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Wave reflection effects within a harbour O’Sullivan, Enda Joseph 1992-03-03

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Wave Reflection Effects Within a HarbourByEnda Joseph O'SullivanB.E., National University of Ireland, 1990A THESIS SUBMITT ED IN PARTIAL FULFILMENT OFTHE REQUIREMENT 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 ColumbiaMarch 1992© Enda O'Sullivan, 1992In presenting this thesis in 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 the 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 Engineering,The University of British Columbia,2324 Main Mall,Vancouver, B.C.,Canada, V6T 1Z4.Date : iiAbstractThis thesis summarizes a numerical model used to predict the wave field in a harbour of constantdepth with partially reflecting boundaries, and describes laboratory tests undertaken to assess thenumerical model and the importance of partial reflection effects. The numerical model is based onlinear diffraction theory, and involves the application of a partial reflection boundary condition.The extension to general harbour configurations that includes breakwaters is made by utilizing awave doublet representation of the fluid boundaries instead of the usual wave sourcerepresentation. The numerical model is initially compared to closed-form results for thefundamental case of a straight impermeable offshore breakwater, and the method is found tocompare well for this case. Further comparisons are made for a semi-circular harbour with a pairof symmetrical protruding breakwaters, and for a rectangular harbour with a pair of symmetricalprotruding breakwaters. The boundaries of the semi-circular harbour were perfectly absorbing andthe numerical model predicts the wave field within the harbour realistically. For the laterconfiguration cases which are considered include perfectly absorbing, perfectly reflecting andpartially reflecting harbour boundaries, and in all cases the numerical model predicts the wave fieldwithin the harbour realistically.Experiments were conducted at the Ocean Engineering Centre at BC Research, Vancouver. Duringthe experiments the wave field within a model harbour was measured under different conditionscorresponding to changes in the wave period, incident wave direction, incident wave height, andreflection coefficients of the harbour boundaries and breakwaters. The experimental results arecompared to those of the numerical model and agreement is generally good. In general the waveheights within the harbour are slightly underpredicted, while the wave heights outside the harbourare slightly overpredicted. Overall, the numerical model is found to provide a reasonably reliablemeans of predicting the wave field within a harbour of constant depth and arbitrary shape withpartially reflecting boundaries.iiiTable of ContentsPageAbstract	 i iTable of Contents	 iiiList of Tables	 vList of Figures	 v iList of Photographs	 i xList of Symbols	 xAcknowledgment	 xi i1	 Introduction	 11.1	 General	 11.2	 Literature Review	 31.3	 Research Objectives	 42	 Numerical Model	 62.1	 Mathematical Treatment	 62.1.1 Governing Equations	 62.1.2 Extension to Partial Reflection	 72.1.3 Green's Function Representation	 92.2	 Numerical Approximation	 113	 Physical Model	 133.1	 Experimental Facilities	 133.2	 Model Harbour	 14iv3.3	 Dimensional Analysis	 153.4	 Wave Elevation Measurement	 163.5	 Reflection Analysis	 173.6	 Wave Generation, Data Acquisition and Analysis 	 183.7	 Test Program	 204	 Results & Discussion	 2 24.1	 Comparison of Numerical Results with Exact Solutions	 224.1.1 Straight Offshore Breakwater	 224.1.2 Breakwater Gap	 234.2	 Effects of Reflection Coefficients	 244.3	 Numerical and Experimental Analysis of Harbour	 264.3.1 Experimental Results	 264.3.1.1 Effect of Wave Period	 264.3.1.2 Effect of Incident Wave Direction	 274.3.1.3 Effect of Boundary Reflection Characteristics	 274.3.1.4 Effect of Incident Wave Height 	 284.3.1.5 Measured Reflection Coefficients	 294.3.2 Comparison of Numerical and Experimental Results 	 295	 Conclusions & Recommendations	 3 25.1	 Conclusions	 325.2 Recommendations for Further Study	 33Bibliography	 34Tables	 3 7Figures	 4 0Photographs	 7 8VList of TablesTable 3.1	 Wave conditions and reflection coefficients for each of the 20 test runs of theexperimental model.Table 4.1Table 4.2Table 4.3Effect of the number of segments N on the computed wave force on a vertical plate.Measured reflection coefficients for vertical plywood, rock at slope 1:1.5, andsand at a slope of 1:2.5 overlain with horsehair.Wave conditions and reflection coefficients for the 4 test runs of the numericalmodel. (Note: Test number corresponds to that for the experimental model inTable 3.1)List of FiguresFigure 1.1	 Definition sketch of general harbour.Figure 2.1	 Geometry of Green's function representation.Figure 3.1	 Sketch of the wave basin at BC Research.Figure 3.2 Location of Comox, BC.Figure 3.3 Details of Comox harbour.Figure 3.4 Sketch of the model harbour showing principal dimensions.viFigure 3.7Figure 3.8Figure 3.9Figure 4.1Figure 4.2Figure 4.3Sketch of the boundary configurations for the 3 Phases of the laboratory tests.Dimensions of the breakwater and harbour boundaries. (a) breakwater for Phases 1and 2, (b) breakwater for Phase 3, (c) harbour boundary for Phase 1, (d) harbourboundary for Phases 2 and 3.Sketch of the wave probe frame.Location of the wave field measurements.Sketch of experimental layout for reflection coefficient measurements.Rigid vertical plate used in the numerical example.Wave height distribution along a straight offshore breakwater with B/L = 2.0,0 = 0°. (a) upwave face, (b) downwave face. 	 N = 10, 	 N = 20,	 exact solution.Diffraction coefficient contours in the vicinity of the offshore breakwater with B/L= 2.0, 0 = 0°. (a) N = 10, (b) N = 20, numerical solution,   exactsolution.Figure 3.5Figure 3.6Figure 4.4	 View of surface elevation (at t = 0) in the region of the offshore breakwater forB/L = 2.0 and 0 = 0°.Figure 4.5	 Sketch of the semi-circular harbour used as a numerical example.viiFigure 4.6	 Diffraction coefficient contours in the vicinity of the gap for the semi-circularharbour with B/L = 1.0, 0 = 0° and Kr = O.	 numerical solution,	 exact solution.Figure 4.7	 View of surface elevation (at t = 0) in the region of the breakwater gap for the semi-circular harbour with B/L = 1.0, 0 = 0° and K r = O.Figure 4.8	 Sketch of the rectangular harbour used as a numerical example.Figure 4.9	 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = O.Figure 4.10 View of surface elevation (at t = 0) within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = O.Figure 4.11 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = 1.0.Figure 4.12 View of surface elevation (at t = 0) within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = 1.0.Figure 4.13 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = 0.2.Figure 4.14 View of surface elevation (at t = 0) within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = 0.2.Figure 4.15 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0°. (a) Kr = 0, (b) Kr = 0.1, (b) Kr = 0.2.Figure 4.16 Diffraction coefficient contours for Phase 1 tests with 0 = 0° and H = 30 mm,showing the effect of wave period. (a) T = 0.94 sec, (b) T = 1.07 sec,(c) T = 1.2 sec.Figure 4.17 Diffraction coefficient contours for Phase 2 tests with 0 = 0° and H = 30 mm,showing the effect of wave period. (a) T = 0.8 sec, (b) T = 0.94 sec,(c) T = 1.07 sec, (d) T = 1.2 sec.viiiFigure 4.18Figure 4.19Figure 4.20Figure 4.21Figure 4.22Figure 4.23Figure 4.24Figure 4.25Figure 4.26Diffraction coefficient contours for Phase 3 tests with 0 = 0° and H = 30 mm,showing the effect of wave period. (a) T = 0.8 sec, (b) T = 0.94 sec,(c) T = 1.07 sec, (d) T = 1.2 sec.Diffraction coefficient contours for Phase 2 tests with T = 0.8 sec showing theeffect of incident wave direction. (a) 0 = -30°, (b) 0 = 0° (c) 0 = +30°.Diffraction coefficient contours for T = 0.94 sec, 0 = 0° showing the effect ofboundary reflection characteristics. (a) Phase 1, (b) Phase 2, (c) Phase 3.Diffraction coefficients along a cross-section at y = 2.1 m for Phase 2 tests withT = 0.8 sec and 0 = 0°.Diffraction coefficients along a cross-section at y = 2.2 m for Phase 3 tests withT = 0.8 sec and 0 = 0°.Diffraction coefficient contours for Phase 1 tests with T = 1.2 sec, 0 = 0°.(a) experimental results, (b) numerical results.Diffraction coefficient contours for Phase 2 with T = 0.8 sec, 0 = 0°.(a) experimental results, (b) numerical results.Diffraction coefficient contours for Phase 2 with T = 1.2 sec, 0 = 0°.(a) experimental results, (b) numerical results.Diffraction coefficient contours for Phase 2 with T = 0.8 sec, 0 = +30°.(a) experimental results, (b) numerical results.List of PhotographsPhotograph 3.1	 Experimental layout in the wave basin at BC Research.Photograph 3.2	 Wavemaker calibration in the wave basin at BC Research.Photograph 3.3	 Wavemaker calibration in the wave basin at BC Research.Photograph 3.4	 Experimental layout for Phase 1 tests.Photograph 3.5	 Experimental layout for Phase 2 tests.Photograph 3.6	 Experimental layout for Phase 3 tests.Photograph 3.7	 Measurement of the reflection coefficient of the sloping sand.Photograph 3.8	 Measurement of the reflection coefficient of the vertical plywood.Photograph 3.9	 Close-up of the measurement of the reflection coefficient of the rocks.ixList of SymbolsThe following symbols are used in this thesis:A = - 411 ,(0C = wave celerity,d = water depth ,g = gravitational constant,G(x,t) = Green's function for a wave doublet,H = wave height,Hi = incident wave height,fl(m1) = Hankel function of the first kind and order m,i =,k = wave number,L = wavelength,n = surface normal vector at segment centre (see Figure 2.1),n = distance in the direction of normal,N = number of segments,r = distance between x and t,S = fluid boundary,S i = one-sided boundary,S2 = two-sided boundary,xxit = time,T = wave period,x = position vector given by (x, y) in the two dimensional problem(see Figure 1.1),z = vertical coordinate measured upwards from the still water level,a = angle between r and surface normal n (see Figure 2.1),R = angle between r and the doublet axis (see Figure 2.1),AS = segment length,clo = velocity potential,4) = velocity potential function,4)w, 4)s = incident and scattered wave components of 4),ri = free surface elevation,0 = wave direction measured anti-clockwise relative to the x axis (see Figure 1.1),(I) = wave angular frequency,t = position vector given by (4,i) giving the location of Green's function on thefluid boundary (see Figure 2.1).xiiAcknowledgmentThe author would like to thank his supervisor Dr. Michael Isaacson for his guidance andencouragement throughout the preparation of this thesis.The author would like to express a special word of gratitude to Mr. John Baldwin for his help inexplaining and running the numerical model and to Mr. Sundar Prasad for his help with thecomputing facilities at UBC. Thanks also to the staff of the Ocean Engineering Centre atB.C. Research, Vancouver, in particular Mr. M. Shaver, for their help throughout the experimentalsequence carried out at their facilities. A special acknowledgment is made to Naomi, Reddy, Dave,Mike, Andrew, and Amal for volunteering their time, to help in the construction of theexperimental layout. Without their help, the experimental sequence would surely have taken twiceas long.Finally, the financial support of a Research Assistantship from the Natural Sciences andEngineering Research Council of Canada is gratefully acknowledged.1Chapter 1Introduction1.1 GeneralA primary consideration in the design of harbours is the degree of protection afforded to vesselswithin the harbour. Consequently the prediction of the wave field within a harbour is of majorconcern to the design team planning the harbour. In developing such predictions, it should beborne in mind that the wave field within the harbour may be influenced by a combination of wavetransformation effects, including wave shoaling, wave diffraction, wave refraction, wavereflection, wave breaking and wave run-up.Over the years both experimental and theoretical approaches to the solution of wavetransformations in a harbour have been developed. The most fundamental theoretical approachhas dealt primarily with the problem of wave diffraction around breakwaters in a harbour ofconstant depth. However, these calculations have either ignored wave reflection off interiorboundaries of the harbour, or else have treated all boundaries as fully reflecting. Since partialreflections are not accounted for, these approaches may give rise to incorrect predictions of thewave field. Closed-form solutions for wave diffraction around a straight semi-infinite breakwater,and a gap between a pair of co-linear straight, semi-infinite breakwaters are often used in marinadesign for estimating short wave diffraction into marinas. In order to treat the more general case ofharbours and/or breakwaters of arbitrary configuration, as indicated in Fig. 1.1, research into thedevelopment of suitable numerical approaches has been underway for some time. Bearing in mindthe ultimate objective of treating the general case of a harbour of arbitrary shape, with breakwatersand with partial reflection, Isaacson and Baldwin (1991) recently presented a numerical method ofpredicting the wave field in a harbour using a wave doublet distribution along the fluid boundaries.The wave doublet distribution is not as widely used as the conventional wave source distribution.2The method is based on linear diffraction theory and gives rise to an integral equation based ona wave doublet distribution along the fluid boundaries. Advantages of a doublet distribution over asource distribution are:(i) breakwaters can be modelled. Unlike a doublet distribution, a source distribution cannotmodel a thin breakwater because the boundary conditions on both sides cannot be modelledsimultaneously when the breakwater is treated as very thin.(H) partial reflection along the boundaries can be modelled. For one-sided boundaries along aharbour contour the doublet distribution can be extended to include the case of partialreflection. In the case of a two-sided boundary, corresponding to a thin breakwater, thescattered potential may be represented as due to a distribution of both wave sources and wavedoublets.The use of a doublet or dipole distribution is well known from classical hydrodynamics (Lamb,1932) but little has been reported on its use to solve problems of water wave diffraction (Mei,1978, Yeung, 1982, Hess, 1990).Isaacson and Baldwin (1991) compared their formulation to the fundamental case of a straightimpermeable offshore breakwater for which a closed-form solution is available and the methodcompared well. Further comparisons were made with more general diffraction problems such as acircular cylinder, rectangular harbour, and a circular harbour with protruding breakwaters, and inall cases excellent agreement with known solutions was obtained.The lack of previous work based on the wave doublet representation of the fluid boundariesprompted the present research to try to verify or calibrate the numerical model using experimentalresults. A harbour of arbitrary shape with both two-sided fully-reflecting boundaries, and one-sided partially-reflecting boundaries is modelled by using wave doublet representations of the fluidboundaries.31.2 Literature ReviewA number of theoretical analyses have been carried out to investigate wave diffractionphenomena neglecting the effects of interior boundary reflection. Penny and Price (1952)published a solution of the boundary value problem for small amplitude waves impinging on asingle semi-infinite straight breakwater, based on the equivalent problem in optical diffractionwhich had earlier been solved rigourously by Sommerfeld. This was verified by experimental datapresented by Putman and Arthur (1948). Wiegel (1962) used Penny and Price's theoreticalapproach to study wave diffraction around a single breakwater and presented tables of diffractioncoefficients and corresponding diffraction diagrams. These are also given in the 'Shore ProtectionManual' (1984), and in other texts.The theory of diffraction of water waves which are incident normally through a gap between apair of coliner, straight semi-infinite breakwaters was also described by Penny and Price (1952).Their theoretical work was in agreement with the experimental work carried out earlier by Blue andJohnson (1949). Johnson (1951) developed an approximate analytical solution to obtaindiffraction patterns for waves approaching a gap between a pair of semi-infinite colinearbreakwaters from various wave directions. More recently, this solution was extended to non-colinear breakwaters by Memos (1980). Sobey and Johnson (1986) investigated narrowbreakwater gaps, typical of smaller gaps where available results were sparse, and extended thetechnique to angled incidence for wide breakwater gaps and generally to non-aligned breakwaters.Kos and Kilner (1987) carried out a set of experiments dealing with pure wave diffractionthrough a breakwater gap. They eliminated the effects of reflected waves, cross waves and basinresonance effects.The above cases all relate to short wave diffraction into a harbour so that wave reflections offthe harbour boundaries are neglected. On the other hand, long wave resonance in a harbour is4governed by complete reflection along the boundary. For this case Hwang and Tuck (1970)presented a numerical procedure by representing the harbour boundary as a distribution of wavesources with strengths chosen to ensure that the full reflection boundary condition is satisfied alongthe boundary. An alternative approach proposed by Lee (1971) involves dividing the fluid intotwo regions, one within the harbour and the other exterior to the harbour, and applying matchingconditions at the boundary between the regions. Mattioli and Tinti (1980) extended this method toharbours with a projecting breakwater or headland. In a variant of the method, Chen and Mei(1974) used a finite element solution for the interior region matched to a boundary integral solutionfor the exterior region. The special cases of resonance in a rectangular harbour has beeninvestigated by Miles and Munk (1961), Garrett (1970) and Mei (1983).The problem of partially reflecting boundaries was treated by Berkhoff (1976). The boundariesare schematized as vertical and a mixed boundary condition is used instead of the full reflectioncondition. Chen (1986) introduced partial reflection and bottom friction refinements to a hybridelement model of wave behaviour in a harbour. Isaacson and Qu (1990) presented a generalsolution for wave behaviour in a harbour of arbitrary shape and constant depth, based on theapproach indicated by Berkhoff (1976) with a matching boundary and taking partial reflections intoaccount. Isaacson and Baldwin (1991) used a wave doublet representation of the harbourboundaries, for harbours of arbitrary shape and constant depth, taking partial reflections intoaccount.1.3 Research ObjectivesThe objectives of the present investigation are:(i) to carry out laboratory tests with a model harbour using different interior reflectioncoefficients in order to investigate the wave field within the harbour, described by contoursof wave height and variations of water surface elevation (1) and wave height (H) alongtraverses of the harbour interior.5(ii) to streamline the numerical model of Isaacson and Baldwin (1991) and to compare itspredictions to corresponding results of the physical model.6Chapter 2Numerical Model2.1 Mathematical Treatment2.1.1 Governing EquationsThe general case of a harbour of arbitrary configuration and with one or more breakwaters isshown in Fig. 1.1. It is assumed that all topographical irregularities lie within the contour C wherethe depth is constant and that the coastline is otherwise straight and coincides with the y-axis. Atrain of regular small amplitude waves approaches the harbour as shown and the wave field in thevicinity of the harbour is to be determined. A coordinate system (x,y,z) is defined with x and yhorizontal and z measured vertically above the still water level. The fluid is assumedincompressible and invisid and the flow is irrotational, so that the flow may be described by avelocity potential (130 which satisfies the Laplace equation within the fluid region. Provided that allbarriers are considered vertical and to extend from the seabed (or deep water) up to the freesurface, the velocity potential is represented as:= A 0(x cosh[k(z+d)] cto(x,y,z)	 exp(-iwt)	 [2.1]cosh(kd)where t is time, d is the still water depth, (0(x) is a two-dimensional potential function which is tobe determined, and x represents a general point (x,y) in the horizontal plane. Also A = -igH/2co,i = H is the incident wave height, k is the wave number, and co is the angular frequencywhich is related to the wave number by the linear dispersion relation:CO 2 = gk tanh(kd)	 [2.2]Eq. [2.1] directly satisfies the seabed and free-surface boundary conditions. In addition, thepotential function itself must satisfy the Helmholtz equation within the fluid region and is also7subject to a boundary condition along the fluid boundaries and to a radiation condition. In the caseof complete reflection at impermeable boundaries, the boundary condition corresponds to that ofzero normal velocity along the fluid boundary:a(I) — 0an — [2.3]where n is the distance normal to the fluid boundary as indicated in Fig. 1.1.It is convenient to express the potential function 4) as a superposition of a known incident wavepotential,,, and a scattered wave potential 4s:(I) = Ow ± Os	 [2.4]The incident wave potential O w is known and may be expressed as:Ow (x) = exp [ ik (xcos0 + ysin0)]	 [2.5]where 9 is the incident wave direction measured from the x axis as shown in Fig. 1.1.Solving for 4)s is the crux of the problem, since a solution for 4)s then directly provides 0. Anyrequired property of the wave field may then be obtained. In summary, 4) s satisfies (i) theHelmholtz equation, (ii) the radiation condition, and (iii) the reflection boundary condition.2.1.2 Extension to Partial ReflectionFor the more general case of wave diffraction in harbours, neither the assumption of fullyabsorbing or fully reflecting boundaries is really appropriate since in practice partial reflectioninvariably occurs within a harbour. A boundary condition corresponding to partial reflection maybe introduced in the manner used by Chen (1986), and Isaacson and Qu (1990). This takes theform of a mixed boundary condition :& I) ±ocko = 0Dn [2.6]8in which n is distance into the fluid region measured normal to the boundary and a ( = at + ia2) isa complex transmission coefficient. This coefficient may be interpreted in a number of differentways as summarized by Isaacson and Qu (1989). These relate to;(i) its relation to the rate of transfer of energy at the boundary,(ii) its relation to the height and phase of the wave field at the boundary,(iii) its relation to the conventional reflection coefficient.In particular, the transmission coefficient a ( = al + ia2) may be related to the reflectioncoefficient Kr and a phase shift p associated with the reflection, and the angle y which the incidentwave train makes with the normal to the boundary. Assuming the wave train undergoes obliquereflection from a vertical wall located at x = 0, the total potential of the combined wave fieldcorresponds to a three-dimensional wave pattern and may be written as the sum the incident andreflected wave potentials;(1) = A [ exp[ik (x cos 7 + y sin 7)] + Kr exp[-ik (x cos y - y sin y) + 43]) [2.7]Here Kr is defined as the ratio of the reflected wave height to the incident wave height.Substituting Eq. [2.7] into Eq. [2.6], the transmission coefficient a is given as:al — 1 + IC? + 2 Kr cos(1 - IC?) cos y 1 + Ki. + 2Kr cos 13For the particular case of normally incident waves (7 = 0°) and 13 = 0° may be expressed interms of the conventional reflection coefficient Kr as:a 1 = 01 - Kr a 2 — 1 + Kra; —2K r sin p cos y[2.8][2.9]92.1.3 Green's Function RepresentationThe boundary value problem which has been specified is solved by expressing the scatteredpotential at any point x in the fluid domain as due to a distribution of wave doublets on the fluidboundary S:Os(x) = —14ic sJr 1-L(t) G(r,t) dS	 [2.10]where g(t) represents the doublet strength distribution function, t = (40)) is the doublet locationalong the fluid boundary S and G is a known Green's function for a wave doublet. Thiscorresponds to a fundamental solution of the Helmholtz equation which satisfies the radiationcondition, and is given as:G(X;t) = i IC H(P(kr) cos[ 	 [2.11]where r = Ix - tl = '\/ (x- )2 + (Y-T1)2, R is the angle at the doublet location which the point xmakes with the doublet axis taken normal to the surface contour as shown in Fig. 2.1 and HT isthe Hankel function of the first kind and order one.The application of the boundary condition Eq. [2.3] for the general case of partial reflectiongives rise to the following integral equation for t():f	 ;t)	 a(p.(4) aG 	 dS +	 a(i)wu.(4) G (x; 4) dS – - ,,, ( x) - k a(x) O w(x) [2.12]4n	 an	k 4nx) iSwhere x is the point on S at which the boundary condition is applied, and n is the normal vector toS at x. In the case of a fully reflecting portion of the boundary, a = 0 so that the second integral inEq. [2.12] is then absent. In the case of a fully absorbing portion of the boundary, a = i and theradiation condition is satisfied directly so that this portion of the boundary can then simply beomitted from Eq. [2.12]. Along one-sided boundaries Si, the Green's function may be taken as10either a wave source or wave doublet, although it is customary to use the wave source (eg. Hwangand Tuck, 1970). In Eq. [2.12] the Green's function has been taken to be a wave doublet and itwill be shown here that this provides a practical alternative to wave source methods. Along two-sided boundaries S2, the wave source representation is no longer appropriate since a sourcedistribution involves a velocity discontinuity across the contour of the distribution. A wavedoublet avoids this difficulty and may be used to simultaneously satisfy the boundary condition offull reflection on both sides of a breakwater, Hunt (1980). The extension to partial reflectionimplies that the velocity potential must be represented as due to a distribution of both wave sourcesand wave doublets. This refinement gives rise to numerical difficulties and has not yet beendeveloped so that in the present study two-sided boundaries are taken to be fully reflecting. Thusa is then zero and the second integral in Eq. [2.12] is omitted for x on S2.In evaluating the integrals in Eq. [2.12], the derivative of the Green's function aG/an isrequired. This may be expressed as:aGan	H (1x;4) = - in {	 'r)(1cr) cos (y+(3) — k H(P) (kr) cosy cos13}	 [2.13]where HW is the Hankel function of first kind and order zero, and y and 13 are indicated in Fig. 2.1and are related to the normal vectors at x and 4:nx (x-4) + 4 (y-11) cos y —	 [2.14]rri (x-4) + n ); (y-i) cos 13 — x r [2.15]where nx and ny are the direction cosines of the normal vector n with respect to the x and ydirections, and the superscript denotes the location at which the direction cosines are evaluated.1 12.2 Numerical ApproximationThe integral equation is solved by a discretization process in which the fluid boundary isdivided into N short straight segments, and the doublet strength distribution is assumed constantover each segment . In this way the integral equation is transformed into a matrix equation: NI Bii jai = bij=1 [2.16]whereNowbi = -	 --.1 (xi) - k a(xi) cp w(xi)Bid  1 1 a9— ' '(x' 4i) dS + k a(xi) 4-17i AfsGi (xj;4j) dS' 4ir	 anASj[2.17][2.18]and ASj is the length of the j-th segment, xi denotes the centre of the i-th segment, and theintegrations apply to the moving point t.The evaluation of the matrix coefficients Bii is carried out separately for the first and secondintegrals in Eq. [2.18]. For i#j, the second integral is evaluated by the usual mid-pointapproximation. However, for the first integral the normal velocity induced by one segment on aneighbouring segment is not small, so that a mid-point approximation is unsuitable and a numericalintegration is then necessary. This has been carried out using a 4 point Gaussian quadrature rule(e.g. Brebbia and Walker, 1980), with care taken to include the variation of y and 13 along thesegment length. When i = j, a singularity occurs in both integrals so that an analytic integration isthen used. Retaining the leading terms in expansions for aG/an and G near the singularity asuitable approximation for Bii is given as:B	 2 	- kAS { In (kAS ) - 1 } + i kA8 S + aii —		 2- nkAS	 4n	 47c [2.19]12where the higher order terms, though not strictly required for convergence, have been found togreatly increase the performance of the method.Once the matrix coefficients Bij have been evaluated, Eq. [2.16] can be solved by a standardcomplex matrix inversion procedure to provide the doublet strengths pj. The potential function, O sat a general point x, may then be obtained from a discretized version of Eq. [2.10]:1Os(x) =	 G(x;j) ASi47c j= 1[2.20]where, as a consistent approximation, G is assumed constant over the segment length and a mid-point approximation is used. If the point x lies on the boundary, a singularity occurs when x is at jand an integration of the leading singular term gives:(	 =	 [2.21]where the positive sign corresponds with the definition of the surface normal.Once Os and hence 4:• are known, then any required property of the wave field may be obtained.In particular, the water surface elevation ri at time t=0 is useful in obtaining a general view of thewave field at a particular instant, and the diffraction coefficient Kd describes the variation of thewave heights within the harbour. These are given as:tl = Re (0)	 [2.22]Kd =14)w +	 [2.23]13Chapter 3Physical Model3.1 Experimental FacilitiesA set of laboratory experiments relating to the wave field within a harbour were conducted at theOcean Engineering Centre (OEC) at BC Research, Vancouver during November 1991. The Centreis operated by the British Columbia Research Corporation, under an agreement between BCResearch, the University of British Columbia, and the National Research Council of Canada(NRC).The wave basin at OEC measures 30.5 m x 26.5 m (100 ft x 87 ft) with a maximum operatingdepth of 2.4 m (8 ft) deep (see Fig. 3.1). The basin is equipped with a unidirectional wavemaker,and a modern VAX computer system. The dimensions of the wavemaker are 15 m x 1.8 m (50 ftx 6 ft). It may be relocated within the basin in order to provide for the propagation of waves froma number of different directions. The wavemaker may be operated in a number of different modeswhich can be selected by adjusting a mechanical pivot point. The wavemaker may be operated ineither a piston mode, a hinged flapper mode, or a combined mode with equal contributions ofpiston/flapper. This allows for the accurate simulation of shallow, intermediate and deep waterwaves.Wave absorber modules may be positioned around the basin so as to minimise the corruption ofthe measured wave field by wave reflections from the basin walls . These wave absorbers aremade from two sheets of 0.6 m x 3 m perforated metal fixed 0.3 m apart by timber blocks. Theabsorbers are portable and work by dispersing the wave energy as it passes through the perforatedmetal.14The wave generation, data acquisition and analysis is carried out using the GEDAP softwaresystem. GEDAP was developed at the NRC Hydraulics Laboratory in Ottowa and is an acronymfor Generalized Experiment control, Data acquisition and data Analysis Package.3.2 Model HarbourThe hypothetical model was conceived on the basis of a harbour at Comox, B.C., Canada (seeFig. 3.2). Fig. 3.3 shows the harbour at Comox in detail, including depth soundings andbreakwater lengths. It should be noted that the marina located to the left of the breakwater hassince been relocated to the right of the breakwater (i.e. inside the new harbour).In planning the model harbour layout, a length scale ratio of 1:50 was found to be suitable, andthe water depth was kept constant at 450 mm. On the basis of Froude scaling, the time scale ratiois 1: 4-5-0. A view of the physical model is shown in Photograph 3.1 and a sketch of the model isgiven in Fig. 3.4.The model experiments can be divided into three different phases as indicated in Fig. 3.5. InPhase 1, both the breakwater and the harbour interior represented fully reflecting vertical walls.This was accomplished by constructing the harbour boundaries from vertical sheets of plywood(see Fig. 3.6 (a), (c)), with hardboard used for the curved portions of the boundaries as shown inPhotograph 3.4. In Phase 2, the breakwater remained fully reflecting while the interior harbourboundaries were changed to represent a partially reflecting beach. In order to achieve this, theplywood and hardboard were replaced by a beach of slope 1:2.5, which was comprised of sandoverlain by a layer of artificial horsehair as shown in Fig. 3.6 (d) and Photograph 3.5. In Phase 3,the breakwater was changed to represent a partially reflecting rubblemound breakwater (seeFig. 3.6 (b)) and the harbour interior remained a partially reflecting beach (see Photograph 3.6).The rubblemound breakwater was represented by placing rocks, of mean diameter 60 mm, againstthe existing plywood breakwater at a slope of 1:1.5. While running the tests for Phase 1 aconsiderable interval was required to allow for dissipation of the wave energy.15The wavemaker at BC Research has four sections on the wave board. In order to minimise theamount of wave energy produced, the two end sections of the wave board were disconnected. Asa result the wavemaker had to be calibrated. To do this calibration a 2-dimensional wave flume,3 m wide, was placed orthogonal to the wave board face. This enabled the measurement of thewave produced by the wavemaker without interference from wave reflections, diffraction and otherdistortions.3.3 Dimensional AnalysisIn planing the model tests and presentation of results, it is useful to carry out a dimensionalanalysis of the problem in order to identify the governing parameters so that controlled variables inthe model could be suitably varied. For regular incident waves and a specified harbourconfiguration, the wave height H at any location within the harbour may be expressed in the form:H = f (Hi, 0, d, L, g, x, y)	 [3.1]whereHi is the incident wave height,9	 is the angle of wave incidence,d	 is the still water depth,L	 is the wavelength,g	 is the acceleration due to gravity,(x,y) is the position inside the harbour.Note that the wave celerity c or wave period T are not specifically identified since these may beexpressed in terms of d, g and L by linear wave theory.The application of dimensional analysis to Eq. [3.1]then provides:Hi = f	 H	 [3.2]16The wave steepness Hi/L may be omitted if nonlinear effects are ignored. In Eq. [3.1] H/Hicorresponds to the diffraction coefficient Kd. Consequently, the diffraction coefficient mayexpressed in the form:Kd =g=f{Lf. , dre} [3.3]Thus, in carrying out tests for a specified harbour configuration subjected to regular waves,contour plots of the diffraction cofficient may be obtained for different incident wave directions 0,and different values of d/L corresponding to changes in the wave period. In addition, since theprimary focus of the present study is an examination of the effects of the reflectivity of the harbourboundaries, contour plots would also be required for different degrees of reflectivity.3.4 Wave Elevation MeasurementCapacitance wave probes were used to measure the instantaneous water surface elevation.Water level measurements made with these probes are accurate to within ± 1.0 mm, and are notinfluenced by spray above the continuous air/water interface.The wave probes were calibrated by using the GEDAP calibration called RTC_NPCAL. Thecalibration is based on a fourth order polynomial relating the wave elevation to the correspondingsensor signal measured in volts. The corresponding fine calibration constants were computed andstored in the GEDAP port file. They were subsequently used in processing the data (see section3.6).It was essential to place wave elevation probes at a sufficiently large number of grid positions inorder to measure the wave field throughout the interior of the model harbour. The grid spacingwas small enough to provide an acceptable resolution of wave elevation information, while notresulting in too cumbersome an amount of data. The grid spacing was also prescribed by theshortest wavelength to be tested. A 0.6 m spacing was deemed appropriate.17The wave probe apparatus is shown in Fig. 3.7. It consists of an array of 14 wave probes setas two rows of 7 probes supported on a rigid frame which could be moved to the requiredlocations with relative ease. One reference probe was located outside the harbour as indicated inFig. 3.8. The measurement of the wave field within the harbour was achieved by moving therectangular array of fourteen probes to three and four positions for Phases 1 & 2 and Phase 3respectively. The corresponding areas covered in the three phases the area shown in Fig. 3.7.3.5 Reflection AnalysisIn order to ensure accurate prediction of the wave field using the numerical model, it is essentialto be able to specify values of the reflection coefficients of the fluid boundaries. To this end aseries of reflection analyses were carried out on the physical model.The experimental layout for these tests is shown in Fig. 3.9. A set of two plywood guidewalls, 1 m apart, were positioned perpendicular to the inner harbour boundary so as to create atwo-dimensional wave field without any influence from reflection or other wave distortions, thusensuring accurate measurement of the reflection coefficients. Three colinear wave probes were setup, with the nearest located 3 m from the boundary. A series of three tests were carried outcorresponding to the three categories of boundary located at the end of the guide walls:(i) sand at 1:2.5 overlain with horsehair, (see Photograph 3.7),(ii) a vertical plywood sheet, (see Photograph 3.8),(iii) rocks at 1:1.5 positioned against the plywood, (see Photograph 3.9).For each boundary, wave reflection tests were carried out for the five wave conditions used in theexperiments.The data was analysed using the program REFLM. This program separates the incident and thereflected wave from a measured wave field on the basis of a least squares analysis using data from3 probes. The accuracy of the method decreases when the spacings between the two pairs ofadjacent probes is equal (Isaacson, 1991), and consequently probe two was placed slightly off18centre such that L12 = 0.95 L23, where L12 and L23 are the distances between the two pairs ofadjacent probes. Isaacson (1991) found that this relative spacing should have good accuracy.3.6 Wave Generation, Data Acquisition and AnalysisThe GEDAP software package includes a program category denoted WAVE_GEN whichcontains a comprehensive set of programs for two-dimensional wave generation in laboratoryflumes, towing basins and wave basins. A program RWREP2 computes the wave machine controlsignal for a regular wave train corresponding to a wave height and period specified by the user.The wave heights and periods may be specified in either full scale units or model scale units, sinceRWREP2 automatically converts these to model scale units when calculating the control signal forthe wave machine. The duration of the control signal is always set to an integer number of waveperiods, so that the signal can be continuously recycled when driving the wave machine. Thecontrol signal file produced by program RWREP2 is sent to the wave machine controller through aD/A output channel by using the real-time control program RTC.The software package RTC (Real Time Control) Single User System was used in all stages ofthe experimental procedure. RTC consists of a main hardware execution program and a commandentry program that allows the user complete control over data acquisition, control loops and signalgeneration.Wave generation was carried out by first loading the control signal file into an RTC buffer fileand then enabling the buffer to start the wave machine. When the enable command was given, theoutput signal was smoothly ramped up from zero amplitude to full amplitude over a period of10 sec. This automatic ramping was a carried out in order to protect the wave machine from beingsubjected to sudden transients in its control signal.The program RTC was also used to measure the wave train produced by the wave machine.The wave probes were sampled at a rate of 20 samples per second for a duration of 45 seconds.19The resulting data file had was demultiplexed by running the program PDMULT2 before themeasured wave train could be analysed.This program is used to demultiplex a GEDAP Primary Data File produce by the GEDAP DataAcquisition System. The demultiplexing produces individual GEDAP compatible data files thatmay then be analysed or plotted by existing GEDAP programs. The output data is converted tocalibration units using calibration factors stored in the GEDAP Port File. The demultiplexing isbase upon the polynomial function:z = A + Bx + Cx2 + Dx3 + E x4 [3.4]From PDMULT2 there is one output file for each wave probe. The signal from each probe canbe inspected visually using GPLOT. From the plot of wave height vs time one can inspect thewave train and choose the segment or subrecord to be analysed.SELECT1 is used to select a sub-record from a longer time series input record. The sub-recordis defined by specifying T1 and T2, where T1 is the initial time of the sub-record and T2 is thefinal time. The selected sub-record will match T1 and T2 as closely as possible subject to theresolution limit imposed by the time step of the input record. The selected sub-record is stored in aGEDAP output file.The sub-record can now be analysed using ZCA (Zero-Crossing Analysis of a Wave ElevationTime Series Record). ZCA performs a time-domain zero-crossing analysis on a time series signal.It is designed primarily for wave elevation records but it may also be used to analyse other typesof data such as force records. ZCA performs both zero up-crossing and zero down-crossinganalyses.The program ZCA checks the time spacing of the input signal to ensure that the sampling rate ishigh enough for accurate zero-crossing analysis. If the input signal contains fewer than 50 pointsper average zero-crossing period, then it is automatically resampled using cubic spline interpolation20so that the time spacing is small enough to meet this criterion. In addition to resampling, ZCA alsouses local parabolic curve fitting to define the peaks and troughs in the signal. The zero up-crossing and down-crossing times are calculated by linear interpolation so they are not limited bythe sampling rate of the input signal. The parameter of interest is average wave height which istaken as the average of the average zero up-crossing wave height and average zero up-crossingwave height.The average wave height from each of the 15 wave records are now collected using theprogram COLLECT, which collects individual header parameter values from several differentGEDAP input files and stores them in a single data vector in a GEDAP output file. The number ofinput files is equal to the number of program cycles. One output file is generated for eachparameter name selected.The wave heights are then exported to the UBC main-frame computer in order to generatecontour plots. This is achieved using the program EXPORT which converts one or more binaryGEDAP data files to a single ASCII file with a simple format. This program is normally used toconvert GEDAP data for processing by non-GEDAP programs or for transfer to non-VAXcomputers such as Apple Macintosh or main frame terminals. Each GEDAP input file is stored in asingle column of the ASCII output file.The contours plots are generated using the program DISSPLA on the UBC main-framecomputer. Once generated, the plots are transferred to an Apple Macintosh IIx computer forprinting.3.7 Test ProgramIn view of the foregoing, the purpose of the experiments was to measure the wave field withinthe harbour under different conditions corresponding to changes in the wave period (andconsequently the length), incident wave direction, incident wave height, the reflection coefficients21of the harbour boundaries and finally the reflection coefficients of the breakwaters. Table 3.1 liststhe characteristics of the wave conditions tested.The conditions were selected with respect to a base case corresponding to the followingparameters:0 = 0°, T = 0.803 sec, H = 30 mm.Thus Table 3.1 corresponds to the following set of tests:(i) the effects of wave period were examined by changing the wave period to includeT = 0.803, 0.937, 1.068, 1.20 sec, while keeping harbour boundaries and breakwaters fullyreflecting and the other parameters constant (tests 2 - 5).(ii) the effects of wave direction were investigated by changing the wave direction to include0 = +30°, 0°, -30°, (tests 1, 2, 7).(iii) the effects of wave height were examined by changing the wave height to includeH = 30 mm, 15 mm (tests 1, 6).(iv) the series of 7 tests were repeated with the low reflecting harbour boundary and fullyreflecting breakwaters (tests 8-14).(v) finally 6 of these tests were repeated (wave direction 0 = -30° was omitted), with the lowreflecting harbour boundary and the partially reflecting breakwaters (tests 15-20).As can be seen from Table 3.1, all of the test conditions correspond to intermediate depth waves(i.e. 0.05 < d/L < 0.5), and are in fact close to the deep water wave region (i.e. d/L 0.5).Therefore the wave generator was used in the flapper mode, which simulates deep water waveconditions, in order to produce the required wave conditions as closely as possible.22Chapter 4Results & Discussion4.1 Comparison of Numerical Results with Exact SolutionsThe wave doublet representation of the fluid boundaries was examined to verify its suitability.Preliminary results are presented here for cases corresponding to:(i) a straight offshore breakwater for which an exact solution is available,(ii) a breakwater gap represented by a semi-circular harbour with straight fully reflectingbreakwaters and totally absorbing interior harbour boundaries. The results may be comparedto the exact solution for a breakwater gap between a pair of straight, fully reflecting, colinear,semi-infinite breakwaters (Sobey and Johnson, 1986).4.1.1 Straight Offshore BreakwaterIn order to investigate the convergence of the wave doublet representation, a comparison wasmade to results for the wave force on a rigid vertical plate, since this force corresponds to a suitableaveraged value of the velocity potential difference across the breakwater. The force F is given as:F _ pgHd tanh(kd) r2	 kd	 J (Ow -F(0s) nx dS e -kot [4.1]Table 4.1 indicates the number of segments necessary to reproduce the closed-form solutionadequately and the corresponding degree of accuracy, for the particular case of the plate subjectedto a uni-directional incident wave train of unit height, and propagating orthogonal to the plate(0 = 0°) as indicated in Fig. 4.1. A plate length to wave length ratio of B/L = 2.0, and a still waterdepth to wave length ratio of d/L = 0.4 were chosen.The ratio of the maximum horizontal force computed using N segments to the correspondingclosed-form solution is tabulated for the various numbers of segments used. The table indicates23that as little as 10 segments per wave length is adequate to predict the force. A 4 point Gaussianintegration of the matrix coefficients in Eq. [2.15] was used to obtain the results. Furtherimprovements corresponding to either 8 or 16 point are not shown here, but were found to giveonly a marginal increase in accuracy and therefore were not warranted. The diagonal componentsof the matrix equation were approximated to the second order.For breakwater applications it is the wave height distribution around the breakwater that is ofpractical interest rather than wave force. Fig. 4.2 shows a comparison of the distribution along thebreakwater contour for the same conditions as before: B/L = 2.0, d/L = 0.4 and 9 = 0°. Thebreakwater extends along the y-axis from y/L = ±1. The distributions of the wave height along theupwave (exposed) and downwave (sheltered) sides of the breakwater are shown in Fig. 4.2 (a)and 4.2 (b) respectively. Numerical solutions for N = 10 and N = 20 are compared to the closed-form solution, and the solution obtained using N = 20 is seen to show excellent agreement with theclosed-form solution.Of more general interest is the wave height distribution in the vicinity of the breakwater, and thecorresponding contours of the wave height are shown in Fig. 4.3 for the same conditions asbefore, and with N = 50. Once more excellent agreement with the closed form solution isobtained. A three-dimensional view of the water surface elevation in the vicinity of the breakwaterat the particular instant t = 0 is shown in Fig. 4.4, and serves to confirm that the general form ofthe wave field in the region near the breakwater is as anticipated. The figure clearly shows thewave build-up in the upwave region and a wave height reduction in the leeward (sheltered) regionof the breakwater.4.1.2 Breakwater GapIn order to investigate the accuracy of the wave doublet representation of the harbourboundaries, numerical results for a semi-circular harbour with protruding breakwaters as shown inFig. 4.5, with fully reflecting breakwaters and fully absorbing harbour boundaries, was compared24to the closed-form solution for the case of a gap between a pair of straight, fully reflecting,colinear, semi-infinite breakwaters. The harbour has a radius of 350 m and a gap width of 50 mbetween the pair of symmetrical breakwaters.The harbour was subjected to a uni-directional incident wave train of unit wave height,propagating orthogonal to the breakwater gap (0 = 0°) as indicated in Fig. 4.5. The wave periodof T = 5.7 sec, depth of d = 20 m corresponds to a wave length L = 50 m which gives abreakwater gap to wave length ratio B/L = 1.0.Fig. 4.6 shows a comparison of the contours of the diffraction coefficient in the vicinity of thebreakwater gap. The wave field is symmetric so that only one half is shown. The breakwater gapextends along the y axis between y/L = IF 0.5. The solution obtained corresponds to 10 segmentsper wave length. The wave doublet representation is seen to show very good agreement with theexact solution.A three-dimensional view of the surface elevation at the particular instant t = 0 in the vicinity ofthe breakwater gap is shown in Fig. 4.7. The general form of the wave field is as expected,exhibiting the expected features of wave crests which approximately form concentric arcs centred atthe middle of the breakwater gap, and wave heights which noticeably decrease in the shadow zonebehind the breakwaters and which are close to the incident wave height outside of the shadowzone. This figure serves to confirm visually that the wave doublet representation of the wave fieldyields satisfactory results.4.2 Effects of Reflection CoefficientsThe wave doublet representation of the fluid boundaries enables the representation of one-sidedpartially reflecting boundaries along with two-sided fully reflecting boundaries. In an effort toinvestigate the performance of this method, the numerical model was applied to the fundamentalcase of a rectangular harbour with protruding breakwaters as shown in Fig. 4.8. The harbour hasa length of 300 m, a width of 300 m, and a gap width of 50 m between the pair of symmetrical25breakwaters. The harbour was subjected to a uni-directional incident wave train of unit waveheight, propagating orthogonal to the breakwater gap (0 = 0°) as indicated in Fig. 4.8. The waveperiod of T = 5.7 sec, depth of d = 20 m corresponds to a wave length L = 50 m which gives abreakwater gap to wave length ratio B/L = 1.0. The fully reflecting breakwaters and the partiallyreflecting harbour boundaries are represented by wave doublets.The wave field within the harbour predicted by the present method for the case of fullyabsorbing boundaries and impermeable breakwaters is shown in Fig. 4.9 and 4.10. Fig. 4.9shows the contours of diffraction coefficients which are compared to the predictions of theanalytical solution for a pair of colinear breakwaters. As can be seen the wave field is similar to theanalytical solution for example the wave heights diminish along the breakwaters. The wavedoublet representation of the rectangular harbour is seen to show very good agreement with theexact solution. Fig. 4.10 shows a view of the computed free water surface elevation at time t = 0.This exhibits the expected features of wave crests which approximately form concentric arcs withcentres at the middle of the breakwater gap.In comparison to this case, Fig. 4.11 and 4.12 show corresponding results for the identicalconditions, except that the boundaries of the harbour are fully reflecting, Kr = 1.0. This case offull reflection was considered by Miles and Munk (1961), Garrett (1970), and Mei (1983) in thecontext of harbour resonance. The diffraction coefficient contours within the harbour are shown inFig. 4.11, while Fig. 4.12 shows the computed free surface elevation at time t=0, indicating agenerally confused, three-dimensional wave field within the harbour.The more genera'. case of boundaries with partial reflection corresponding to a reflectioncoefficient of Kr = 0.1 is shown in Fig. 4.13 and 4.14. For the relatively low value of reflectioncoefficient adopted here, the results are not too different from the case of fully absorbingboundaries as already indicated in Fig. 4.9 and 4.10. The diffraction coefficient contours show aslight increase in the wave energy in the harbour (Fig. 4.13). However Fig. 4.14 shows nosignificant differences from the corresponding results for fully absorbing boundaries (Fig. 4.10).26Fig. 4.15 compares the wave height contours for the example problem being considered, butwith the reflection coefficient along the harbour boundaries taken as Kr = 0, 0.1 and 0.2 in turn.The figure shows an increasing irregularity in the diffraction coefficient contours as a transition tothe more confused state of full reflection, indicated in Fig. 4.11 is being approached.4.3 Numerical and Experimental Analysis of Harbour4 A3xExperimental ResultsltThe purpose of the experiments was to measure the wavefield within the harbour underdifferent conditions corresponding to changes in the wave period (and consequently the length),incident wave direction, incident wave height, the boundary reflection characteristics of the harbourboundaries and finally the reflection coefficients of the breakwaters. The experimental layoutshown in Fig. 3.4 was subjected to a total of 20 test runs (see Table 3.1).4.3.1.1 Effect of Wave PeriodFour wave periods were used in the experiments as indicated in Table 3.1 (T = 0.8, 0.94, 1.07,1.2 sec). The reference probe (Fig. 3.8) did not function during one of the tests in this set (test 2,T = 0.8 sec) so that the corresponding plot is absent. Fig. 4.16 shows the diffraction coefficientcontour plots for the different periods (except one), used for Phase 1 of the tests, corresponding tohighly reflective harbour boundaries and breakwaters. The contours indicate a confused wave fieldcorresponding to standing waves within the harbour. On close examination, the contour plotsshow some unacceptable features, including contours crossing and relatively jagged contours.These effects may be due in part to an imperfect contouring program and in part to the relativelycourse spacing of the wave probes used to measure the wave field. (A spacing of 0.6 Lmin =600 mm, was chosen). The wave lengths of the standing waves in the harbour appear to havebeen considerably shorter than this value. This combined to seems to have given rise to the poorquality of the contour plots. It is difficult to extract any useful results from these plots.27Fig. 4.17 shows diffraction coefficient contour plots for the four periods used for Phase 2 ofthe tests, corresponding to highly reflective breakwaters and partially reflecting beaches. In allcases the wave heights now decrease significantly in the shadow region as expected. The generaltrend of increased wave heights in the shadow region with increasing wave period is in agreementwith standard results (e.g. Shore Protection Manual, 1984).Fig. 4.18 shows the diffraction coefficient contour plots for the four periods used for Phase 3of the tests, corresponding to partially reflecting breakwaters and partially reflecting beaches. Oncemore the wave heights decrease significantly in the shadow region. There is little change in theplots with increasing period except for the largest period (T = 1.2 sec) which shows a slightdecrease in the wave heights near the tip of the breakwater. This may be due to inaccuratemeasurement, rather than indicating a general trend.4.3.1.2 Effect of Incident Wave DirectionThree incident wave directions were used throughout the experiments: 0 = -30°, 0°, +30° (seeFig. 3.4). Fig. 4.19 shows the diffraction coefficient contours for the three directions withT = 0.8 sec for the Phase 2 set of tests with increasing angle of incidence. As expected, a greaterlevel of wave energy enters the harbour, indicated by the contours extending further into theharbour. A comparison of Figs. 4.19 (a) and 4.19 (c) indicates that the difference is quiteappreciable emphasising the importance of adequately accounting for the incident wave angle.In Fig. 4.19 (c) the contours may not be as accurate as one would like (the 0.8 contour shouldnot change orientation). This may be due to the reference probe miss-reading the wave height. Itis probable however that the general appearance of the contours is relatively accurate.4.3.1.3 Effect of Boundary Reflection CharacteristicsA total of three combinations of reflection coefficients were used throughout the experimentscorresponding to the three Phases of the tests. Phase 1 corresponds to a highly reflecting28breakwater and fully reflecting harbour boundaries; Phase 2 corresponds to a highly reflectingbreakwater and partially reflecting harbour boundaries; and Phase 3 corresponds to a partiallyreflecting breakwater and partially reflecting harbour boundaries. The three correspondingconfigurations are sketched in Fig. 3.5.The diffraction coefficient contours for each of the three Phases and for T = 0.94 sec are shownin Fig. 4.20. The diffraction coefficient contours for Phases 2 and 3 show a pattern similar to thatfor the case of a semi-infinite breakwater (Shore Protection Manual, 1984) since the interiorharbour boundaries have a relatively low degree of reflectivity. On the other hand the contours forPhase 1 show a more confused wave field associated with the presence of standing waves whichwere observed. The highly reflecting boundaries prevented the dissipation of wave energy at theharbour boundaries.The difference in diffraction coefficient contours between Fig. 4.20 (a) and (c) emphasises theimportance of accounting for the reflectivity of the harbour boundary.4.3.1.4 Effect of Incident Wave HeightTwo incident wave heights were used throughout the experiments. The nominal incident waveheight was 30 mm for most of the tests, whereas this was reduced to 15 mm test 6, 13 and 20,corresponding to T = 0.8 sec for each of the three phases. In order to indicate the influence ofwave height, Fig. 4.21 shows a comparison between the results of test 13 and 9 corresponding tothe two wave heights but otherwise identical conditions (Phase 2, T = 0.8 sec, 0 = 0°). The figureshows the diffraction coefficient along a traverse at y = 2.1 m. In both cases the wave heightsdecrease significantly in the shadow region and the diffraction coefficients are very similarindicating that the diffraction coefficient is independent of wave height. As indicated earlier thereference probe did not function during test 2, therefore the comparison between test 6 and test 2cannot be made.29Fig. 4.22 shows a corresponding comparison between the results of test 20 and 16corresponding to the two wave heights but otherwise identical conditions (Phase 3, T = 0.8 sec,0 = 0°). The diffraction coefficient along a traverse at y = 2.2 m is shown, and once more thewave heights decrease significantly in the shadow region and the diffraction coefficients are verysimilar indicating that the diffraction coefficient is independent of wave height.4.3.1.5 Measured Reflection CoefficientsThe reflection coefficients of the vertical plywood, sand covered by horsehair at a slope of1:2.5, and the rocks at a slope of 1:1.5 were measured during the physical experiments. Theresults are summarized in Table 4.2. These results did not exhibit any trends (i.e. no apparentrelationship between reflection coefficient and either wave period or wave height).4.3.2 Comparison of Numerical and Experimental ResultsThe harbour configuration used for the numerical modelling is shown in Fig. 3.4. In setting upthe numerical model the fluid boundaries were divided into short segments of length of 0.1 mcorresponding to one tenth of the shortest wave length (1 m) as recommended in section 4.1.1. Totest the harbour using the numerical model, the coordinates and reflection coefficients of eachsegment was specified. All the fluid boundaries (coastline, breakwaters and harbour interior) wererepresented by wave doublets. Table 4.3 shows the four conditions and the correspondingexperimental tests for which the numerical results were obtained.Test 5 corresponds to the case of highly reflecting breakwaters and interior boundaries (Phase1) with waves propagating orthogonal to the breakwater (8 = 0°) and a period T = 1.2 sec.Fig. 4.23 shows a comparison of the diffraction coefficient contours in the region behind thebreakwater. Unfortunately there seems to be little correlation between the numerical andexperimental results. This is probably due to the difference in grid size used and to theshortcomings in the experiments as indicated in section 4.3.1.1. For the experimental model a gridof 8 x 7 points was used to create the contour plot, whereas for the numerical model a grid size of3020 x 30 was used. The fluid boundaries were all highly reflecting giving rise to standing wavesand therefore a large number of contour intervals. Lack of resolution with the experimental gridmay have caused the omission of many contour lines thus changing the superficial appearance ofthe plot. However on close examination of Fig. 4.23 it becomes apparent that, at any given pointthe actual values of the diffraction coefficients are similar on both plots while the contours takedifferent routes due to the lack of precision.Test 9 corresponds to the case of highly reflecting breakwaters and interior boundaries (Phase2) with waves propagating orthogonal to the breakwater (0 = 0°) and a period of 0.8 sec.Fig. 4.24 shows a comparison of the diffraction coefficient contours in the region behind thebreakwater. The agreement between the experimental results and the numerical predictions is nowgenerally good, with all of the important features of the wave field within the harbour beingreproduced. The lack of resolution of the experimental grid may again have contributed to theomission of some contour lines.Test 12 was for the case of highly reflecting breakwaters and interior boundaries (Phase 2) withwaves propagating orthogonal to the breakwater (0 = 0°) and a period of 1.2 sec. Fig. 4.25 showsa comparison of the diffraction coefficient contours in the region behind the breakwater.Agreement is generally quite good, with the numerical model underestimating the wave heightswithin the harbour. This may be due to radiating secondary waves generated at the breakwater tipas indicated by Pos and Kilner, 1987. The occurence of secondary waves has been described byBiesel (1963) who states that "any local surface pressure fluctuations of a given frequency willgive rise to a circular wave of the same frequency and radiating energy in all directions."Test 14 was for the case of highly reflecting breakwaters and interior boundaries (Phase 2) withwaves propagating at an angle to the breakwater (0 = +30°) and a period T = 0.8 sec. Fig. 4.26shows a comparison of the diffraction coefficient contours in the region behind the breakwater.Agreement between the numerical and experimental contour plots is generally quite good, with the31numerical model underestimating the wave heights within the harbour. The lack of resolution ofthe experimental grid may again have caused the omission of some contour lines.In all cases the wave heights decrease significantly in the shadow region. In general the waveheights within the harbour are slightly underpredicted. This phenomenon has been reportedpreviously by Pos and Kilner, 1987 and may be due to radiating secondary waves as describedearlier in this section.32Chapter 5Conclusions & Recommendations5.1 ConclusionsA numerical method developed by Isaacson and Baldwin (1991) for predicting the wave field ina harbour of constant depth and arbitrary shape which contains partially reflecting boundaries hasbeen verified as predicting the wave field accurately. The approach used is based on lineardiffraction theory and uses a wave doublet representation of the harbour boundaries.Numerical results are presented for the wave field due to a specified incident wave trainapproaching a straight offshore breakwater, a semi-circular harbour with a pair of symmetricalprotruding breakwaters, and a rectangular harbour with a pair of symmetrical protrudingbreakwaters. The wave field in the lee-side of the straight offshore breakwater predicted by thenumerical model agrees closely with the exact solution. For the semi-circular harbour case, theboundaries were considered to be perfectly absorbing, and the numerical model predicts the wavefield within the harbour realistically. For the rectangular harbour, cases which are consideredinclude perfectly absorbing, perfectly reflecting and partially reflecting harbour boundaries. In allof these cases the numerical model predicts the wave field within the harbour realistically.The numerical model using a doublet representation for both one-sided and two-sided fluidboundaries is compared to an experimental model for different wave period, incident wavedirection, incident wave height, reflection coefficients of the harbour boundaries and reflectioncoefficients of the breakwaters. The agreement between the experimental results and the numericalpredictions is generally good. As a rule the wave heights within the harbour are slightlyunderpredicted, while the wave heights outside the harbour are slightly overpredicted.33The numerical model gives an accurate and reliable means of predicting the wave field within aharbour of arbitrary shape with partially reflecting boundaries. It is relatively easy to use and ausers guide is available to help in the execution of the program.5.2 Recommendations for Further StudyThe scope of the present study was limited to partially reflecting one-sided boundaries and fullyreflecting two-sided boundaries. The obvious progression from here is to investigate the case ofpartially reflecting two-sided boundaries. Isaacson and Baldwin (1991) made initial efforts toextend their method to the case of two-sided partially reflecting boundaries with limited success.The closed form solution for wave scattering around a partially reflecting straight breakwaterexhibited noticeable wave height reduction along the surface with reduced reflection coefficients.However when the numerical method was applied the results exhibited a discontinuity in thevelocity potential at the breakwater tip and convergence was not apparent. Research into this areawould be another step along the road towards a numerical model which realistically predicts thewave field within an arbitrarily shaped harbour.34BibliographyBeisel, F. 1963. "Radiating Second-Order Phenomena in Gravity Waves". Recent Research inCoastal Hydraulics, International Association of Hydraulic Research, 10th Congress,London, England, Vol. 1, pp. 197-203.Berkhoff, J.C.W., 1976. "Mathematical models for Simple Harmonic Water WaveDiffraction and Refraction". Delft Hydraulics Laboratory, Rpt.No. 168.Blue, F.L. and Johnson, J.W., 1949. "Diffraction of Water Waves Passing Through aBreakwater Gap". Transactions of the American Geophysical Union, Vol. 30, pp. 705-718.Chen, H.S. and Mei, C.C., 1974. "Oscillations and Wave Forces in a Man-Made Harbourin the Open Sea". Proceedings of the 10th Symposium on Naval Hydrodynamics,Cambridge, Mass., pp. 573-596.Garrett, C.J.C., 1970. "Bottomless Harbours". Journal of Fluid Mechanics, Vol. 43, pp.432-449.Harms, V.W., 1979. "Diffraction of Water Waves by Isolated Structures". Journal of theWaterway, Port, Coastal and Ocean Division, ASCE, Vol. 10 (WW2), pp. 131-147.Hess, J.L., 1990. "Panel Methods in Computational Fluid Dynamics". Annual Review ofFluid Mechanics, Vol. 22, pp. 255-274.Hunt, B., 1980. "The Mathematical Basis and the Numerical Principals of the BoundaryIntregral Method for Incompressible Flows Over 3-D Aerodynamic Configurations", InNumerical Methods in Applied Fluid Dynamics, ed.B.Hunt, Academic Press, pp. 49-135.Hwang, L.S. and Tuck, E.O., 1970. "On Oscillations of Harbours of Arbitrary Shape".Journal of Fluid Mechanics, Vol. 42, pp. 447-464.Isaacson, M. 1991. "Measurement of Regular Wave Reflection", Journal of the Waterway,Port, Coastal and Ocean Engineering, ASCE, Vol. 117, No. 6, pp. 553-569.Isaacson, M. and Qu, S., 1990. "Waves in a Harbour with Partially ReflectingBoundaries". Coastal Engineering, Vol. 14 , pp. 193-214.35Isaacson, M. and Baldwin, J., 1991. "Numerical Prediction of Wave Diffraction using aDoublet Distribution". Coastal/Ocean Engineering Report, Department of Civil Engineering,University of British Columbia, Vancouver, Canada.Isaacson, M. and Baldwin, J., 1991. "Numerical Prediction of Wave Diffraction using aDoublet Distribution". Proceedings of the Tenth Annual Conference of the Canadian Societyfor Civil Engineering, Vancouver, British Columbia, Canada, pp. 76-85.Johnson, J.W., 1951. "Generalized Wave Diffraction Diagrams". Proceedings of the 2ndInternational Conference on Coastal Engineering, Houston, Texas, pp. 6-23.Lamb, H., 1932. "Hydrodynamics", Cambridge University Press, Cambridge.Lee, J-J., 1971. "Wave induced Oscillations in harbours of Arbitrary Geometry". Journal ofFluid Mechanics, Vol. 45 , pp 375-394.Mattoli, F. and Tinti, S., 1979. "Discretization of the Harbour Reasonance Problem".Journal of the Waterway, port, Coastal and Ocean Division, ASCE, Vol. 105 (WW4),pp. 464-469.Mei, C.C., 1978. "Numerical Methods in Water Wave Diffraction and Radiation". AnnualReview of Fluid Mechanics, Vol. 10, pp. 393-416.Memos, C.D., 1980. "Water Waves Diffracted by Two Breakwaters". Journal of HydraulicResearch, Vol. 18, No. 4, pp. 343-357.Miles, J.W. and Munk, W.H., 1961. "Harbour Paradox". Journal of Waterway andHarbours Division, ASCE, Vol. 87, pp. 111-130.Penny, W.G. and Price, A.T., 1952. "The Diffraction Theory of Sea Waves byBreakwaters and the Shelter Afforded by Breakwaters". Philosophical Transactions, RoyalSociety, Series A, Vol. 244, London, pp. 236-253.Pos, J.D. and Kilner, F.A., 1987. "Breakwater Gap Diffraction: An Experimental andNumerical Study". Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE,Vol. 113, No. 1, pp. 1-21.Putman, J.A. and Arthur, R.S., 1948. "Diffraction of Water Waves by Breakwaters".Transactions of the American Geophysical Union, Vol. 29, pp. 481-490.36CERC, 1984. "Shore Protection Manual", 1, 4th Edition. Coastal Engineering ResearchCentre, U.S. Army Corps of Engineers, Vicksburg, Mississippi.Sobey, R.J. and Johnson, T.L., 1986. "Diffraction Patterns near Narrow BreakwaterGaps". Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 112,No. 4, pp 512-528.Wiegel, R.L., 1962. "Diffraction of Waves by a Semi-infinite Breakwater". Journal of theHydraulics Division, ASCE, Vol. 88, No.HY1, pp. 27-44.Yeung, R.W., 1982. "Numerical Methods in Free-Surface Flow". Annual Review of FluidMechanics, Vol. 14, pp. 395-442.37TablesTest run e(deg)H(mm)T(sec)L(mm)dgT2 dL KrBreakwaterKrShorelinePhase1 +30 30 0.803 1,000 0.072 0.450 0.92 0.92 12 0 30 0.803 1,000 0.072 0.450 0.92 0.92 13 0 30 0.937 1,333 0.052 0.3375 0.92 0.92 14 0 30 1.068 1,666 0.040 0.270 0.92 0.92 15 0 30 1.200 2,000 0.032 0.225 0.92 0.92 16 0 15 0.803 1,000 0.072 0.450 0.92 0.92 17 -30 30 0.803 1,000 0.072 0.450 0.92 0.92 18 -30 30 0.803 1,000 0.072 0.450 0.92 0.21 29 0 30 0.803 1,000 0.072 0.450 0.92 0.21 210 0 30 0.937 1,333 0.052 0.3375 0.92 0.21 211 0 30 1.068 1,666 0.040 0.270 0.92 0.21 212 0 30 1.200 2,000 0.032 0.225 0.92 0.21 213 0 15 0.803 1,000 0.072 0.450 0.92 0.21 214 +30 30 0.803 1,000 0.072 0.450 0.92  0.21 215 +30 30 0.803 1,000 0.072 0.450 0.45 0.21 316 0 30 0.803 1,000 0.072 0.450 0.45 0.21 317 0 30 0.937 1,333 0.052 0.3375 0.45 0.21 318 0 30 1.068 1,666 0.040 0.270 0.45 0.21 319 0 30 1.200 2,000 0.032 0.225 0.45 0.21 320 0 15 0.803 1,000 0.072 0.450 0.45 0.21 3Table 3.1	 Wave conditions and reflection coefficients for each of the 20 test runs of theexperimental model.N F(N)Fexact10 1.036020 1.018150 1.0071100 1.0036200 1.0020Table 4.1	 Effect of the number of segments N on the computed wave force on a vertical plate.38Material KrPlywood 0.92Rock (1:1.5) 0.45Sand (1:2.5) 0.21Table 4.2	 Measured reflection coefficients for vertical plywood, rock at slope 1:1.5, andsand at a slope of 1:2.5 overlain with horsehair.39Test run e(deg)T(sec)L(m)dL KrBreakwater KrShoreline5 0 1.20 2.00 0.225 0.92 0.929 0 0.803 1.00 0.450 0.92 0.4512 0 1.20 2.00 0.225 0.92 0.4514 +30 0.803 1.00 0.450 0.92 0.45Table 4.3	 Wave conditions and reflection coefficients for the 4 test runs of the numericalmodel. (Note: Test number corresponds to that for the experimental model inTable 3.1)Incident wavedirectionFigures40Figure 1.1	 Definition sketch of general harbour.0 Doublet Axisn tboundary conditionapplied at point x(4,1i)Surface normalxFluid boundary Swave doubletdistribution at point t41Figure 2.1	 Geometry of Green's function representation .Removable (End) WaveAbsorber Modules Ramp	 26.5 m 	42Figure 3.1	 Sketch of the wave basin at BC Research.S KAG-WAYa •YJUN EATAKU R.SITKA0 e • GEL NASS	PRINCE 0	 --cOFWALES	aKETCHIKANIS.	 aO PRINCERUPERTP1AS SETCOASTAL B.C.KITIMATnQUEEN	 SKIIDEGATECH ARLOTTEOCEANpusQUEENCHARLOITESOUNDFRASER R .Comox VAN COUVE R1S.•C	4SKEENA R.43Figure 3.2 Location of Comox, BC.44Figure 3.3	 Details of Comox harbour.CoastlinerCoastline6 mHarbourBoundary3 m45if 4/ /0 = —30°	 /0 = 0°kI0. +30°Figure 3.4	 Sketch of the model harbour showing principal dimensions.Rock (1:1.5)HorsehairSand (1:2.5)Harbour BoundaryPartial ReflectionPlywoodBreakwaterPartial ReflectionPlywood Plywood461:7Breakwater	 Harbour BoundaryFull Reflection	 Full ReflectionPhase 1Plywood	 HorsehairV	 TSand (1:2.5)Breakwater	 Harbour BoundaryFull Reflection	 Partial ReflectionPhase 2Phase 3Figure 3.5	 Sketch of the boundary configurations for the 3 Phases of the laboratory tests.PlywoodIT	 V600 mm , ;450 mm(a)Rock (1:1.5)Plywood600 mm 450 mm/	 /.7	 7 /	 777	 /7/X(b)Plywood450 mm	 600 mmV (d)Sand (1:2.5)A A=450 mm 600 mmV	 V7,/	 //////////(e)HorsehairFigure 3.6	 Dimensions of the and harbour boundaries. (a) breakwater for Phases 1 and 2,(b) breakwater for Phase 3, (c) harbour boundary for Phase 1, (d) harbourboundary for Phases 2 and 3.473600 mm•	 an	 .1	 sr	 w	 INa	 -	 a	 a	 -	 - I	Wave Probes	III	 i>I	 >I	 >1	 >II48600 mmElevationPlanFigure 3.7	 Sketch of the wave probe frame.600 mmff. Phase 3Phases 1 & 2a = (0.5,0.3) a = (0.4,1.0)b = (0.5,3.9) 13 = (0.4,4.0)c = (-3.7,3.9) y= (-3.2,4.0)d = (-3.7,0.3) S = (-3.2,1.0)Reference Probe49Figure 3.8	 Location of the wave field measurements.Incident wavedirection---)0- 450 mmIncident wavedirection1500 PlywoodGuide WallsWave ProbesPlanElevationFigure 3.9	 Sketch of experimental layout for reflection coefficientmeasurement.BAIncidentwavedirection	 YVertical plate51Figure 4.1	 Rigid vertical plate used in the numerical example.0.0-1.0 -0.5 0.0y/L(a)0.5 1.03.0(b)0.0y/L- 1.0 -0.5 0.5 1.0Figure 4.2 Wave height distribution along a straight offshore breakwater with B/L = 2.0, 0 = 0°.(a) upwave face, and (b) downwave face. 	 N = 10, 	 N = 20,	 exact solution.524.03.0(a)1-1 2.0--IAN— 1.00.0530.0	 1.0	 2.0	 3.0	 4.0	 5.0x/L0.0	 1.0	 2.0	 3.0	 4.0	 5.0x/L4.03.01-1s.,	 2.0—31m- 1.00.0(b)Figure 4.3	 Diffraction coefficient contours in the vicinity of the offshore breakwaterwith B/L = 2.0, 0 = 0°. 	  exact solution, 	 numerical solution.(a) N = 20, (b) N = 50.54Position of the BreakwaterFigure 4.4	 View of surface elevation (at t = 0) in the region of the offshore breakwater forB/L = 2.0 and 0 = 0°.IncidentwavedirectionL=50m 55	).-	 B=50mFigure 4.5	 Sketch of the semi-circular harbour used as a numerical example.563.01 .00.0 2.00.22.0_ ————————	 •• _0.43.0	 4.0x/L5.0	 6.0T.oFigure 4.6 Diffraction coefficients in the vicinity of a breakwater gap with B/L = 1.0, 0 = 0° andK = O. -  numerical solution for semi-circular harbour,  exact solution forbreakwater gap (Sobey and Johnson, 1986).Figure 4.7	 View of the surface elevation (at t = 0) in the region of the breakwater gap forthe semi-circular harbour with B/L = 1.0, 0 = 0° and K r = 0.IncidentwavedirectionL=50m 57B=50m 300m300mFigure 4.8	 Rectangular harbour used as a numerical example.583.02.0....,..„.... _,,,/'----------,.._______°♦4 -1.0	 _--------.° 4(---,,n310.- -e. ,0.0	 1.0 2.0	 3.0x/Lsi . 0	 5.0Figure 4.9	 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0° and Kr = 0. 	 numerical solution, 	 exact solution for breakwatergap (Sobey and Johnson, 1986).Figure 4.10	 View of the surface elevation (at t = 0) for the rectangular harbour withB/L =1.0, 0 = 0° and K r = 0.4.0	 5.0itCP'a—IP.- -8.6-, 	0.0593.02.01, 01.0	 2.0 3.0x/LFigure 4.11	 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0° and K = 1.0.Figure 4.12	 View of the surface elevation (at t = 0) for the rectangular harbour withB/L = 0, 0 = 0, and Kr = 1.0.A\-) V V yr,3.01.0	 2.0 3.0	 4.0X/1_,5.0	 8.00.02.0 -1.0Figure 4.13	 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0° and K =0.2.Figure 4.14	 View of the surface elevation (at t = 0) for the rectangular harbour withB/L = 1.0, 0 = 0°, and Kr = 0.2.600 01 .0 ----------------- 0.40.43.02.00.6•-.).- -13.8-, r-N4.0	 5.00.0 -"T6.0r\f-l'° 2o2.0 -----------------41111.1r3.06.0../..,\__..../..1-----n--.7_./,_	 ..____„---- 0.6/..-\-).- -G.9-,0.0	 1.0 2. 0 3.0x/L(b)1.0-61Figure 4.15	 Diffraction coefficient contours within the rectangular harbour with B/L = 1.0,0 = 0°. (a) Kr = 0.0, (b) Kr = 0.1 and (c) Kr = 0.2.6.2) (i5. 04. 0 6.0v V0 C Vy2.0-3.00toC'	 /*N7Ais.1.0 -_./N7--°n.,__.--.-- -6--)...- —0.9—,I0.0	 1.0(c)n 0 i2.062Figure 4.15	 Continued.(a)(b)63Figure 4.16	 Diffraction coefficient contours for Phase 1 tests with 0 = 0° and H = 30 mmshowing the effect of wave period. (a) T = 0.94 sec, (b) T = 1.07 sec,(c) T = 1.2 sec.Figure 4.16	 Continued.(c)64a?0o0.5—0.2 4(b)Figure 4.17 Diffraction coefficient contours for Phase 2 tests with 0 = 0° and H = 30 mmshowing the effect of wave period. (a) T = 0.8 sec, (b) T = 0.94 sec,(c) T = 1.04 sec, (d) T = 1.2 sec.3.93.0-0 2a-'5',	 .1.2-0.3 	 ,—3.7 -1.6x(m)—0.9—2.3-3.065(a)3.93.00iV—0.2	 0.53.9663.0-.to (c)0 2.1-0,t31.2-0.3 	 ,—3.0	 —2.3 -0.9	 -0.2—3.7 —1.8x(m)0.50.41.2-{ /\ —1.8	 —0.9x(m)0.3—3.7(d)- 3.0 —2.3Figure 4.17	 Continued.opd/ —2.3	 4.4	 —0.5X(11-1)4.0(b)3.0-2.0-1.0—-3.2\\0.4404.0—	3.0—67(a)2.0—1.0 	—3.2Go.—2.3	 —1.4x(m)—0.5	 0.44Diffraction coefficient contours for Phase 3 tests with 0 = 0° and H = 30 mmshowing the effect of wave period. (a) T = 0.8 sec, (b) T = 0.94 sec,(c) T = 1.04 sec, (d) T = 1.2 sec.Figure 4.18 2.0-1.0-2.3-3.2(c)684.03.0-(d)0)/-0.5	 0.4fFigure 4.18	 Continued.3.969I3.0-2.1->.11.2-(a)r—2.3	 —L6	 —0.9x(m)/I	 I—0.2	 0.50.3-T	 I—3.7 —3.03.93.0 o0 2.1— (b)cc!01.2-0.3 	 ,-4.0	 —2.3—3.7 -0.9	 -0.2 0.5ikFigure 4.19	 Diffraction coefficient contours for Phase 2 tests with T = 0.8 sec showing theeffect of incident wave direction. (a) 0 = -30°, (b) 9 = 0°, (c) 0 = +30°.703.9-1.6	 -0.9X(r11)o1.2020.3-1	-3.7Figure 4.19	 Continued.-3.0	 -2.3 -0.2\ 0.53.0- \(c)(a)(b)71Figure 4.20	 Diffraction coefficient contours for T = 0.94 sec, 0 = 0° showing the effect ofboundary reflection characteristics. (a) Phase 1, (b) Phase 2, (c) Phase 3.4.072(c)-3.22.0-1.0-	....-------- 0.2-2.33.0-I0.44Figure 4.20	 Continued. Shadow Region ••1.0— — H = 30 mm	  H = 15 mm0.8 —I y = 2.1 ml0.6 —0.4 —........................................................................................0.2 —0 . 0-4	 -3	 -2	 -1	 0	 1Figure 4.21 Diffraction coefficients along a traverse at y = 2.1 mfor Phase 2 tests with T = 0.8 sec and 9 = 0°.1.0— — H = 30 mm	  H = 15 mm0.8 —Shadow Region y = 2.2 ml0.6—........730.4— ........................................0.2 —0 . 0-4	 -3	 -2	 -1	 0	 1Figure 4.22 Diffraction coefficients along a traverse at y = 2.2 mfor Phase 3 with T = 0.8 sec and 8 = 0°.(**-----___---./3.93.0 -(a)1.2 -0.3-3.73.9 -	3.0-2.1-1.2 -0.5(b)0.3-•-3.7 -3.0	 -2.3	n-1.8x(m) -0.9 -0.2 f	0.5Figure 4.23	 Diffraction coefficient contours for Phase 1 tests with T = 1.2 sec, 0 = 0°.(a) experimental results, (b) numerical results.743.0 01—3.9	—0.2 0.5if3.9(a)/Figure 4.24	 Diffraction coefficient contours for Phase 2 tests with T = 0.8 sec, 0 = 0°.(a) experimental results, (b) numerical results.752.1-›,1.2-—3.0	 —2.30.3 	 ,-1.8x(m)—3.7(b)(a).\/3.93.0 -760.41.2 - 0iS30N-3.7 -3.0	 -2.3 -0.9 -0.2 0.50.33.9(b)3.0 -2.1-1.2 -0.3-3.7 -0.9	 -0.2 4	 0.5Figure 4.25	 Diffraction coefficient contours for Phase 2 tests with T = 1.2 sec, 8 = 0°.(a) experimental results, (b) numerical results.3.9\3.0 -(a)(b)77Figure 4.26	 Diffraction coefficient contours for Phase 2 testswith T = 0.8 sec, 0 = +30°.(a) experimental results, (b) numerical results.PhotographsPhotograph 3.1	 Experimental layout in the wave basin at BC Research.78Photograph 3.2	 Wavemaker calibration in the wave basin at BC Research.79Photograph 3.3	 Wavemaker calibration in the wave basin at BC Research.Photograph 3.4	 Experimental layout for Phase 1 tests.80Photograph 3.5	 Experimental layout for Phase 2 tests.Photograph 3.6	 Experimental layout for Phase 3 tests.81Photograph 3.7	 Measurement of the reflection coefficient of the sloping sand.Photograph 3.8	 Measurement of the reflection coefficient of the vertical plywood.82Photograph 3.9	 Close-up of the measurement of the reflection coefficient of the rocks.

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