SHALLOW WATER

WAVE CO M PO NE NT IDENTIFICATION FROM SHALLOW WATER WAVES

by

©Gracie Watts,8.Eng.

A thesissub mitt ed to the

Schoo lofGraduateStudies

in partial fulfillmentof therequi rem entsforthedegree of

Masterof Engineering

Facultyof Engineering and Applied Science Memorial Universityof Newfoundl and

2012

St. John ' s Newfoundl and

Abstrac t

ABST RACT

Theaimof this workwastodevelop anumericalpro gramtodecomp ose sha llowwater wavesinorder toprop erly ana lyzethewaveinducedforces onoceanvessels.The program wasusedinconjunctionwithwaves genera ted inamod eltank enviro nme nt.The program identifiedthe kno wnwavesinthetank andremove d unwant ed waveelevation s created by impe rfect mod eltank cond itions.Thiswillenable modeltesting and numericalmodelin gto more accuratelyana lyzewave-ship interaetion ,thus improv ing design andpredictionfor vesselsoperatingin sha llow waterregions.

Aliterature review discusses the theories require d inorder to createa wave splitting tool as wellasthetheoryusedto sim ulate thetheoretical set-dow n wave.Theresultswere validatedin thetime and frequ encydom ain againstarepo rtfromMaritim eResear ch Inst ituteNeth erland s (MARIN)and the Nationa l Research Counci l(NRC) results, respec tive ly. Result s weregene rallyacceptable;test casesbecam eless favorableas wave freque ney decreased.Potenti alfutur e workonthis top ic includes usin gimprovedinput wave datato computewave-ind uced fo rces on vessels.

Acknow ledge me nts

ACKNOWLEDGE MENTS

Theauthorwishesto expresssincereapprec iationto Dr.HeatherPeng andDr.Wei Quifor their assistaneeinthe preparation of this thesis.Iwouldalso liketothankthe CREATE program fortheirassistance in funding.

As well Iwould liketo thank Dr. Zama nforhis assistanceinobtai ningthe datatousefor mytrial cases in my thesisas wellas resultswhic henabled meto valida te my results.

Iwouldalso liketothank SteveForward forhishelp ful suggest ionsand preparation of my thesis.Hisconstantsupport throughoutthe entire process ofthisresearchhasbeen a tremendoushelp.Finally Iwould liketothank my familyfortheir patienceandsupport duringthe lengthofmy study.

Tabl e ofContents

TABL E OF CONTE N TS

Abstrac t i

Acknow ledge me nts ii

Tableof Contents ....

ListofTables...

. iii

ListofFigures vi

Listof Appe nd ices x

I. Introduct ion I

Review ofLiteratu re 6

Stateme ntof Problem 1

1.1.

1.2.

1.3.

Thesis Struc ture ... . 4

2. Theo ries 12

2.1. Digit alMethod s 12

2.2. Four ier Se ries 21

2.3. Spec tral Anal ysis 36

2.4. Wave Splitti ng Theory .44

2.5. Ana lytica l Mod el of Wave Set-dow n 52

2.6. Power Spec trumEstima tion 56

3. Wave Gene ratio n...

3.1. WaveGene ratio n Introducti on...

. 59

. 59

3.2. PistonType WaveMaker 60

3.3. Basin TestConfig uration ... ...62

TableofContents

3.4. PhysicalModeling 64

4. Computational Method... .. 65

Wave Splitting Program 75

Band-pass Filter 65

FastFourier Transform 67

Newton-RaphsonMethod 72

4. 1.

4.2.

4.3.

4.4.

4.5.

SingularValue Decomposition.... . 73

5. Results 78

5.1. Validation 78

5.2. OEB Results ... .. .. 85

5.3. FirstOrderVersusSecondOrder Wave Generation 97

5.4. Discussion 106

6. Conclusionand Recommendations... . 112

References... . .... 114

ListofTables

LIST OFTAB LES

Table5. 1-Parameters of Model ScaleandFullScale 85

Table5.2-Percent DifferencebetweenMeasured and Theoretical Set-down Wave 107

ListofFigures

LISTOF FIG UR ES

Figur e1.1-Wave Set-do wnCom pa riso n betwe en Sha llowand DeepWater(VanDijk 2007).... ... ... 2 Figur e1.2-Wav e group versu sthe wave set-down (Hansen et.at. 1980) 7 Figure2.1-Sinuso ida l Wave (Bea ucham pand Yen,1973,p.3) 13

Figure2.2-Wa ve Spec tru m(Voog t 2005) 36

Figu re2.3-A two sided line spe ctru m (Nation alInstrument, 2009) 42 Figur e 2.4- Flow Diagram of Step sfor Usc of Wave Sp litting (Dijk200 7) 51 Figur e 3.1-Pistontype wavemak er(Wavemaker).... ...60

Figur e 3.2-NRCOcean Eng inee ring Basin(O EB ) 62

Figu re3.3-Positionsof theFourteen WaveProbesWithinthe OEB. 63 Figu re 4.1-Jon sw ap Spec tral DensityPlot(W aveSpec tra,FORMSY S) 65 Figur e4.2-Decomp ositi onofa16PointSigna l (Smith1997) 67 Figure 4.3-Bit reversin g algo rithm (Smith 199 7)... .. 68 Figur e 4.4-Recon structin g signalsin thetimeand frequen cydom ain(Smith1997 ) 69

Figure4.5-TheButterfl yCalcu lation (Smith1997) 70

Figur e4.6-Tot al FFT Procedur e(Smith1997)... . 71

Figure4.7-Wav e Sp litt ingCode FlowCha rt... ... 77

Figure5. 1-OEB versusMARI N:Incid entWav e,TimeSeries,Probe1 80 Figure5.2-OEB versusMA RIN:Incid entWave,TimeSeries,Probe I(300-5 00s) 80 Figure5.3 -OEBversusMARIN:Reflect edWave,TimeSeries,Prob eI XI Figure5.4 - OEBversusMARIN :Refle ctedWave,TimeSeries,Prob eI(300-500s) XI

Listof Figures

Figure5.5-OEB versus NRC:FOMeasu redWave, FrequencySeries,Prob eI 82 Figure5.6- OEB versusNRC:SOMeasu red Wave, FrequencySeries, Prob eI 82 Figure5.7- OEBversus NRC: FOMeasur edWave, FrequencySeries, Probe 2 83 Figure5.8- OEBversus NRC:SOMeasu red Wave, Frequ ency Series, Probe 2 83 Figure 5.9- OEB versus NRC:FOMeasur ed Wave, FrequencySeries,Probe 9... ...84 Figure5.10 -OEB versus NRC:SOMeasur edWave,Frequ ency Series,Prob e 9 84

Figure5.11-TimeSeriesWave Splitting,Prob eI 0I

Figure5.12 -TimeSeriesWave Splitting, Prob eI(100- 150s) 87

Figure5.13 - Time Series Wave Splitting,Prob e 2 88

Figure5.14 -TimeSeries Wave Splitting, Probe 2 (100- 150s) 88

Figure 5.15 - Time SeriesWave Splitting, Probe 3 89

Figure5.16 -TimeSeriesWave Splitting, Prob e 3(I00-150s) 89 Figure5.17 -TimeSeriesWave Splitting, Prob e 8... . 90 Figure5.18-Time Series Wave Splitting, Prob e 8(I00-150s) 90

Figure5. 19 - Time SeriesWave Splitting,Prob e 9 91

Figure5.20-TimeSeriesWave Splitting,Prob e 9 (100- 150s) 91 Figure5.21-Frequency DomainWave Splitting,ProbeI 7<-

Figure5.22-Low Freque ncy Dom ain Wave Splitting,Prob e1 92 Figure5.23-Freq uency Dom ainWave Splitting,Probe2 93 Figure5.24-LowFreq uency Dom ainWave Splitting, Probe 2 93 Figure5.25- Frequency Dom ainWave Splitting, Prob e 3 94 Figure5.26-Low Freq uency Dom ain Wave Splitti ng,Probe 3 94 Figure5.27-Frequency Dom ain Wave Splitting, Prob e 8.... ...95

List ofFigures

Figure5.28-LowFrequencyDom ainWave Splitting, Probe 8 95 Figure5.29- Frequ encyDom ainWave Splitting,Prob e 9 96 Figure5.30 -LowFrequencyDomainWave Splitting, Prob e 9 96 Figure5.31-FO versusSO:Time series, IncidentWave,Probe1, Case 1 97 Figure5.32- FOversusSO:Timeseries,Incid entWave,ProbeI,Case 2 98 Figure5.33- FOversusSO:Time series,IncidentWave,ProbeI,Case 3 98 Figure5.34 - FOvers usSO: Time series,ReflectedWave,ProbeI, CaseI 99 Figur e 5.35- FO versus SO:Timeseries, ReflectedWave,ProbeI,Case2 99 Figur e 5.36-FOversusSO:Time series, Refle ctedWave,Probe1, Case3 100 Figur e 5.37 - FO versus SO:Time series, TheoreticalSet-d own,ProbeI,CaseI 100 Figur e 5.38- FOversusSO:Time series,Theoret ical Set-do wn,ProbeI,Case2 101 Figure5.39-FOversusSO:Time series,Theoreti cal Set-do wn,Prob eI,Case3 101 Figure5.40-FOversusSO:Frequency Dom ain ,Incid entWave,ProbeI,CaseI 102 Figure5.41 -FOversusSO:Frequency Dom ain ,Incid entWave,ProbeI,Case 2 102 Figure5.42-FO versus SO:Frequency Domain,IncidentWave,ProbeI,Case 3 103 Figure5.43-FO versusSO:Frequ encyDom ain,ReflectedWave,Probe1, CaseI 103 Figure5.44 -FO versusSO:Frequ encyDomain,Refle ctedWave,Probe1,Case2 104 Figur e 5.45 - FO versusSO:Freq uency Dom ain ,ReflectedWave,ProbeI, Case 3 104 Figure5.46- FOversusSO: Frequ encyDom ain,Set-do wn Wave,Prob eI,Case3 105 Figure5.47-SecondOrder Theoretical Set-dow n WavePeakPer iod Compa rison 106 Figure5.48-Measur ed versus Theor etical Set-down Wave:CaseI,ProbeI 109 Figure5.49-Measured versus Theoretical Set-do wn Wave:CaseI,Prob e 2 109 Figure5.50-Measur ed versus Theoretical Set-do w n Wave:Case2,Prob e1 110

List ofFigures

Figure5.5 1-Meas ured versusTheoretical Set-do wn Wave: Case 2,Probe 2 110 Figure5.52-Measured versus Theoretical Set-do wn Wave:Case3,ProbeI III Figure 5.53-Measured versus Theoreti cal Set-do wn Wave:Case 3, Probe2 III

Listof Appendiees

LIST OF APPE N DICES

Append ixA-Cosine Deeomp ositon... . 118

AppendixB-Model Seale Results 123

AppendixC-FullSeale Result s... . 144

Stateme ntofProblem

I. INTRO D UCT ION

1.1. Statemen tofProbl em

Having various meanstopredi cttheperform ance ofa vessel's interaction with the ocean env ironme nt isimportantforshipstruc ture design.Fullscale trialtestingis expensiveand time consumi ng but can be oneof themost reliab le form s of predi ction.Modeltest ingis less expens ive than fullscale trialtestin gbut is conduetcd inaman-made environme nt, which canproduce inaccur acies.Numerical simulation istypicall ytheleast expensive alternative,which canpredi ct vessel perform anceusing empirica lana lysisor computationa l method.However,numerical simulationalso has associatedinacc urac ies in partduetothe difficu ltyof usin g ana lyt ica lexpress ions to simulate the rando mnessof oceanconditions.

Formost companiesfullscale test ing isnot a viableoption duetofinancial constraints and lim itedresources,there foremod eltestin g and numerical simulationsarc favorable.

However,inorder todetermin etheperf orm ance ofa vesselinamodeltankenviro nme nt, it isimportanttohave aprop erunderstandin g of what types of waves arc produ cedin the tan k.Whenwaves arcmeasur edina tank, thetotal wave eleva tionismeasur edby an instrume ntsuc has awaveprob e.The measur edwave eleva tionismadeupofseveralwave components:incident,reflected,and boun d waves.Incid ent waves arecreated bya mechanical wavemakingdevice.Reflected waves arc created dueto the constraintsofthe mod el enviro nme nt inwhich theincid ent waves arc reflec tedfromtheoppos iteendof the lan k inwhichtheywerecrea ted . The inte nsityofthereflec tedwavescan beredu cedifthe

Statementof Problem

tan kis equipped with man-m adewave absorbers,which simulatesataper edbeach shoreline. Bound waves ,also known aswaveset-downandset-up,werea phenomen onfirst reportedbyLonguet-Hi ggins andSteward(1964). Theyintroducedaconceptof radi ation stresswhichexplained that an individu alwave component fromawave groupcanexert an internal compress iveforce in thedirection of thepropagatedwave.Theforceis balancedby the water leveldecrea singin regions of longerwavers,known aswave set-down,and the water levelincreasinginshorter waves,known aswave set-up.This conce pt wasproven experime ntally byBowen et.al. (1968).

Theincident and reflectedwavesarc comprised up fromfirst orderwavecomponents whereastheboundwaves compose dofsecondorder wavecomponent sthat arc more signi ficantinshallow waterdepth.VanDijk (200 7)illustrat estheincreasein wave set- down in shallow waterregion s, as shown by Figure I-I,whereWD is water depth .

Figure 1-1-Wave Set-down Comparison between Shallow and Deep Water (Van Dijk2007)

Statementof Problem

Incident,reflected,and bound waves arc the waves known wavestypesthatarcproduced in amodeltank environme nt. However,imperfect conditions withinthetank createextra unwanted waves and needtobedistin guished and removedfromthedeliberately generated wavetypes.Iftheunwant edwaves arcnot removed fromthe known waves,large discrepancies canoccurfromthe modeltesting resultsand fullscaletrial data.

Once themeasuredwaves arcsplitinto thedifferent wave components in thetank,amore accurateanalysiscan be completed for vessel design.Mooredliquefiednatu ral gas (LNG) caniers, in particular,have afundam entall yweakl y damped nature and when combined withthesecondorder waves can produ ce significant resonantmotion andassociated loads (Naciri,2004).Therefore modeling ofmoorin gdesignandship-wa ve interaction inshallow water regions improvessubstantially bytheuse of thiswave splitting technology.

Thes isStruc ture

1.2. ThesisStruc ture

This thesisdescrib estheprocessof develop ing anumericalprogramthat willidentifyand split the various wave compo nents within measur edwaves.The resultsof aliteratur eand theo ry reviewarc also discu ssed.

Chapter Iintroduces thetop ic and present s aliteraturereview.The probl em and theories developedby differentauthorsandexam ines why wave decomp ositionisneeded espec ially when deal ingwith amodeltankenvironm ent.

Chapter2present sthetheoriesusedtodevelopthe eq uationsthatare usedforthe wave splitting program .Thisincludesdetails on digitalmeth ods, Fourierana lysis,spec tra l ana lysis,as wellas the wave splitting theory.

Chapter3describ esthe environme nt inwhich thewavedata was generatedand used for the wave splitting program.Thetype ofwavetank andwave mak er are describ edaswellas equat ions usedto simulate theseco ndorde rwave components.

Chapter4describ es thecomputational methodsthatwereincorp oratedintothewave splitting program.The band-p assfilter, fastfour ier transform,newt onraph son, and also sing ularvalue decomp ositionweredescribed.

ThesisStructure

Theresultsaredisplayed and discussedinChapter5.Theresult s arevalidatedagainst published results fromMARINand NRC.Theresult s areshown forthelongestwave period case of 2.145s for theOffshor e Enginee ring Basin(O EB)wave generation data, since that case shows thelargerpeaks.A comparison betw eenthefi rst orde randsecond orde rwave generation isdoneto show thecontributionofthe second orderwave component.Adiscussionisdoneto compare themeasured wavedatatothelowfrequ ency

Chapter6 consists of a summaryof theresults and findingsfrom theresearchand future work is recommended.Theresearch can be continued insuchaway asto find thewave inducedforces,from theproperly splitwave data, aswell asthewaveinducedmotion ona ship.

Review ofLiteratur e

1.3. Review of Literature

The need forthisresearchhasbeenwelldocument edby variousauthors.Hansen et.al.

(1980) discusshowresonance conditionscancreate unacceptablylargcmovem ents of mooredvessels andcan resultin moorin g failure in harbours and bays.This phenom enon can resultin difficulti esfor the operationofship tcnnin als.Harbourresonan ce can be caused byavarietyof occurrences but in particularlongperiod wavesfrom distant storms orwave groups(Hansenet.aI., 1980).Hansen et. al.(1980) further state how generating the long period waves can bedifficult in physicalmodel ,whichisreiteratedbymany authors, suchas Sand(1982)whostates theimportanceof longwavesbeing correctly representedin thephysicalmodelduetoitsinfluence on mooringforces andslow drift oscillations.

Recently, ahigherdem and for LNG carriers(Nac iri,2004)and in tum ahigher developm ent of LNGtermin als (Voogt,2005) hasbeennoted.Duetothis increase, there is a greater needforhydrodyn am icmodelingtools forfloaters,particularlyinshallow water, asmost of theterminalsfor theLNG carriers arclocatedncar shoreor in relatively shallow water. Low frequency excitationof the wave set-downincreasesinshallowwater(Voog t, 2005).LNG carriers haveinherentweaklydamp ednature(Naciri,2004)andwhen combined withthelowfrequencywaves cancausesignifica nt resonantmotionsand related mooringloads (Voogt,2005).

Low frequency bound waves,alsoreferred to as set-downwaves,foundinshallow water environ ment havebeendiscussedin detailby Longuet-HigginsandStewa rt(1964), Hansen

Review of Literatur e

et.al. (1980), andSand(1982),amo ngothers.Longuet-Hi ggins andStewa rt(1964)stated that short periodwavesinducelongperiodwaveswith period s equal tothatof the wave group.Theydescrib e a'wave group' as "...wavetrainsof nearl ythe samefrequencyand wavelengthpropagatedin the same direction ,resultin gin theform ation of'group'of waves".These longperiodwaves,which can alsobe termedlowfrequencywaves,arc bound tothe group of wavespropagatin gwithgroup-velocitycg(Sand ,1982).Hansenet.

al.(1980) describethelongwaves as"wave set-down"asthetroughs of the longwaves arc found at the regionsofthe largerwavesin the group,as shown inFigure 1-2.

SET-DOWN PRO PORTIONAL TO H 2

_H2

Figure 1-2 - Wave group versus the wave set-down (Hansen et. at. 1980)

The set-do wn longwaveswerediscovered from radiation stressand themomentum equation (Longuet-HigginsandStewart, 1964)but were alsoobtained from perturb ation analysisof theLaplaceequation (Hansen et. al.,1980) .Thelatterwill be explored in this work.

Hansen et.al. (1980)further describesthe problem sthat canoccur in thephysicalmodel tank environment.Although modeltestinghasimprovedwith the generation ofnatural, irregular waves,aproblem stilloccurs whenthepaddle genera tes wavesusingthetypical first orde r motion.Since thelongwaves areofsecondorde rapprox imation, thefirst orde r

Review ofLiteratur e

approxi ma tion doesnotrep rodu eethedrift veloeities(Ha nsenet.al.,1980).The paddl e will then prod ueenaturaldriftveloc ities thatwillbe of equa l magnitud e andoppos itesign, which isaprogressivelon gwavethat is free and notbound edtothewave group incontras t tothe set-dow nwavethatisbound edtothewave group. Hansen et. al.(1980)also term s thefree wave as aparasiti clongwave,which causesthelongwave effects tobe exaggerated in term s ofharb orresonance andslow drift ship motion s.Sand(1982)furt her elabora tes thatthe freelongwa vesdonotfollowthedispersionrelation.Thisparasiti c wavemustbeidentified andremove dfromthe modeltestin g enviro nme nt inorder to produ ce accura te result s.Hansen et. al.(1980)sugges t that a secondorder longperiod signa l be impose donthe first orde rsigna ltoproducethedriftveloc ities requi redfor the set-dow n wave.Hansen et. al,(1980 ) expand thebounda ry conditions to apply tothe secondorde rapprox ima tionas wellas applying theLaplace eq uationwith thenonlin ear sur face cond itions (Sand 1982).Thustheposit ionforthepaddl e can be corrected to apply thesecondorderapprox imations.

The long waves are found by a summa tionof differencesbetween each pair of frequ encies ofthe short,or first orde r, wave spec trum(Hanse net. al.,1980 ).Han sen et al.(1980) descri bethelon gperiodwaves asthe sumof subhanno nics.They state that aregularwave groupwithwatersurfaceeleva tion11nm andfrequencyD.f,Ullthat consis tof tworegular waveswith watersurfaceeleva tions1111,11mand fr equ enciesI'll,f""wherenandmareindices for the number ofwavesbeing considered.Thewave groupeleva tionand frequ encyis equa lto:

1]nm

<n:

+1]m

Review of Litera ture

!J.inm=in- im

Sinceweknowthatthewave groupgenerates wave set-dow n,wave group'lnl11willgenera te wave set-do wn~nl11'Hanson et.al.(1980)state that each pair ofn,mcomponentsof the spectrum will contribute tothe set-do wn. Thusby summingall the contributionsofall the pairs willgive:

wherem*=f*/fo,f*isthelowest frequencyof theregula rwave spec trumand 10isthe interval of discretization oftheregular wave spec trum.Further theory on thecalculationis discussed in thetheory section in thisThesis.

Hansen etal.(1980)and Mansard (\99 1) brieflydiscuss supcrharmo nicscorresponding to the secondordercorrectioncontaining a number of term s that pertainin gtothe sumofthe a pair offrequenciesin theshort wave spec trum.Since thisThesisis concerned with inves tigating low frequencywaves,superharmo nicsarc not considered.

Voogt (2005) describes aprocedu re for splitting theindividu al wave componentsinorde r to identify thelow frequency boundwave andcomparing the analyticalwave set-do wn to the separated wave set-downfromthemeasur edwave.Theanalytica l wave set-dow n modelisdescrib edby Sharmaand Dean (198 \)and isfrequ ently referred to when calculatingthe theoretical wave set-do wn.Thewave splitting program created fo rthis research usedthe methods describedbyboth Voogt(2005)andShermanand Dean (191\1) since thcirm ethods arcwell defined.

Review ofLiteratur e

Stransbe rg(2006)present snewresult sfo r ide ntifyi ng thelow frequencywaves frombi- chroma ticwaves (waveshavingtwofrequ en cies) withand without a correc tion tothe wave makertoremovethe freewaves.Heuses anitera tive process tofind the optima lcorrec tion signa l throughrepeatedtestsinalarge wavebasin.Hisfindings showimprove me nt in the correc ted result s inwhic hthefree waveswerereduc ed.In thepresentworkanattempt is madeto comparea correc tedsecondorde r wave genera tion but notinthe iterati ve sense that Stransbe rg present s,butasproposedby Sand(1982)which is previously describ ed.A large wavebasinwasutilized, although theNRCbasinisnot asbig astheMARINT EK basin in whichStransbe rgconducted his expe rime nts, butitis cons ideredalargebasin.

Masard andFunke(1980) describ e ameth od toidenti fythe incidentand reflected spec tra from themeasu red spec trafroma wave tankusingtheleast square method.Itrequ iresthree simultaneo us measur em ent swithinthetank that arc in reasonabl eproxim ity to eachothe r andarc inparall elto eachother inaline from thedirect ion of the waveprop agation.Ithas shown to give goodagree me ntincompariso n toincid ent spec tra measur ed concurre ntly in a sidechannel.Voogt(2005) states thattheleast square meth odwill givebest results lor three wave compone nts:incid ent,reflected , and wave set-dow n.For this workonly two componentsarcsplit fromthemeasur ed wave and thusthe singularvalue decomp osition form ulationwasused.

Mansard (199 1) developed anumericaltechniq ue,basedon thepreviou slydescr ibed interactionsbetweenfree and bound waves, that illustrates the posit ion in the wavetank

Review ofLiterature

where waves ofthe samefrequencywouldinterac tandeithereance lor reinforce each other.The results of hisresearch confirmthe use ofseco ndorder wave generation techni ques reduce theparasiticfree wave which produ cesmore realisticwaveprofiles.He states that even when inco rporatingsecondorde rwavegeneration techn iquesfor an irregular spectrum, ratherthanusingmonochromati c orbichrom atic waves,there can be high oscillationsat thetail endofthe spectrum duetothe interactio noffree and bound components.These oscillationscancause differences inwave param eters suchas significantwave height andcrestfrontsteepness(Mansard, 1991).

The experimentscarriedout by Zama net.al.(20II)weredoneinorde rto ide ntify the spuriouswavesinshallow waterusingthe OEB, NRC.They generated multi-chromati c wavesusingfirst orderandsecondorde r techniqu es.Theyshow thedifferences betwee n the firstorderandsecondorde rwavegeneration.The resultsfromthisresearch are validatedagainst their results.

DigitalMethods

2.THEO R IES

2.1. DigitalMeth od s

A signa l is anyconv ers ionofsome propert y, suc h astemp eratur e orpressur e,fromits origi na l physicalformto arelat ed elect rica l quantit y.A sig nalca n beclass ified intotwo categor ies:det ermini sti c and rand om. Adeterm ini stic signalcan bepredi ctedby math em at icalrelati on s andarand om sig na l isnotpredi ctabl e,however itcan be estima ted through sta tistics and probabili ties. Ameth od ofdistin gui shin gbetweenthetwo categories isto compareseveralsetsofdata obtained underidenti cal condi tionsovera reason ableperiod of time. Some timesa signa l may appea r tobe rando m but is in fact det ermin isti c, andcan beprovedtobedetermini sticusin g,forexample , anauto- correlatio n process (Beauc ham pand Yen ,1973).

Thefollow ingsections describ efundam ent alprincipl es of digital signa lana lys iswith reference toDigit alMeth od sfor Signa lAna lys is(Bea uc hampand Yen,1973).

DeterministicSignals

Adetermini stic signa lcan bedeterm ined from time-hi story and is genera lly repr odu cibl e underiden tical cond itions. Themathemati calform ofadeterm inist ic signalcan be described asperi odi c or tran sient signa ls. Period ic signa lscontinuous ly repeat at regul ar intervals whe reas tran sient signa ls decayto a zero valueafte rafinite lengthof time.

Peri od ic signa lsarccompr isedof oneormor e sinuso ida lsigna ls havin g an integr al relation sh ip with thisperiod overtime.Thebasic sinuso ida leq uatio nisdescrib edin

DigitalMeth ods

Equa tion [2.1],where Aisthe eons tant repr esentin gthe peak amplitudeof thewav eform ,

(1)0isthe angular frequ ency and 0 istheinitialphase ang le with resp ecttothetime orig in.

Equa tion [2.2)describ esthemath em aticalexpressi onforthe angular frequ en cywo,wher e 1'0isthe cyclical frequ en cyin Hz. Eq uation [2.3)descr ibestheperiod,T, of the wave form.Figure2- 1 displa ys graphica lly the sinusoi da lequa tion.

x(t)=Asin(wo t

+

8)

Wo =2nfo

1 T= - fo

Figure 2-1-Sinusoidal Wave (Beauchamp and Yen,1973) (2,11

12.31

TheFour ierseries is amean sto expressa genera lrelations hipfor period ic signa ls,in whichharm oni c compo ne nts repeat exac tly for allvaluesoft,describ edinEquat ion [2.4).

Digital Methods

(2.4J

whereAo: the meanlevel of thesigna l An:thepeak amplitudeofthen'hharm oni c On: thephase ofthen'hharmon ic

Dete rm inist ic signa lscan be a combina tionof severalsinuso ida lcleme ntsand maynotbe harmoni call y related. Thedeterm in istic signal will have simi larspectra lcharac ter isticsas aperiodi c signa l.

RandomSign als

Arand om signa l isuse fulfor obta ining new and unpredi ctable values that would notbe guessedor estima tedfrom previou sdat a, althoug hthisdoesnotmean tha t the valuesare uns truc tured,Sincerando msigna lscanno t bedetermin ed explicit ly, prob abilities and stat isticsarethe primarymeans of estimat inga solution.

An ensem b leis one rand omprocess whic hwill produ ce a set oftime-histo ries.An exampleof anensem ble maybe anexpe rime ntwhic h produ ces rando m dat a andis repeatedN times(BeauchampandYen,1973).Thestatis tica lvaluefor this ense mb le is obtained by conside ring record s take nat specificinstantsintime.Theaveragecan be calculatedat a spec ific timelj,showninEquation [2.5], as wellas theaverage valueof

DigitalMethods

the productsoftwodiff er ent samplesat differ enttim es, sayt)andt2,known asthe auto - correlatio n functi on ,describ edby Equa tio n [2.6].Not ethatT=trtl.

N (X(t l ))

= J~ ~ L

^Xk(tl)

k=l

(2.51

12.61

where Nisthelen gth ofthe record,and kistheindex ofsumma tion forthe ense mb le.

Thesigna lis statio na ry if<xf tj) and R(T) are consta ntforallvaluesoft,andalsoifR(T) reliesonlyon timedispl acem ent (T).Arecord can bepartiti on edinto eq ua lsectionsin order to calculate the average valueofanysectio n,as shown inEquat ion [2.7]:

T XM

=

_{T~ ooT}lim

~f

^{XM(t)· dt}

o

where Mi s a section ofthe record.

(2.7)

Propertiesof the Signal

Theproperti es ofa signalcan bedefin edinaprob ab ilist ic sensein term s ofamplitude, timeandfreque ncydoma in.

AmplitudeDoma in

• Rootmean- squ a re value:shows the amplitude'seffec tofthe signa l but is

DigitalMethods

insuffi cient inshowing the variablenature of theprocess.Therefor ethe probabilit y of the amplitudeexceed ing,or lyingbetw een spec ified levels,mustbe determin ed.Thecalculation fortherootmean square is given by Equatio n [2.8]

for continuous data and by Equation [2.9] for discretedata.

T

XM

= ~ ^[

^{x2(t) .dt (2.81}

12.91

• Probab ilitydensit yfunction:determin es the probabilitythatthe signa l willbe foundwithin a givenrange.Itismore suitable for sma ller possibl enumbers of discrete valuesofx.

• Probabilit ydistributionfunction:describ estheprobab ilitythatthevariabl ewill assumeavalu elessthan orgrea ter thanx.This caseismore suitable forlarger domai nssuch as when thepossibl enumber ofdiscrete valuesof x is great.

• Auto-co rrelationfunction:used to statistica lly determin einform ation about the periodicbehaviourbytakin gmeasur em ents oftheamplitudeof the signa lattwo

DigitalMethod s

instances,separa ted byT,and findingthei rproduct andaverag ingover thetime of the record. Themainvalu e ofauto- corre lation isto exposeany hidden peri odi cit y within the signa l.

• Crosscorrelation functi on:calculates therelati onbetweenthetwo signa ls.

Freque ncy Dom ain

• Fourie r Tran sform:Fourierseriesana lys is can provid epeak amplitudesand relat edharm onic s conta ined withina signal. But itcanno t beappliedtorand om signa lsasthecomp on ent s arc notnecessaril yhann oni eall yrelated . There forea Fourier Tran sformisusedtomeasur etherelati veamplitudes ofthefrequency compo ne nts.

• Pow er spec tra l den sity:calculated byfindingthemean- squ arevalu e ofthe instantaneo us pow er at agive n frequ en cy over timeT.

• Weiner-Kh intchin eRelati on sh ip:relatesthepow er spec tra ldensi ty functi on and theauto- corr elationfunct ion andisimport antforpracti calmeasur em entmeth od s.

RequiredLength of Record

Thelength of record isimport an t for accuracyof the statis tica lestima tes.Each meth od of ana lys is defin esits own min imumlen gth.Ingenera l,therecord lengthisinversely prop or tion altotwice ofthebandwi dth (B) multipliedby anestima tionerro r(c)as well as beingprop ortion alto aprop ort ion al constant(K)which isdep end ent on themeasur ed property.Foramean squareand powe r spec tra l densit y estimates, K is oftenassumed to be equa l to 2.

DigitalMethods

DigitizationofContinuousInformation

Inorder to conve rta continuo usana logsigna l to adiscr ete formof values,samp ling must be comp letedin thetimedom ain,quantizat ion in the amplitude dom ain , andthencoding thefinalresults intodigitalfo rm.Errorscanoccurfromthe limitation s ofthese procedur es.

Samplingcanbe completedata cyclic rate samplingwhic h is a sinuso ida lfunc tion that samp les in accordan ce withalinear functio noftime.Anoth erme thod of samplingcan be doneat equa llyspace d timeinterval s, kno wnasuniform sampling.Apert ure isthelength oftime over whic h datais average dandsho uldbe sma llcompa red to thesampling period inorder toprevent erro r.

Oneofthemost significa nt difti culti es arisesfromalias ingeffec ts in which the measur ed signa lis indistinguis hable from othe rsigna ls.Forexa m ple if ahighfrequ ency signa l is measur ed at tooIow aratethenthe signa lcan beinterpr eted as alowfrequ en cy signal.

Thusthehighfreq uencysigna l look slikethelow freque ncysignalandcanno t be ide ntifiedfrom one ano the r,as shown byFigur e 2-2.

DigitalMeth ods

Ahigh frequenqrsig na l

_ MNMNWMMm

111111111 .111111111 .11111

scrople d ct too lov c r cta

cloverIrequen cysiqnol

Figure2-2- Aliasing Effectsfrom Signal Processing (©BORESSignal Processing 2009)

Ano the rim porta nt param eter insigna l processin gistheNyqu istFreque ncy(fN).Itis de fined ashal fthe samplingfreq ue ncyof a discre tesigna l. When a continuo ussignalis sam pled,allthe alias ingeffectsoccurabove theNyqui st Frequency.Addi tiona lly, if

1: ,

is the funda me nta lfreq ue ncyofthetrue signa l,alias ingwill not occurwhe n frequ enci es range fromzeroto fN .

Accord ingtotheRaleigh Theore m,timeTmustbe grea teroreq ualto the recip ro cal of twi cetheband w idth,8:

T> -- 281

Signa ls mustbe subjec ted toband-p ass filters prio rtodigitization.Insumma ry,signa ls ha veto be afinitebandwidthup to8 Hzandareseparated by1/(28 ) seconds.

DigitalMethods

Thesamplingrate mustbemad elargeinorder tohave apracti calfilterin reality,and thus maybehigh erthan the sampling theorem sugge sts.Properfrequ en cyreduc tionmaynot be achieved until two octav es high erthanthe cut-o ff frequ encies. Often,the sam pling rate is set to1.25timesthe filte rcutofffrequ en cy.

Aspart of thedigiti za tionprocess,quanti zation allowsa setofcontinuo usvalues tobe repr esent ed as alimited seriesofdiscretenumbers.Thisis onlyanapproxi ma tionsince the origina l number,whichhas an infinitenumberof states, mustbetruncatedinorde r to berep resenteddigitall ywith alimitednumbera bits.Thisprocessisnon-line arwhen repr esentin g aphysicalquantitynumericall y. The resultis expresse das an integer value corres po ndi ng tothenear est who le number ofunits.

Fourier Series

2.2. Fourie rSeries

Fou riertransformisthe mean stowhich data can betransform edinto thefrequency dom ain from the time domain.Usin gthecommontrigon om etric ide ntityshown inEquation [2.10], the Fourier 's theorem series shown inEquation [2.4]can be rewritten into an expanded form shown inEquation [2.11].

sin (A

+

B)=sinA cos B

+

cosAsinB

xU)

=

Ao

+

Alsin

e

l.coswot

+

Alcos e_l'sinw ot

+...

+

Ansinen . cosnwot

12.10)

(2.111

We cansimplifyEquation [2.11]bylettingthesine andcosine term sequal to variabl es which can belaterderivedtheoreti call yusin g various integrals.The variable s can be writtenas:

Uk

=

An sinen bk=An cosen

Notethat the constant termaoiswritten astwicetheoriginalconstantwhichisdonefor simplicitylateronin computationsince the constantisarbitrary.Inputtingthe above term s

FourierSeries

into Eq uation [2.11]we obtainEq uation [2.1 2]whichcanbe simplified intothe Fouri er seriesequation notation , as shown by Eq uation [2.1 3].

x(t )

= -T +

al coswot

+

b,sinwot

+..

(2.12]

n n

x(t)=-T+

~akcoskwot+~bksinkwot

^(2.131

Using thepropert y of orthogonalit yforsinuso ids, theonly finitevalue that can be obtained forthe sineandcosine functionsis eq ual to1/2.Thisis obtained by squaring the sineand cosine respectiv ely.Inaddition, frequen ciesand phase shifts mustbeequal,otherwise the produ ctwill bezero (Beau champ and Yucn ,1973).Orthogo na lity is an importantprincipal fortheFouri er seriesasit willsimplify the equat iongrea tly. We cansecthatfromEq uation [2.12]all theterm swillthusredu cetozeroexcept forthe firsttermand therefor ethe integral ofx( t)over theperiodT will be:

J T JT

ao ao

ox(t) dt=0Z"dt=Z"T

Theconstant termistherefor e equa l to:

T

ao

1f

- = - x( t ) d t

2 T

o

Four ierSeries

(2.141

We cansolve forthevariabl es akand b,bymultiplyin gthe x(t) functionby bothsin kwot andcoskwot.Theterm swill allvanis hexeept forthe sin²kwotandcos²kwot term s which are equa l to'12aspreviou sly stated. Therefore wehave:

T T

[x(t) sinkw otdt=[bksin²kwotdt=

», ~

T T

[x(t)coskw otdt

= [

akcos?kw otdt

=

ak

~

Therefore wenowhave equations forthe variableswhich repr esentthe amplitudesof the harm on ics foundin the orig inalfunctionx(t)andare known asthe Four iercoefficie ntsand are showninEq uations [2.15] and [2.16].

T bk=

~

^{[x(t)sinkw otdt}

T ak

= ~ [

x(t)coskw otdt

(2.151

12.16]

FourierSeries

Complex rep resent ation of theFourierseriesand integral allows lorfurtherdevelopment of the Fourier transform.Using thefollowin g identities we canexpand thenotation sin Equation [2.13] whcrc j-v-I :

coskwot=

2"

1[exPCikwot)

+

exp(-j kw ot) ] 12.171

jsinkwot

=

1

2"

[exPCikwot )-exp(-jkwot)] (2.181

Incorp orat ing Equations [2.17] and [2.18]into Equation [2.13] gives:

=-T[eXpCik wot )

+

exp( -jkwo t) ]

+

f[exPCikwot )-exp( - j kwo t) ] (2.19]

=AkeXPCikwo t)

+

Bkexp(-jk wot )

whereA, and Bkrepresent complexconj uga teamplitudecoefficientsandarcequal to:

12.201

FourierSeries

(2.211

UsingEquations [2.15] and [2.16], aswellasincorporatingthe ide ntitiesofEquation [2.17]

and [2.18],we can re-writ e Equations [2.20] and [2.21]intotheexpand edintegralform, which usesthe original function x(t)as shown by Equations [2.22] and [2.23].Notethatt has beenchangedtopfor theintegrations since t needstobe maintainedforthe exponcntial function.

T

Ak

= ~

^[X(P)[CO Sk Wop-jSinkWoP]dP

T

= ~

^[X(P) exp( -jkwo p) dp

T

e,

=

~

^[x(p) [cos kwo p

+

jsinkwop]dp

T

= ~

^[x (p ) exP(jk woP)dP

12.221

12.231

Wc can now expandtheFourierseries into complex notationbyincorp orating Equations [2.14],[2.19],[2.22]and [2.23] intoequation [2.13].

Four ierSeries

x(t)

12.241

Express ion [2.24] can be sim plifiedas the rearc anumber ofseque nceofte rm sthat arc summedfromk=I tok=nas wellask=-Iand k=-nfor the thi rd term (by incorpor atin gthe negati veintheexpo ne ntial term ) so tha t we canjoin theterm stobethe sum ma tio ns from k=-nto k=nwhichincludesthefirstterm at k=O.The re foreEqua tion [2. 24] can be rewritte n

Fourier Integral Transform

SinceFourier series'haslim itation s in thatitassumes thetimefunctionisinfi nite and that thedata isper iodi c, wh ich in practiceisnotreali ty,Four ier Integraltran s formisused for most signal data.Thedata can nolon gerbe assume d that itwill rep eat in fin itely,therefor ek

Four ierSeries

andT arcextended toinfinity and thefundam ental freq uencywilllead toward szero. The separation betweenharm oni csalsotendsto zerowhichwill causethe Fourier eoelli cicnt s tobecome continuousfunctio nsof frequ ency.A coupleofnotew orth yrelation s arc:

y=f

k

w=2rrf

There fore we can restate Equation [2.25) in term s offrequ encyin the complexformofa Fourier integral as shownby Equation [2.26).

x(t)=

J [J

^x(p)ex p( -jwp)"dP]ex P(jw t)<d] (2.26(

The contentswithin thebracket s inEq uation [2.26)representtheamplitude of thecomplex Fouriercoellic ients for a continuoustimeseries.Isolatin gthebracketed express ion we have afrequ en cyfunction as sho wn inEqua tion [2.27)"

X(f) =!x (p ) exp(-jWP)" dP 12.271

X(f)=

J

^{x(t) exp(-j wt )· dt}

to show inits origina ltime form.

FourierSeries

12.281

Equation [2.28] is thecomplex Fou riertrans fo rm of the time series x(t).The absolut e valuesofX(f) gives the freq uencyamplitudeandtheargume ntgives thephase.Thus Equation [2.26]becom es:

x(t)=

J

X(f) exp(jwt).df 12.291

whichis know nastheinverse complexFourier tran sform.In discreteterm s Equation [2.29]

becom es:

x(t)

= f

Xk(f) exp(jwt).df 12.301

Often(Jlisusedin place off, for examplewhena scalingfactorof 1/2rr is app lied to a comp lex transfo rm.Alterna tiveformsof writingFourier transform areused toquanti fy them intermsof cosineandsine.Ifthex(t)functionis even,ie.x(t)is symme tricabout the t=Oaxis,the Fourier transformbecom esknown asthe Fourie rcosine transformXc(f)andif thefunctionis odd the Four ier transform becomes known as theFouriersine transform XsCf), as show nby Equations [2.31] and [2.32].

Xc(f)=

J

^{x(t)cos wt-dt}

Xs(f)=

J

^x(t)sinwt-dt

Fourie rSeries

(2.311

(2.32(

Discret eFourierSer ies

Fordiscretedigitize d data, afinite formof Fourierseriesisneededtoderive a discrete fo nn oftheFourier transform.Computat iona llythediscrete Fouriertransfo rm isidenti cal to the complexreprese ntationofFour ier series described in thepreviou s section butthe theoreticalderivationis quite different.There arequiteafewmorelimitations to discontinuousdiscretedatain contras t to continuous,whic hoccurfromanattempt to preserve informa tionwhenconve rting between continuo usanddisco ntinuous data.

A numbe rof conditions mustbe inc luded inorde r toderivethe Four iertransfo rm,Fora sample record oflength T seconds,itis dividedintoN equa llyspace d pointsbyh length, alsoknow nas the time step.Thesamplingrate,fs,of therecord is1/h.Therearealimited number of frequ encies thatcanrep resent the time series andis estab lished bytheNyquist frequency,fN.TheNyquistfrequency ishal fthe samplingfrequencyand isthus equa l to:

FourierSeries

Since the record lengthforthe time serieshasN number ofvalues to compute, but in the frequ encydomainthe Fouriercoefficientcontainsa rea landimag inaryvalue,thefrequency components mustbelimitedtoN/2discrete values.

Now to expand the Fourierseriesshown inEquation [2.13],wewill usei todescrib ethe time seriessummationand nforfrequeney domain .Therefore:

t=i·h (i=1,2,3...,N)

N f=fn(n=1,2,3··'2)

ReplacingWoby 21t!Tand k byn,Equation [2.13]will becom e Equation [2.33].

NI Z NIZ

ao "\' n2rrih "\' .n2rrih

Xi

=2+

~an cos-T-+~bnSm-T- 12.331

SinceT=h*Nand to show the spec ialcasewherethe constant terminEquation [2.33] can beremoved,whenconsidering themean valuecasewhenn=O,we canrewriteEquation [2.33]to:

NI Z NI Z

"\' n2rr i "\' n2rri

Xi

= fa

^anCOS

_N + fa

^bn

s inN

^(2.34)

Fourier Series

Equation [2.33] and [2.34] are the mainexpress ions usedtodefin ediscrete Fourierseries.

Sim ilarly, thecoefficientscan bederi vedfromEquations [2.14],[2.15]and [2.16]to be:

2"\:'

N n2rri an=N~XiCOSN

2"\:'

N n2rri bn=N~Xi SinN

Discrete Fourier Transform

Since there are limitation simplem ent edon discret edata,weknow that integralsmustbe repla cedwithsumma tionsand that thelimit s on the summa tionscannot beinfinite.In order tousethediscreteFouri er series derived in theprevious section,the transformX(!)mustbe complexsothatit wouldcontainbothnegativ eand positi vefrequencies suchas:

wherejsignifiesthecomplex comp onent.There forewecan rewrit e Eq uation [2.34] as

N

xi=

IN ^x,

^exp

en~rri)

⁽ⁱ⁼1,2, 3, ... ,N) 12.35) n=-2

FourierSeries

Due to thefact thattwo spec trumcomponentsarc generatedforeach real frequency, the sum mationofthetwo componentswillcausea doublingof amplitudeof thespectralseries produced.There forea scaling factor ofliNmustbeinclud edtoproperly account forthe increase,whichallows thetransformto be writtenas:

N 2

1~ (

jn2rri ) x;

=N.~xiexP ---;:;-

1=-2

12.361

Equatio ns [2.35] and [2.36] canbefurther simplifiedbynotingthatthey arc symmetrica l forthepositive and negative valuesofN . There foreXiandX,becom e:

Xi=

%, X

nexp

en~rri) ^{0, n}

= 0,1,2, ...,(N-1)) 12.37]

1~N- l

(jn 2rri)

x; =N

~^Xiexp ---;:;- (2.381

Equa tion [2.37] and [2.38] arc the discreteFourie r transform andinverse discrete Fourie r transform, respec tive ly.

Fast Fourier Transform

Duetothe slowcompu tationaltimeof theFourier transform , whichis aI: Iconversionof a data sequence,amethod called the fastFourier transform (FFT)is often usedinstead.Itwas describ edby CooleyandTurkey in 1965 (Beauchampand Yuen,1973)inamethodto case

Four ierSeries

machin e calcu lationandalso independ entl ydiscoveredbyDan ielson and Lan ezosin 194 2 (Presset. al.,1992)whower enot able to secmuchusc ofit when the most adva nce technology was ahand ca lculator.

The Cooley -T ukey meth odredu ces alargeprocess ofmat rixmultiplicat ion andadditio n to a seriesof spa rse matri ces.This elim ina tes much oftheredundancythat occurs usin g disc re teFourier tran sform s (OFT),whereman y ofthe produ cts arcrepea ted.The matri x multiplicationrequ iresN²com plex multipli cat ion sfor aOFT buta FFT can bedon eusin g Nlog-Nwhich mak es alargediffer en cein term s of computation time.For exam p le (Press et.aI., 1992 ) N=10⁶will tak e roug hly30 seconds forNlog-Nand2 weeks forN² com putatio n time onamicro second cycletimecom puter.

Dan ielson andLanezos(Presset. aI., 1992)provi de oneoftheclearestderivati on s ofthe FFT algo rithm.They describ ethelen gth T of a signa lXican be sp lit intotwodiscrete FourierTra nsfo rmsoflen gthT/2,one beinganeven- indexe d(i=2m) point s ,X2m,and the other bein g odd-indexe d(i=2m+ I),X2m+l.After thetwoDFTs arccom puted,they arc comb ined toprodu cethe fu llsequenceas show ninEqua tion [2.39] , which is expa nde d fromEqua tio n [2.38] ,notingthatthescaling facto r isnot sho wn fo r sim p licity.

FourierSeries

N/Z-l

\""' ( jn2 rr(2 m) )

= ~oXzm exp -- -N-

12.39]

N/ Z-l

\""' ( jn2rr(2 m + l) )

+

~oX(Zm+l)eXP- - - N - -

Takingouttheeommo n multiplier exp (-~)ofthe odd-i ndexe dsumma tion,usingthe e(,,+b)=e"eb identi ty,and simplifyi ngwe eanclearl y see theevenandoddindexed parts that makcupthc x.funct ion:

~-l

^{( 2rrjnm)}

=

L xzm exp

-~

m=O 2

~-l

^{( )}

2rrjn 2rrjnm

+exP(- ~) LX(zm+1)exp ":»:

m=O 2

=Feven

+

exp (-

2~n).

Fod d

12.401

One of the easiest ways to eomp utetheFFTistouse a radix of 2 for the numberof inputs N,ie.N=2^Pwhere P=1,2,3...co,Theadvantageoftheradix of 2, asdescribed by

FourierSeries

Beauchamp and Yuen (1973),isdueto someoftheterm s ofthe commo n multiplier,shown inEquation [2.40],redu ceto either 1or -I which avoidseven further comp lexarithme tic.

The struc tureof the computationof the FFT isdescribedinsection 4.2.

Spectra lAna lys is

2.3. Spec tra lAnalysis

Theprevious section describ esthe valueof working in thefrequ encydom ainusing a series of values.This section developstheusefulness ofthefrequencydom ainin term s ofan electro nicsigna l.

Amplitudeand thepower ofthe signa larcthemost frequently used and useful charac teristicsof asigna l in the freq uency dom ain and weredefin edinsection2.1.Each charac teristic pertainsto a certain frequ en cy and when combinedcrea tea wave spec trum as showninFigure2-3.Thecharac teris ticsof a signa lcreatewhatisknown asa power spec trumandisanaverage quantit y which describesthe energy that is alloca tedatassorted frequencies.

Figure2-3-Wave Spectr u m (Voogt 2005)

Spec tra lAna lys isallowscertaincharac ter isticsof a signa l to be calculatedwhichwould otherwise bequitedifficult.For example,asnotedby Beauc hampandYuen(1973), the difficult y of calculating thetotal energyof signa lscan beredu ced by integ rating thepower spectra ldensitysince thepower at individu alfrequ encies can be easilyfound.Another

Spectral Analysis

useful application ofspec tral analysisisdeterminin g the relation shipbetwe en correlation andspectral density,in which one quantitycanbefoundfromone anoth er,even ifthe signa l isimmeasurablein the timedom ain.Thelink betw eenthe two quantiti eshasbeen usefu lforfindinganalogiesof theoriesin the subjec tofsignalanalysis.Beauchamp and Yuen (1973) notethatwecannot measure visible light as atime series, butit ispossibleto measurethe spec trum of alight source.Thistraitbecom esparticularlyusefulwhen incorporatin gfast Fourie r transform algorithms.

Beauchamp and Yucn(1973)note several techniqu esusedfordeterminin gthepower spectral density:

• Dircet Fourier transform:Computes Fouriertransform of thetime seriesand then themean-square valu ei s ealculatcd.

• Indirectmethod:TheFouri ertransform of the auto-correlation functionis calculated and thenthe spec tral densityfunctionisderived.

• Band-p assfiltering:A method of filtering out unwanteddata froma rangeof frequencies.

The direct methodwasusedin thisresearch asismore commonafter thedevelopm ent of the FFTsubroutinessinec this allowed for amuch fastercomputational time.Aband-pass filter wasusedto smoo th the spectral density curve toproduecreliable estima tes.

SpectralAnalysis

PowerSpectru m Estimation

Thesquareof a rando msignalwillcalculateitsenergycontentand the squareof Fourier trans formata frequencyfwillgive the energycontentatf.Sincewe are dealin g witha rando m process,the valuesatthe Fourier transformwillfluctuate atanyfrequency.

Therefore twopieces of arand om signalat differenttimeswouldproducedifTerent Fourier trans form s,thusthe squareofa Fourier transform cannot deseribeits energycontent in general. But theuse ofanensembleaverage,asdescrib edinsection2.1,ofaFourier transform squareean beusedtodescrib etheimportanc e of the Fouriertransformatany frequ ency.There fore we ean describ ethepower spec trum,or auto-spect rum,as:

S(f)=<

IX CfW

>

whereX(I) isthe Fourier trans form asderived in thepreviou s section.Inorde r toprovethat thetotalpower contained inX(I)is equal tothat of x(t),Parsevaltheoremisderived.Real functionsx(t)andyet)are multiplied andare expresse d in term s of Fourier transformusing Equation [2.30]:

x( t)y( t)=x(t)

f

^{Y( f) exp(j wt).df}

Takingthe integralof Equatio n [2.41]we get:

12.411

f

^x(t)y(t)dt=

f {f

^{x(t) Y( f)exp(jw t)df }d t 12.421}

And rearran ging Equation [2.42]we get:

Spec tra lAna lys is

J

x(t )y(t) dt=

J

Y( f)

U

^x(t)exp(jw t) dt} df 12.43)

whe rethe bracke ted partinEqua tion [2.4 3]isthe com p lexconj uga teof X(I) andthus Eq uat ion [2.43] can berewr itten as Equat ion [2.44].

J

x(t) y(t) dt=

J

X' Cf )Y Cf )df Lettingx(t)=y(t)andX(I)=Y(f)then equa tion becom es:

12.441

J

x²⁽t)dt

= J

X'Cf)XCf) d f

= J IXCf)1

²df (2.45J

When the ense mb leaverages are tak en on both sides inEqua tion [2.4 5], we get:

J

^{(x2( t )) dt}

= J

^{S(f)df 12.461}

Eq uatio n [2.46]pro ves thatthe power spec tra ldens ityaddcd upforallfreq uencieseq ua ls the average powerof x added upovertime.

Auto-S pectrum and AmplitudeSpectrum

Theauto-s pec tru mand theamplitude spect ru mare imp ort ant propert iesof signa ls thatuse thc DFT coefficientsto calculateoneof the othe r.Theauto-spec tru mwasdescrib ed earlier as thepow cr spec trumand the am plitudespectru m is thespec tru mofmodulu s ofthe coefficie ntsobta ine dfromthe FFT. TheFFTprodu ces a two-side dspec tru mincom plex formand mustbe conve rted to polarandscaled by the len gth oftherecordinorde r to obtainthemagni tudeand phase (Nationa l Instrum ent,2009).Toillustrate this relation ship

Spect ralAna lysis

thewaveprofil e,~p,obse rvedatawaveprobeis shown inEquation [2.4 7] (Mansa rdand Funke,1980).

f

^{[2rrkt ]}

C;p

=

~Ap,k'sin----r-+ap,k (2.47)

whichlooks sim ilar to Equation [2.37],exceptit isden otedusin gthe sine notat ion.ApJis theFou rier coefficient for frequency, T isthelength of the waveprofile,Up.kisthephase rela tivetothetime origi n,andNistheupperlimitofthe summation whichis related tothe maximum significant frequen cy compo nent.TheFouriercoefficientsand theirphases can be expresse din polar or rectan gul arform asindicatedin equation s andarestatedasin Equation [2.48] and [2.49],respecti vely:

Bp,k=Ap,k.cosap,k

+

iAp,k'sinap,k

12.481

12.491

whereBpJisthe FourierTrans formof~p.k.The FFT subro utine that isusedforthis research uses the rectangularform,and thususingtheprop erty ofcos²0+si n²0=I,scaling by T,and renam ingBp.ktobetheamplitud e spec traAs(k,t) we get:

The phase spect rum is calculatedas:

_ -1[(Ap,k)imag]

Es- ta n

- ( A )

C-rr...rr) p,k rea1

SpectralAnalysis

(2.501

Themagnitud e and phase spec trum arcin radians.The auto-spec tracan now beusedto calculate the auto-spec traS.(k,f) using Equation [2.50]:

(2.51J

or using angular frequ ency cowehave

Whenthe auto-spec trumis calculated through alternative means,for example usin gthe indirectmethod,themagnitude spec trumcan be calculated bytheinverse of Equation [2.51] oras shown by:

AsCk,w)=~2.SaCk ,w).dw

Since the FFT isprodu cing atwosidedspectrum, theamplitude spectrum is onlyshowing thehalfpeak ofthe energyspreadoutover thepositiv e and negative frequencies toprodu ce amirrorimage,as shown inFigure2-4.

Spec tra l Analysis

Figure 2-4-A two sided line spectrum (National Instrument, 2009)

To correct themirrorin g effect,the amplitude mustbe multipliedby 2forhalf of the freq uencies,ie. freq uencyofi=I toN/2-1,andamplitudesat i=Oand N/2totheirorigin al amp litudesand discardtherest of theamplitudesbylettingthem equa l to 0 (Na tional Instrum ent,2009).

The spec tra lcharacte risticscan berelatedtothetime series withthe usc ofthepreviou sly menti oned root mean square,shown by Equation [2.52],and the spec tral mom ent ,mil.The spec tral mom ent tothenthpower hasthefollowin g formulation (Wave Spec tra , FORMS YS):

12.521

Thezerothmom ent ,mn,isthemostimportant as itgivesthe equiva lent tothe areaunderthe wavespect rumcurveandit is equa l tothe varianceof the time series,orthe root mean square,as show n by Eq uation [2.54]. Equation [2.52]becom es Eq uation [2.53] for mo.

Spec tra lAna lysis

rmso=~ (2.541

Wave Splitting Theo ry

2.4. Wave Splitting Theory Introduction

Wave splitting isdon ebyfirstmeasurin gthe wavesatanassortme ntof locationsina modelbasin and thenusing ofawavemodel withregress ion techn iquestodistingu ishhow much energyis coming fromthewavemakertothebeach andviceversa.Thelow frequ ency wave,associatedwith the secondorderwave,can beidentifi edbythe wave model. The deviceusedtorecordthewaveis awaveprobeinwhich there can bemany placed at various location sin thebasin.

The follow ingenv iro nme ntalbasineffectscause lowfrequ ency waves (Voog t,2005):

I) Generat ingwavesusing a Flat Wave Flap.This type ofwave flap cancause amism atchbetween generated water veloc ities ina genera tedwave at the wave flap.Thiserro rcan beminimizedby applyinga secondorder correc tion tothewave which isfurtherdiscussed inChapter3.

2)Reflection s ofthewavesduetothefinite sizeofthebasin.Minimi zedby inputtinga parabo lic beachbut docsnot suffic ientlydampen thelong wave compo nents.

3) Shoa lingeffectscreated bythebathymetry ofthebasinbottom.Significant in modelin g sha llowwatersituationsas wave set-dow nwill reflect asfree wavesfromtheparaboliebeaeh .

Thewave can be split into four different categories:inc identfree, incidentbound,reflected fr ee wave,and reflectedboundwave,as shown by Eq uation [2.55]. The reflectedbound

Wave SplittingTheory

waveisneglecteddue toitsinsignifi cant sizeas compared withotherwave catego ries (Voog t,2005).

C;=C;incJ ree+C;inc,bou nd+ C;refJ r ee+C;ref,bou nct (2.551

where:

C;:Thetotalwave elevation.

C;inc.frcc:Incidentfree waves,propagatin gfrom the waveflapstothebeach.

C;inc.bollnd:Wavefound bound to the incident free waves, also knownas wave set-down.

C;rcr.rrcc:Reflectedfreewavefrom the waveflaptothebeach.

C;rc( bollnd:Waveboundedto thereflectedwave.Aspreviouslymentionedthisterm is

neglectedin the final waveclevationcalculati on.

The speed of the free wavesdiffersfrom the bound waves and theyhavedifferent directions,This allows thewavecomponent stobedistinguishedfromone another.

Equations[2.56] and [2.57]describethewave velocities forthe free and bound waves, respectively,in whichthebound usesthedifferenceof the frequency(to)and wavenumber (k).

<-r:

w ^12.561

12.571

Wave Splitting Theor y

where

erree :Spee dofthefreewaves

Cbound:Speedoftheboundwaves w: Wave frequency k:Wavenumber bW:Parti alwavefrequ ency Sk:Partial wavenumber

Splitting thewavesmustbedon efor each wave freq uencyand thewaveelevationfor at least 3 spatial position s(Voogt, 2005).The incid ent and boundfreewaves at onespeci fic frequencyare show n inEquations [2.58],[2.59] , and [2.60].

C;inc,free=C;if

cos (

wt-kincx

+

Eif)

C;ref,free=C;rfcos(wt

+

kref x

+

Erf)

C;inc,bound

=

C;ibcos ( flw t - tskx

+

Eib)

whereLrefers to arand omphase ang le betw een-J[andJ[for eachwavetype.

(2.581

[2.591

(2.60)

Inorde r tofindthe frequ ency,ui,and wav enumber,k,the dispersionrelationship can be used as show n by Equation [2.61].Notethatthedifference frequency,flw, and difference

WaveSplittingTheor y

wavenumber,Ak, donot satisfy thedispersion equa tion.Thetermgrefersto gravity

w²

=

gktankh 12.611

TheNewton-R aph sonmeth odwasthenumericalmeth odusedto solve the wavenumber fro m thedispersionequation,whichisdescribedinsection 4.3.Since theinformati on pertain ingtothewave components islimited , it is assumed thattheboundwavepropagates witha groupspee d thatcorre spond stothepeakperiodof theenergy spectru m(ie.(t)=(t)p).

Thisisreasonabl ewith thefurther assumptionofanarr owbanded spec trum (Voogt, 200 5).

AFouriertrans form willgive theindividualfrequen cy compo nentsof thewave.Each frequeney eomponentean bewrittenas Eq uation [2.62]whereAand Bare known coefficients.

SubstitutingEquations [2.62],[2.58],[2.59] , and [2.60] intoEquation [2.55] ,thetotalwave eleva tionequation isfound:

Wave Splitting Theory

(2.631

Thetenu sinEquation [2.63] can becollected and the unknownterms can beisolated and show n inEquation [2.64]asvector

y :

12.641

Collectingthecosineandsine tenu s,two equationscan be found for A and Bas shown by Equations [2.65] and [2.66] and isprovedinAppendixA.

Wave Splitting Theory

12.661 .y

Equations [2.65]and [2.66] arc well conditioned thenthe solutioncan be found bymatrix inversion.Singular ValueDecomp osition wasthe computational methodusedformatrix inversion and isdescribedinsection 4.4.Thedistanceto which the waveprobes arc separated relativetothe wave length has great influence on the condition number,the case of digitalcomputation. The conditioningworse ns(ie.the conditionnumberwill increase) when thedistancebetweentwoprobes arc integermultipleof halfthe wavelength.Voogt (2005) discuses thatthe optimumspacingofthewaveprobesis 0.25timesthe wavelength.

Whenthere arc largedistancesbetweentheprobes, aliasingeffectscanoccurwhere the signalfromthe probesbecom esindistinguishable.Thereforethedistancesbetweenthe probesmustbelimitedbut must also belarge enough to separate thefree and bound waves travellingin the same direction.

Wave Splitting Theory

ApplyingWave Splitting Theory

When applying the wavesplitting theoryto amodeltestbasin,thereis aprocedurethatcan be followedasdiscussedbelow shown inFigure2-5 (Dij k2007).

Step I:Install waveprobesinoptimum locations

Step 2:Comme nce measurement of wavedataduring calibration;including startof wavemarkerto some timeafterwavermakeris shut down Step 3:Prepare input forwave splitting tool

Step 4:Runwavesplitting tool Step 5:Check datafrom wavesplitting result s

Step 6:Attainwave forces of incid ent and reflectedwaveswhicharcbased on reflected database

Step 7:Solveequationsof motionin thetimedom ain

Step 8:Do compariso n betweennumerical and measured vesselresponse in the basin

Wave SplittingTheo ry

Step1·

S~"Cljon ci w""" prOOe po.sifun5

In cident

Free Reflected

Free

Figure2-5 - Flow Diagram of StepsforUseofWave Splitting (Dijk2007)

Itshouldbenotedthatattention mustbepaidtothedamp ing valuesofthefloat ing vessel to ensure thatthenumericalresultsmatchupwith themeasu red results.

AnalyticalModelofWave Set-do wn

2.5. Analytical Model ofWave Set-down

Ananalytica l modelisusedto calculate thetheoretical set-do wn in the wavesas an alternate method of calculating set-downandcanalso be compared tothe set-do wn that was decomposed fromthemeasured waveusingthewave splitting tool.

As previouslystated, the secondorde rcomponentofthewavescontainsalowfrequ ency component known aswave set-down.Itisa quad raticfunction ofthewave amplitudesand can befoundusing aquadratic transfe r function whichgives the wave amplitudesand phases of the set-down inatravellin g wave group(Huijs ma ns,2002).Inorde r todevelop the analyticalequations, potential theory is assumed.

The velocity potenti als offluid flowin waves arca summationofthedifferent approx imationorde rs thatmakeupa wave andcan bedescribed as:

The first orde rapprox imationofthe equation,</>(1 ),isusuallythemost significa nt component,espec ially in deepwater situations. Huijsman s (2002) describesthebound ary conditions forthefirst orderapproximatio n:

• Thecontinuityequationstates thatthegradie ntof the firstorderapprox imationis equal to zero everywherein thefluid domain.

• The free surfacecondition:thebottompotent ial addedwith thetoppotentialmust be equal tozero at thefree sur face.

Ana lytica l Model of Wave Set-dow n

• The botton condition:

Using the prev ious lystated bound ary conditionforaregul arplane wave prog ress ionwe can get anexpress ion forthefirstorder wavepotenti al(Huijsm an s, 2002),show nby Eq uatio n [2.67].

For thesecondorde rapproxi ma tion, Huijsm ans (2002 )states thebound ary conditionsas:

• The continuityequationstates thatthe gradientof thesecondorde rapprox ima tionis equal to zero everywhere in thefluid dom ain.

/::"rfyCZl =0

• The freesurfacecondition,shown by Eq uatio n [2.68], shows thatthe secondorder potential isdepend ent on thefirst order potent ial.

• Thebottomboundarycond ition:

rfJY

^l

=

O, at z=-h

To use theboundaryconditions to deriveanexpress ion forthe secondorder potential,we canfirst deri ve the first order potent ial associatedwitha regularwave group that cons istsof

Ana lytica l Model of Wave Set-dow n

tworegularwaves com ingfrom twodiffe rentdirect ions (Huijs ma ns,2002) show n by Equation [2.69].

where icannotequa l k.

lIuij sm ans (2002)thendescribesthelow frequencycomponentof thesecondorder potent ialbetweenthetworegul ar waves from differentdirections as show n by Equation [2.70].

2 2 2 2 2 (I)(I )_ cosh (K; -Kj)(Z

+

d) . cP()=

IIII

^{(;k (jl Aijk1 cosh (K'-K.)d expz((K; COSJ.lk}

;; l j; l k; l l; l I J

12.701

The unknownAijklterm canbefoundbyutilizingthe free surfacecondi tion for thesecond orde r potential,show n inEquat ion [2.68].ThusforAijkl we get:

1 Bil k1+Cil k1 2

2(Wi-Wj)2- (Ki-Kj)g ta n h(K; -Kj)h9

Ana lytica lModelofWavc Set-down

ZKiKj(Wi - Wj)(COS(Ilk-Ila

+

tanhKihtanhKjh) WiWj

Thewave set-downisthelow frequencycompo nentofthe secondorderwaveheight on the free surface,given by Equat ion[2.71].

Therefo rethewave set-down is obtainedas Equation [2.72].

2 2 2 2

((2)=

LLLL

^{ZCJ(;/lJDil klcos((KiCOSIlk}

i=lj=lk =ll=l

-KjCOSlll )X

12.711

12.721

whereDilk/isa transferfunctio nthatincl udesAijkl'Bijkl' and[ilklcompo nents.Dilklis expressedas:

Pow er Spec trumEstima tion

2.6. Power Spec trumEstima tion

Apower spec trumestimation theorywasusedinorder toprodu ce smoo th pow er spectra since thewave splitting methodprodu cednoisy spec tragraphsand madetherealproperti es of the graphs difficultto distin gui sh.Ameth odtoredu celarge variancefrom a

I xol

setof dataisto multiplyit by a weightingfunction W(l) (Bea uchampand Yucn ,1973). The processistypicall y calledwindo w ingand theweight ingfunctionis called thewindow, which produ cesasmoo thedversionof a spec trum,The type of wind owisdiscussedlaterin this section.

The meth odusedtoproducethe spec traisbased on theWelchmethod (Welch 17).The length ofthe record was sectioned into overlappingsegme ntsandamod ifiedperiod ogram was foundforeachsection bywindowin gthe origina l record.Then the mod ifi ed periodog ramwasthen average d.This methodrequir esIcssiterati onsthan other meth ods (We lch 17),duetothe sho rter record segme nts for computations.

Forthismeth od,a record has alength of:

X(J),j =0•...,N-1

whereN isthetotalnumber ofrecordpoint s.Overlappingsegme nts,withalength ofL, are separate d bydistanceM.There areatotal ofKsegme nts.The firstfew segmentswill have the form:

Xl (J)=X(J),j=0,...•L-1 Xz(J)

=

X(J

+

M).j

=

0, ...•L-1 X3(J)

=

X(J

+

2*M), j

=

0,..,L - 1

Pow er Spectru mEstima tion

Therefo re agenera lform ula canbe stated as

XK(j)=XU

+

(K-1) M),j=0,..,L-1 NotethatL+(K-I)M=N.

Foreachofthe segme nts, themod ifiedpcriodogram , another termfor estima ting the spec tra l density ofatimeseries,can be found bymultiplyin git byawindow ing funct ion.A harmingfunction , orwindow ,wasusedwhichhas acosine shapeand hasthefollow ing formul at ion:

W(t)=

~+~coset)

where Nisthetotalnumber ofpoint s (Beauchampand Yucn ,1973).

Inordertodeterm inethetotal number ofsegme nts K,thelen gthofM wasfirst calculated, which wasbased on the Nyq uistfrequencyand the frequency resoluti ondw .Thefrequ ency resolutionwasassum edinorde r for the resolution tobelow enoughto smoo th the fluctuation sbuthigh enoug h nottounderestim atetheproperti es of the spec tra.The Nyqui st frequency,aspreviouslymenti oned,isbased on thetime stepand^7[.Thus Mwas calculated to bethe Nyq uistfreq uency dividedbytheresoluti on (Dij k,2007 ). Kcan thenbefound as N/M- I.The overlapping len gthL is calculatedas 2*M.

For eachsegme ntof lengthL themodifi edperiod ogramis calculatedand thenthefinite Fourie r transform s arefound as show ninEq uation[2.73].

Power Spec trumEstima tion

L- l

AK(n)=

~ I

^{XKU)WU) e-2kijn/L}

j=O

We can thus calculate themodifiedpcriodograms as:

12.731

whichis amodifi ed equation based on Welsh(J976)and Dijk(2007).Thespectralestimate isthe averageof theperiodograms as show n inEquation [2.74].

where8/3is anaveraging function asrecomm end edbyDijk(200 7).

(2.74]