerged and emerged smooth breakwaters. he complete text of the paper is available at

2 40 D. Carevic, G. Loncar, M. Prsic 1. Introduction It is usual to use the characteristic periods and heights of incoming irregular waves for calculating run up, overtopping, morphological changes and reflection from perforated seawalls. If a coastal structure is defended by a smooth submerged breakwater, it is important to calculate the modified wave parameters behind it. When waves cross a breakwater, wave breaking and nonlinear interactions occur between the components of wave spectra. hese interactions cause a transition of wave energy from primary harmonics to higher harmonics of the wave spectra. he amount of energy transferred depends on the incoming wave parameters, breakwater geometry and water depth. Beji& Battjes(1993) observed high frequency wave energy amplifications as waves propagate over a submerged bar in a laboratory experiment. hey found that the bound harmonics were amplified during shoaling and released in the deeper water region after the bar crest. Wave breaking itself is a secondary effect in this process, dissipating the overall wave energy without significantly changing its relative spectral distribution. Generally speaking, knowledge of the impact of breakwater geometry and incoming wave parameters on wave spectrum deformation is insufficient. hetransferofenergytohigherharmonicsofthewavespectraleadsto a transformation in statistical and spectral wave periods. he present study addresses the problem of the impact of breakwater geometry and incident wave parameters on wave period transformation. Breakwater geometry refers to the submersion of the breakwater crown, the wave parameters to wave height and length. he general conclusion of the works of Goda et al.(1974), animoto etal.(1987)andraichlenetal.(1992)isthatwhenwavescrossalowcrowned breakwater, mean spectral wave periods are reduced by 60% in relationtoincomingmeanwaveperiods. VanderMeeretal. (2000) conducted tests on smooth emerged breakwaters and found that the transmitted mean spectral wave period was reduced by up to 40% compared totheincidentone. heyalsoconcludedthatthemeanandpeakwave periods were reduced by the increase in the wave height transmission coefficients. Briganti et al. (2003) studied the impact of wave height transmission coefficients on the transfer of energy from lower to higher harmonics. hey established that the deformation of the wave spectra when waves cross breakwaters differs for low crested structures with smooth surfaces and with rubble mound armour. Wang et al. (2007) studied theimpactoftheangleofincomingwavesonthetransformationofthe mean spectral centroid period. ests were conducted for breakwaters with submergedandemergedcrowns,aswellaswiththecrownlevelwiththe

3 ransformation of statistical and spectral wave periods crossing water surface. It was established that in the case of approximately normal incident waves approaching the breakwater, the mean centroid periods were reducedbyupto25%inrelationtotheincomingperiod. 2. Material and methods Laboratory tests were conducted with a piston wave generator using the AWACS system(anti-reflecting system). A dissipation chamber was situated at the end of the channel, which gives a maximum reflection coefficient0.2forthelongestwavelengthscitedinable1intheempty able 1. Wave parameters used in laboratory testing and results of measurements of transmitted wave parameters, standard JONSWAP spectrum (g = 3.3, s 1 =0.07, s 2 =0.09), K r measuredreflectioncoefficientfromthe submerged breakwater measured incident R c2 = 0.055m measured transmitted est H m0 i 0.2 i p i H m0 t 0.2 t p t K r [m] [s] [s] [m] [s] [s] [1] measured incident R c1 = 0.10m measured transmitted est H m0 i 0.2 i p i H m0 t 0.2 t p t K r [m] [s] [s] [m] [s] [s] [1]

4 42 D. Carevic, G. Loncar, M. Prsic channel(without a breakwater). he wave channel width was 1 m, the height1.1m,andthedepthsofwaterinthechannelwered 1 =0.44mand d 2 =0.4m.hesubmergedbreakwatermodelwasmadeofwood,thecrest widthbeingb =0.16mandtheslope1:2(Figure1). hemeasurements wereperformedfortwosubmersionsofthewavecrown(r c1 = 0.055mand R c2 = 0.101m),achievedbychangingthedepthsofwaterinthechannelto d 1 =0.4mandd 2 =0.446m.Measurementswereperformedinconformity withable1foreachdepth,yieldingatotalof18measurements.he durationofanexperimentwas 5min.,whichisequivalenttoapprox. three hundred waves per experiment, according to the recommendations by Journée& Massie(2001). G1 G2G3 incident wave parameters transmitted wave parameters G4 G5G6 H m 0-i, max-i, 1/10-i, s-i, m-i, wave gauges 0.2-i, p-i R = m and c 1 R = m c 2 H m 0-t, max-t, 1/10-t, s-t, m-t, 0.2-t, p-t wave gauges 0.5 m 1:2 1:2 d = 0.4 m and d 1 2 = m Figure 1. Details of the wave flume and measured incident and transmitted wave parameters Capacitive gauges G1 G6 were used for measuring surface elevation. he measured data were processed according to spectral and statistical(zero up-crossing) methods. According to the spectral principle, the spectral wave parameterswereestablishedash m0, 0.2 and p (seethelistofsymbolsat the end of the paper). he incident wave parameters were determined by separating the incoming and reflected spectra on gauges G1 G3, and the transmitted wave parameters by separation on gauges G4 G6. he Zelt & Skjelbreia(1992) method was used for separating the incident spectrum from the reflected one. hestatisticalwaveparametersh max, H 1/10, H s, H m, max, 1/10, s and m weredefinedbyzeroup-crossingforincidentandtransmitted wave time series(see list of symbols). Incident and transmitted wave time series were determined by inverting FF of the incident and transmitted spectrum defined by the procedure described in the previous paragraph. o avoid the influence of wave reflection from the breakwater and dissipation chamber,thepositionsofthegaugeswerechosentobeaminimumofone wavelength away from the structure, thereby preventing spatial variation of the statistical parameters(goda 2000).

5 ransformation of statistical and spectral wave periods crossing Results and discussion 3.1. Reduction in wave periods as a result of wave transmission over a breakwater he process of non-linear interaction can be explained from the point ofviewofphysicsinthefollowingway: whenalongerwavefroman irregular wave train crosses the breakwater, waves of shorter periods are superimposed on its wave profile, thereby reducing the statistical wave train periods. he phenomenon is evident in waves of considerable length but is lessnoticeableinshorterwaves. Figure2presentsanexampleofatime series(for est 8, able 1) with a considerable incident mean wave period m. hisisanevidentoccurrenceofsuperimposedshorterwaves,which generate a larger number of waves behind the breakwater(calculated by the zero up-crossing method). Figure 2. Incident and transmitted wave time series for raw data recorded on gaugeg1(inc)andgaugeg4(trans),(est8,able1) Figure3showsanexampleofatimeseries(forest2,able1)with ashorterincidentmeanperiod m. hereisnosignificantoccurrenceof superimposed shorter waves. he phenomenon is therefore more pronounced in wave trains with asmallerr c /L s i ratio. hereductioninthestatisticalwaveperiods ( 1/10 t, s t and m t )ofthewavetrain,behindthebreakwater,thus dependsontherelativesubmersionr c /L s i (Figure4). hegreatestreductionoccursat m,becauseitcoversallthewavesfrom therecord,includingthenewlyformedshortperiodwaves.significantly, s andonetenth 1/10 waveperiodsindicateasmallerreductioninrelation tothereductionofthemeanperiod. hemaximumperiodisrelatedto thewaveofthegreatestwaveheightinthewavetrain.asthisvalueisnot subject to statistical averaging, it causes extreme oscillations of relations max t / max i,andonlylimitedconclusionscanbedrawn.

6 44 D. Carevic, G. Loncar, M. Prsic Figure 3. Incident and transmitted wave time series for raw data recorded on gaugeg1(inc)andgaugeg4(trans),(est2,able1) 1.0 max-t / max-i, 1/10-t / 1/10-i, s-t / s-i, m-t / m-i max-t 1/10-t s-t / / / / s-i m-t m-i, max-i 1/10-i R c / Ls-i Figure 4. Reduction in statistical wave periods crossing a smooth submerged breakwater; max i, 1/10 i, s i, m i incidentwaveperiods; max t, 1/10 t, s t, m t transmittedwaveperiods In general, representative statistical wave periods are correlated, whereas theempiricalinterrelationsweredefinedbygoda(1974,2008)as max 1/10 s m. Consideringthatstatisticalperiodsdependon the form of the wave spectrum(goda 2008), and that deformations of the wave spectrum occur when waves cross the breakwater, the question arises in what way the above relations between statistically representative periods change when the waves cross the breakwater. Figure 5-left shows

7 ransformation of statistical and spectral wave periods crossing s-i, 1/10-i, max-i [s] m-i [s] y = 1.2x y = 1.1x y = x s-i 1/10-i max-i s-t, 1/10-t, max-t [s] m-t [s] y = 1.5x y = 1.1x y = x s-t 1/10-t max-t Figure 5.Ratioofstatisticalwaveperiods max i, 1/10 i and s i infrontof (left)and max t, 1/10 t and s t behind(right)asmoothsubmergedbreakwater withmeanstatisticalperiods m i and m t (ordinate-y,abscissa-x) therelationsbetweenthestatisticalperiods max, 1/10 and s andthe meanperiod m infrontofthebreakwater. Allthevaluesarepositioned between lines y=1.1x and y=1.2x, which corresponds approximately to the above empirical interrelations. Figure 5-right illustrates the relations of the same statistical parameters behind the breakwater. here is a change in these relations when in higher periods the relationship tends to max 1/10 s 1.5 m. hishappensbecausewhenthewavescross thebreakwater,amoresignificantreductioninthemeanperiod m occurs (Figure4)inrelationtotheotherperiods max, 1/10 and s. m is more significantly reduced by the appearance of high frequency harmonics (short waves), which are not so important from the engineering point of view because of their small height. So one should be careful when applying themeanperiod m toengineeringpurposesinthecaseofsubmerged structures. As a consequence of wave spectrum deformation, i.e. wave nonlinearity effects in shallow water, an error could occur when estimating the mean spectralperiod, 0.2 (seelistofsymbols),whichmaybeunderestimated byasmuchas70%ofthestatisticalmeanperiod m (Longuet-Higgins 1983). Since wave spectra are deformed when waves cross a breakwater, the question arises whether a similar mistake might be expected in the estimationofthetransmittedmeanspectralperiod 0.2 t. Figure 6 illustrates the ratio of mean statistical and spectral wave periods for incident and transmitted waves: mean spectral 0.2 i is comparedwith m i forincidentwaves,and 0.2 t iscomparedwith m t for transmitted waves. It can be concluded that wave spectra deformation does not influence thecalculationaccuracyofspectralmeanperiods 0.2 t.

8 46 D. Carevic, G. Loncar, M. Prsic statistical: m-i, m-t [s] spectral: 0.2-i, 0.2-t [s] incident transmitted y = x y = 1.1x y = 0.9x Figure 6.Ratioofspectral( 0.2 )andstatistical( m )meanwaveperiodsfor incident and transmitted waves at a smooth submerged breakwater It has already been mentioned that in the process of wave transmission over a breakwater, the wave energy is transmitted to higher frequencies, alongwiththeincreaseinthetermm 2 (secondmoment), resultingin areductioninthemeanspectralwaveperiodoftransmittedwaves 0.2 = m0 /m 2 andthereductionofthe 0.2 t / 0.2 i ratiosinthefunctionof relativesubmersionr c /L 0.2 i (Figure7).hedatafromVanderMeeretal. 0.2-t / 0.2-i p-t / p-i 0.2-meas, (this meas.), (Van der Meer et al. 2000) 0.2-meas p-meas p-meas, (this meas.), (Van der Meer et al. 2000) incident: 0.2-i p-i ,2 -R c R / L c 0.2-i +R c transmitted: 0.2-t p-t Figure 7.Dependenceofparameters 0.2 t / 0.2 i and p t / p i onrelative submersionr c /L 0.2 i, forasmoothbreakwaterwithsubmergedcrown(from thispaper)andanemergedcrownaspervandermeer s(vandermeeretal. 2000)measurementswithcrownwidthB b =0.13and0.3m;H m0 = m, wavesteepnesss op =0.03, waterdepthd= m, breakwaterslope1:4, R c /H m0 =0 1.0

9 ransformation of statistical and spectral wave periods crossing (2000) for smooth emerged breakwaters with a similar breakwater geometry andsimilarwaveparametersasinthispaperareusedforcomparison. In suchaway,thereductionofthemeanspectralwaveperiods 0.2 forawider rangeofrelativesubmersionr c /L 0.2 i,namelyfrom 0.15to 0.06,is obtained. Itcanbeseenintheabovefigurethattheratio 0.2 t / 0.2 i tends toavalueof 0.68whentherelativesubmersionR c /L 0.2 i tendstozero, takenfromeitherthepositiveorthenegativeside. heresultsofvan der Meer s measurements for the emerged breakwater are closer to this value,sincethemeasurementsweremadeforthelowerparameterr c /L 0.2 i. heobviousdependenceofparameter 0.2 t / 0.2 i ontherelativesubmersion means that the transfer of energy from lower to higher frequencies in a non-linear interaction process depends on the relative submersion (R c /L 0.2 i ). heimpactofrelativesubmersionr c /L 0.2 i onpeakperiod p for smooth breakwaters with submerged and emerged crowns is also presented. he investigations conducted so far suggest that the transmitted peak periodisveryclosetotheincidentperiod(vandermeeretal.2000, 2005). hese conclusions have been confirmed here, namely, that parameter p t / p i forasubmergedbreakwater(figure8,left)rangesfrom1.0to 1.15.Withregardtoemergedbreakwaters(Figure8,right), p t / p i was foundtodependontherelativesubmersionr c /L 0.2 i. hetransmitted peakperiodincreasedinrelationtotheincomingperiodby 35%forthe shortest waves. S (f) [m s] 2 η test f [Hz] meas-inc meas-trans teor-inc-jonswap S (f) [m s] 2 η test 6 f [Hz] meas-inc meas-trans teor-inc-jonswap Figure 8. Comparison of the measured incident, theoretical incident(jonswap) andtransmittedspectrafortests17and6 he figures above present measured incident and transmitted spectra. he theoretical incident JONSWAP spectrum is also shown for comparison.

10 48 D. Carevic, G. Loncar, M. Prsic he agreement between measured and theoretical incident spectra is satisfactory. he same conclusion can be drawn for the other tests from able 1. he area of the transmitted spectra is reduced because wave breaking and the transition of energy to higher frequencies are evident An empirical model for estimating the reduction in the meanspectralperiodafterawavehascrossedasmooth breakwater he equation for reducing the coefficient of the mean spectral wave period(k R 0.2 )afterawavehascrossedasmoothbreakwaterreadsas follows: K R 0.2 = 0.2 t 0.2 i = m0 t /m 2 t m0 t = m0 i /m 2 i m 0 i m2 i m 2 t. (1) he first term in the above equation represents the transmission coefficient of the significant wave height over the breakwater: K Hm0 = H m 0 t = 4 m0 t H m0 i 4 m0 t =. (2) m 0 i m 0 i If equation(2) is inserted in equation(1), the following is obtained: K R 0.2 m2 i =. (3) K Hm0 m 2 t Inpractice,theequationofVanderMeeretal.(2003)isusuallyusedfor estimating K Hm0 : K Hm0 = [ 0.3R c /H m0 i [1 exp( 0.5ξ op )]] (4) withaminimumof0.075andamaximumof0.8(seelistofsymbols).his paperusestherangeoftheaboveequationfrom0.075to1.0.hesecond term in the above equation regulates the impact of wave steepness and breakwaterslopeoverthebreakerparameter ξ op.fortheusualbreakwater slopeof1:2,itisfoundthatequation(4)variesintherangedk Hm0 =0.15, owingtothechangeofwavesteepnessh m0 i /L op i =1/10 1/30.herefore, thevariabilityofthesecondmemberwillbeneglectedandthevalueof0.51, estimatedforthesteepnessh m0 i /L op i =1/20,canbetakeninstead.he influenceofsuchareductiononthefinalaccuracyoftheempiricalmodelis minor; in any case we shall simplify the model. he following equation is obtained: K Hm0 = [ 0.3R c /H m0 i ]. (5)

11 ransformation of statistical and spectral wave periods crossing Coefficient K may be defined from equation(3) and equation(5): K = K R 0.2 [ 0.3R c /H m0 i ]. (6) hecoefficientrepresentstheratio m 2 i /m 2 t andcontainsinformationonhowmuchthewavespectrumareahasbeenreducedasaresult ofthewavescrossingthebreakwater,butalsoonhowmuchenergyhas been transformed from lower to higher harmonics, since the definition of themomentis m 2 = this paper, however. 0 f 2 S η (f)df. heseimpactswillnotbestudiedin Derivation of an empirical model for a submerged breakwater he values of parameter K(eq.(6)) are estimated for every measured K R 0.2,(whichcanbeestimatedforeachtestfromable1).heordered pairs( R c /L 0.2, K)areinsertedinthediagram,sothepointspresentedin Figure 9 are obtained ( ) K = K 0.2 / K H m R / H = 0.5 c m 0-i R / H = 0.8 c m 0-i R / H = 1.0 c m 0-i R / H = 1.6 c m 0-i K = * (R c / L 0.2-i) K = * (R c / L 0.2-i) K = * (R c / L 0.2-i) K = * (R c / L 0.2-i) R c / L0.2-i Figure 9. Data fitting of equation (7), for four categories of parameter R c /H m0 i =0.5,0.8,1.0and1.6,forasubmergedsmoothbreakwater,coefficient B =0.87

12 50 D. Carevic, G. Loncar, M. Prsic hepointsarearrangedaccordingtotheparameterr c /H m0 i,sothat fourdatagroupsareformedforparametervaluesofr c /H m0 i =0.5,0.8, 1.0and1.6.MeasuredvalueswithsmallerparameterR c /H m0 i havelarger valuesofcoefficient K = m 2 i /m 2 t becauseofthesmallerwavetransmissioncoefficients K Hm0.Smallervaluesof K Hm0 meanalargerdifference betweenm 2 i andm 2 t. Allmeasuredvaluesforeachgrouparereduced whenwavelengthsgrowbecauseofincreasing K Hm0.heinfluenceofthe periodcoefficients K 0.2,whicharereducedwithincreasingwavelengths,is minor. Inotherwords,themainreasonwhy Kdecreasesisbecausethe influenceofspectralsurfacereduction(includedin K Hm0 )islargerthan thatofnon-linearinteractions(includedin K 0.2 ).Whenvaluesof Kreach 1, and below 1, this means that non-linear interactions play a significant role. he function in the form of equation (7) was fitted to each data group: K = A ( Rc L 0.2 i ) 2 + B. (7) Itispresumedthatallthemeasureddatainthediagram(Figure9) passthroughthesamepointontheordinate,whichmeansthatthevalue ofthecoefficient Binequation(7)willbethesameforeverydatagroup R c /H m0 i =0.5,0.8,1.0and1.6. hisassumptionisnecessarybecauseofthelackofmeasureddatainthe areaaroundthevaluer c /L 0.2 i =0.heconsequenceofsuchanassumption isthatthefinalmodelisnotreliablenearr c /L 0.2 i =0.hecoefficient B hasbeendeterminedundertheconditionthatwhenl 0.2 i,thefirst termofequation(7)tendsto0,andisobtainedbyequalizingequation(6) with equation(7), that is: B = K R 0.2 [ 0.3R c /H m0 i ]. (8) If Biscalculatedaccordingtoequation(8)forfourvaluesofR c /H m0 i = 0.5, 0.8, 1.0 and 1.6, and for the approximate value K R (Figure7),thenthevalues B =1.03,0.91,0.84and0.69areobtained.For thefinalvalueofcoefficient B,themeanvalueofcalculatedvaluesistaken, whichis B =0.87. hecoefficient Aisobtainedbyfittingthefunction(eq.(7))withthe constantvalueof B =0.87tothedatagroups,aspresentedinFigure9.For thedatagroupsr c /H m0 i =0.5,0.8,1.0and1.6,thecoefficients A =207.4, 61.9, 40.7and8.8areobtained. heorderedpairs(r c /H m0 i, A)are

13 ransformation of statistical and spectral wave periods crossing insertedintothediagramandthecurveintheformasindicatedbelow isfittedtothem: A = A1 exp[b1 (R c /H m0 i )]. (9) Coefficients A1 =1280.6and B1=3.65areobtained. By equalizing equation(6) and equation(7), in which the function of coefficient A (eq. (9)) has been included, with coefficients A1 and B1, the empirical equation for estimating the reduction coefficient of the mean spectral wave period when crossing the submerged breakwater is obtained: [ K R 0.2 = 0.3 R ] c (10) H m0 i [ ( ) 2 Rc exp (3.65 (R c /H m0 i )) ]. L 0.2 i he above equation is valid, provided that the following limitations aretakenintoaccount: maximum K R 0.2 =1; 1.6 R c /H m0 i 0.5; 0.15 R c /L 0.2 i 0.02;0.034 H m0 i /L s i Derivation of an empirical model for an emerged breakwater he empirical model presented below was derived for an emerged smooth breakwater, based on measurements conducted by Van der Meer etal. (2000)inthewavechanneloftheDelftHydraulicscompany. he measurements are given in able 2, and the measured reduction coefficients ofthemeanspectralwaveperiod K R 0.2,whichdependontherelative submersionr c /L 0.2 i areshowninfigure7. Foreachmeasured K R 0.2,thevaluesofparameter Kareestimated accordingtoequation(6). heorderedpairs(r c /L 0.2, K)areinserted into the diagram and the points presented in Figure 10 obtained. he function of the form equation(7) should be fitted to the points, assuming thatthecoefficient A =const,i.e.thatitdoesnotdependontheparameter R c /H m0 i. Inthecase ofanemerged breakwater, thecoefficient B isdefined differently than in the case of a submerged breakwater. In data for the emerged crown, the measured coefficient K R 0.2 is very close to theordinate, withr c /L 0.2 =0.003andparameterR c /H m0 =0.05. B is determinedprovidedthatl 0.2 inequation(7),sothatthefirstterm ofequation(7)tendsto0,andequation(8)isobtained. Byinserting

14 52 D. Carevic, G. Loncar, M. Prsic able2.waveparametersusedformeasurementsinthestudybyvandermeer etal.(2000)andmeasurementoftransmittedwaveparameters;b b crownwidth, water depth d = m, breakwater slope 1:4 B b=0.13 m measured incident measured transmitted est R c H m0 i 0.2 i p i H m0 t 0.2 t p t [m] [m] [s] [s] [m] [s] [s] B b=0.03 m measured incident measured transmitted est R c H m0 i 0.2 i pi H m0 t 0.2 t p t [m] [m] [s] [s] [m] [s] [s] thevaluesr c /H m0 i =0.05andmeasuredvalues K R (Figure7) in equation (8), B 1.35 is obtained. Equation (7) with coefficient B =1.35isfittedtothepointsinFigure10sothatcoefficient A=810.6is obtained. By equalizing equation(6) and equation(7), into which the coefficients A=810.6and B =1.35areinserted,theempiricalequationforestimating the reduction coefficient of the mean spectral wave period when crossing the emerged breakwater is obtained: [ K R 0.2 = 0.3 R ] [ ( ) 2 c Rc ]. (11) H m0 i L 0.2 i he above equation is valid provided that the following limitations are takenintoaccount:maximum K R 0.2 =1;0.05 R c /H m0 i 1.1;0.003 R c /L 0.2 i 0.06;0.043 H m0 i /L s i 0.053,thefirsttermofequation(11) canbeaminimum [ 0.3R c /H m0 i ] =0.075.

15 ransformation of statistical and spectral wave periods crossing Van der Meer et al. (2000), B b = 0.13 m Van der Meer et al. (2000), B b = 0.30 m K = (R 0.2-i 2 c / L ) / K Hm0 ) 3.5 K = K 0.2 ) R c / L0.2-i Figure 10. Data fitting of equation(7), for an emerged smooth breakwater, with coefficient AindependentofparameterR c /H m0 andcoefficient B = Verification of the empirical model for submerged and emerged breakwaters Figure 11 shows the verification of the empirical models for estimating the reduction coefficients of the mean spectral periods when waves cross the submerged and emerged breakwaters(eq.(10) and eq.(11)). he measured coefficient (K R 0.2 ) meas and the calculated coefficient (K R 0 ) calc are comparedinthefigurebymeansofequation(10)andequation(11). he agreement of measurements with the empirical model results for the submergedbreakwaterisgoodwhenthemajorityofdataareintheregion ofdk R 0.2 = ±0.05. heempiricalmodelforanemergedbreakwateris formed on the basis of fewer measurements. herefore, there is weaker agreement between the estimated and measured values than for the results of the submerged breakwater. Bothequations(eq.(10)andeq.(11))werederivedonthebasisof asmallnumberofmeasureddata: thisisthemajorweaknessofthese equations. Nevertheless, we have presented a new approach for calculating thereductioninmeanperiod,whichcouldbeagoodbasisforfurther investigations of these issues.

16 54 D. Carevic, G. Loncar, M. Prsic 1.1 submerged breakwater emerged breakwater 1.0 (K R-0.2 ) calc ( ) K R- 0.2 meas Figure 11. Diagram comparing the measured reduction coefficients of the meanspectralwaveperiod (K R 0.2 ) meas andthevaluescalculatedbymeans ofequation(10)andequation(11) (K R 0.2 ) calc forsubmergedandemerged breakwaters he application of this empirical model to the design of low-crested structures is limited. It is important to stress that this empirical model wasdevelopedfromadatasetrecordedinawaveflume.inreality,athreedimensional wave transformation occurs across a breakwater, which means oblique, short-crested incident waves. Piling-up behind the submerged breakwater is also specific to wave flume tests, which is not the case for real submerged breakwaters with wide gaps along the structure where offshore directed flows occur. Martinelli et al. (2006) compared piling up at breakwaters with narrow gaps(3d laboratory model)withpilingupinthewaveflume.hoseauthorsfoundthatpiling up was approximately 50% smaller when narrow gaps were present. he influence of piling up on measurement accuracy was not tested. he piling up measured in the laboratory investigations conducted in this work is presentedinable3. hevalueswerecalculatedinthesamewayasthe average surface oscillations. he first parts of the time series, which are statistically unsteady, were cut off. heuseofthemeanspectralperiod 0.2 = (m 0 /m 2 ) 0.5,basedonthe2nd order spectral moment, could be questionable, because it is very sensitive to

17 ransformation of statistical and spectral wave periods crossing able3.pilingupmeasuredinthewaveflumetests est Measured pilingup [mm] high frequency disturbances. he EU CLASH Project suggested employing either 0.1 = (m 0 /m 1 )or 0, 1 =(m 1 /m 0 )asthemoststableindexfor the period. herefore, the same calculations as those presented for Figure 7 wereconductedbutwithsuggestedperiodsof 0,1 and 0, 1.Astheresults areverysimilartothosepresentedinfigure7,theperiod 0.2 waschosen because of the clear comparability with statistical periods. 4. Conclusion Experimental investigations in a wave channel were conducted with a smooth submerged breakwater. ests showed, in general, that when wavescrossthebreakwaterthestatisticalwaveperiods 1/10, s and m are reduced. he reduction of wave periods depends on the relative submersion, i.e. on the ratio of the breakwater crown submersion and the incoming wavelengthr c /L s i.hereisagreaterreductioninwaveperiodsforlower relativesubmersionvalues,sothatthemeanwaveperiod m isreducedby asmuchas25%inrelationtotheincomingmeanperiod.hemeanwave periodsarereducedthemost,and 1/10 ( 5%)theleast. Consequently, the interrelations of representative statistical periods also change. Instead ofthestandardrelations max 1/10 s m,theserelations behindthebreakwatertendto max 1/10 s 1.5 m. It was also concluded that the mean wave periods calculated by both thestatisticalapproach(zeroup-crossing) m andthespectralapproach 0.2 have approximately the same values behind the breakwater, i.e. wave spectral deformation does not affect the calculation of the mean spectralperiod 0.2.hemeanspectralperiod 0.2 dependsontherelative submersionr c /L 0.2 i andisreducedassubmersionapproacheszerofor both submerged and emerged breakwaters. It is estimated that the greatest reductioninperiod 0.2 whenwavescrossthesmoothbreakwateroccurs whentherelativesubmersionisr c /L 0.2 i 0andamountsto 70%ofthe value of the incoming mean period. hepeakperiod p increasesorremainsthesamewhenthewavescross the smooth submerged breakwater. As far as the emerged breakwater is concerned,thereisadependenceofthepeakperiod p,andtherelative

18 56 D. Carevic, G. Loncar, M. Prsic submersionr c /L 0.2 i. By increasing the relative submersion, the peak period p increasesbyupto35%inrelationtotheincomingpeakperiod. he empirical model is formed for estimating the reduction in the mean spectral period when the waves cross submerged and emerged smooth breakwaters. For the incoming wave parameters and the depth in the breakwater crown, the model provides the values for the reduction coefficients K R 0.2.Asthemodelwasderivedfromarestrictednumberof measurements, additional measurements will be necessary, particularly in thezoneofrelativesubmersionr c /L 0.2 i 0. Measured reduction coefficients of the mean period agree well with the calculated values. List of symbols: H max H 1/10 H s H m max 1/10 s m maximumwaveheight,[m],(zeroup-crossing), 1/10thwaveheight,[m],(zeroup-crossing), significantwaveheight,[m],(zeroup-crossing), meanwaveheight,[m],(zeroup-crossing), maximumwaveperiod,[s],(zeroup-crossing),whichcorresponds to the maximum wave height, 1/10thwaveperiod,[s],(zeroup-crossing),whichcorresponds toonetenthofthegreatestwaveheights, significantwaveperiod,[s],(zeroup-crossing),whichcorresponds toonethirdofthegreatestwaveheights, meanwaveperiod,[s],(zeroup-crossing), H m0 significantwaveheight,[m], H m0 = 4 m 0, 0.2 meanwaveperiod,[s], 0.2 = m 0 /m 2, m 0,m 2 zeroandsecondspectralmoment, m 0 = S η (f) p m 2 = 0 f 2 S(f)df, wavespectra,[m 2 s], peakwaveperiod,[s], 0 S(f)df, L 0.2 meanwavelengthatthetoe,[s], L 0.2 = (g /2π) tanh(2πd/l 0.2 ), L m meanwavelengthatthetoe,[s], L m = (g m 2 /2π) tanh(2πd/l m ),

19 ransformation of statistical and spectral wave periods crossing s op wavesteepness,[1], s op = H m0 /L op, L op K Hm0 R R 0.2 offshoreincidentwavelength,[m], transmissioncoefficientofsignificantwaveheight,[1], reductioncoefficientofmeanspectralwaveperiod,[1], ξ op breakerparameter, ξ op = tan α s op, a breakwaterslopeangle,[ ], K B b R c ratio of measured reduction coefficient of wave period and calculated transmission coefficient of wave height,[1], breakwatercrownwidth,[m], crownheightinrelationtowaterlevel,positiveifemerged,negative if submerged,[m]. References Beji S., Battjes J. A., 1993, Experimental investigation of wave propagation over a bar, Coast. Eng., 19(1 2), , doi: / (93)90022-z. Briganti R., Van der Meer J., Buccino M., Calíbrese M., 2003, Wave transmission behind low-crested structures, Proc. Coast. Struc., ASCE, Portland, Goda Y., 1974, Estimation of wave statistics from spectral information, Proc. Int. Symp. Ocean Wave Measurement and Analysis, waves 74, ASCE, New Orleans, Goda Y., 1979, A review on statistical interpretation of wave data, Rept. Port Harbour Res. Inst., 18(1), Goda Y., 2000, Measurement of the reflection coeffcient in a wave flume,[in:] Random seas and design of maritime structures, 2nd edn., Adv. Ser. Ocean Eng., World Sci. Publ., Goda Y., 2008, Overview on the applications of random wave concept in coastal engineering, Proc. Japan Acad., Ser. B 84, Goda Y., Suzuki Y., Kishira Y., 1974, Some experiences in laboratory experiments with irregular waves, Proc. 21st Japanese Conf. Coast. Eng., ,(in Japanese). Journée J. M. J., Massie W. W., 2001, Offshore hydromechanics, 1st edn., Delft Univ. echnol., Longuet-Higgins M. S., 1983, On the joint distribution of wave periods and amplitudesinarandomwavefield, Proc.RoyalSoc.Lond.,Ser.A., Martinelli L., Zanuttigh B., Lamberti A., 2006, Hydrodynamic and morphodynamic response of isolated and multiple low crested structures: experiments and simulations, Coast. Eng., 53(4), , doi: /j.coastaleng

20 58 D. Carevic, G. Loncar, M. Prsic RaichlenF.,CoxJ.C.,RamsdenJ.D.,1992,Innerharborwaveconditionsdue to breakwater overtopping, Proc. ASCE Coastal Engineering Practice, ASCE Long Beach, animoto K., akahashi S., Kimura K., 1987, Structures and hydraulic characteristics of breakwaters the states of the art of breakwater design in Japan, Rep. Port Harbour Res. Inst., 26(5), Van der Meer J., Briganti R., Zanuttigh B., Baoxing W., 2005, Wave transmission and reflection at low-crested structures: design formulae, oblique wave attack and spectral change, Coast. Eng., 52(10 11), VanderMeerJ.,RegelingH.J.,deWaalJ.P.,2000,Wavetransmission:spectral changes and its effect on run-up and overtopping, Proc. ICCE, ASCE, Sydney, VanderMeerJ.,WangB.,WoltersA.,ZanuttighB.,KramerM.,2003,Oblique wave transmission over low-crested structures, Proc. Coast. Struc., ASCE, Portland, Wang B., Otta A. K., Chadwick A. J., 2007, ransmission of obliquely incident waves at low-crested breakwaters: theoretical interpretations of experimental observations, Coast. Eng., 54(4), , doi: / j.coastaleng Zelt J. A., Skjelbreia J. E., 1992, Estimating incident and reflected wave fields using an arbitrary number of wave gauges, Proc. 23rd ICCE, ASCE,