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Oblique wave reflection from a model rubble-mound breakwater Papps, David Arnold 1992

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OBLIQUE WAVE REFLECTION FROM AMODEL RUBBLE-MOUND BREAKWATERbyDavid Arnold PappsB. E. (hons.), University of Canterbury, 1990A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Civil EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJanuary 1992© David Arnold Papps, 1992In presenting this thesis in the partial fulfilment of the requirements for an advanced degree at theUniversity of British Columbia, I agree that the Library shall make it freely available for referenceand study. I further agree that permission for extensive copying of this thesis for scholarlypurposes may be granted by the head of my department or by his or her representatives. It isunderstood that copying or publication of this thesis for financial gain shall not be allowed withoutmy written permission.Department of Civil EngineeringThe University of British Columbia2324 Main MallVancouver, CanadaDate:irciettAcs7 2_.?, /792AbstractThis thesis investigates the characteristics of oblique wave reflection from a model rubble-moundbreakwater. An experimental investigation of the reflection of obliquely incident regular, irregularand multi-directional waves, undertaken at the Hydraulics Laboratory of the National ResearchCouncil of Canada in Ottawa, is described. A method of analysis, which uses a least squares fit tomeasurements from three wave probes to estimate the reflection coefficient and the reflected phaselag for regular wave tests, is extended to the case of oblique wave reflection. In the regular wavetests, the wave height, wave period, and angle of incidence were varied in order to determinerelationships between the reflection characteristics and parameters describing the incident wavecharacteristics. Results show that both the reflection coefficient and the reflected phase lag aredependent on the depth to wavelength ratio and the angle of incidence. Reflection coefficientsestimated from the analysis of irregular wave tests are also presented and are compared to reflectioncoefficients measured in regular wave tests, showing good agreement. For multi-directionalwaves, directional spectra of the incident and reflected wave fields were obtained using a maximumlikelihood fit to measurements from five wave probes. Directionality results are presented for onemulti-directional wave test showing increased directional spreading of the reflected wave field ascompared to the incident wave field.iiTable of ContentspageAbstract ^ i iList of Tables vList of Figures ^ v iList of Photographs i xAcknowledgement ^ x1. INTRODUCTION 11.1 General^ 11.2 Literature Review^ 21.3 Scope of Present Investigation ^ 52. METHODOLOGY^ 72.1 Theoretical Background^ 72.1.1 Regular Uni-directional Waves ^ 72.1.2 Irregular Uni-directional Waves 92.1.3 Irregular Multi-directional Waves ^ 112.2 Laboratory Facilities ^ 122.2.1 Wave Basin 122.2.2 Wave Generator ^ 132.2.3 Wave Generation and Data Acquisition ^ 142.3 Test Set-up ^ 152.3.1 Basin Layout ^ 152.3.2 Breakwater 172.3.3 Wave Measurement ^ 182.4 Test Programme^ 223. ANALYSIS TECHNIQUES ^ 2 5iiiiv3.1 Regular Uni-Directional Reflection Analysis ^ 253.1.1 Derivation of Least Squares Method 253.1.2 Application of Least Squares Method ^ 293.2 Irregular Wave Analysis ^ 313.3 Analysis of Wave Directionality 324. RESULTS AND DISCUSSION^ 3 44.1 General Observations 344.1.1 Wave-Breakwater Interaction ^ 3 64.2 Regular Uni-Directional Wave Tests 374.2.1 Performance of Sinusoidal Fitting Program ^ 374.2.2 Accuracy of the Least Squares Method 404. 2.3 Reflection Coefficient ^ 414.2.4 Reflected Phase Lag 474.3 Irregular Wave Tests ^ 5 34.3.1 Spectral Density Functions ^ 534.3.2 Comparison of Irregular and Regular Reflection Coefficients ^ 564.4 Directionality Results ^ 595. CONCLUSIONS AND RECOMMENDATIONS^ 6 35.1 Reflection of Regular, Obliquely Incident Waves 635.2 Reflection of Irregular, Obliquely Incident Waves^ 645.3 Directionality of Incident and Reflected Wave Fields 665.4 Recommendations for Further Study ^ 66References ^ 6 7List of Tablespage2.1 Test Programme ^ 22VList of Figurespage2.1 Definition sketch of uni-directional wave reflection ^ 82.2 Plan view of the wave basin ^ 122.3 Basin layout for 0 = 0° 162.4 Basin layout for 0 = 60° ^ 162.5 Breakwater cross-section 182.6 Approximate dimensions of the wave probe array ^ 203.1 Location of wave probes ^ 304.1 Sample of water surface elevation for test 3 (T = 1.6 s, H = 10 cm) and 0 = 45°,with and without the breakwater in place ^ 344.2 Sample of beginning of wave record showing portion selected for analysis,test 3 (H = 10 cm, T = 1.6 s), 0 = 60° 354.3 Results of the sinusoidal fitting program APHRES, test 1 (H = 10 cm, T = 1.0 s),0 = 45° ^ 394.4 Results of the sinusoidal fitting program APHRES, test 3 (H = 10 cm, T = 1.6 s),0 = 45° 394.5 Results of the sinusoidal fitting program APHRES, test 5 (H = 10 cm, T = 2.4 s),0 = 45° ^ 404.6 Reflection coefficient vs. wave steepness, d/gT2 = 0.020, 0 = 0°, 45° ^ 434.7 Reflection coefficient vs. d/gT2, 0 = 0° ^ 434.8 Reflection coefficient vs. d/gT2, 0 = 30° 444.9 Reflection coefficient vs. d/gT2 , 0 = 45° ^ 444.10 Reflection coefficient vs. d/gT2, 0 = 60° 454.11 Reflection coefficient vs. d/gT2 ^ 454.12 Reflection coefficient vs. angle of incidence, high d/gT 2 ^ 46vipage4.13 Reflection coefficient vs. angle of incidence, low d/gT2 ^ 464.14 Sketch showing effect of phase lag on partial standing wave position ^ 474.15 Reflected phase lag vs. wave steepness, d/gT 2 = 0.020, 0 = 0°, 45° 494.16 Reflected phase lag vs. d/gT2, 0 = 0° ^ 494.17 Reflected phase lag vs. d/gT2, 0 = 30° 504.18 Reflected phase lag vs. d/gT2, 0 = 45° ^ 504.19 Reflected phase lag vs. d/gT2, 0 = 60° 514.20 Reflected phase lag vs. d/gT2 ^ 514.21 Standing wave shift vs. d/gT2 524.22 Incident and reflected spectral density, test 10 (Hs = 6 cm, Tp = 1.6 s) and0 = 0° ^ 544.23 Incident and reflected spectral density, test 11 (H s = 12 cm, Tp = 1.6 s) and0 = 0° ^ 544.24 Incident and reflected spectral density, test 10 (H s = 6 cm, Tp = 1.6 s) and0 = 45° ^ 554.25 Incident and reflected spectral density, test 11 (H s = 12 cm, Tp = 1.6 s) and0 = 45° ^ 554.26 Comparison of regular and irregular reflection coefficients, test 10 (H s = 6 cm,Tp = 1.6 s) and 0 = 0° ^ 574.27 Comparison of regular and irregular reflection coefficients, test 11 (H s = 12 cm,Tp = 1.6 s) and 0 = 0° ^ 574.28 Comparison of regular and irregular reflection coefficients, test 10 (H s = 6 cm,Tp = 1.6 s) and 0 = 45° ^ 584.29 Comparison of regular and irregular reflection coefficients, test 11 (H s = 12 cm,Tp = 1.6 s) and 0 = 45° ^ 584.30 Incident and reflected directional spreading functions for test 13, Op = 30° ^ 61viipage4.31 Definition sketch showing directions expressed as angles ^ 614.32 Standard deviation of incident and reflected spreading functions for test 13,Op = 30° ^ 62viiiList of Photographspage2.1 View of wave basin showing breakwater and guidewall set up for A = 0°^ 172.2 View of test 17 with 0 = 45°^ 19ixAcknowledgementThe author would like to thank his supervisor Dr. M. Isaacson for his help and encouragementthroughout this project, in particular for his suggestions and guidance in the planning of theexperiments, and his advice in the preparation of the thesis.The author would also like to express his immense gratitude to the Hydraulics Laboratory of theNational Research Council of Canada for giving him the opportunity to carry out the experimentalwork in their excellent facilities. The support of the entire staff at the Hydraulics Laboratory isgreatly appreciated. In particular, the author wishes to thank Dr. E. P. D. Mansard, Dr. 0.Nwogu and Mr. E. R. Funke for their advice on the experimental work and their efforts in theorganisation of the experiments, and Mr. B. Atkins and the technicians at the HydraulicsLaboratory for their assistance in the operation of the experimental programme.Financial support in the form of a fellowship from the University of British Columbia and ascholarship from the J. Templin Fund is gratefully acknowledged.xChapter 1INTRODUCTION1.1 GeneralWhen a train of waves approaches a shoreline or coastal structure a number of physical processeschange the form of the waves to produce a particular water surface profile in the vicinity of thewater boundary. These processes include wave diffraction around barriers, wave refraction andshoaling over sea bottom topography, wave breaking, and wave run-up on and reflection byboundaries. The combination of these effects, each changing the waves in a particular manner andeach to a varying extent, defines the resulting wave condition. Wave reflection contributes greatlyto the final wave condition in situations where a body of water is bounded by an artificialbreakwater rather than a gently sloping beach. In coastal waters, waves reflected by a shorelinestructure or steeply sloping beach can result in agitated sea states which makes the navigation ofvessels difficult and which may increase sediment erosion. Within harbours, wave reflection bybreakwaters and harbour boundaries can give rise to confused and agitated sea states which maymake navigation through entrance channels difficult, and which have an undesirable effect on themotions of moored vessels.In the design of coastal engineering works it is often desirable to be able to predict the wavecondition resulting from a particular breakwater configuration without resorting to a laboratorymodel. One method of analysing wave conditions in the coastal region, favoured for its efficiencyof time and money, is that of numerical modelling. A numerical model can have varying degreesof accuracy, depending on the number and combination of the above processes that are included,and in order to predict wave conditions in a harbour accurately, wave reflection effects shouldgenerally be taken into account.1When a regular wave train strikes a boundary it will give rise to a reflected wave train of certainheight, direction and phase. For waves reflected by rough sloping surfaces there may be somescattering of direction in the reflected wave, and the reflected wave energy may be shifted to higherfrequencies, caused by wave breaking for example. By assuming that a regular wave train reflectsfrom a boundary at an angle equal to the angle of wave incidence and that directional and frequencyspreading does not occur, the height and phase of the reflected wave train are sufficient to describeit completely. A more detailed approach would not make the above assumptions but would requirethat the directional spectrum of the reflected waves be known. The present study is concerned withthe characteristics of reflected waves from a straight boundary. Both the above approaches havebeen considered, with the characteristics of oblique wave reflection from a model rubble-moundbreakwater investigated by means of physical modelling.1.2 Literature ReviewThe action of normally incident regular waves on rubble-mound breakwaters has been studied quiteextensively. Wave reflection under such conditions is generally described by the reflectioncoefficient Kr, which is defined as the ratio of reflected wave height H r to incident wave height Hi :HrKr =HiA number of researchers have carried out physical model tests and developed numerical models todetermine the reflection coefficient under different conditions.The use of the surf similarity parameter to determine the reflection characteristics of waves wasfirst suggested by Battjes (1974). This was initially introduced in the context of waves incident ona rigid, plane, impermeable slope extending to deep water and is defined as:1.4 - ^1.0^(1.2)Hicot a2where a is the angle of the slope above horizontal and Lo is the deep water wavelength.Seelig and Ahrens (1981) have argued that this parameter may be re-defined so as to apply to arubble-mound breakwater, such that a is taken instead as the angle of the breakwater face abovethe horizontal. On the basis of laboratory tests, Seelig and Ahrens presented a curve of K r as afunction of the surf similarity parameter. Seelig and Ahrens argue that the calculated value of Krshould be adjusted to account for the relative water depth in front of the breakwater, the thicknessof the armour layer and underlayer and the relative size of the armour units.A review of wave reflections from coastal structures by Allsop and Hettiarachchi (1988) presentsplots of Kr as a function of for rubble-mound breakwaters armoured with different types of rockor concrete units.For the case of reflection of normally incident irregular waves, methods have been proposed(Thornton and Calhoun, 1972, Goda and Suzuki, 1976 and Mansard and Funke, 1980) to measurethe spectral density of incident and reflected wave trains in model tests. These spectra can in turnbe used to estimate the reflection coefficient as a function of wave frequency, K r(f):Sr(f)Kr(f) = All Siff)where Sr(f) and Si(f) are the reflected and incident spectral density respectively. An averagereflection coefficient K r can also be calculated from the spectra:(mo)r—Kr = Al (mo)i(1.4)where (mo )r and (mo)i are the zeroth moments of the reflected and incident wave spectrarespectively.Kobayashi et al. (1990) used an extension of the methods given by the above authors to separateincident and reflected wave trains in model tests of irregular wave reflection on rough impermeable3(1.3)slopes. The measured incident and reflected spectral densities were compared to those obtainedusing a numerical model, showing good agreement for the three tests carried out. Kobayashi usedthe relationship given in Equation (1.3) to plot reflection coefficient as a function of frequency forhis experiments. Measured values of Kr compared well with values provided by the numericalmodel. Kobayashi also measured and numerically predicted the phase difference between theincident and reflected wave trains at the toe of the slope.Different studies have concentrated on related effects such as wave transmission through abreakwater and wave run-up on a structure. For example, Sollitt and Cross (1972) presented astudy of wave reflection from and transmission through permeable breakwaters. They linearizedthe unsteady equations of motion for flow through a porous medium to set up a potential flowproblem. Their solution to this provides values of Kr and transmission coefficient, K t. Thepredicted values of Kr showed reasonable agreement with experimental results for waves of lowsteepness.As stated previously, it is necessary to provide details of the reflection of oblique waves in order todescribe more fully the process of wave reflection from a structure. Tautenhain et al. (1982)performed experiments to investigate the influence of incidence angle on wave run-up for waveincidence angles between 0° and 35°. It was found that for waves of small angles of incidence theamount of run-up was slightly higher than that for normally incident waves. No results wereavailable on the reflection coefficients occuring in these tests.Scale effects in rubble-mound breakwater models have been studied by Delmonte (1972) andWilson and Cross (1972). Delmonte presents experimental results showing the effect of Reynoldsnumber on the transmission coefficient for permeable breakwaters. Reynolds number effects areshown to diminish with increasing wave steepness. Wilson and Cross examine scale effects onreflection and transmission. Scale effects in rubble-mound breakwater models are generallyunderstood to be more important when considering wave transmission rather than wave reflection.4Reynolds number effects in wave reflection tests on rubble-mound breakwaters can be minimisedby using a minimum size of breakwater armour corresponding to a critical value of Reynoldsnumber.1.3 Scope of Present InvestigationA review of the literature has revealed a lack of experimental data on the reflection of obliquelyincident waves by coastal structures. This apparent void in the understanding of wave interactionwith coastal structures is a major hindrance in the analysis of wave conditions in harbours andmarinas. In view of this deficiency the objective of the present investigation is to study thereflection characteristics of waves obliquely incident on a model rubble-mound breakwater.In particular, the primary objective of this research is to measure the reflection coefficient Kr andphase lag 13 of the reflected waves due to regular uni-directional incident waves, and to determinerelationships providing Kr and 13 as functions of the incident wave characteristics and the angle ofwave incidence.The secondary objective of this research is to use measurements of the frequency and directionalspectra of the reflected wave field to provide comparisons between the characteristics of regularand irregular wave reflection, and to determine the amount of frequency and directional scatterpresent in the reflected wave field.Only one particular breakwater type was able to be tested within the scope of the present project.Since rubble-mound breakwaters are the most commonly used form of breakwater used in harbourconstruction and coastal engineering works, the experiments were carried out with a breakwater ofthis type. Rubble-mound breakwaters are comprised of one or more layers of large armouringrock over a core of finer stone material, and the particular breakwater characteristics employed inthis study were selected in order to model the reflection characteristics of breakwaters commonly5used in practice. It is hoped that the knowledge gained here will be of use in the design andanalysis of all breakwater structures.6Chapter 2METHODOLOGY2.1 Theoretical BackgroundIn this section, the background theory describing the reflection of oblique regular, irregular andmulti-directional waves is presented.2.1.1 Regular Uni-directional WavesFigure 2.1 provides a definition sketch for the case of regular uni-directional wave reflection.Using the notation defined in this figure, the water surface elevation tl adjacent to a breakwatersubjected to obliquely incident regular waves may be expressed as:Hi^ Hr= 2 cos ( kxcosOi + kysinei - o)t ) +^cos ( -kxcosej + kysin8i - cot + (3)^(2.1)where k is the wave number, co is the wave frequency, and Oi and Oi represent the incident andreflected wave angles.This equation assumes that linear wave theory holds and that there is no directional spreading orfrequency scattering of the reflected wave train. If it is further assumed that the angle of incidenceis equal to the angle of reflection, then the reflection characteristics of the breakwater may bedescribed by two parameters; the reflection coefficient K r, and phase lag 13. The assumption thatlinear theory holds is an assumption that is often made in the analysis of water waves and will notbe tested in this study, however the validity of the remaining assumptions made in this section willbe examined later in this thesis. On the basis of a dimensionless analysis the two quantities K r and(3 can be written as functions of independent dimensionless variables, as follows:d Hi 0.Kr, = f ( r—f2g '^"cot, type ) (2.2)7water surfaceintersectionreflected.wavesincidentwaveswhere d, g, T and L represent the water depth, the acceleration due to gravity, the wave period,and the wavelength respectively, and the reflection coefficient K r is defined as: Kr = Hr / Hi.The dimensionless groups are representative of the depth to wavelength ratio, the wave steepness,the angle of incidence, the breakwater slope and the breakwater type (taken to describe rock sizeand placement, breakwater permeability etc.) respectively.(a) Plan(b) ElevationFigure 2.1 - Definition sketch of uni-directional wave reflection82.1.2 Irregular Uni-directional WavesAn irregular signal, such as the water surface elevation signal due to an irregular wave train, maybe considered as periodic over a sufficiently long duration and hence may be represented by aFourier Series which contains components at multiples of the fundamental frequency, f o . Thismay be written as follows:00ii(t) = E An cos (2nnfot - On)^ (2.3)n=1where An are Fourier coefficients representing the amplitudes of each individual frequencycomponent, On are the random phase angles of each component, and fo is the fundamentalfrequency given by fo = 1/Tr, where Tr is the record length. The variance of the signal may bewritten as:0.01 ' 2 . 1 IA n12n=1(2.4)Thus I IAn 1 2 represents the contribution to the variance which is associated with the frequency2 component nfo . If the signal period is increased such that fo —> 0, with the signal now consideredto contain a continuous range of frequencies rather than discrete harmonics, the above summationmay be replaced with an integral and the variance may be written as:0067 = f S i (f) df^ (2.5)0where Sn (f) is known as the spectral density of r. S i (f) df represents the contribution to thevariance of the signal due to its content within the frequency range f to f + df. This may beexpressed as:fn+dfS11^2(f) df = I L IAn 1 2^(2.6)9From Equation (2.6), a relationship between the Fourier amplitudes and the spectral density can beobtained:An = -V 2S 1 (f)10^(2.7)For the case of a reflected wave field, Equation (2.3) may be extended to describe the water surfaceelevation due to the superposition of incident and reflected wave trains:11(t) = IAn cos (2mnfot - On)n=100^ 00= I, Ain cos (27cnfot - Oin) + I Arn cos (27mfot - Ord^(2.8)n=1 n=1It is apparent that, by separating the time series into contributions due to incident and reflectedwave trains, the water surface elevation may be described by incident and reflected spectraldensities, Si(f) and S r(f). If the incident spectral density and reflected spectral density are known,the reflection coefficient as a function of frequency Kr(f), can be found using the result given inEquation (2.7):Ar(f)Kr(f) — Ai(f)-v  S r (f)=Si(f)(2.9)In addition, an average reflection coefficient Kr can be calculated as follows:(mo)rKr = (mo)i(2.10)where (mo) r and (mo)i are the zeroth moments of the reflected and incident wave spectra10respectively.2.1.3 Irregular Multi-directional WavesThe water surface elevation due to an irregular multi-directional wave train may be expressed as thelinear summation of waves in a range of individual directions and a range of individual frequencies:CO^COi(t) =^I aid cos(kixcosOi + kiysinei - wit +^ (2.11)This sea state may be described using a directional spectral density in a similar way to thedescription of an irregular, uni-directional sea state by the spectral density. The variance of thewater surface elevation signal is considered to be made up of contributions from each frequencyrange f to f+df, and each direction range 0 to 0+0:2ir^0 = J^ S (f dO df110^0The spectral density of the directional spectrum Son (t0), can be expressed as follows:S11 (f = S (f) • D(f 0)(2.12)(2.13)where D(f,0) is the directional spreading function which is a function of frequency as well asdirection.The characteristics of the directional spreading function can be described by the mean direction ofwave propogation 0 and the standard deviation of the directional distribution cri. These are definedas follows:(f) = f D(f,0) e (113^ (2.14)110CLwave generator/ate30.0 mwave absorberEOEOOC,)126-FIC/20(1) = fD(f,0) (0-0)2 d0^ (2.15)The standard deviation of the directional distribution provides an indication of the degree ofdirectional spreading. As the value of 64 increases the directional spreading becomes greater.2.2 Laboratory FacilitiesThe experiments were conducted in the Hydraulics Laboratory of the National Research Council(NRC) of Canada in Ottawa, during July and August of 1991.2.2.1 Wave BasinThe wave basin used is 50 m wide, 30 m long, and 3 m deep with a 30 m segmented wave50.0 mFigure 2.2 - Plan view of the wave basingenerator located along the 50 m wall of the basin, as shown in Figure 2.2. The working area ofthe basin has been reduced to the 30 m width of the generator and a length of 19 m by thepositioning of a partition wall across the basin and the placement of wave absorbers around theperiphery of the basin. The absorbers are made up of vertical rows of perforated galvanised steelsheets. The porosity of the sheets decreases closer to the wall in order to extract more and moreenergy from the wave as it passes through each sheet. This design results in a reflection coefficientof less than 5% for the wave types used in this study (Jamieson and Mansard, 1987).The normal water depth used in this basin is 2.0 m. However, a shallower depth of 0.5 m wasused in the present study in order to reduce the height of the model breakwater.2.2.2 Wave GeneratorThe wave generator is capable of producing regular or irregular, multi-directional or uni-directionalwaves which can be directed at an angle normal or oblique to the generator face. In this study uni-directional regular, uni-directional irregular, and multi-directional irregular waves were generatedover a range of principal directions.The generator consists of 60 segments, each able to move independently of any other in a specifiedmanner so as to produce the desired wave field. The individual wave boards measure 0.5 m inwidth and 2.5 m in height. Each board is driven by a hydraulic actuator with a stroke length of 0.2m and rated static force of 45 kN. The actuators are connected to the wave boards by mechanicallinkages which provide stroke amplification and can be adjusted to operate the boards in eitherpiston mode, hinged flapper mode, or a combination of both. The 60 actuators are driven by ahydraulic power supply consisting of six pumps, each rated at 50 US gal/min and driven with a 75kW motor. The elevation of the wave generator paddles above the basin floor can be varied to suitthe particular water depth used. In this study, because of the small water depth, the paddles werelowered to the basin floor.132.2.3 Wave Generation and Data AcquisitionWave generation and data acquisition were carried out using the GEDAP software on a microVAXcomputer. GEDAP is a general purpose software system developed by the NRC HydraulicsLaboratory for the analysis and management of laboratory data, and includes real-time experimentalcontrol and data acquisition functions.The wave generator was controlled with the use of the GEDAP program SWG. The SWGprogram allows the user to initialize and calibrate the wave generator, to download the drivingsignals, to adjust the span of the paddle motion, and to start and stop the machine. The 60individual driving signals must first be synthesized by using one of a number of GEDAP signalgeneration programs For uni-directional regular waves program DWREP2B provides the drivingsignals after requiring the user to select the wave height, wave period, water depth, angle of wavepropagation from the generator and also the portions of the generator where the paddles are to beheld stationary. For the generation of driving signals for irregular waves and multi-directionalwaves, alternative GEDAP programs are used, and require the user to specify frequency anddirectional wave spectra. In order to operate the wave generator the driving signals aredownloaded to the generator and the START command is given. The driving signal is smoothlyramped up from zero to full amplitude over a 10 second period by the program SWG in order toprotect the wave machine from any sharp transients in the control signal . The generator runs untilthe STOP command is given.Data acquisition was carried out on the same VAX computer using the GEDAP programDAS_CMD and a Neff Series 500 data acquisition system. Use of the Neff system allowed 16signals to be sampled simultaneously through a rack of 16 circuit cards. The signals were eachsampled at a rate of 20 samples per second for a duration of 7 minutes, converted from analogsignals to digital data, then stored as one GEDAP data file for subsequent analysis. Use of the14program DAS_CMD allows the data acquisition to be synchronised with the start of wavegeneration.2.3 Test Set-up2.3.1 Basin LayoutIn order to vary the angle of wave incidence relative to the breakwater, the breakwater was fixed inposition and the wave approach direction was varied using the directional wave generator. Toattain the required range of incident wave angles of 0 to 60 degrees, the breakwater centreline wasaligned at an angle of 45 degrees to the generator face. The basin layout is shown in Figures 2.3and 2.4 for angles of incidence of 0 and 60 degrees respectively. Photograph 2.1 shows a view ofthe basin set up for tests with 0 degrees angle of incidence.These configurations were obtained by considering incident waves with the various directions ofapproach, and examining regions of reduced incident wave height due to diffraction, and regionswhere waves reflected off the breakwater are then re-reflected off the generator face. Guidewallswere placed parallel to the direction of wave approach extending from each end of the breakwaterto the generator face so as to prevent the incident wave height from being affected by diffraction.Additional wave absorbers were placed in front of the longer guidewall to prevent waves reflectedfrom the breakwater from re-reflecting from this guidewall into the test region. The guidewallswere re-aligned with the direction of incident wave approach for each angle of incidence for theuni-directional wave tests. During the multi-directional wave tests the guidewalls were removedfrom the basin.15111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111116.....0Sa)as3wave absorberFigure 2.3 - Basin layout for 0 = 0°1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111wave absorberFigure 2.4 - Basin layout for 0 = 60°SO,^ee.^mw• woe ••■• ••••Sr SO 04117Photograph 2.1 - View of wave basin showing breakwater and guidewallset up for 0= 0°2.3.2 BreakwaterThe model rubble-mound breakwater was designed using methods set out in the Shore ProtectionManual (U.S. Army Corps of Engineers 1984) to withstand 0.30 m waves without any movementof the armour units or any over-topping, although the largest wave height used in the testprogramme was less than 0.25 m. A secondary design constraint was that the armour units on thebreakwater were to be sufficiently large for Reynolds number scale effects to be reasonably small.Personal communications with staff at the National Research Council indicated that a suitableminimum stone weight for this was approximately 200 g.A sketch of the cross-section of the breakwater is shown in Figure 2.5. As indicated in the figure,the breakwater consists of a core, a filter layer and an outer armour layer. The core of finermaterial was used to reduce energy loss due to wave transmission. The core material was coveredby a filter layer consisting of angular rock weighing a minimum of 100 g. The outer layer was twostone sizes thick and consisted of angular rock weighing a minimum of 1 kg.armour (W > 1 kg )fitter (W > 100 g )core material0.25 m1Figure 2 .5 - Breakwater cross-sectionThe breakwater characteristics were as follows:Breakwater height : 0.75 mWater depth : 0.50 mCrest width : 0.25 mLength : 18 mFront and back slopes : 1: Wave MeasurementThe water surface elevation was recorded at a number of locations in front of the breakwater usingtwin wire capacitance wave probes. An array of nine probes was used in order to provide data forthe directional analysis. The dimensions of the nine probe array are shown in Figure 2.6. Workcarried out by Goda and Suzuki (1976) indicates that measurements should be made at a distanceof not less than one approximate wave length from the breakwater so as to provide accurateestimates of reflection coefficients. For this reason the wave probe array was placed such that thedistance from the intersection of the water surface with the breakwater face to the nearest wave18probe in the array was 3 m, which corresponds to one average wavelength. The approximatepositions of the wave probe array for angles of incidence 0° and 60° are shown in Figures 2.3 and2.4. A view of a test in progress, showing the position of the wave probe array, is given inPhotograph 2.2. In this photograph the incident wave parameters are: 0 = 45°, T = 1.6 s, H =0.15 m.Photograph 2.2 - View of test 17 with 0 = 45°The reflection analysis used for uni-directional, regular waves required water surface elevationsignals measured at three locations aligned perpendicular to the breakwater face. Consequently,the array was aligned such that the three wave probes nearest the breakwater, which in Figure 2.6are labelled "1", "2", and "3", lay precisely on a line perpendicular to the breakwater face. Thesethree probes were used for the uni-directional, regular wave analysis and for the uni-directional,irregular analysis.19Apart from measurements using the wave probe array, measurements were also made using asingle wave probe placed in front of the breakwater, as indicated by Figure 2.4, so as to measurethe incident wave height on the wave orthogonal that was reflected through the probe array.Figure 2.6 - Approximate dimensions of the wave probe arrayMeasurements for the directionality analysis were made using the nine-probe array shown inFigure 2.6. The dimensions of this array were chosen to match the array used by Nwogu (1989)so that the directional analysis methods used by Nwogu in his multi-directional wave tests could beused in this study. The location of the array is shown in Figures 2.3 and 2.4. In addition to thewave probe array, an alternative set of directional wave measurements were made using a Delft bi-axial current meter, with water surface elevation measurement, located adjacent to the wave probearray, (see Figures 2.3 and 2.4). This was mounted on a separate tripod stand which was located20some distance from the wave probe array and any other metallic objects to minimise the possibilityof signal noise due to electromagnetic interference.All the wave probes were calibrated at the start of each day of testing by raising and lowering theprobes by a known distance and recording the signal voltage at these two positions as well as theoperating position. A linear relationship between signal voltage and water surface elevation wasobserved, with very small error, which was typically less than 0.5 mm, and there was goodrepeatibility of calibration coefficients from calibration to calibration.212.4 Test ProgrammeThe waves used in the test programme are outlined in Table 2.1 below:testnumberwaveperiod(s)waveheight(m)angle ofincidence (deg.) digT2(or d/gTi2))H/L(or Hs/Lp)wave type1 1.0 0.10 0, 30, 45, 60 0.051 0.066 regular2 1.2 0.10 0, 30, 45, 60 0.035 0.049 regular3 1.6 0.10 0, 30, 45, 60 0.020 0.032 regular4 2.0 0.10 0, 30, 45, 60 0.013 0.025 regular5 2.4 0.10 0, 30, 45, 60 0.009 0.020 regular6 1.6 0.05 0, 45 0.020 0.016 regular7 1.6 0.15 0, 45 0.020 0.049 regular8 1.6 0.18 0, 45 0.020 0.058 regular9 1.6 0.19 0, 45 0.020 0.062 regular10 T^= 1.6p H = .S^0.06 0, 45 0.020 0.019 irregular11 TP = 1.6 Hs = 0.12 0, 45 0.020 0.039 irregular12 Tp = 1.6 Hs = 0.15 15, 30 0.020 0.048 uni-directional,irregular1 1.6TP =—  Hs = 0.15 15, 30 0.020 0.048 multi-directional,irregularTable 2.1 - Test ProgrammeIt was expected that the wave parameters that would have the greatest influence on the reflectioncharacteristics would be the wavelength L, the wave height H, and the angle of incidence O. Forthis reason, the water depth was held at a constant value of d = 0.5 m for all waves generated, andthe wave period, the wave height and the angle of incidence were varied.2216 fp (fifp)5 exp^4^Sr) (0 =    (ff )-45 H s 2^1^5 (2.16)Analysis of regular wave reflection was intended to reveal relationships between the parameters K rand (3 and the two dimensionless wave parameters representing wave steepness and depth towavelength ratio. With this in mind, the following set of tests were undertaken, corresponding totests 1 to 9 in Table 2.1:a) The effects of depth to wavelength ratio were examined in tests 1 to 5 bychanging the wave period and keeping other parameters constant. Values ofwave period ranging from T = 1.0 s to 2.4 s were used which provided valuesof depth to wavelength ratio ranging from d/gT 2 = 0.009 to 0.051.b) The effects of angle of wave incidence were examined by repeating these testsfor the four angles of incidence: 0 = 0°, 30°, 45°, 60°.c) The effects of wave steepness were examined in tests 6 to 9 by keeping thewave period constant at T = 1.6 s and changing the wave height. Values ofwave height ranging from H = 5 cm to 19 cm were used which provided valuesof wave steepness ranging from H/L = 0.016 to 0.066. These tests were carriedout with two angles of incidence: 0 = 0°, 45°Tests 10 and 11 in Table 2.1 represent the phase of testing with irregular, uni-directional waves.These tests were undertaken in order to examine how adequately the results obtained from theregular wave experiments described the characteristics of irregular wave reflection. The irregularwave trains were approximately described by a Bretschneider spectrum given by:23where Hs is the significant wave height and fp is the peak frequency of the spectrum (fp = 1/Tp ).The characteristics of the irregular waves were chosen so as to allow comparisons to be madebetween the irregular wave results and results from selected regular wave tests. Tests 10 and 11,with significant wave heights H s = 6 cm and 12 cm and peak period Tp = 1.6 s, were generatedwith two angles of wave incidence: 0 = 0° and 45°.One of the objectives of this research was to search for and quantify any directional spreading inthe waves reflected from the breakwater. As an extension of this a series of multi-directionaltesting was undertaken. Both uni-directional (test 12) and multi-directional (test 13) wave trainswere generated with the same frequency spectrum to compare the spreading of the reflected wavefield for uni-directional and multi-directional incident waves. These waves were generated in twoprincipal directions, Op = 15°, 30°.In order to provide data on the incident wave field in the absence of breakwater reflections, theentire test programme for all angles of incidence was performed twice; firstly without thebreakwater in place and secondly with the breakwater in place. During the first phase of tests theguidewalls were left in place and reflections were suppressed by wave absorbers around theperiphery of the basin. Full measurement of the water surface elevation and water particlevelocities, as described in Section 2.3.3, was made for both phases of testing.24Chapter 3ANALYSIS TECHNIQUES3.1 Regular Uni-Directional Reflection AnalysisFor the case of regular, uni-directional wave reflection, a method of analysis was required thatprovided three wave reflection parameters: the incident wave height, the reflection coefficient, andthe reflection phase lag, from measurement of water surface elevations. In this project a methodwas used which applies a least squares technique to measurements from three wave probes. Theapplication of this method to normal wave reflection has been described by Isaacson (1991) andMansard and Funke (1980) and its extension to oblique wave reflection has been indicated byIsaacson (1991).3.1.1 Derivation of Least Squares MethodThe water surface elevation in front of the breakwater is assumed to correspond to thesuperposition of sinusoidal incident and reflected wave trains. The reflected wave train is assumedto reflect away from the breakwater at an angle of reflection equal to the angle of incidence. Todescribe the wave field, Equation (2.1) can be re-written as:ri = ai cos (kx cos() + ky sine, - (it) + ar cos (-kx cost) + ky sine, - wt + (3)^(3.1)where al and ar are the amplitudes of the incident and reflected wave trains respectively and p is thephase difference between the two trains at the position x = 0. The origin of the x axis is defined asthe intersection of the still water level and the breakwater face. 0 is the angle of wave incidence onthe breakwater and k and w are the wave number and angular frequency, related by the dispersionrelation:25co2 = gk tanh(kd)^ (3.2)26where g is the acceleration due to gravity and d is the water depth. Assuming co is known frommeasurements of the wave period, there are three unknowns: ai, ar, and D.Equation (3.1) is applied at a series of known probe positions on the x axis; xn , yn , n = 1, 2, 3 ...and yn = O. The location along the x axis may be written in terms of the location of the first probex i and the intervals between the probes:xn = x i + X,n^ (3.3)where An is the distance in the x coordinate between the nth probe and the first probe, and X i = O.It is convenient to write this in dimensionless form as:kxn cos8 = kx 1 cose + An^(3.4)where An is the dimensionless distance between the nth probe and the first.An =^cos()^ (3.5)Applying Equation (3.1) at each of the probe locations gives:rin = ai cos (kxncos0 + kynsin8 - cot) + ar cos (-kxncos9 + kynsin0 - wt + R )= ai cos (kxicos8 + A n -^) + ar cos (-kx icosE) - An - cot + 13 )^(3.6)The actual measurements at the probe locations will provide corresponding amplitudes and relativephases, such that the measured elevation at the nth probe may be written as:11(nm) = An cos(cot - 1:1)n)= An cos(cot - (1) i - Sn)^ (3.7)where An is the measured amplitude of the water surface at the nth probe, 0 1 is the absolute phaseof the first probe which need not be measured, and 8n is the measured phase of the nth waverecord relative to the first, so that 8 n =^n — ch.27It is convenient to describe the water surface elevation in complex notation in order to simplify thealgebra, with the real parts of complex expressions corresponding to the physical quantitydescribed. Equation (3.6) expressing the water surface elevation in terms of the unknown incidentand reflected wave parameters can be written in complex form as:in = { ai ei(kxicos0+An) + ar ei(-kxicos0-An+13) } e-kot^(3.8)The measured water surface elevation can be written in complex form as:inm) = An ei(4) 1+8n) e-icot^ (3.9)Equations (3.8) and (3.9) can be re-written in terms of complex amplitudes:in = { bi eiAn + br ciAn } cicot^ (3.10)rinm) = Bn e-icot^ (3.11)whereIN = al ei(locicos0 )^ (3.12)br = ar ei(-kxicos8+(3) (3.13)Bn = An ei(41 4-8n)^ (3.14)The sum of the squares of the error between the complex amplitudes of the assumed and measuredelevations may be written from Equations (3.10) and (3.11) as:3E2 =E obi eiAn + b  ciAn - B nen=1 (3.15)The error of fit is minimised with respect to the required complex unknowns bi and br by settinga(E2)/abi and a(E2)/abr in turn to zero. This gives rise to two complex equations for bi and hr:3E eiAn [bi eiAn + bre-ion - Bn] = 0n=13^.E e-IAn [bi eiAn + bre-jAn - Bn] = 0n=1Solving these two equations provides bi and b r in terms of Bn and thereby provides ai, ar and 13 interms of An . This solution was expressed by Isaacson (1991) as :a . =1X.1i^1 (3.18)ar =1Xr1 (3.19)x = Arg(Xi) - Arg(Xr)^ (3.20)13 = 2kx i - x I- 27cm (3.21)where m is any integer, usually chosen such that 0 S 13 < 27r, and whereX i — s2s3 5- 3s4 (3.22)SS1S45- 3 S3 Xr —^s (3.23)3si = E ei2An^ (3.24)n=i3s2 = E e-i20n^ (3.25)n=i3s3 = E An ei(8n + An)^ (3.26)n=i3S4 = E i An ei(8n - An)^ (3.27)n=s5 = sis2 - 9^ (3.28)Simple substitution of measured quantities into Equations (3.24) to (3.28) and trivial calculationsusing Equations (3.18) to (3.23) will yield the incident and reflected wave heights and thereflection phase lag. Hence from measurements of the amplitudes and the phases of three waveprobes aligned perpendicular to the breakwater the basic reflection characteristics K r and 13 are ableto be calculated.3.1.2 Application of Least Squares MethodThe least squares method was applied to signals measured at locations on a perpendicular line tothe breakwater at distances from the breakwater of approximately 3.00 m, 3.46 m, and 3.89 m.Extensive numerical testing of this method by Isaacson (1991) shows that the accuracy of themethod decreases when the spacings between probes one and two and probes two and three areequal. Therefore, in this project the centre probe was not placed exactly mid-way between the twoexterior probes, but was located such that 2■,2 = 0.52 73. For this ratio of probe spacings, Isaacsonfound that the method should have good accuracy.The amplitude and phases of the measured signals were obtained using the GEDAP programAPHRES. This program performs a nonlinear optimization to fit a sinusoidal curve to the data. Inthe program, three parameters: the amplitude A, phase angle f3, and frequency f, are optimized inan iterative process. Convergence is deemed to have been obtained when the the change inparameters between successive iterations falls below a user specified limit.This method was used in preference to a Fourier analysis method for two reasons:1) Fourier analysis methods are sensitive to record length and the number ofsample points, whereas the program APHRES can be used on short lengthrecords with confidence. The eventual record length used in the analysis of theregular waves was only 10 seconds.29reflected  waveorthogonalpr obearrayk^ 3.00mincident waveprobe^incident waveorthogonal2) Accurate measurement of phase angles is needed for the least squaresreflection analysis. An initial comparison of sinusoidal fitting programs withFourier analysis programs indicated that the former provides more accuratephase angle results.It was found that APHRES did not converge to give reasonable results when the mean water levelfluctuated significantly with time due to low frequency waves. This was not the case when shortrecord lengths were used in the analysis.Once the three amplitudes A i , A2, A 3 and the two phase differences 5 1 , 52 were known for eachwave record, these five quantities, together with the angle of incidence 0, wave period T, waterdepth d, and probe positions x i , x2, x3 were specified as input to a simple FORTRAN programwhich used Equations (3.18) to (3.28) to calculate ai, a r and 13.Rather than use ai to calculate the reflection coefficient K r, the incident weave height measured at30Figure 3.1- Location of wave probesthe incident wave probe in the absence of the breakwater has instead been used. This probe waslocated on the wave orthogonal that was reflected through the probe array, as is shown by Figure3.1. It is expected that there is generally a spatial variation in the incident wave height, and that theuse of the height measured at the incident wave probe is more appropriate.3.2 Irregular Wave AnalysisAnalysis of the irregular wave tests required that the spectral density of the incident and reflectedwave trains be obtained. The spectral density, described by Equations (2.5) and (2.6) in Section2.1.2, was calculated using the GEDAP program VSD. The reflection coefficient as a function offrequency was then calculated by dividing the reflected spectral density function by the incidentspectral density function, as is described by Equation (2.9). Equation (2.10) was used to calculatethe average reflection coefficient Kr, from the spectral density.VSD is a general purpose program which uses a Fourier analysis technique to calculate the spectraldensity of a signal. Before taking a Fourier Transform of the data signal, the program firstmultiplies the signal by a trapezoidal window in order to reduce leakage. The Fourier Transform isthen taken and the periodogram resulting from this operation is smoothed using a simple movingaverage filter to provide the spectral density function. The length of this filter is set to obtain eithera specified filter band width or a specified number of degrees of freedom per spectral estimate. Inthis project 100 degrees of freedom were specified in order to obtain a smooth spectral densityfunction. Output from the VSD program consists of the spectral density function as well as manyspectral parameters including the peak frequency and the zeroth moment of the spectrum.Portions of the water surface elevation signals of length 5 minutes were analysed using the VSDprogram. The incident spectral density was obtained by analysing the water surface signals fromthe tests without the breakwater in place. Signals from three probes were analysed and the incidentspectrum was taken as the average of the three measured spectra. The probes labelled "1" "2", and"3" in Figure 2.6 were used and the probe array was located in the same positions used for the31regular wave tests, as is indicated in Figures 2.3 and 2.4. To obtain the reflected spectral densityfor each test, the water surface elevation signals without the breakwater in place were subtractedfrom the signals measured with the breakwater in place. This operation provides the reflectedwater surface elevation signal directly, provided that the incident and reflected wave trains can beassumed to be combined by linear superposition. Program VSD was run on the reflected wavesignals obtained in this manner from the three wave probes to get an average reflected wavespectrum. Dividing the reflected by the incident wave spectra provided the reflection coefficient asa function of frequency for each test.3.3 Analysis of Wave DirectionalityIn order to examine the multi-directional tests, the directional spectra of the incident and reflectedwave fields were estimated using the GEDAP program MLMWP.The directional spectrum of a wave field can be calculated from measurements of water surfaceelevations at a number of locations, or from the water surface elevation and water surface slope orhorizontal water particle velocities. Program MLMWP read water surface elevation signalsmeasured at five probes and used a maximum likelihood method to calculate the directionalspectrum. The maximum likelihood method, MLM provides an estimate of the directionalspectrum which maximises the likelihood of obtaining the observed data set. The MLM wasdescribed in further detail by Nwogu (1989). Another directional analysis program MLMVL wasused to check the results of the MLMWP program. The GEDAP program MLMVL is differentfrom the program MLMWP in that it analyses a water surface elevation signal and horizontal waterparticle velocity signals rather than five water surface elevation signals.The directional spectra for the incident and reflected wave fields were obtained using a similarprocedure to that used to obtain the spectral density functions. To get the reflected directionalspectrum for each test the program MLMWP was applied to the signal resulting from thesubtraction of the incident from the combined incident and reflected wave signals. The incident32directional spectrum was obtained by applying MLMWP directly to the incident wavemeasurements, i.e, measurements made without the breakwater in place.33incident waveincident plus reflected waves806040E 200-20-40Chapter 4RESULTS AND DISCUSSION4.1 General ObservationsThe generation of the entire programme of test waves in the open basin before the placement of thebreakwater allowed the incident wave field to be examined in the absence of any reflections fromthe breakwater. This period of testing showed that the incident wave profiles were of satisfactoryquality. The incident wave heights were measured and checked against those specified. Duringthe tests without the breakwater in place, the probes were located in positions identical to thoseused with the breakwater in place. A sample of the measured water surface elevation signals isgiven in Figure 4.1 showing signals from tests with and without the breakwater in place. This20^22^24^26^28^30time (s)Figure 4.1 - Sample of water surface elevation for test 3 (T = 1.6 s, H = 10 cm)and 6 = 45 °, with and without the breakwater in place34V1I^140 5030time (s)portion of recordanalysed^ tbfirst incident waveI-80 110^20EE806040200-20-40-60reflected wave fully developedincident wave fully developedI35figure shows that the combined incident and reflected wave signal is cyclic with the samefrequency as the incident signal. No other frequencies are present in the reflected signal indicatingthat the assumption made in equation (2.1), that there is no scattering of frequency in the reflectionprocess, is valid.At each wave probe slight variation in incident wave height was observed with time over the sevenminute record length. This was attributed to a gradual build up of reflected wave energy from thewave absorbers around the basin sides and also to possible basin resonance. Basin resonance canbe encountered when regular waves are generated in a wave basin and it is accepted as being moreprominent in regular rather than irregular wave tests. The regular wave analysis was carried outusing an initial portion of the record, which was measured before undesired extra reflected waveenergy and resonance effects built up, in order to avoid contamination of the results.Figure 4.2 - Sample of beginning of wave record showing portion selected foranalysis, test 3 (H = 10 cm, T = 1.6 s), 0 = 60°The propogation of the waves across the basin and away from the breakwater needed to beexamined in order to ensure that the selected portion contained the fully developed reflected wavefrom the breakwater as well as the incident wave. Travel times for the wave to travel to thebreakwater and to reflect back to the probe array were calculated from the wave group celerity foreach different wave type. A 10 second long portion of record, beginning at the instant that thefully developed reflected wave first reached the probe array, was then selected for the analysis ofeach test. Figure 4.2 shows a 40 second sample of the water surface elevation record for one testincluding the 10 second portion selected for analysis. The first incident wave was calculated fromthe wave group celerity to arrive at the probe position at t = 16 seconds, the fully developedincident wave and the fully developed reflected waves were calculated to arrive at t = 26 and t = 31seconds respectively. Figure 4.2 shows these predictions to be accurate. The addition of thereflected wave, at t = 26 to t = 31 seconds, decreased the amplitude of the total wave signal, whichindicates that this probe must have been located near a node in the partial standing wave pattern.More significant variation in incident wave height was recorded between different waveorthogonals. Spatial variation in wave height was expected in this wave basin due to diffractionand reflection of the generated wave. Such variations were measured in this basin by Shaver(1989) and wave height variations in the basin can also be predicted using a linear diffractionmodel. In this project the unwanted diffraction of the incident wave was reduced with the use ofguidewalls, however some variation was still observed. Differences in incident wave heightbetween the wave probe array and the incident wave probe of up to ten percent were recorded.Differences were generally less than 5 percent and allowance was made for this in the reflectionanalysis, as described in the previous section, so that the results would not be affected.4.1.1 Wave-Breakwater InteractionDue to the principle of conservation of energy, the sum of the energy reflected from, transmittedthrough, and dissipated by the breakwater must be equal to the total wave energy incident upon the36breakwater. Therefore, the amount of wave reflection is directly affected by the amounts of wavetransmission and wave energy dissipation.Wave transmission was observed to be negligible in this project. Wave energy dissipation due towave-breakwater interaction was observed in the form of wave breaking and run-up on thestructure. Very different wave-breakwater interactions were observed for the range of wavestested. Longer period waves appeared to lose less energy in turbulence when they broke on thebreakwater when compared to the steeper short period waves. Also, waves with more obliqueangles of incidence tended to show less vigorous action on the breakwater face.4.2 Regular, Uni-Directional Wave Tests4.2.1 Performance of Sinusoidal Fitting ProgramThe performance of the curve fitting program APHRES, which fitted sinusoidal curves to the 10second portions of water surface signal for each test, was checked visually by plotting thesinusoidal curve superimposed on the measured signal. Samples of these plots are shown inFigures 4.3 to 4.5. From these plots it can be seen that there is no perceptible difference in phasebetween the measured and fitted signals, but that there is some difference between the amplitudesof the two signals. Also shown on each of the plots is the difference between the measured watersurface elevation and that given by the fitted curve. In these figures the mean of the differencesignal is not equal to zero, implying that the wave probes were not zeroed before this test andhence the mean of the measured signal is not zero. These difference curves are observed to consistmainly of the second harmonic to the fundamental wave.Second harmonic wave activity is commonly encountered when regular waves are generated in awave basin or flume and is made up of two main components:1) One component of second harmonic waves is bounded to the fundamentalwave train, and will exist in all except deep water conditions. This component37propogates at the same celerity as the fundamental wave train and gives the wavea non-linear profile.2) Another component of second harmonic wave activity may be presentdepending on the wave generation method used. Most wave generationtechniques, including the method used in this study, use first order theory tocalculate paddle motions. This theory does not satisfy the necessary secondorder boundary conditions and therefore undesired free waves of frequencytwice that of the fundamental are produced. These propogate at a different speedto that of the fundamental wave train and so will lead to different wave profiles atdifferent locations.Another component of second harmonic wave activity may be present due to the wave reflectionprocess. When a portion of the fundamental wave train is reflected from the breakwater a certainamount of bounded second harmonic wave must also be present in the reflected wave for it toretain its non-linear profile. Any excess in reflected second harmonic wave energy over thisamount will be released as free second harmonic wave activity. Higher harmonic activity was notof interest in this study and therefore no analysis of these components was undertaken. A moregenerous treatment of the reflection of second harmonic waves is given by Mansard et al. (1985).383980I^I— measured ----^fitted^ difference...I^I^I 30 32 34time (s)604020E0-20-40-60I28 36 38^40Figure 4.3 - Results of the sinusoidal fitting program APHRES, test 1 (H = 10 cm,T= 1.0 s),0= 45°measured ^fitted^ differenceI^I20^22^24^26^28^30 32time (s)Figure 4.4 - Results of the sinusoidal fitting program APHRES, test 3 (H = 10 cm,T= 1.6 s), 0 =45°806040t"-- 20Eo-20-40-60302822 2624time (s)18 20measured------ fitted^--..-^ difference806040--E— 20Ep- 0-20-40-60Figure 4.5 - Results of the sinusoidal fitting program APHRES, test 5 (H = 10 cm,T= 2.4 s), 6=45 °45°4.2.2 Accuracy of the Least Squares MethodAs a test of the accuracy of the least squares method, incident wave amplitudes derived fromanalysis of wave records measured with the breakwater in place were compared to incidentamplitudes measured without the breakwater in place. In general, good accuracy was obtained,with a maximum error of 8% of the measured amplitude and an average error for all the tests of3%.A further check on this method of analysis was made by analysing data from the normal incidencetests with a computer program based directly on Mansard and Funke's least squares method ratherthan the method used in this project. Comparison of the results from the two methods showed thatthe reflection coefficients obtained by the two methods agreed to within one percent accuracy,although this is generally expected since the present method is essentially a variant of that given byMansard and Funke.404.2.3 Reflection CoefficientThe most important parameter describing wave reflection is the reflection coefficient, Kr. Resultsfrom the analysis showed that for the entire range of wave types tested, and for all the angles ofincidence, Kr ranged from 10% to 59%. The highest reflection coefficient of 59% was recordedfor a wave height of 10 cm and period of 2.4 s with angle of incidence of 30°.The dimensional analysis in Equation (2.2) indicates that for a given breakwater and water depthand for regular, uni-directional incident waves, the reflection coefficient K r and phase angle (3 eachdepend on the three parameters, the wave steepness H/L, the depth to wave length ratio d/gT 2 andthe incident direction 9.The dependence of Kr on H/L is shown in Figure 4.6, in which K r is plotted as a function of H/Lfor a constant value of d/gT 2 = 0.020 and with 0 = 0° and 45° in turn. The reflection coefficientKr is seen to decrease very slightly with increasing wave height, such that K r varies from 10% to14% for 9 = 0° and from 32% to 39% for 9 = 45° over the range of wave heights encountered.This small variation indicates that Kr has a very slight dependence on wave height and suggeststhat the depth to wave length ratio d/gT2 and the angle of incidence 9 may be more important.Figures 4.7 to 4.10 show Kr plotted against d/gT2 for each angle of wave incidence. In thefigures, values of H/L vary from 0.014 to 0.073, however this is not expected to have muchinfluence on the relationships on account of the observations taken from Figure 4.6. As expected,Kr was found to decrease with increasing wavelength parameter, that is, with decreasingwavelength. For angles of incidence 9 = 0° and 30° Kr decreased dramatically as d/gT2 increasedfrom 0.01 to 0.02. However, there was no further decrease in K r as d/gT2 was increased furtherfrom 0.02 to 0.05, instead Kr remained approximately constant at values of K r = 0.15 and 0.23for 0 = 0° and 30° respectively. This plateau in the Kr versus d/gT2 relationship may be associatedwith a maximum level in the proportion of energy able to be dissipated by the breakwater. Asd/gT2 is increased from 0.01 to 0.02 the proportion of wave energy dissipated by the breakwater41increases as the wave breaking on the structure becomes more vigorous. It is expected that, due tothe principal of conservation of energy, the amount of energy reflected from the breakwater musttherefore decrease as d/gT2 is increased, as is shown by Figures 4.7 to 4.10. At a value of d/gT2of 0.02, where the proportion of incident wave energy dissipated is approaching unity, thisproportion is observed to reach a maximum as Kr decreases no further. This abrupt change inwave reflection is probably associated with a transition in the type of wave breaking on thestructure from a surging type of wave, with low energy dissipation characteristics, to a plungingtype of wave, with high energy dissipation characteristics and a low reflection coefficient. Thisspeculative proposition may be used to explain the disjoint shapes of the curves in Figures 4.7 and4.8. For angles of incidence of 9 = 45° and 60°, because the waves are approaching at an obliqueangle, which means that at different positions along each wave crest the waves are at various stagesof breaking, a different transition between the types of wave breaking can be expected. For higherangles of incidence the relationship between Kr and d/gT2 can therefore be expected to change to aless disjoint, more continuous function, as is shown by Figures 4.9 and 4.10.Figure 4.11 combines the results shown in Figures 4.7 to 4.10 and emphasizes the effect of angleof incidence on the reflection coefficient over the range of wavelength parameter d/gT 2 examinedIn Figure 4.12 Kr is plotted against angle of incidence 0 for intermediate and larger values ofd/gT2 . Figures 4.12 shows that the plateau in Kr observed in Figures 4.7 to 4.10 occurs at valuesof Kr that increase significantly with increasing angle of incidence. This indicates that as 0increases, the consequent decreasing vigour of the wave action on the breakwater that wasobserved in Section 4.1.1 leads to a decrease in the proportion of energy dissipated by thebreakwater and therefore an increase in the minimum value of Kr. For the smaller values of d/gT 2,Figure 4.13 shows that the effect of 0 on Kr is different and that the largest values of Kr weremeasured for 0 = 30°.420.6—0—L=3.00m, 0= 0 deg.—0—L=3.00m, 0=45 deg.0.50.4'-' 0.03 4.6 - Reflection coefficient vs. wave steepness, dIgT2 = 0.020, 0 = 0 °,45 °^0.02^0.03^0.04^0.05d/gT2Figure 4.7 - Reflection coefficient vs. d/gT2 , 9 = 0°43440. .0 1^0.02^0.03^0.04^0.05d/gT2Figure 4.8 - Reflection coefficient vs. d/gT2 , 0 = 30° 10 .00.01^0.02^0.03^0.04^0.05d/g T2Figure 4.9 - Reflection coefficient vs. d/gT2 , 0 = 45 ° .0145-X- 0= 0 deg.-0- 9=30 deg.-x-- 0=45 deg.-0- 9=60 deg. 4.10 - Reflection coefficient vs. d/gT2 , 0 = 60 °0.01^0.02^0.03^0.04^0.05d/gT2Figure 4.11 - Reflection coefficient vs. d/gT20.^15^30^45^60eFigure 4.12 - Reflection coefficient vs. angle of incidence, high d/gT2Y -^15^30^45^609Figure 4.13 - Reflection coefficient vs. angle of incidence, low d/gT2reflectedwavesantinodes atx = -nL/2 + x', n = 1,2,3...incidentwaves4.2.4 Reflected Phase LagThe reflected phase lag (3 is the reflection characteristic which determines the lateral shift of thepartial standing wave pattern from the reflecting boundary. 13 = 0° corresponds to anti-nodes atinteger half wavelengths from the origin of the x axis. If p > 0° then this standing wave pattern isshifted in a direction towards the breakwater. Results of this study gave p as well as the moreimportant wave reflection variable K r. These results showed that 13 can be as large as one quarterof a cycle. For the entire range of regular wave tests Pranged from 9° to 116°.From Equation (24), and using the notation defined in Figure 4.14, it can be shown that therelationship between f3 and the shift in the standing wave position x' is:, .^0it = 2k= L47c(4.1)Figure 4.14 - Sketch showing effect of phase lag on partial standing wave position47In similar manner to the reflection coefficient analysis, analysis of the reflected phase lag wascarried out to examine relationships between 13 and the parameters describing the wavecharacteristics, the wave steepness H/L, the depth to wavelength ratio d/gT2 , and the angle ofincidence 0.The dependency of [3 on the wave steepness is seen in Figure 4.15. In this figure 13 is plottedagainst wave steepness H/L for constant values of d/gT 2 , and d = 0.5 m, d/gT2 = 0.020 and 9 = 0°and 45° in turn. From this plot it is seen that 0 has only a very small dependence on wave height,as was found for the relationship between K r and wave height.Figures 4.16 to 4.19, in which 13 is plotted against d/gT2 for each angle of incidence in turn, showthat has a strong dependency on d/gT2 . These plots indicate that f3 increases with increasingd/gT2 , i.e: decreasing wavelength. However, for 0 = 60°, 13 is seen to be independent of d/gT2 .This difference in the behaviour of 13 may be due to to the fact that the angle of incidence of 0 = 60°is very extreme. The results shown in Figures 4.16 to 4.19 are combined in Figure 4.20, in which13 is plotted against d/gT2 for all angles of incidence. This plot shows that 13 is also dependent onangle of incidence 0, and indicates that 13 increases with increasing 0. The dependency of 0 ond/gT2 may be explained by examining the shift in position of the standing wave pattern x', givenby Equation (4.1). Figure 4.21, in which x' is plotted against d/gT 2 for all angles of incidence,shows that for all angles of incidence except 0 = 60°, x' is approximately constant. Therefore,from Equation (4.1), 13 must increase with increasing d/gT 2 .48xIFigure 4.15 - Reflected phase lag vs. wave steepness, d/gT2 = 0.020, 0 = 0°,45 °12010080--:....0 ^60,x1..402001 201 008 0..-:,CMa)1::)^6 0CCL40200 49-0- 9=45 deg.-0- 0= 0 deg.0.01^0.02^0.03^0.04^0.05^0.06H/L0.050.01^0.02^0.03^0.04d/gT2Figure 4.16 - Reflected phase lag vs. d/gT2 , 0 = 0°12010080---.-0)a)-0^60=,.40200500.01^0.02^0.03^0.04^0.05d/gT2Figure 4.17 - Reflected phase lag vs. d/gT2 , 0 = 30°12010080--7-aa)-0^60=,..402000.01^0.02^0.03^0.04^0.05d/g T2Figure 4.18 - Reflected phase lag vs. dIgT2 , 0 = 45°—x— 0= 0 deg.—o— 0=30 deg.--x— 0=45 deg.—0— 0=60 deg.1 201 008 06 04 02001201008060402000.01^0.02^0.03^0.04^0.05d/gT2Figure 4.19 - Reflected phase lag vs. d/gT2 , 0 = 60°0.01^0.02^0.03^0.04^0.05d/gT2Figure 4.20 - Reflected phase lag vs. dIgT2510.60.50.4I-x 4.21 - Reflected phase lag vs. d/gT24.3 Irregular Wave TestsThe purpose of the phase of testing with irregular waves was to determine whether the resultsobtained from the regular reflection tests could be applied accurately to reflection of irregularwaves. This has been done by estimating spectral density functions for the incident and reflectedwave fields and thereby calculating the reflection coefficient as a function of frequency. Thisreflection coefficient function can be directly compared to the reflection coefficients for each waveperiod measured in the regular tests.4.3.1 Spectral Density FunctionsFigures 4.22 to 4.25 show estimates of spectral density for the four particular tests, Tp = 1.6 s,Hs = 6 cm and 12 cm in turn, and 0 = 0° and 45° in turn. The use of program VSD in the analysismeant that smooth frequency spectra were able to be obtained. This smoothing was needed toobtain meaningful results when the reflected spectra were divided by the incident spectra to obtainthe reflection coefficient function. From these plots it is apparent that the reflected wave energy isof the same approximate frequency range as the incident wave energy. The reflected frequencyspectra do, however, show a disproportionately high amount of reflected wave energy forfrequencies in the range f > 1.5 Hz. Also, the peaks of the reflected spectral density appear to be ata lower frequency than the peaks of the incident spectral density. Results from the regular wavetests, indicating that the lower frequency waves would have more energy reflected than the higherfrequency waves, also predict this shift in the peak frequency of the reflected wave field.536005007,,I 400"E--- 300co2001000 .0 0.5 1.0^1.5Frequency (Hz)252.054— incident spectrum^ reflected spectrum.....( ----___ _ - .. I ..... .... iI— incident spectrum^ relfected spectrum-1---- ' -.. ....... I............... UGE.1;;W.25002.52.00.500.050020007.,I"E 1500EE- 1000co1.0^1.5Frequency (Hz)Figure 4.22 - Incident and reflected spectral density, test 10 (H s = 6 cm, Tp = 1.6 s)and 0 = 0°Figure 4.23 - Incident and reflected spectral density, test 11 (H s = 12 cm, Tp = 1.6 s)and 0 = 0°0.5 1.0^1.5Frequency (Hz)— incident spectrum^ reflected spectrum500400N-.;I--.300E"=- 200cn100s ....,...•^...... S.-1 ---------- ---- +00.0 252.0— incident spectrum^ reflected spectrumFigure 424 - Incident and reflected spectral density, test 10 (Hs = 6 cm, Tp = 1.6 s)and 0 = 45°552000I:, 1500INEE100050000.0^0.5^1.0^1.5^2.0^2.5Frequency (Hz)Figure 4.25 - Incident and reflected spectral density, test 11 (Hs = 12 cm, Tp = 1.6 s)and 9 = 45 °4.3.2 Comparison of Irregular and Regular Reflection CoefficientsFigures 4.26 to 4.29 show the average reflection coefficients and the reflection coefficients asfunctions of frequency for the irregular wave tests. These figures show that the estimatedreflection coefficient functions for the irregular wave tests exhibit a reasonably close fit to thosemeasured in the regular wave tests.For 0 = 0°, estimated reflection coefficients are higher than those measured in regular tests above afrequency of 0.6 Hz, and for frequencies greater than 1.0 or 1.2 Hz in all irregular tests, highvalues of reflection coefficient were measured. This is thought to be due to scattering of waveenergy from lower to higher frequencies caused by turbulent wave-breakwater interaction. Theresult of this action is that wave energy, incident at low frequencies, is reflected at higherfrequencies. This is more apparent for the tests with 0 = 0°, which indicates that the morevigorous action of the normally incident waves, as observed in Section 4.1.1, has lead to morefrequency scattering.Values of average reflection coefficient Kr of 31.2% and 31.5% were measured for 0 = 0° andHs = 6 cm and 12 cm respectively, and for 0 = 45° the measured values of Kr were 39.0% and37.5% with the same respective significant wave heights. The dependency of K r on 0 and Hs canbe examined using these results. For the limited number of irregular tests undertaken, it is apparentthat Kr varies more with 0 than with Hs. This is a similar result to that obtained in the regularwave tests, indicating that the average reflection coefficient has very little dependence on thesignificant wave height, while having a stronger dependency on 0. Higher values of Kr wererecorded for 0 = 45° in comparison to 0 = 0°. This indicates that Kr is dependent on 0 and that Krincreases with increasing 0, as was found for the regular wave tests with high d/gT 2 .560.4 1 .2 1 .40.8^1.0Frequency (Hz)0.6^ irregular waves, Hs =6cm—41— regular waves, H=10cm0.60.00.5Ea).5'. .E 0.4m000c 0.30a)c1) 0.2cc0.10.4Ztj--,a)o0^ irregular waves, H s=12cm^,-----.,-- regular waves, H=10cm0.60c 0.3=017) 0.2cc0.10.00.4^0.6^0.8^1.0^1.2Frequency (Hz)1 .4 1 .657Figure 426 - Comparison of regular and irregular reflection coefficients,test 10 (Hs = 6 cm, Tp = 1.6 s) and 0 = 0°Figure 427 - Comparison of regular and irregular reflection coefficients,test 11 (Hs = 12 cm, Tp = 1.6 s) and 0 = 0°^ irregular waves, Hs=6cm--1,- regular waves, H=1 Ocm0.60.5-6a)•_00.4a)00oc 0.30a)6 0.2cr0.10.0580.4^0.6^0.8^1.0^1.2^1.4^1 .6Frequency (Hz)Figure 428 - Comparison of regular and irregular reflection coefficients,test 10 (Hs = 6 cm, Tp = 1.6 s) and 0 = 45 °0.6^0.8^1.0^1.2^1 .4^1 .6Frequency (Hz)Figure 429 - Comparison of regular and irregular reflection coefficients,test 11 (Hs = 12 cm , Tp = 1.6 s) and 0= 45°4.4 Directionality ResultsThe phase of testing with multi-directional waves was undertaken to investigate two characteristicsof reflection from the breakwater. These were:1) The principal direction of the reflected waves and whether the angle ofreflection is equal to the angle of incidence.2) The amount of directional spreading in the reflected wave field.Tests were analysed using the analysis program MLMWP. This program was developed by theHydraulics Laboratory and the performance of this program has been described in detail by Nwogu(1989). Further development of the program to make it more suitable for wave fields containingincident and reflected wave energy was considered to be beyond the scope of this study. For thisreason, as was previously mentioned, the incident wave field was extracted so that the methodcould be applied to the reflected wave field alone. However, errors can be introduced in thisprocess when the incident wave signal is subtracted from the combined incident and reflectedsignal.Tests were performed with multi-directional waves generated in two different principal incidentwave directions. Good results appear to have been obtained in the multi-directional test for Op =30°, Tp = 1.6 s, Hs = 15 cm. Figure 4.30 shows the directional spreading functions at the peakfrequency fp and 1.5 fp. Definition of the directions with respect to the breakwater is shown inFigure 4.31. The incident wave field appears to have a principal direction of 35° rather than the30° angle of incidence specified. This discrepancy may be the result of errors in the wave signalgeneration, or, more probably, the result of errors in directional measurement.The principal reflected direction appears, from Figure 4.30, to be between 130° and 170°. Areflected direction of approximately 145° was expected, corresponding to the measured incident59direction of 35° and assuming the angle of reflection to be equal to the angle of incidence. Thisplot shows that the waves at frequency f = fp were reflected with a principal reflection angle of50°, which is a more oblique angle than the angle of incidence. For the waves of frequency f = 1.5fp the principal angle of reflection was 10°, which is almost normal to the breakwater. These twoangles differ considerably from but are centred around the expected reflection angle of 35°.The amount of directional spreading in the reflected wave field is shown by Figure 4.30 to behigher than the spreading of the incident wave field. The reflected directional spectra show a lot ofleakage of energy away from the principal direction, which may, or may not, indicate that theanalysis program had difficulty in determining the shape of the spectra. Figure 4.32 shows thestandard deviation of the incident and reflected directional spectra as a function of frequency.From this plot it is shown that the average ai of the incident spectrum is approximately 20° whilefor the reflected spectrum the average q is approximately 45°. This indicates that the reflection ofthe incident wave field by the breakwater leads to a considerable amount of directional spreading.The results of this test suggest that:1) For intermediate and lower frequency waves, the principal angle of reflectionis more oblique than the angle of incidence; while for higher frequency waves,which may or may not be present due to frequency scattering, the principal angleof reflection is less oblique than the angle of incidence.2) Some directional spreading may be expected in the reflected wave field.However, the analysis of the multi-directional test for Op = 15° did not provide meaningful results,possibly due to errors made in the operation of the experiment. Also, the analysis did not convergeto give meaningful directional spectra for either of the uni-directional tests, however, the analysisprogram MLMWP was expected to give better results when applied to data with some spreading indirection rather than the uni-directional data used in these tests. This meant that the directionality60—————...."- --.....incident spectrum at f=fpincident spectrum at f=1.5 fpreflected spectrum at f=fpreflected spectrum at f=1.5 fpreflectedwavesincidentwavesanalysis was not able to be applied to measured signals from the regular uni-directional wave testsor the irregular uni-directional wave tests.610.050.04.••••••■6)cy 0.0313CE3*0 -135 -90^-45 ^0^45^90^135^1800 (deg)Figure 4.30 - Incident and reflected directional spreading functions for test 13, Op = 30°Figure 4.31- Definition sketch showing directions expressed as angles0.6^0.8^1 .0^1 .2Frequency (Hz)Figure 432 - Standard deviation of incident and reflected spreading functions for test 13,Op = 30 °62Chapter 5CONCLUSIONS AND RECOMMENDATIONS5.1 Reflection of Regular, Obliquely Incident WavesTests were undertaken using a range of regular, uni-directional waves obliquely incident on amodel rubble-mound breakwater. A constant water depth of d = 0.5 m was used and waves weregenerated in four angles of incidence, 0 = 0°, 30°, 45° and 60°, with periods ranging from T = 1.0s to 2.4 s, and heights of H = 5 cm to 19 cm. The selection of these wave characteristics resultedin a range in wave steepness parameter of H/L = 0.016 to 0.066, and depth to wavelength ratio ofd/gT2 = 0.009 to 0.051.The model rubble-mound breakwater used was impervious, armoured with large size rocks, W > 1kg, and front and back slopes of 1 : 1.5.The reflection coefficient and the reflected phase lag, which describe the reflection characteristics ofthe breakwater, were estimated using a least squares method and measurements of water surfaceelevations from three probes. The least squares method, described by Mansard and Funke (1980)and Isaacson (1991), was extended in this study to analyse oblique wave reflection in a mannerindicated by Isaacson (1991). This analysis method assumed that the wave field was described bylinear wave theory and also assumed that there was no frequency or directional scattering in thereflection process and that the angle of reflection was equal to the angle of incidence. A frequencyanalysis and examination of the reflected wave records indicated that the assumption made aboutthe frequency of the reflected wave field was valid. Insufficient evidence was found in thedirectionality analysis to prove or disprove the assumption that the angles of reflection andincidence were equal. Some evidence of directional scattering was found, however this is notthought to greatly affect the results from the regular wave analysis. The accuracy of the regularwave analysis method was examined and it was found to give good accuracy.63Results from the regular wave tests indicate that the reflection characteristics, Kr and 13, arefunctions of the parameters depth to wavelength ratio d/gT 2 and angle of incidence 0. Kr and 13were shown to be independent of wave height.The measured reflection coefficients ranged from 10% to 59% for the range of incident wavesgenerated. For each angle of incidence 0 , K r was presented as a function of d/gT2 . These resultsindicated that relationships which describe the reflection of normally incident waves do notadequately describe the reflection of obliquely incident waves. In general, K r was shown toincrease with increasing angle of incidence 0, especially for waves with higher values of depth towavelength ratio d/gT2. Results also indicated the presence of a minimum value of Kr related to amaximum proportion of wave energy dissipated by wave breaking on the breakwater. The lowerproportion of energy dissipation observed in tests with more oblique angles of incidence wasspeculated to be the reason that K r was observed to increase with 0.Measured values of reflected phase lag 13 were in the range 13 = 9° to 116°. 13 was shown to be afunction of d/gT2, and increasing d/gT2 was shown to lead to increased values of P. The shift inposition of the standing wave pattern x' was observed to be approximately constant with changingvalues of d/gT2 for all angles of incidence except for 0 = 60°. The study also indicated that p andx' increase with increasing angle of incidence 0.5.2 Reflection of Irregular, Obliquely Incident WavesFour irregular wave test were undertaken in this study with the objective of determining whetherresults obtained from regular wave tests would adequately describe irregular wave reflection.Waves were generated in two angles of incidence 0 = 0° and 45° with constant water depth d = 0.5m, constant peak frequency corresponding to Tp = 1.6 s, with spectral density approximatelydescribed by a Bretschneider spectrum, and Hs = 6 cm and 12 cm.64The water surface elevation signals measured without the breakwater in place were subtracted fromthe signals measured with the breakwater in place to get the reflected wave water elevation signals.Estimates of the incident and reflected spectral density were then obtained by analysing the incidentand reflected wave elevation signals with a Fourier analysis program. Estimates of spectral densityfrom three probes were smoothed using a moving average filter, then averaged. The reflectioncoefficient function Kr(f) was estimated by dividing the reflected by the incident spectral densitythen taking the square root. The average reflection coefficient K r, was estimated for each test asthe square root of the ratio of the zeroth moments of the reflected and incident spectral density.The spectral density of the incident and reflected wave trains were compared and were seen to beclose in shape, however the peaks of the reflected spectra were generally at a lower frequency thanthose of the incident spectra. Also, some incident wave energy appeared to be scattered to higherfrequencies by the breakwater, resulting in higher amounts of reflected wave energy at frequenciesof approximately twice the peak frequency than was expected from results of the regular wavetests.The irregular reflection coefficient function Kr(f) compared closely to the reflection coefficientsmeasured for the regular wave tests. This comparison was not as close for waves of frequency f >0.6 Hz and angle of incidence 0 = 0°. In general, the results from the regular wave tests closelydescribed the reflection of the irregular waves.Values of average reflection coefficient Kr, ranging from 31% to 38% were measured. K r wasshown to be independent of significant wave height H s , but dependent on angle of incidence 0.For angles of incidence 0 = 0° and 45°, values of average reflection coefficient of K r = 31% and38% were measured. These results indicate that in general K r increases with increasing 0,reiterating the results of the regular wave tests.655.3 Directionality of Incident and Reflected Wave FieldsThe conclusions that were able to be taken from the directionality analysis were limited due to theinadequacy of the directional analysis method in analysing wave fields containing reflected waveenergy. Reasonable results appear to have been obtained for test 13 with principal incident wavedirection Op = 30°. The results from this test indicate that the multi-directional incident wave fieldwas reflected from the breakwater with an amount of directional spreading greater than that of theincident wave field. For this test, the principal angle of reflection was not equal to the principalangle of incidence for all frequencies, rather, results indicated that the angle of reflection was afunction of frequency. Higher frequency waves were reflected at angles more normal than theangle of incidence, and lower frequency waves were reflected at angles more oblique than the angleof incidence.5.4 Recommendations for Further StudyA review of the literature on wave reflection, undertaken during the initial stages of this study,revealed a lack of understanding of the reflection characteristics of oblique waves. This study hasincreased the level of understanding in this area. Relationships between the important parametersKr, (3, d/gT2 , and 0 have been determined and discussed for the conditions chosen in this study,however there is a need for more experimental work to be done. In particular, experimental data isneeded on oblique wave reflection from structures with different types of construction and faceslopes over a range of incident wave angles.More information is also needed on the directionality of reflected wave fields, includinginformation on the amount of directional spreading in the incident and reflected wave fields andinformation on the principal direction of the reflected wave field. For this purpose, an improvedmethod of directional wave analysis is needed which will estimate the incident and reflecteddirectional spectra in wave fields containing significant amounts of reflected wave energy.66ReferencesAllsop, N. W. H. and Hettiarachchi, S. S. L. (1988) "Reflections from CoastalStructures", Proc. 21st Coastal Engineering Conference, June, Costa del Sol-Malaga, Spain,pp.782-794.Battjes, J. A. (1974) "Surf Similarity", Proc. 14th Coastal Engineering Conference, June,Copenhagen, Denmark, pp.466-480.Delmonte, R. C. (1972) "Scale Effects of Wave Transmission through Permeable Structures",Proc. 13th Coastal Engineering Conference, July, Vancouver, Canada, pp.1867-1872.Goda, Y. and Suzuki, Y. (1976) "Estimation of Incident and Reflected Waves in RandomWave Experiments", Proc. 15th Coastal Engineering Conference, July, Honolulu, Hawaii,pp.828-845.Isaacson, M. I. (1991) "Measurement of Regular Wave Reflection", J. Waterway, Port,Coastal and Ocean Engineering, ASCE, vol.117, no.6, pp.553-569.Jamieson, W. W. and Mansard, E. P. D. (1987) "An Efficient Upright Wave Absorber",Proc. ASCE Speciality Conference on Coastal Hydrodynamics, Newark, Deleware.Kobayashi, N., Cox, D. T. and Wurjanto, A. (1990) "Irregular Wave Reflection andRun-up on Rough Impermeable Slopes", J. Waterway, Port, Coastal and Ocean Engineering,ASCE, vol.116, no.6, pp.708-726.Mansard, E. P. D. and Funke, E. R. (1980) "The Measurement of Incident and ReflectedWave Spectra using a Least Squares Method", Proc. 17th Coastal Engineering Conference, March,Sydney, Australia, pp.154-172.Mansard, E. P. D., Sand, S. E. and Funke, E. R. (1985) "Reflection Analysis of Non-Linear Regular Waves", Technical Report No. TR-HY-011, Hydraulics Laboratory, NationalResearch Council of Canada, Ottawa, Canada.Nwogu, O. (1989) "Analysis of Fixed and Floating Structures in Random Multi-DirectionalWaves", Ph.D. Dissertation, Department of Civil Engineering, University of British Columbia,Vancouver, Canada.67Seelig, W. N. and Ahrens, J. P. (1981) "Estimation of Wave Reflection and EnergyDissipation Coefficients for Beaches, Revetments, and Breakwaters", report no.TP 81-1, CoastalEngineering Research Centre, U. S. Army Engineer Waterways Experiment Station, Vicksburg,Mississippi.Seelig, W. N. (1983) "Wave Reflection from Coastal Structures" Proc. Conference on CoastalStructures, ASCE, Arlington.Shaver, M. D. (1989) "Regular Wave Conditions in a Directional Wave Basin", M.A.Sc.Thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada.Sollitt, C. K. and Cross, R. H. (1972) "Wave Transmission through PermeableBreakwaters", Proc. 13th Coastal Engineering Conference, July, Vancouver, Canada, pp.1827-1846.Tautenhain, E., Kohlhase, S. and Partenscky, H. W. (1982) "Wave Run-up at SeaDikes under Oblique Wave Approach", Proc. 18th Coastal Engineering Conference, November,Capetown, South Africa, pp.804-810.Thornton, E. B. and Calhoun, R. J. (1972) "Spectral Resolution of Breakwater ReflectedWaves", J. Waterway., Port, Coastal and Ocean Engineering, ASCE, vol.98, no.4, pp.443-460.U. S. Army Corps of Engineers (1984) "Shore Protection Manual", Coastal EngineeringResearch Center, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Mississippi.Wilson, K. W. and Cross, R. H. (1972) "Scale Effects in Rubble-Mound Breakwaters",Proc. 13th Coastal Engineering Conference, July, Vancouver, Canada, pp.1873-1884.68


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