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Regular wave conditions in a directional wave basin Shaver, Mark D. 1989

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R E G U L A R W A V E C O N D I T I O N S I N A D I R E C T I O N A L W A V E B A S I N B y M a r k D . Shaver B . E n g . (Mechanical) Carleton Univers i ty A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August , 1989 © M a r k D . Shaver, 1989 In presenting this thesis i n part ia l fulfilment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia , I agree that the L i b r a r y shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publ icat ion of this thesis for f inancial gain shall not be allowed without my writ ten permission. C i v i l Engineering The Univers i ty of B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5 Date: Abstract Thi s thesis represents a small step toward improved generation of realistic sea states i n laboratory wave basins. Experiments were conducted in an offshore directional wave basin equipped w i t h a segmented wave generator. Regular waves were generated for several periods and propagation directions, and the resulting wave elevations were mea-sured throughout the basin. Motions of the 60 wavemaker segments were based on the snake principle of directional wave generation. Results are summarized and compared w i t h predictions of a boundary element numerical method. Findings encourage further development of this linear diffraction numerical technique. It should be used to correct generator segment motions as prescribed by the snake principle, so as to account for diffraction and reflection effects which affect the wave field. The experiments were conducted at the Hydraulics Laboratory of the Nat iona l Research Counc i l of Canada, i n Ottawa. A six metre square array of wave probes was used to measure wave elevations at discrete points spaced on a 2m grid around the wave basin. Three wave periods were investigated, w i t h waves directed normal to the generator face, as wel l as i n three oblique propagation directions. For a l l but the most severe propagation direction, tests were run a second time w i t h a large surface piercing cyl indr ica l structure positioned i n the test area. M a x i m u m waveheight results were plotted, and compared w i t h numerical model predictions at the same locations around the basin. A linear diffraction computer program based on the boundary element method was used to predict the wave field at the same points around the basin. B y this method, generator segment faces and reflecting walls are represented by a dis tr ibution of discrete sources around the basin boundaries. i i Measured elevation time series were analysed using a mult iple regression screening program which extracts a prescribed number of sinusoidal components from the signal of interest. T h e program was modified to accommodate harmonic analysis. Funda-mental and second harmonic components were synthesized from each t ime series. The second harmonic component was generally an order of magnitude smaller than the fundamental component. Discrepancies between these results and the linear numeri-cal model predictions are attr ibuted to nonlinear effects, and to basin resonance. The linear diffraction computer model is seen to predict the wave field to a high degree of accuracy, even though imprecise boundary definition was necessary at the current level of program development. i i i . . . M r . Palomar now tries to limit his field of observation; if he bears in mind a square zone of, say, ten meters of shore by ten meters of sea, he can carry out an inventory of all the wave movements that are repeated with varying frequency within a given time interval. The hard thing is to fix the boundaries of this zone, because if, for example, he considers as the side farthest from him the outstanding line of an advancing wave, as this line approaches him and rises it hides from his eyes everything behind it, and thus the space under examination is overturned and at the same time crushed. In any case, Mr . Palomar does not lose heart and at each moment he thinks he has managed to see everything to be seen from his observation point, but then something always crops up that he had not borne in mind. If it were not for his impatience to reach a complete, definitive conclusion of his visual operation, looking at waves would be a very restful exercise for him and could save him from neurasthenia, heart attack, and gastric ulser. A n d it could perhaps be the key to mastering the world's complexity by reducing it to its simplest mechanism. . . . Now, in the overlapping of crests moving in various directions, the general pattern seems broken down into sections that rise and vanish. In addition, the reflux of every wave also has a power of its own that hinders the oncoming waves. A n d if you concentrate your attention on these backward thrusts, it seems that the true movement is the one that begins from the shore and goes out to sea. Is this perhaps the real result that M r . Palomar is about to achieve? To make the waves run in the opposite direction, to overturn time, to perceive the true substance of the world beyond sensory and mental habits? No, he feels a slight dizziness, but it goes no further than that . . . - Italo Calvino, Mr. Palomar iv Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgment xiv Nomenclature xv 1 Introduction 1 1.1 General 1 1.2 Literature Review 3 1.3 Scope of the Project 5 2 Theoretical Background 7 2.1 Two-Dimensional Plane Wavemaker Theory 7 2.2 The Snake Principle of Wave Generation 10 2.2.1 Refinement B y Use of Reflective Par t i a l Side Walls 12 2.3 The Diffraction Principle of Wave Generation 16 2.3.1 Analogy W i t h a Breakwater Gap 16 2.3.2 Point Source Representation of Generator Mot ions 17 3 Experimental Setup 21 3.1 The N R C Offshore Wave Bas in 22 v 3.2 The Wave Probe A r r a y 25 3.3 The Grav i ty Base Structure 28 4 The Test Program 28 4.1 Synthesis of D r i v i n g Signals 31 5 Data Analysis 32 5.1 Modifications to the S C R E N Program 35 5.2 Types of Plots Generated 36 6 Results and Discussion 39 6.1 Synopsis 40 6.2 Effect of Prescribed Waveheight 42 6.3 Repeatabil i ty Test 44 6.4 M a x i m u m Waveheight Plots 44 6.4.1 Plots For Normal ly Propagat ing Wave Trains 47 6.4.2 Plots For Oblique Wave Trains 49 6.5 Phase Plots F r o m the Experimental Results 54 6.6 Numer ica l M o d e l Predictions of Wave Condit ions 56 6.6.1 The Linear Diffraction Computer Program 56 6.6.2 Waveheight Plots of Numerica l M o d e l Predictions 58 6.6.3 Numerica l M o d e l Phase Predictions 63 6.6.4 Numerica l M o d e l Surface Elevation Plots 64 6.7 Further Considerations 65 7 Conclusions and Recommendations 65 7.1 Conclusions 69 vi 7.2 Recommendations 8 References 9 Tables 10 Figures List of Tables 1 Tests Conducted and Pertinent Parameters 76 2 Side W a l l Lengths Specified For Drive Signal Synthesis, and Resultant Generator Mot ions 77 v m List of Figures 1 Elevat ion V i e w of the Wavemaker Configuration and Coordinate System 78 2 Definit ion of Coordinate System, P l a n V i e w 78 3 Oblique Wave W i t h Interference Wave, Conventional M e t h o d of Oblique Wave Generation 79 4 Superposition of Direct and Reflected Indirect Waves 79 5 V i e w of the N R C Offshore Wave Bas in 80 6 Wave Probe A r r a y R i g i n Cal ibrat ion Posi t ion 80 7 Bas in Layout , W i t h the Four Probe A r r a y Positions Indicated 81 8 Use of Bas in Symmetry to Measure Wave Condit ions Over O n l y Hal f the Bas in 82 9 Diagram of the Probe A r r a y Frame 83 10 V i e w of the Four Probes Facing the Generator 84 11 V i e w F r o m Wave Generator As Bas in F i l l s 84 12 Closeup of 20cm Spacer i n Place For Ca l ibra t ion of Probe A r r a y . . . . 85 13 Closeup of Brass F i t t i n g Holding a Wave Probe '. . 85 14 The Shortest Wave Tra in Propagating A t + 3 0 ° 86 15 The Shortest Wave Tra in Propagating A t - 3 0 ° 86 16 Intermediate Length Waves Propagating A t —45° 87 17 V i e w Showing the Probe Array In the F i n a l Posit ion 87 18 Wave Generator Segment Drive Signals for Test 4, T = 1.25sec A t 0 = + 3 0 ° 88 19 Dr ive Signal Plots Il lustrating Dependence of Paddle Stroke 90 ix 20 K e y Operat ional Steps of the S C R E N M u l t i p l e Regression Screening Program 91 21 Examples of Plots Generated For Each Test, For the Case of One A r r a y Posi t ion of Test 5, T = 1.75sec, 9 = 30° 92 22 Hal f Waveheight Test: Hsy Plots Over Hal f the Bas in of the T = 1.25sec, 9 = 30° Results 98 23 Hal f Waveheight Test: Phase Plots Over Hal f the Bas in of the T = 1.25sec, 9 = 30° Results 99 24 Repeatabil i ty Test: Hsy Plots For a Single A r r a y Posi t ion of the T = 1.25sec, 9 = 30° Results 100 25 Repeatabil i ty Test: Phase Plots For a Single A r r a y Posi t ion of the T = 1.25sec, 9 = 30° Results 101 26 Au A2 and Hsy Plots For Test 7: T = 1.25sec, 9 = 45° 102 27 Ax, A2 and Hsy Plots For Test 5: T = 1.75sec, 9 = 30° 103 28 Au A2 and Hsy Plots For Test 20: T = 1.75sec, 9 = 30 ° , C y l i n d r i c a l G B S i n Place 104 29 Hsy Plots For Normal ly Propagating Wave Trains 105 30 Hsy Plots For T = 2.25sec, 9 = 0 ° , W i t h and W i t h o u t the G B S 106 31 Hsy Plots For Wave Trains Propagating Obliquely A t 9 = 30° ; Results For the Three Wavelengths Investigated 107 32 Hsy Plots For the Intermediate Length Wave Tra in Directed A t the Three Oblique Angles Tested 108 33 Hsy P lo t For Test 12, T = 2.25sec, 9 = 60° 109 34 Hsy P lo t For Test 15, T = 2.25sec, 9 = 30° , Involving the Simple Snake M e t h o d of Generation W i t h o u t Corner Reflection 110 35 Hsy Plots For T = 1.75sec, 9 = 30° , Wave F i e l d and G B S Cases I l l x 36 Phase P lo t For Test 4, T = 1.25sec, 0 = 30° 112 37 Phase P lo t For Test 7, T = 1.25sec, 0 = 45° 113 38 Phase Plot For Test 8, T = 1.75sec, 0 = 45° '. . 114 39 Rectangular Bas in Configuration For Use W i t h the S E G E N Linear Diffraction Computer Program 115 40 Use of the Arb i t r a ry Bas in Configuration For the S E G E N Linear Diffraction Computer Program 116 41 Exper imental Hsy and Numerica l M o d e l M a x i m u m Waveheight (Ai) Results Compared For T = 1.25sec, 0 = 0° 117 42 Exper imental Hsy and Numerica l M o d e l M a x i m u m Waveheight (Ay) Results Compared For T = 1.75sec, 0 = 0° 118 43 Exper imenta l Hsy and Numerica l M o d e l M a x i m u m Waveheight (Ay) Results Compared For T = 2.25sec, 0 = 0° 119 44 Exper imental Hsy and Numerica l Ay Results Compared For T = 1.25sec, 0 = 30° 120 45 Exper imental Hay and Numerica l A\ Results Compared For T = 1.75sec, 0 = 30° 121 46 Exper imental Hsy and Numerica l Ay Results Compared For T = 2.25sec, 0 = 30° 122 47 Exper imenta l Hsy and Numerica l Ai Results For the Simple Snake M e t h o d of Generation, T = 2.25sec, 0 = 30° 123 48 Exper imental Hsy and Numerica l A\ Results Compared For T = 1.75sec, 0 = 45° 124 49 Experimental Hsy and Numerica l A-y Results Compared For T = 2.25sec, 0 = 60° 125 x i 50 Exper imental Hsy and Numerica l Results Compared For T = 1.25sec, 9 = 0 ° , W i t h the G B S In Place 126 51 Exper imental Hsy and Numerica l A\ Results Compared For T = 1.75sec, 9 = 0 ° , W i t h the G B S In Place 127 52 Exper imental Hsy and Numerica l A\ Results Compared For T = 2.25sec, 9 = 0 ° , W i t h the G B S In Place 128 53 Exper imenta l Hsy and Numerica l Ai Results Compared For the Simple Snake M e t h o d of Generation W i t h the G B S In Place, T = 2.25sec, 6 = 30° 129 54 Exper imental Hsy and Numerica l Ax Results Compared For T — 1.25sec, 9 = 30 ° , W i t h the G B S In Place 130 55 Exper imental Hsy and Numerica l Ax Results Compared For T = 1.75sec, 9 = 30 ° , W i t h the G B S In Place 131 56 Exper imental Hsy and Numerica l Ax Results Compared For T = 2.25sec, 9 = 30° , W i t h the G B S In Place 132 57 Exper imental Hsy and Numerica l A\ Results Compared For T = 1.25sec, 9 = 45 ° , W i t h the G B S In Place 133 58 Exper imental Hsy and Numerica l A\ Results Compared For T = 1.75sec, 9 = 45 ° , W i t h the G B S In Place 134 59 Exper imental Hsy and Numerica l Ax Results Compared For T = 2.25sec, 9 = 45 ° , W i t h the G B S In Place 135 60 Numerica l ly Derived Phase Plot Corresponding to Test 2, T = 1.75sec, 0 = 0° 136 61 Numerica l ly Derived Phase Plot Corresponding to Test 4, T = 1.25sec, 0 = 30° 137 x i i 62 Numerica l ly Derived Phase Plot Corresponding to Test 8, T = 1.75sec, 0 = 45° 138 63 Three-Dimensional Perspective Block Diagram Showing Instantaneous Water Surface Elevat ion From the Numerica l Result Corresponding to Test 5, T = 1.75sec, 9 = 30° 139 64 A m p l i t u d e vs. Frequency Line Spectra For Selected Probe Locations Perta ining to Test 10, T = 1.25sec and 6 = 60° 140 xin Acknowledgment I wish to thank my supervisor, D r . Michael Isaacson, for his guidance and good counsel throughout the preparation of this thesis. I wish also to extend my deep gratitude to the staff of the Hydraul ics Laboratory of the Nat iona l Research Counc i l i n Ottawa, for permitt ing me the use of their facilities during a busy period. I would like especially to thank Senior Research Officer E d Funke for generous assistance i n modifying the S C R E N program for harmonic analysis, and A n d r e w Cornett for help w i t h the scheduling of tests and instal lat ion of apparatus. Thanks also to D a n Pelletier, B r i a n W i l s o n and R o n Gi ra rd , who accommodated a large analytical demand upon the computer resources during a period of transi t ion, and to M i k e Miles for occasional insightful tips along the way. xiv Nomenclature a - wave amplitude a 2 - two-dimensional wave amplitude A - constant used i n derivation of 2-D generator transfer function Ay - amplitude of the fundamental synthesized wave component A 2 - amplitude of the second harmonic synthesized wave component A-wave - primary, or direct wave t ra in b - generator horizontal displacement b2 - two-dimensional generator horizontal displacement B - generator segment w i d t h Bpn - segment source strength matr ix coefficients B-wave - secondary, or indirect wave tra in which joins i n phase w i t h the A-wave upon reflection from the side wal l Cn - constant associated w i t h evanescent wave components T> - diameter of cyl indrical surface piercing structure / - frequency fa - fixed fundamental frequency value supplied as input to S C R E N f(z) - dimensionless form of generator displacement amplitude as a function of elevation z / ( £ ) - source strength distr ibution function F(kh) - 2-D transfer function relating wave amplitude to generator displacement, as a function of the depth parameter kh T - 3-D transfer function relating wave amplitude to generator displacement xv g - gravitational constant —* G, G(x, £) - Green's function G B S - gravity base structure G E D A P - Generalized Experiment control, D a t a acquisition and data Analysis Package h - s t i l l water depth (using the convention suggested by the International Associat ion for Hydraul ic Research (1986)) H - wave height HQ 1^ - Hankel function of the first k i n d and zero order i - V^T J 0 - Bessel function of the first k i n d k - wave number k' - wave number along generator face, i n y direction L - wave length L' - wave length along generator face, i n y direction m - number of generator segments involved i n B-wave generation M - number of point sources used to represent a generator segment n - a counter, or normal direction when writ ten as dn N - total number of point sources Ng - number of wave generator point sources M - non-integer number of generator segments equal to length L' r - radia l distance from any point to a source location: r = 1^- £1 S - sidewall length S - horizontal contour along al l non-absorbing surfaces, at s t i l l water level S C R E N - mult iple regression screening program x v i t - t ime T - wave period u - horizontal fluid particle velocity V, V(x) - velocity amplitude along surface S x - horizontal coordinate direction normal to generator face x - a point (x,y) i n the wave field y - horizontal coordinate direction along generator face z - vertical coordinate direction, measured from basin bo t tom A - generator source facet length e - horizontal stroke amplitude of generator paddle at the base of the paddle (for flapper mode, e = 0 ) £ - wavemaker displacement function rj - water surface elevation 8 - wave t ra in propagation angle, measured from the x axis v - vertical distance between the basin bot tom and the generator base —* —* £, £(x , y) - location of a source point i n the x , y plane r - factor to modify generator transfer function to account for finite segment widths ( r = \k'B) <f), <f>(x) - two-dimensional velocity function $ - three-dimensional velocity function if - segment phase angle u> - angular frequency xv i i 1 Introduction 1.1 General A number of laboratory wave basins worldwide are capable of generating directional wave trains to simulate naturally occurring short-crested sea states. M a n y facilities employ the serpentine, or snake principle of wave generation. Suitabi l i ty of this method is l imited by effects associated wi th the finite size of these basins. A short-crested, or multi-directional random sea state may be considered a super-position of many infinitely long-crested, regular wave trains of various frequencies, and travelling i n numerous directions. Reproduction of such sea states by this concept relies on an abil ity to generate waves which propagate obliquely away from the wave generator. M a n y laboratories use segmented wavemakers which consist of a straight or curved bank of individual ly controlled, narrow wave board segments, forming one or more basin wal l . Oblique regular wave trains are produced using the snake principle, introduced by Biesel i n 1954. The generator motion is sinusoidal both i n time and spatially along the generator face, such that segment displacements themselves take on a travelling waveform along the row. Waves generated by the snake method are subject to a diffraction effect due to the finite length of the generator. In a rectangular wave basin, waves which emanate from the generator segments furthest 'upstream', i n a propagation sense, are drawn by diffraction into the area adjacent to the region which is i n the direct path of propaga-t ion. Th i s effect reduces the size of the region over which a homogeneous wave field is produced. Diffraction of energy into the quiescent region causes attenuation of wave 1 1 Introduction 2 heights w i t h i n the direct wave field region. In addit ion, the snake method is unable to account for waves reflected back into the pr imary wave field from basin walls. These secondary waves produce an interference pattern which contaminates wave conditions throughout the test area. A n alternative approach has been proposed to alleviate the shortcomings which characterize the snake method of generation (Isaacson, 1984, 1985). Referred to as the diffraction method, the wave field created by the motion of a single generator segment moving i n isolation is considered analogous to the diffraction pattern formed by waves propagating through a breakwater gap of the same w i d t h as the oscil lating wave board. Development of this principle by a point source representation of generator segments w i l l enable segment control which accounts for wave reflection and diffraction effects wi th in the basin. W h i l e the snake principle method has been implemented and involves relatively straightforward programming, the diffraction method is comparatively com-plex, and untried i n practice. For each frequency component to be superposed to form a directional sea state, a series of large matr ix equations must be solved. The appeal of the method is enhanced by present trends i n computer technology, toward increasingly powerful array processing machines at diminishing cost. Experiments were conducted in the directional wave basin of the Nat ional Research Counc i l ( N R C ) of Canada i n Ottawa. Elevation time series of normal ly and obliquely propagating, long-crested regular wave trains were recorded at discrete locations w i t h i n the test area. A modified form of the snake principle of generation was used to deter-mine generator segment motions for a desired oblique wave height, period and angle of propagation. The wave height was held constant between tests. The period and direc-t ion of propagation were varied to investigate wave conditions over ranges frequently encountered i n research endeavours. The test program also included a series w i t h a surface piercing impervious cylinder placed i n the test area, to model a large gravity 1 Introduction 3 base structure. A random wave analysis program was modified to evaluate the fundamental and harmonic wave components present i n the data. Plots derived from this information enable visual inspection of diffraction and reflection effects i n the basin, and also provide comparisons w i t h predictions generated by a numerical model developed using the diffraction principle. Examinat ion of the time series stripped of fundamental and second harmonic components offers insights into the significance of higher harmonic and other effects occurring i n the test area. 1.2 Literature Review T h e prediction of forced waves generated i n a laboratory flume or basin remains largely based on linearized, or small-amplitude theory, as put forth by Havelock (1929). Two-dimensional theory for a piston wavemaker is reviewed by Ursel l , Dean and Y u (1960), who affirm empirically the use of sinusoidal generator motions for wave trains of mod-erate steepness. They discuss effects not predicted by simple theory. These include the influence of wave reflection from a wave flume beach, attenuation of wave height along the flume and higher harmonic effects associated w i t h wave generation. Non-linear terms i n the equations of motion are negligible for small amplitude waves. Discrep-ancies between empirical results and theory not due to the effects just mentioned are attr ibuted pr imar i ly to nonlinear, finite-amplitude effects. Recent efforts have been made to extend wave generation to include second order components which are bound to the fundamental wave. H i g h harmonics distort the wave profiles, sharpening crests and flattening troughs. Linear wave generation can-not reproduce these components, instead producing free harmonic components which diminish the realism of generated sea states. Sand and Mansard (1986) address this 1 Introduction 4 problem, and provide a second order theory for satisfactory reproduction of irregular sea states. The first treatment of the theoretical relationships between segmented generator motions and the resulting wave field was that of Biesel (1954). One aspect considered was the generation of oblique waves by a straight, segmented generator. He developed the snake principle method, and established a practical l imi t to the mean direction of propagation attainable without spurious secondary wave effects. Subsequent maturat ion of two-dimensional wave generator theory enabled Gilbert (1976) to extend Biesel's results to relate generator displacements to wave field eleva-tions. Thi s transfer function for a specified wave field has been rederived and extended by Sand (1979). Sand and Mynet t (1987) discuss the practical l imitat ions of generating directional waves i n a finite basin by this method. They examine the l imi ta t ion , by diffraction and reflection, of the area over which a homogeneous wave field may be produced. They also indicate that local disturbances immediately i n front of the wave boards further decrease the homogeneous test area. Funke and Miles (1987) present a technique of using part ia l side walls adjacent to the wave generator ends, incorporating reflected waves into the direct wave field, at one end. These indirect waves appear to emanate from a v i r tua l extension of the generator, and effectively enlarge the homogeneous test area. A t the opposite end of the generator, reflections from the side wal l are suppressed by holding a sector of generator segments adjacent to the wal l motionless. A n alternative method of relating generator displacements to the wave field was introduced by Naeser (1979). He drew an analogy between a single generator segment oscil lating i n a straight wal l , and a gap of the same width i n a straight breakwater, through which waves pass and diffract. A directional sea state would be produced 1 Introduction 5 by superposition of addit ional segment and frequency contributions. Naeser (1981) subsequently described how this 'diffraction method' might be applied i n practice. For smaller ratios of segment width to wave length, B/L, the assumptions upon which Naeser developed this concept become unrealistic. Isaacson (1984) presents a more general analysis, based on the diffraction principle, but unconstrained by these restrictions. The breakwater gap analogy is reproduced by a point wave source rep-resentation, using potential flow theory. Isaacson treats the case of a single, straight generator bank, and also covers L- and U-shaped, mult iple generator configurations not pertinent to this study. A single oscillating generator segment may be approximated by a point wave source, for small values of B/L. As B/L increases, a single segment produces wave crest contours which are non-circular, i n agreement wi th breakwater gap results. The point source representation may be extended for this case by distr ibuting addit ional sources along the segment face. Reflections from basin walls and structures situated w i t h i n the wave field may also be accounted for by point source representation of these surfaces. Isaacson demonstrates the validity of this approach by numerical comparisons w i t h Naeser's breakwater analogue. Subsequent comparison of wave fields predicted by the snake principle and the diffraction principle (Isaacson, 1985) indicates that the diffraction method appears to reproduce the desired wave field realistically. 1.3 Scope of the Project The present study is confined to the generation of single regular wave trains. Mea-surements were made, i n the N R C offshore wave basin, of wave trains directed at four propagation angles, including the case of waves propagating normal to the generator face. Three wave lengths typical of the range encountered i n laboratory studies were 1 Introduction 6 investigated. A pr incipal goal of the project was to acquire sets of wave elevation data spanning the entire basin test area. In addit ion, a large cyl indr ica l structure was placed i n the test area to investigate its interaction w i t h the incident wave field. M a n y undesirable effects are present i n the three-dimensional flow field of a large test basin. These exert negative influences on the desired wave field. Such effects as wave diffraction, and reflection from solid boundaries are qualitatively understood. Measured wave conditions are complicated further by harmonic factors, resonance effects wi th in the basin, and by nonlinear effects generally attr ibuted to wave generation aspects and the energy dissipation processes occurring w i t h i n the artif icial beaches situated i n front of solid basin boundaries. A n existing analysis program was modified to enable the removal of synthesized fundamental and harmonic wave components from measured wave elevation time se-ries. The synthesized time series remaining after extraction of these components are composed of higher harmonics and other effects. V i s u a l inspection of these residual series provides some indicat ion of the magnitudes and spatial extent of these effects for various wave conditions. Long-crested regular wave tests provide the simplest data from which to examine underlying phenomena. Fundamental and second harmonic amplitude and phase plots were produced to facilitate inspection of diffraction and reflection effects w i t h i n the wave field. These plots are compared w i t h the predictions of a two-dimensional, linear diffraction numerical model developed by Isaacson. Appl icab i l i ty of the diffraction method is thereby verified, by confirming the abil i ty of the method to account for diffraction and reflection effects. 2 Theoretical Background To date most laboratory wave tanks are equipped w i t h some form of wave making machine along one boundary of the tank. Such facilities include long, narrow flumes for two-dimensional studies, and broader coastal and offshore basins. L inear wave theory has generally proven sufficient i n predicting resultant wave mot ion and generator power requirements, although nonlinear factors are increasingly being addressed as analytical methods evolve. 2.1 Two-Dimensional Plane Wavemaker Theory Detailed development of two-dimensional wavemaker theory using a paddle is presented by Dean and Dalrymple (1984, pp. 172-178). The boundary value problem corresponds to two-dimensional waves propagating i n an incompressible, inviscid fluid, and due to a prescribed wave generator motion. A Cartesian coordinate system is defined i n Figure 1, w i t h x measured horizontally orthogonal to the generator face, and z measured vertically upward from the flume bottom. The flow is assumed to be irrotat ional , and may be described by a velocity potential <j> which satisfies the Laplace equation w i t h i n the fluid region. Th i s potential is also subject to various boundary conditions on the flume bot tom, at the free surface and the generator face, and to a radiat ion condit ion at large distance x f rom the generator. The horizontal displacement £ of the wavemaker is given by ( = b2f(z)smujt (2.1) where f(z) is a dimensionless function of the generator displacement amplitude as a 7 2 Theoretical Background 8 function of elevation z, such that f(z) — 1 at z = h; b2 is the amplitude of the generator displacement at the st i l l water level z = h. The subscript signifies the two-dimensional case. Horizontal displacement is sinusoidal i n time t, w i t h angular frequency u>. The displacement amplitude is assumed small w i t h respect to s t i l l water depth h and the characteristic wave length of the resulting wave motion. Because of this, the free surface boundary conditions may be linearized, and the kinematic boundary condit ion at the generator face may be applied at the generator equi l ibr ium location x = 0: u = ^(0,z,t) = b2f(z) cocosut (2.2) ox A solution to the boundary value problem provides an expression of the velocity po-tential as (p = Acosh kh sin (kx — out) CO + 53 Cne~knX c o s knh cos ut (2.3) n=l where A and Cn are constants. The first term represents a progressive wave tra in produced by the generator. The series of terms represents standing wave components whose amplitudes decay w i t h distance away from the wavemaker. The progressive wave number k and standing wave numbers kn are defined by their corresponding dispersion relations: gktanhkh = u? (2.4) and gkn tan knh = — u> 2 for n — 1 , 2 , 3 , . . . (2-5) i n which g is the gravitational constant. Equat ion 2.4 has a single solution for k, whereas Equat ion 2.5 has an infinite number of solutions for kn. The decaying standing waves are referred to as evanescent waves. These are caused 2 Theoretical Background 9 by an energetic adjustment of the waveform from its in i t i a l profile adjacent to the wave-board, w i t h its linear shape and motion characteristics, to the downstream nonlinear progressive wave profile. Examinat ion of the series terms i n Equat ion 2.3 reveals that the heights of these evanescent waves decay to less than one percent of their in i t i a l values w i t h i n a horizontal distance approximately three times the water depth h from the generator face. The progressive wave component of the corresponding free surface elevation is given as rj — cosh kh cos (kx — ut) 9 = a 2 cos (kx — cot) (2-6) The general two-dimensional transfer function relating this wave amplitude a 2 to the generator displacement amplitude b2 is given by s a 2 Aksinhkh fh ,. . , , . F(kh) = - 1 = — :—r—7T- / f(z)coshkzdz (2.7) v ! b2 2kh + sinh2Jfc/» Jo  w y J Using the notat ion presented in Figure 1, Isaacson (1984) elaborates: In the ideal case of a generator whose displacements correspond to the horizontal particle displacements given by linear theory, we have f(z) = cosh kz/cosh kh and the transfer function is given s imply by F(kh) = tanh kh. As a more practical case, many generator motions are included i n the gen-eral representation ,of f(z) given as { 0 for z < v (2.8) c + ( l - c ) ( * = £ ) f o r z ^ Thi s corresponds to a fixed vertical plate from z — 0 to z = v and a linear variation of generator amplitude from z = v to z = h w i t h f(z) = e at z = 2 Theoretical Background 10 v. For this generalized generator motion, the transfer function F(kh) is given as 4 sinh. kh f F ( k h ) = 2tt+sinh2tt { £ ? ( s i n h k h ~ s i n h *") + llz^(kh sinhfc/i — fci/ sinh kh + c o s h — cosh fc/i)j (2.9) For a paddle extending to the flume bottom, simplified expressions for piston, flapper and combined piston/flapper modes are presented by Isaacson (1984, p. 11). 2.2 T h e Snake Principle of Wave Generation It is possible to generate i n a laboratory basin three-dimensional waves which propa-gate obliquely from the generator face, using a segmented wavemaker. The direction of propagation depends on the wave machine segment motions. Lineari ty of the problem allows superposition of mult iple generator motions to generate realistic sea states in a basin. A t most research facilities capable of directional wave generation, the program-ming of segment motions has to date been based on the snake, or serpentine principle. First consider a straight, infinitely long bank of generator segments, each of in -finitesimal wid th . The coordinate system used is defined in Figure 2. Horizonta l displacement of each segment is sinusoidal in time. The snake principle takes its name from the fact that segment motions are also sinusoidal i n the y direction, w i t h wave number k', such that they undergo a travelling wave or 'snake-like' monochromatic mo-tion. Th i s produces a regular wave tra in which propagates obliquely from the generator face. A superposition of these motions for regular wave trains of different frequencies and directions produces desired regular or irregular, mult i-directional wave fields. The wavemaker displacement, sinusoidal i n both time and distance, is given for 2 Theoretical Background 11 infinitesimal segment widths as ( = -bf(z) sin (k'y - ut) (2.10) where b is the generator displacement at s t i l l water level z = h, for this three-dimensional case. The generator shape function f(z) is as defined i n Section 2.1, such that at s t i l l water level f(h) = 1. W r i t t e n in complex notation, Equat ion 2.10 becomes ( = ibf(z) e « ( ^ - " < ) (2.11) where i = . Th i s type of generator motion results i n an oblique monochromatic wave t ra in of amplitude a, wave number k and direction 0, w i t h the surface elevation given by rj — n (x,y,t) = a cos (kx cos 9 + ky sin 9 — ut) (2.12) The velocity potential is ua cosh kz <p — ——:—-—— sin (kx cos 9 -\- ky sin 9 — ut) k s inn kh ga cosh kz . = — — sm (kx cos 9 + ky sm9 — ut) (2.13) u coshfc/i The relationship between the propagation direction 9 and the wave numbers k and k', and corresponding wave lengths L and Z/, is given by s i n 6 = T  = T>  ( 2 - 1 4 ) Recal l ing the two-dimensional transfer function F(kh) relating wave amplitude a2 to generator displacement amplitude 62 at s t i l l water level, the three-dimensional transfer function is o cos 9 2 Theoretical Background 12 The problem may be extended to the case of a finite generator segment w i d t h B. The rfi 1 segment mid-point location is given by xn = 0 y„ = (n - 1)B (2.16) The instantaneous displacement of the n segment, at z = h , is („ = bsin(k'(n - 1)B - ut) = 6s in( (n - l)kB sm6 - ut) (2.17) The wave t ra in generated is as before, but w i t h a different wave amplitude. The wave amplitude to generator displacement amplitude transfer function is now modified to a F(kh) sinr b cos 6 (2.18) where r = \kB sin6> = \k'B , If the segment w i d t h to wave length ratio B/L is not sufficiently small , secondary propagating wave components w i l l also be generated. Biesel st ipulated that i n order to avoid these undesired components, f s TTilhT«i < 2 1 9 ' The theoretical l imi t corresponding to 6 = 90° gives B/L — 0.5, but cannot be realized; there is then no preferred wave propagation direction. Biesel therefore proposed a more practical B/L l imi t as T ^ -T-^i ^7 ( 2 - 2 0 ) L v/2 + | s in0 | V ' Th i s gives B/L = 0.41 when 6 = 90° . 2.2.1 Refinement By Use of Reflective Partial Side Walls A major problem i n reproducing multi-directional sea states i n a finite basin concerns the small size of the test area over which a homogeneous wave s imulation occurs. Thi s 2 Theoretical Background 13 aspect has been considered by Sand and Mynet t (1987) and subsequently by Funke and Miles (1987). Diffraction and reflection effects significantly influence the effective test area. The existence of evanescent waves which decay exponentially away from the wave boards prohibits use of the region wi th in about three water depths from the generator face. In the N R C offshore basin, it has been found that side walls adjacent to the wave generator ends are needed to maintain uniform crest elevations when running regular, long-crested waves normal to the generator face (8 = 0 ° ) . However, wave absorbing structures, referred to as 'beaches', are needed to minimize reflection of waves from the basin sides when generating multi-directional sea states. Th i s reflected energy contaminates the wave field in the test area by its interaction w i t h incident waves. The performance of side beaches is easily appreciated by noting the rap id dissipation of such 'cross-waves' after a test. Depending on the predominant sea conditions i n a test program, the side wal l lengths are adjusted as a compromise between these two conflicting goals of m i n i m a l 'downstream' attenuation of wave energy propagating at close to 8 = 0° on the one hand, and reflections from such side walls on the other. As i l lustrated i n Figure 3, using the simple snake method it is not possible to generate waves directly into the region bounded by side wal l ' R ' and the line ' A - A ' , which projects from the corner at angle 8 to the side wal l . Th i s boundary represents the ideal edge of the plane waves propagating directly away from the generator i n the desired direction. The region adjacent to the direct wave field is referred to as the diffraction zone, into which some wave energy is drawn by the process of diffraction, which involves bending of wave crests and waveheight attenuation due to lateral transfer of energy along wave crests. The result is a reduction of waveheights i n the direct wave field, thereby reducing the area in which a homogeneous wave t ra in is produced. 2 Theoretical Background 14 Funke and Miles (1987) present a technique which enlarges the homogeneous test re-gion using waves reflected from one corner, and by keeping generator segments adjacent to the opposite side wal l motionless to reduce unwanted reflections. They demonstrate the rationale for this approach over the simple snake method. Referring to Figure 4, the 'indirect reflection' principle consists of generating a secondary wave toward the side wal l so that, when reflected, it joins i n phase wi th the direct, or primary, wave. Th i s has the effect of extending the effective test area by translating boundary line A - A so that it projects from the end of the left side wal l . L ine ' B - B ' , drawn at angle 9 to the reflecting side wal l R , and extending from the side wa l l outer end to the generator face, marks the boundary of the secondary wave tra in . The point of intersection of this line w i t h the generator face indicates the number of segments m to be involved i n this secondary wave generation. Thi s is given s imply as „ tan 9 m = S—— (2.21) B where S is the length of the side wal l , and m is rounded up to the nearest integer value. For these m segments the secondary, or 'B-wave', mot ion is superposed on the pr imary 'A-wave' mot ion. The B-wave motion mimics the mot ion of m v i r tua l segments which would extend the generator to the left of the side wal l . Assuming perfect reflection of the B-wave from the wal l , the reflected indirect wave is of the same amplitude (and frequency) as the A-wave, and it joins i n phase w i t h the A-wave to form a wider wave crest of uniform height. O n the other side of the basin, line ' C - C , drawn similarly to B - B , defines the region in which the propagating A-wave would reflect from the wal l , heading back into the direct wave field at an angle of —9° to the side wal l . Therefore, the segments which lie w i t h i n this region are held motionless, so that the only energy which reflects back into the wave field is due to diffraction of the direct wave t ra in into this 'quiescent zone'. 2 Theoretical Background 15 T h e phase relationship between adjacent wave boards, moving i n snakelike fashion, is given by <P = j? (2-22) where M is the (non-integer) number of segments comprising one wavelength L' along the generator face: M = -—— (2.23) B sin 9  v ; th The phase angle of the n segment is 1 27T <Pn = (n - Jf (2-24) and the n ^ n wave board displacement is given by Co(n, t) = jsm(ut + (n-±)jf) (2.25) where T is the transfer function given by Equat ion 2.15, and a and u are the desired wave amplitude and frequency, respectively. The subscript D refers to the direct wave component, while V w i l l refer to the v i r tua l , or indirect, component. The displacement of the n'^ v i r tua l segment to the left of the wal l would be given as 1 0 CD(n',t) = jsin(ut + (n'--)^) (2.26) for n' = 0 , —1, —2 , . . . , m — 1 In the absence of these v i r tua l segments, a wave tra in is generated which is directed against the side wal l , and upon reflection produces the same indirect waves as would the v i r tua l segments. Thi s is achieved by prescribing motions for segments 1 to m by substituting n' = 1 — n i n Equat ion 2.26. Then : Cv(n, t) = - sin (ut - ( „ - - ) — ) (2.27) for n = 1, 2 , . . . , m 2 Theoretical Background 16 Combining the direct and indirect wave motions, the first m segments have com-bined displacements given by (D(n, t) + (v(n, t) = y sinut cos ((n - | ) ^ ) (2.28) for 1 < n < m For the segments between the double-wave section and the quiescent section at the other end of the generator, Equat ion 2.25 applies. Th i s equation shows that, for simple snakelike motion, a l l segments move w i t h a phase shift relative to one another. However, as shown by Equat ion 2.28, the first m segments a l l have the same phase, but different displacement amplitudes. The first m segments have displacement amplitudes (at z = h) varying from 0 to which is twice the value of the segments responsible for generating direct waves only. Thi s is the case when, for single frequency oblique waves, the normal velocities of the direct and indirect waves are i n phase. Thus, the stroke l imi t of the mave machine w i l l be exceeded first by these segments. For long-crested oblique waves, whether regular or irregular, these are the segments closest to the corner. Th i s problem is less severe for short-crested wave generation, both because the component wave amplitudes from the extreme directions are typically small , and because for some components the corner segments are quiescent. Hence, the total displacement of corner segments w i l l generally be less for short-crested than for long-crested oblique waves. 2.3 T h e Diffraction Principle of Wave Generation 2.3.1 Analogy W i t h a Breakwater Gap The wave field produced by a single generator segment moving i n isolation was perceived by Naeser (1979) to be analogous to the diffracted wave pattern produced by a wave 2 Theoretical Background 17 t ra in passing at normal incidence through a gap of the same w i d t h i n a straight section of breakwater. The wave crest pattern beyond the gap generally resembles flattened circles, depending on the breakwater gap width-to-wavelength ratio, B/L. According to the Shore Protection Manual (1984), for B/L > 5 , the diffraction effects due to each wing of the breakwater are nearly independent. In other words, the diffraction pattern is comparable to a mirrored pair of wave patterns each due to a single, semi-infinite breakwater. Over the range of about 0.2 < ^ < 5 , Naeser suggests representing the propagating wave field height by H B i B H = -j= cos25(0) for x > 2B , | cot 9 \ > — (2.29) yLx L where B B S = 2 ( - + 0 .6) 3 w i t h — >0.2 (2.30) Li LI and H0 is the wave height close to the paddle, the same height as would exist throughout a flume of w i d t h B. The cosine-power form of the wave height expression requires a superposition pro-cess of the same form to produce a short-crested sea state. Naeser's approach is found to be of l imited applicabil i ty i n the context of laboratory wave generation. The requirement that B/L > 0.2 is a condit ion which is generally not satisfied w i t h most current segmented wave machines. 2.3.2 Point Source Representation of Generator Motions A more general approach to the diffraction principle was introduced by Isaacson (1984), using a point source representation of the wave field. A single frequency component only is considered; irregular, long- or short-crested wave trains may be obtained by superposing other frequency components. 2 Theoretical Background 18 Consider the wave motion throughout the basin to be represented by a velocity potential given by where x = (x, y), and <f> (x) is the two-dimensional potential function. Complex nota-t ion is used, w i t h the real portion representing the physical quantities described. The corresponding surface elevation is given by n(x,t) = — <t>{x)e- iwt (2.32) 9 The three-dimensional potential $ satisfies the Laplace equation, which requires that <j> must satisfy the Helmholtz equation w i t h i n the fluid: <*•> The potential <p is also subject to boundary conditions along the generator face and al l reflective surfaces, and to a radiat ion condition along energy absorbing sides of the basin. According to potential theory, <f> may be represented as due to point sources dis-tr ibuted along any reflective surfaces, including the wave boards: *(g) = 'hJs /(^ ^'dS (2- 34) Here / ( i f ) is the source strength distr ibution function, S represents the horizontal con-tour along a l l non-absorbing surfaces at st i l l water level, and dS denotes a differential length along S. A l o n g this surface, the boundary condition equates the flow velocity normal to S to the velocity of active generator segment surfaces, or to zero for quiescent segments and reflective walls. The potential at any point x = (x,y) due to a source at point £ = £(x,y) may be —* described by a Green's function G(x,£) which represents a concentric wave tra in ema-nating from the point source. Thi s function must satisfy both the Helmholtz equation 2 Theoretical Background 19 and the radiation boundary condition, and is given by G = inH^ (kr) (2.35) where HQ 1^ (kr) is the Hankel function of the first k i n d and zero order, HQ 1^ (kr) = Jo (kr) + i YQ (kr), r = \x — £J, and J 0 and Y0 are zero order Bessel Functions of the first and second k i n d , respectively. The surface boundary condit ion along S is given by the following line integral equation: i n which V is the velocity at point x on S where the boundary condit ion is applied, and n represents the unit direction normal to <S at point x. Equat ion 2.36 is generally incorrect i n that it assumes the variat ion of generator velocity w i t h elevation to follow that corresponding to Equat ion 2.31, such that the shape function f(z) = cosh kz j cosh kh. For the actual case of any other generator shape function f(z), an approximate generator transfer function expression may be developed analogous to the two-dimensional expression of Equat ion 2.7. The resulting expression for the velocity amplitude V(x) along surface <S, w i t h x on the generator face, is V(x) —  U a — tob ^ ( ^ ) 2^ 37) tanh kh tanh kh The value of V is set to zero for inactive sections, as well as at reflective surfaces such as side walls and structures i n the test area. The integral Equat ion 2.36 for the source strength function f(£) may be solved numerically by discretizing the S contour into N short, straight lengths, and w i t h a constant source strength over each of these segments, or facets. The N source facet —* is of length A „ , w i t h a point source of strength / n (0 a * the facet mid-point location. The tota l number of sources N w i l l generally exceed the number of wave generator 2 Theoretical Background 20 segment sources Ng . Thi s is due to the facets needed to represent reflective walls, and to the fact that, since the source facet length should be small compared to generated wavelengths, for higher frequency waves more than one source per generator segment may be required. For small values of B/L , the effects of an isolated generator segment are adequately represented. For larger values of B/L , however, it is necessary to distribute addit ional sources along the face of the generator segment to more accurately represent an in -creasingly non-circular wave field. Consider each generator segment to be divided into M equal facets each of length A = B/M. A point source is located at the centre of each facet. Computat ions for B/L = 0 .1 , 0.2, 0 .5, 1.0 and 2.0 w i t h a range of M values confirmed the validity of the breakwater gap analogy and led to the condition suggested by Isaacson (1984), that w i t h M rounded up to the nearest integer value. In particular , for B/L < 0.2 , M = 1 is suitable. Isaacson suggests, for instance, that to properly model the case of B/L = 1.0 at least five sources per segment should be used . However, for the conditions investigated, corresponding to laboratory segment widths and basin depths commonly encountered, it was concluded that use of one point source per segment ( M — 1) is usually sufficient. To offset the large number of equations required i n a model consisting of mult iple point sources per segment, Isaacson suggests the use of the asymptotic form of the Hankel function for large values of kr. Thi s is given as T h i s approximation increases computational efficiency at a cost of about two percent i n accuracy. for kr > 2TT (2.39) 2 Theoretical Background 21 —* B y the discretization approach, the integral equation for / ( £ ) is approximated by a matr ix equation for the source strengths at each segment: N £ BpnUO = XP for p = 1,2 . . . , N n=l Here, 'pn 1 for p = n i o r p ¥ z n and A p = < 2ub^tanh2i ^ o r P o n a g e n e r a t o r face 0 otherwise In Equat ion 2.41, MfkM = -cos^ = -ZTr ib c o s ( * r ) an or (2.40) (2.41) (2.42) (2.43) where /? is the angle between the line jo ining fp to £ n and the outward normal vector at the point x*p . In Equat ion 2.42, denotes the displacement amplitude of the generator segment associated w i t h the p^ 1 point source. 3 Experimental Setup 3.1 T h e N R C Offshore Wave Basin Acquis i t ion and in i t i a l analysis of a l l experimental data took place at the Hydraulics Laboratory of the Nat ional Research Counc i l ( N R C ) of Canada i n Ottawa, from June to August of 1988. The experiments were conducted in the Offshore Engineering Wave Bas in , which has since 1986 been equipped w i t h a segmented wave generator capable of producing directional sea states. In recent years, increasing emphasis has been placed on generation of realistic sea states i n laboratory wave basins. Sophisticated wave generation programs exist which enable control over a range of wave parameters i n the time and frequency domain, as well as the space domain. A port ion of the 60 segment, directional wave machine installed i n the N R C offshore basin is seen i n Figure 5. Each of the half-metre wide wave boards is driven by a hydraulic actuator under indiv idua l computer control. T h e actuators are l inked to lever arms for mechanical stroke amplif ication 1 . Th i s increases the range of wave heights which can be reproduced. A change i n the mechanical pivot point results i n a change i n the operational mode of the wave machine segments. These may be operated i n piston mode, hinged flapper mode, or a combined mode, w i t h equal contributions of piston/flapper. Different water depths may be accommodated i n this way, but conversion between modes involves the relocation of 60 linkages. The piston mode is used most often. The waveboards of the N R C segmented generator may be relocated vertically to accommodate different water 1For signal generation purposes, waveboard displacement £(x) at the still water level z = h is pre-scribed by the stroke angle of the lever arm. 22 3 Experimental Setup 23 depths. Operat ing i n piston mode i n an elevated position provides a compromise over the range of waves most frequently encountered i n offshore studies. The wave generator forms one 30 metre long wall of the rectangular offshore basin. Wave absorbing structures are located along the three remaining sides of the basin. These are visible i n Figure 6, behind the large tubular steel structure used to support an array of wave probes. The basin is three metres deep, and is usually operated at a water depth of about two metres. The configuration used throughout the experiments, and depicted i n Figure 7, results i n a working area of 30 by 19.2 metres. Th i s corresponds to the layout most frequently used for offshore conditions. The end wave absorber along the wall opposite the wave machine is 3.5m in breadth. The wave absorbers positioned in front of the three passive walls of the basin are based on a design developed at the Hydraulics Laboratory. K n o w n as a vertical pro-gressive porosity, or upright wave absorber, the design consists of vertical rows of perforated galvanized steel sheets supported rigidly on a tubular steel frame. M o d u l a r design and adjustable fittings allow modifications of the instal lat ion to be made. B y progressively decreasing the porosity of these sheets toward the rear of the absorber, the transparency of the sheets to incident waves diminishes as the waves pass into the structure. Th i s concept enables development of compact absorbers which l imi t reflec-tions to below 10% consistently over a wide range of water depths and wave conditions encountered i n laboratory studies. For details, refer to Jamieson and Mansard , 1987. Solid side walls were installed adjacent to each end of the wave generator face. These are comprised of removable steel panels placed along the front face of the side beaches, and are installed for wave generation purposes, as outl ined i n Section 2.2.1. The side wa l l lengths can be varied to suit particular applications. The 9.2m length for both side walls used i n the present study was prescribed by a separate experiment run concurrently i n the basin. 3 Experimental Setup 24 The experimental program coincided wi th a period of transit ion w i t h i n the labora-tory. A l l aspects of data acquisition and analysis, wave synthesis and generation were i n the process of conversion from a Hewlett-Packard ( H P ) 1000 computer system to a M i c r o V A X network. A t the time of testing, the data acquisition package was s t i l l tied to the H P computers; other aspects were supported by the V A X system. A software system referred to as " G E D A P " was used for the synthesis of wave gen-erator drive signals, the sampling of wave elevation (probe) signals, and for subsequent analysis of data. G E D A P was developed at the N R C Hydraul ics Laboratory, and is an acronym for general ized Experiment control, D a t a acquisition and data Analysis Package. D a t a acquisition was run on an HP1000 computer operating under software control which permits multi-user, foreground/background activities (Funke, et. a l . , 1980). Use of an H P alphanumeric/graphic terminal enables ful l control of data ac-quisit ion from a station at the experiment site. The analog/digital subsystem consists mainly of a N E F F Series 500 serial data l ink wi th remote converters. The G E D A P sys-tem remains independent of the analog/digital hardware; inter-communication software is changed as required. Wave generation is controlled by a dedicated V A X workstation adjacent to the H P terminal . A software package provides commands for various aspects of the segmented wave generator operation, and makes use of drive signals created by G E D A P software for each of the 60 waveboards. G E D A P logically and physically organizes the analog/digital interface into analog input /output ports, each consisting of 16 analog input and two analog output channels. Th i s arrangement is implemented through a card rack of 16 slots. Each measurement instrument is connected to a printed circuit card which fits into one of the 16 slots. Th i s provides a pick-up for power and reference voltages and transmission of the analog signal. Successful experimentation depends upon reliable cal ibration of instruments. 3 Experimental Setup 25 These are freed of l inearization electronics by implementing nonlinear compensation operation through G E D A P ; this allows use of cheaper, more stable and more easily maintained measuring devices. Cal ibrat ion using G E D A P may be carried out i n nu-merous ways; calibration constants are defined by specifying dimensional units , which are then carried through to data analysis and graphic output phases. 3.2 T h e Wave Probe A r r a y It was essential to place wave elevation probes at a large number of gr id positions to measure the wave field throughout the basin. The gr id spacing had to be smal l enough to provide acceptably fine resolution of wave elevation information, while not resulting i n a cumbersome quantity of data. Thi s dimension was also prescribed by the shortest wavelength to be tested. A two metre spacing was deemed appropriate. Capacitive-wire wave probes were used for wave elevation measurements. The off-shore basin is symmetrical about a centreline running perpendicular to the wave ma-chine face. Consequently, the entire wave field was i n effect measured while making actual physical measurements over only one half of the basin. Th i s concept is illus-trated i n Figure 8. T w o similar wave trains generated at propagation angles of positive and negative 6 degrees form a mirrored pair which yields measurements representa-tive of conditions throughout the entire test area, and corresponding to the positive 8 propagation direction. Figures 7 and 8 indicate that measurement over the half-basin gr id was achieved by locating a square array of 16 wave probes i n four positions. These probes occupied one input /output port. A single stationary probe was necessary to reference phase data between tests. Th i s necessitated the use of a second input /output port. The array of probes had to be supported rigidly to minimize instrument v ibrat ion due to wave 3 Experimental Setup 26 action. The structure supporting this array had to be largely transparent to the waves, and light enough to be easily moved between positions. Laboratory staff suggested that this large-scale frame needed to hold the 6 by 6 metre probe array be constructed using steel pipe and fittings which are stocked for construction of wave absorber modules. B y use of aircraft cable cross braces to stiffen the structure, a r ig id yet transparent frame evolved. The probe array support frame is i l lustrated i n Figure 9 and seen i n Figures 6, 10 and 11. Ca l ibra t ion of wave probes is achieved by immersing each probe in s t i l l water to at least three user-prescribed depths. A G E D A P program for calibrating each probe calculates a quadratic polynomial , fit to the voltages registered for each of these depth readings. A small second order term is indicative of an acceptably linear probe cal ibration. Offset values for a l l probes are obtained as required between full calibrations. In order to facilitate calibration of the 16 probes placed on a two metre gr id spacing, the support frame was designed so that the probe array could translate vertically as a uni t , assisted by a winched cable led from an overhead pulley. T h e probes were mounted on a superstructure which could slide up and down the vertical corner members of the core structure. In p lan view, the principal structure is 4 by 4 metres square, while the probe array superstructure covers a 6 by 6 metre region. The probe array was designed so as to minimize the necessary number of placements to acquire measurements throughout the test area. As seen i n Figure 7, the 6 x 6m array meets this goal. The support structure was stiffened w i t h cable bracing. Thi s ensured m i n i m a l v ibrat ion of the large planar superstructure without resorting to heavy reinforcement i n the th i rd dimension. For calibration purposes, support blocks of precise lengths were fabricated. These snapped i n place on the upright tubes when i n use. Th i s arrangement is shown i n 3 Experimental Setup 27 Figure 12. As seen i n Figure 13, inserts were fabricated which fit inside standard fittings to hold probes on the assembly. A l l probes were at least one metre long, and were mounted for about 40% immersion i n st i l l water. R i g and probe dimensions were established i n light of requirements for the separate experiment run concurrently i n the basin. A s seen i n Figure 6, the array r ig also incorporated components provided for another concurrently run experiment. These consisted of an extra pipe length on the super-structure to hold wave probes, and a current measuring assembly inside the rig, below st i l l water level. These additions increased the weight of the structure to 300-350kg. F lo ta t ion balloons on each corner of the r ig were used to facilitate its movement from one measurement location to the next. For each move, several balloons were secured to the r ig at a submerged level. These provided enough flotation that the author was able to singlehandedly move the r ig around the basin, pul l ing at floor level. The services of only one other person were required to assist in moving instrument cables suspended from roof girders. Wave probes were calibrated at the beginning of each of two series of tests, first without and then w i t h the gravity base structure ( G B S ) i n place. A winching cable was suspended i n a position above the first of four array positions. D u r i n g a test series, the r ig was moved i n a circular loop, so that at the end of the wave field series, the r ig was adjacent to the cal ibration position for the subsequent series w i t h the G B S i n place. Offset readings were taken following each move of the r ig , and at intervals during test sets. These readings do not entail vertical movement of probes, but update probe voltage magnitudes at the st i l l water level. The basin water level was monitored for leakage, and offsets taken accordingly. Evanescent waves from the generator, described in Section 2.1, d iminish to less than 10% of their in i t i a l height at a distance of about 4.0m away from the waveboards, and 3 Experimental Setup 28 to less than 1% at a distance of 6.0m. The wave probe array was positioned so that the closest measurements to the generator were along a line 4.0m from the generator neutral position. A s is visible i n Figures 14 and 15, the central test set positions abutted the basin centreline, w i t h the reference probe located between two array probe positions. T h e probe array dimensions also resulted i n some probe positions less than two metres from side and end beaches. 3.3 T h e Gravi ty Base Structure Tests were conducted w i t h a surface piercing, impervious cylinder to simulate the pres-ence of a gravity base structure of large diameter. The cylinder was comprised of three sections of concrete manhole pipe, each of 1.48m outside diameter, 0.91m long and weighing 1090kg. These were stacked to form a massive, yet easily installed surface piercing cylinder. The wave heights chosen were adequate for the measurement and observation of a l l pertinent wave effects, yet d id not cause motion of the structure, nor severely tax the generator hydraulic system, which was required to r u n for long periods during the test program. A s seen i n Figures 11 and 16, the structure was installed along the basin centreline, 12.0m from the generator face. Thi s location was close to the position used i n a previous G B S study. It also corresponded to a corner probe location on the array, when the array was situated in the last location for the test series involving the G B S . Hence, to avoid interference of the cylinder w i t h the placement of the array i n that posit ion, the cantilevered member from which this probe was supported was s imply cut off, and the probe removed for the final tests. Thi s configuration is visible i n Figure 17. 4 The Test Program Thi s study deals solely wi th regular, long-crested wave trains. Tests were grouped according to two series. The principal series was comprised of wave fields without the disturbing influence of a structure. A shorter series of tests involved wave trains from the first series, but w i t h a large, surface piercing, circular cylinder i n place. The tests of this latter series are referred to as ' G B S ' tests, to distinguish them from the wave field tests. Four propagation directions were considered. In addit ion to waves propagating normal to the generator face {9 = 0 ° ) , wave trains were directed at three angles to the normal direction: 30° , 45° and 60°. A pair of tests was run for each of the oblique wave conditions: for each wave t ra in propagated at + 0 ° , a matching test at —9° was conducted. The positive sense of 9 is shown in Figure 2. Bas in symmetry allowed wave conditions over the complete test area to be measured by running tests i n such pairs, while only actually moving the probe array around half the basin. Th i s concept is i l lustrated i n Figure 7. Waves were produced w i t h the generator segments moving i n piston mode. They were elevated 0.48m above the basin floor. Thi s value corresponds to parameter v of Equat ion 2.8, and as seen i n Figure 1. Thi s configuration has proven adequate for deep and intermediate depth tests such as comprise this study. A steel plate prevents passage of energy beneath the generator segments. Three wave periods were chosen as representative of the range commonly encoun-tered i n laboratory studies. These three sets are sometimes referred to by wavelength as short, intermediate or medium, and long wave cases, corresponding to periods of 29 4 The Test Program 30 1.25, 1.75, and 2.25 seconds, respectively. For the side wal l lengths used (9.2m), the 60° wave tests are of l imi ted practical importance. A large proportion of the generated wave energy becomes trapped i n the form of standing waves between the side walls. These tests are of interest, however, for evaluating resonance effects wi th in the basin, and for assessing the predict ion capabil-ities of the numerical model. The G B S series excluded any tests at 60° . Characteristics of a l l tests are presented i n Table 1, along w i t h values of various non-dimensional parameters of interest. A set of tests was run w i t h waves propagating at 30° without use of the corner reflection generation method, for both wave field mea-surements and the G B S case. A l l other tests employed the corner reflection technique. A l l but one test set involved the same prescribed wave height (20cm). To establish whether results are, i n fact, independent of wave height, a set of wave field tests at 30° was conducted at half height. A single test was also conducted twice to assess the repeatability of results. For the short wavelength, the segment width to wavelength ratio B/L — 0.205, a value close to Isaacson's suggested l imi t of 0.2. A numerical model was run to appraise the effect of using two point sources per generator segment (M=2) for this case, compared to only one. The short wavelength provides a deep water condit ion. The medium and long wavelengths fal l w i t h i n the intermediate depth range. W i t h a cylinder diameter V = 1.48m, a l l three wavelengths result in diffraction parameter magnitudes w i t h i n the diffraction range, T>/L > 0.2. Hence for a l l three wave conditions the cylinder behaves as a large structure. Wave steepness ranges from 0.027 to 0.082. The upper l imit corresponds to short wave conditions and can result i n breaking steepnesses when waves superpose. 4 The Test Program 31 4.1 Synthesis of Dr iv ing Signals G E D A P programs are provided for the synthesis of wave generator drive signals. For regular waves directed normal to the generator, a l l segments move i n unison, so that only one signal file is produced. For oblique regular waves, indiv idua l signal files are needed for each of the 60 segments. Each sinusoidal signal is synthesized from 35 or more discrete values, calculated at an interval of 0.1 sec. A t least one complete cycle is produced; this is repeated without truncation as required during operation, thereby avoiding the need for large signal files. Synthesis is executed through a configura-t ion parameter file, which prescribes paddle stroke mode, and other options. For this study, displacement control of segment motions was used, rather than velocity control. Displacement signals are expressed i n terms of the lever arm stroke angle. Plots of selected drive signals are included to illustrate some aspects of generator motions. In Figure 18, drive signals are shown which represent a l l regions of the generator bank, for test 4, T = 1.25sec at 6 = + 3 0 ° . Segments 1 to 11 comprise the combined A - and B-wave region. The simple snake method defines the mot ion of segments 12-46, while segments 47-60 are held motionless. As seen i n Figures 18(a) to (d), some combined-motion segments move w i t h strokes approaching twice that of snake-motion segments. The former move i n a phase-locked standing wave pattern; segment 3 is almost quiescent at a nodal position. A phase shift is evident between adjacent segments i n the simple snake region of the generator. T w o plots are presented in Figure 19. Plot 19(a), of the three drive signals for the 6 — 0° tests, shows that stroke increases directly w i t h wavelength, for a given wave height. P lo t 19(b), of drive signals corresponding to T = 1.75sec tests at a l l propagation angles, illustrates the fact that the stroke required to produce a given wave height varies inversely w i t h propagation angle, as seen by Equat ion 2.15. 4 The Test Program 32 The dr iv ing signal synthesis program utilizes the corner reflection technique, unless zero side wal l lengths are specified. Normal ly the program calculates the number of 'B-wave' segments using Equat ion 2.21. For the purposes of this study, side wal l length values which were prescribed to the synthesis program were altered from the true lengths to improve the wavefield. These values are presented i n Table 2. A l so specified are the numbers of those segments involved i n combined A - and B-wave generation, as well as of the segments which comprise the quiescent region. A l l entries i n Table 2 correspond to positive values of propagation angle 9. It was observed that some wave energy inevitably diffracts into the quiescent zone. Thi s is reflected from the side wall back into the wave field. To lessen this effect, the apparent side wal l length at this end of the generator, and hence the number of stil led segments, was increased. A t the other end of the generator, almost a l l energy propagating toward the side wall reflects to jo in w i t h the pr imary wave t ra in . However, some wave attenuation does occur at the edge of this B-wave tra in . Increasing the apparent side wal l length slightly may send a 'wider' wave toward the side wal l , reducing the degree of attenuation at the outer end of the wal l . A n y 'excess' wave passes into the side beach beyond the wal l . Th i s second change was used to a m i n i m a l extent, setting side wal l lengths at 9.3m rather than the actual value of 9.2m. In only one case, that of the short wavelength at 9 = 6 0 ° , d id this i n fact result i n a larger (integer) number of segments being assigned to B-wave production. 5 Data Analysis Capacitive-wire wave probes were the only instruments used. These provided t ime series elevation records at discrete locations around the basin. Seventeen probes were used, consisting of sixteen probes which formed the movable array, and one reference probe on a stationary, guyed pole situated along the basin centreline. A l l necessary signal conditioning was achieved through analog cards on the electronics racks. The chief analytical tool used was a modified version of a powerful program known as S C R E N . The name refers to its function of multiple regression screening of a given time series. The program was developed i n the N R C Hydraul ics Laboratory for the analysis of irregular wave trains, and was modified to accommodate harmonic components of regular waves. The S C R E N program uses a nonlinear regression routine to systematically extract the best-fitting sinusoidal components of a given time series. In this way, a prescribed number of pr incipal frequency components of a measured wave t ra in may be evaluated. A simplified flowchart is presented i n Figure 20, of the program following modification to analyse harmonic components. A description of sequential steps adopted in the root program are as follows. Before reading the data, the program requests values of certain parameters which define the desired analysis. One parameter concerns the m a x i m u m number of frequency components to be extracted. Others w i l l be introduced i n context. The data file is then read and the mean value removed, which may be reinstated if requested. The presence of a mean value indicates a change in st i l l water level since the most recent probe offset reading. The R M S value of the time series is then determined. 33 4 The Test Program 34 Users specify the number of parameters to be optimized. A three parameter fit optimizes the amplitude, phase and frequency of each fitted sinusoidal signal, whereas a two parameter fit optimizes the amplitude and phase, but not the frequency. In the computation loop for the current frequency component, the t ime series un-dergoes a Fast Fourier Transformation and is scaled. T h e n the m a x i m u m amplitude is found, along w i t h the frequency at which this occurs. Th i s method is sensitive to a sample duration which is not an integer mult iple of the period of the selected wave component. Better amplitude and frequency approximations are next made by quadratic interpolation. These values are then used as in i t i a l guesses i n a nonlinear regression opt imizat ion routine which fits a sinusoidal curve to the data. The regression is performed using the Gauss-Newton algorithm, which is perhaps the most efficient algorithm available for least squares curve f i t t ing 2 . Corrections to starting values of pa-rameters are computed by iterative cycles unt i l the change i n the error sum of squares between successive iterations falls below a user specified tolerance. The phase angle is also determined using quadratic interpolation. Having established the final estimates of these values, the computed sinusoid is removed from the time series, and the pro-cedure recycles for the next frequency, i f requested. The program cycles unt i l either the prescribed number of frequency components have been extracted, or the variance of the amplitude exceeds a l imi t ing percentage of the original t ime series R M S value, whichever comes first. Thi s l imit is also set by an input parameter. After the pre-scribed number of sinusoids have been removed, the time series is synthesized using the computed values of frequency, amplitude and phase. The program outputs four files, each of which contain a l l pertinent parameter values i n a common header. T w o time series are output. The synthesized time function is the 2For detailed coverage of the least squares method, as applied in a wave reflection analysis program which preceded SCREN, refer to Mansard and Funke, 1980. 5 Data Analysis 35 sum of a l l sinusoidal functions at their computed amplitude and phase. Al so output is the residual t ime series remaining after the mean and al l opt imal ly fitted sinusoids have been removed from the original t ime series. The other two files contain line spectra, of the amplitudes and the phases of the sinusoidal t ime functions, each plotted as functions of frequency. 5.1 Modifications to the S C R E N Program The S C R E N program was modified to extract harmonic components from the elevation time series. The flowchart presented in Figure 20 illustrates key operational steps. A n input parameter indicates the number of harmonic components to be analysed. If given a value greater than the default value of one, this parameter causes a two-parameter fit (amplitude and phase) to be performed for the second and higher harmonics. O t h -erwise, the program performs as before modification. A three-parameter fit was avoided i n the harmonic analysis. For some tests, the second harmonic was of sufficient magnitude that S C R E N would treat the second har-monic as the fundamental component. Knowledge of the forcing frequency means that i n order to avoid such occurrences a simple check could be made on the fundamental frequency value chosen by the program. However, it was decided instead to fix the frequency w i t h a user specified value, fo- Thi s approach avoids a further problem of the program closing i n on other values near the forcing frequency, such as could happen if the forcing frequency is close to a resonant frequency of the basin. Th i s condition would cause energy to be shifted away from the forcing frequency, resulting i n a higher energy density at the resonance frequency. Thi s would cause the program to perform a fit at a frequency shifted away from the actual fundamental value. The frequency value assigned to JQ for each of the three wavelengths tested was 5 Data Analysis 36 not the precise value specified for the synthesis of drive signals. A two-parameter fit for the first component only was run for the reference probe channels of a l l tests of the same wavelength. The mode of these values was used as the fixed fundamental value fa. As an i l lustrat ion of this procedure for the case of the intermediate wavelength, the desired frequency is 1/1.75 = 0.57143Hz, whereas opt imizat ion of amplitude and phase gave the following set of results: 1 occurrence of 0.5709, 5 x 0.5710, 14 x 0.5711, 15 x 0.5712, 13 x 0.5713 and 4 x 0.5714. The value used for fG was 0.5712Hz. The S C R E N program was modified to provide basic wave parameters of the fun-damental component, which represents the generated wave t ra in without its bound harmonic component. The user specifies the st i l l water depth. F r o m this and the fun-damental frequency value fo, the wave number, wave length and phase velocity are obtained on the basis of linear wave theory. The synthesized time series consists of superposed fundamental and second harmonic components. In a program loop, maxi-m u m and m i n i m u m values of this nonsinusoidal waveform are stored, yielding the wave height (Hsy). The program stores this and the fundamental wave parameters i n the G E D A P output file headers for subsequent retrieval. 5.2 Types of Plots Generated A versatile set of generalized plott ing routines is an important component of the G E D A P software package. Several pages of plots were generated for each test set of sixteen array probes plus the reference probe. Representative examples are pre-sented i n Figure 21, from a test involving the intermediate wavelength, propagating at 8 = 30° . The test location is indicated i n Figure 21(a), for a positive propagation angle. 5 D a t a Analysis 37 For every test set, one page was devoted to the reference probe (number 17) po-sitioned along the basin centreline, 7.37m in front of the generator neutral position. Thi s page is presented in Figure 21(b), and consists of two plots and a l ist ing of per-tinent parameters. Prescribed values of computational l imits are listed along w i t h the number of parameters optimized (2 = amplitude and phase). Fundamental and second harmonic values of frequency, amplitude and phase are displayed, as wel l as the wave parameters calculated by the modified S C R E N program. Al so shown are the magni-tude of the mean value which was removed by the program, the R M S value of the original measured time series, and of the synthesized and residual t ime series. R M S values offer an in i t i a l measure of the quality of fit of the synthesized series compared to the measured trace. The plot along the top of the page is the entire 50 second record of the time series measured by the reference probe. The larger plot consists of three superposed traces for the first ten seconds of the sample. The solid trace is that of the original measured time series. The dashed line which roughly mimics the solid plot is the synthesized fundamental plus second harmonic waveform generated by S C R E N , w i t h the mean value removed. The remaining dotted trace of smaller magnitude corresponds to the residual t ime series of higher harmonic and other components. Quick inspection of the residual trace affords some insight into the magnitude and nature of effects other than the fundamental plus second harmonic wave, measured at the fixed reference location for a l l tests. For each test, four pages of plots were generated, four plots to a page, of measured elevation time series traces for the 16 probes on the array frame. Examples of these are presented i n Figure 21(c)-(f), which relate to the probe 17 plot of Figure 21(b). Individual probe locations are seen i n Figure 21(a). The vertical scale is constant for the four plots of any one page, but may vary from page to page. Together, these plots 5 Data Analysis 38 provide a visual record of a l l probe wave elevation records for each test set. For any one wave t ra in and propagation direction, data sets from eight tests were united to represent wave conditions throughout the basin test area. Us ing a spreadsheet program, these eight data sets were merged, then manipulated to create data files of fundamental amplitude ( A i ) , synthesized wave height (Hsy), and phase prior to plott ing. The Golden Graphics System, a plott ing package for personal computers, was used to generate a l l plots of basin conditions from these data files. F i r s t , a gridding program creates a regularly spaced gr id from a data file. For ful l basin plots, 90 gr id lines along the longest side were used. A 45 line gr id was specified for half-basin plots. Th i s gr id size represents a six-fold increase i n resolution over the original spacing of data points as set by probe locations. A n increase in the density of the gr id results i n improvements i n the accuracy and smoothness of the plot, at the expense of increased computat ional effort. The smoothness of plots generated is a function of the original data, the gr id density and the value assigned to a smoothing factor which ranges from 0 to 1. A value of 1 corresponds to no smoothing, and yields a plot wi th very jagged extrema. A large degree of smoothing has the effect of flattening peaks and troughs to the extent that important information is lost. The default value of 0.95 was applied throughout this study. Phase plots were produced by manually manipulat ing the phase predictions gener-ated by the S C R E N program. Phase values for a series were related between different test sets through the reference probe. For each set, the phase value corresponding to the reference probe was subtracted from al l other probe values w i t h i n the set. In this way, the reference probe phase for each test set was zeroed, so that phase values of a l l probes i n a series of eight test sets were then related. A t this point , phases ranged 5 Data Analysis 39 from 0° to 3 6 0 ° , which resulted i n problems when a contouring program evaluated neighbouring crest/trough contours. For this reason, phase values were increased or decreased by 180° increments i n proportion to distance away from the reference probe location, i n the propagation direction. The contouring program uses the gridded data to generate a three-parameter plot i n the x - y plane. Lines of equal value are plotted for wave amplitude, height or phase. The user chooses a contour interval appropriate to the complexity of the plot. A contour interval of 360° may be perceived to represent wave crest contours, at the instant that a crest passes the reference probe. For finer resolution, a contour interval of 180° yields a pattern of alternating crests and troughs. Another program which uses the same gridded data produces three-dimensional perspective block diagrams for surface representation. Vert ica l exaggeration of surface features is controlled by a he ight /width ratio parameter. 6 Results and Discussion 6.1 Synopsis Eight sets of test data were combined to form a complete data file for each case of a part icular wavelength and propagation direction. These consist of four test sets corresponding to each of the matching positive and negative propagation directions. For the 9 = 0° cases, basin symmetry enables evaluation over only half the basin, such that only four tests were merged. Table 1 provides a summary of tests conducted, w i t h the corresponding values of various parameters of interest. For the wave field series, tests were conducted for 9 = 0°, 30° , 45° and 60° , while for the G B S tests w i t h the cylinder i n place, the 9 = 60° case was excluded. Experimental m a x i m u m waveheight plots are presented for a l l but tests 9 and 10, which offer no unique features not present i n other plots. Figures 22 to 38 pertain to experimental m a x i m u m waveheight and phase results. Tests not represented i n this group of plots are found i n Figures 41 to 59, which provide comparisons between exper-imental waveheights and numerical predictions. Figures 60 to 62 consist of numerically derived phase plots, and presented i n Figure 63 is a three-dimensional instantaneous surface elevation plot from one numerical run . Included i n the first oblique wave max imum waveheight plot presented for each of the three wavelengths investigated is a scaled representation of the pertinent wave-length, as shown by a group of three wave crests. These are Figure 26(c) for the shortest wavelength, Figure 27(c) for the intermediate length and Figure 31(c) for the longest wave. For a l l other waveheight plots, propagation direction is indicated by an arrow. 40 6 Results and Discussion 41 The x — y coordinate axes are depicted i n Figure 2 w i t h the origin at the midpoint of the first generator segment face. Thi s corresponds to Equat ion 2.16. In a l l other figures showing these axes, the origin is positioned adjacent to the side wal l , which simplifies plotted results and coordinate values prescribed for numerical models. One contour interval (0.1) was used for most normalized waveheight plots. A value of 0.05 increased the resolution of detail i n the plots of Figure 22 of the half waveheight test and i n Figure 24, concerning the repeatability test. Thi s contour interval was also used for the plots of Figures 29 and 30. These figures deal w i t h tests involving waves propagated at 8 = 0 ° , i n which a diminished level of reflected energy throughout the basin results i n sparse contours at an interval of 0.1. It is interesting to note i n the plots of Figures 50-52 that the presence of the G B S i n the path of the wave t ra in significantly elevates the level of reflected energy, so that a contour interval of 0.1 provides adequate detail . Different contour intervals between comparative plots are found i n Figures 26-28, which include second order waveheight plots. Second order and other nonlinear components were consistently an order of magnitude smaller than the fundamental, linear wave components. Two contour intervals are also employed i n the plots of Figures 41-43. These involve comparison of experimental and numerical waveheight results for tests at 8 — 0 ° . These plots illustrate the reduced reflection levels present i n these tests, and that the linear diffraction numerical model underestimates the effects found i n the basin, because no account is made of nonlinear contributions. Th i s results i n shallower waveheight gradients, and hence, sparser contours than for measured results. V i s u a l comparison of experimental and numerical waveheight results reveals strong similarities. E r r o r plots could be produced by subtracting numerical results from ex-perimental plots. However, findings would not likely warrant such effort. Second order 6 Results and Discussion 42 effects are responsible for most differences between experimental and numerical wave-heights. These effects vary i n prominence around the basin, and from one test condit ion to the next. E r ror plots are not expected to reveal trends not already apparent i n the waveheight plots. Energy reflected from the downstream side wall from oblique wave trains usually produces a part ia l standing wave aligned perpendicular to the propagation direction. Longer waves strongly penetrate the downstream beaches, resulting i n a standing wave system aligned w i t h the propagation direction. In the case of the longest wave directed at 6 = 60 ° , a standing wave between the side walls was produced, which corresponds to a near-resonant condition i n this region of the basin. W h e n basin resonance is a predominant effect, it is important to include the up-stream side wall as part of the basin boundary i n numerical models. Boundaries were most precisely modelled for tests involving the simple snake method of generation. Corresponding model predictions agree very well w i t h experimental findings. Th i s i l -lustrates that to achieve opt imum linear model predictions, basin boundaries should be modelled as accurately as possible. Readers making a cursory inspection of plotted results are referred to Section 7.1, which provides a comprehensive summary of findings. For detailed discussion of these plots i n the sequence i n which they are presented, refer to Sections 6.2 to 6.6. Other features visible i n Figures 21 and 64 are considered briefly i n Section 6.7. 6.2 Effect of Prescribed Waveheight Referring to Table 1, tests 4 and 13 both correspond to the case of the short wavelength and the positive 30° propagation direction ( T = 1.25sec, 6 = 3 0 ° ) . In test 13, the prescribed value of the desired waveheight H was reduced relative to a l l other tests. 6 Results and Discussion 43 The measurements from test 13, over half the basin, allow comparison w i t h the ful l waveheight result of test 4. For test 13, a drive signal span of half the value used for test 4 was specified. The resulting waveheights recorded were generally only about 45 percent of the ful l height values, i n consequence of the nonlinearity of the propagating waves. In Figure 22, two separate synthesized wave height (H3y) plots are presented. Refer to Section 6.4 for a definition of Hsy. These plots are normalized to the desired waveheight H, corresponding to either H = 10cm or the ful l waveheight value of H = 20cm. Plots cover only the half of the basin over which measurements were actually taken for the half height test 13. In general, the half height plot bears a strong resemblance to its ful l height counter-part, indicat ing that max imum waveheight findings are independent of the prescribed waveheight. The fact that the smaller waveheight was about ten percent lower than the target value is evident when the two plots are compared. The plots are most no-tably s imilar i n the quiescent zone i n the top left corner of each plot. The only major discrepancy is found i n a high waveheight region indicated by a ' D ' i n the half height Figure 22(b). The bracketed region ' R ' of the ful l height plot of Figure 22(a) denotes the location of the isolated test set pertaining to the repeatability test. Figure 23 consists of the corresponding phase plots superposed. A contour interval of 180° was used, which displays alternating crests and troughs. The zero phase line intersects the basin centreline at the reference probe location. The half height plot is shown dotted. The figure shows that the wavelength of the ful l height wave tra in is greater than that of the half height case. The ful l height crests and troughs fall increasingly ahead of the half height contours, i n the 'downstream' direction. Th i s is another consequence of the nonlinearity of the waves. 6 Results and Discussion 44 6.3 Repeatability Test As seen i n Figure 24, a pair of normalized Hsy plots corresponding to tests 4 and 14 (T = 1.25sec, 8 = 30 ° ) was obtained i n order to assess the repeatability of results. These plots pertain to duplicate sets of tests at a single array posit ion, as indicated by region R i n Figure 22. They show consistent repeatability throughout the region. W h i l e minor discrepancies on the order of ten percent result i n some alteration of contour shapes, s imilar trends are apparent i n both plots. Except at the corner probe indicated by label ' P ' i n the upper plot of Figure 24, measurements at locations along the top row of the plots fall w i t h i n the quiescent zone. These signals are of small amplitude compared to records at other locations, and exhibit relatively strong second harmonic components. The two superposed phase plots are shown i n Figure 25, and display excellent agreement. The test 14 duplicate result is shown dotted. W h e n assessing differences between contours of the two plots it should be borne i n m i n d that the plot magnification is four times greater than i n Figure 23. 6.4 Maximum Waveheight Plots A l l contour plots presented were generated from grids produced w i t h a low degree of smoothing. Thi s results i n negligible loss of information, at the expense of unrealisti-cally jagged peaks at the probe locations. Trends indicated by contours surrounding these peaks are fairly representative of actual levels between probes. Plots may be compared more closely than if greater smoothing had been used. For a l l series of four or eight test sets, contour plots of fundamental amplitude A\ and synthesized waveheight Hsy were produced. Values of Hsy were normalized wi th respect to the desired waveheight H; Ai values were normalized w i t h respect to H/2. 6 Results and Discussion 45 Considering a regular wave tra in as having several superposed harmonic components, the elevation t ime series at any point i n the wave field is given by CO r] = ^2 An cos(nut - <f>n) (6.1) 71=1 The synthesis procedure extracts the first two components of this series, so that the corresponding synthesized elevation rjsy is rjsy = Ai cos(u;t — <pi) + A2 cos(2u;i — <p2) (6.2) Then , the synthesized waveheight Hsy is given by Hsy = 2(r]Sy)ma,x (6.3) where ?7max signifies the max imum value of rj w i th respect to t ime. The residual t ime series contains higher harmonic and other contributions not represented by the fundamental plus second harmonic waveform of height Hsy. Compar ing pairs of A\ and Hsy plots, only minor differences are apparent. Contour plots of the second harmonic amplitude A2 were also generated for a number of series. Having normalized the Ax results to correspond to Hsy plots, Ax = 1.0 is equivalent to the desired amplitude H/2 = 0.1m. Hence, the A2 values were mult ip l ied by a factor of ten before plott ing, i n order to be comparable w i t h the normalized Hsy and Ax plots. It is perhaps appropriate, at this point, to briefly outline second harmonic wave components. A regular wave is never truly linear; l inearity implies that the wave elevation time record is a pure sinusoid. In fact, measured wave profiles are distorted, such that troughs are flatter, and crests steeper than for linear waves. A consequence of this is that troughs are wider, and crests narrower than for sinusoids, so that about 60 percent of the total waveheight lies above the st i l l water level. A regular wave always binds some higher harmonic energy which propagates at the same celerity as 6 Results and Discussion 46 the fundamental wave component. Thi s bound second harmonic component has a frequency which is twice the fundamental value. It is the principal cause of the nonlinear distortion of the wave profile, and is particularly significant i n shallow water situations. In addit ion to the bound second order component, a significant amount of 'free' second harmonic wave energy is also associated w i t h regular wave generation i n a wave basin or flume. In deep water, this component travels at a celerity equal to half that of the fundamental and bound second order components. The result is a beating oscillation of the envelope height i n the flume. The free harmonic is part ly due to linear wave generation, which does not satisfy the second order boundary conditions pertaining to the bound second harmonic. The amount of free second order energy generated i n this way is dependent on the water depth, the wave length and height of the fundamental component, and the mode of operation of the wave generator. Another source of the free second order component is the release, i n the reflection process, of any second harmonic energy i n excess of that which can remain bound to the reflected fundamental. In Figure 26, contour plots of A\, A2 and H3y are presented for test 7, for the wave field corresponding to the short wave propagating at 45° ( T = 1.25sec, 6 = 4 5 ° ) . The magnitude of the A2 component in this case is relatively significant, compared to most other tests. A high A2 component is apparent in region ' A ' of F igure 26(b), one metre away from the 'upstream' side wall i n the corner reflection region. Th i s may be evidence of free harmonics caused by reflection from the side wal l . Alternatively, the high second harmonic reading could be due to a relative phase difference of about 180° between the two crossing wave trains i n this region. The program S C R E N may treat the indirect B-wave t ra in as a second harmonic associated w i t h the direct A-wave tra in . A s imilarly high A2 region is apparent i n the corner indicated by label ' B ' . Thi s 6 Results and Discussion 47 may be due to small reflections from the mutual ly perpendicular beach faces. Another region of high A2 values is indicated by label ' C i n Figure 26(b). Elsewhere i n the basin, the normalized A2 values range from about 0.1 to 0.14, and l ikely represent the bound second harmonic component of this short, but fairly steep wave t ra in . As presented i n Figures 27 and 28, another two sets of A\-A2-Hsy plots depict conditions without and w i t h the G B S , respectively. These correspond to tests 5 and 20, and pertain to the medium wavelength ( T = 1.75sec) propagating at 6 = 30° . Very l i t t le difference is observed between the A\ and Hsy plots i n these two figures, compared to the plots of Figure 26. The ubiquitous bound second harmonic component of this longer, flatter wave t ra in appears in the A2 plot, Figure 27(b), to be about 0.08 i n magnitude, compared to 0.1 or greater for Figure 26(b) of test 7. As i n Figure 26, a region of high A2 values is observed 'downstream' of the cross-wave, corner reflection region. A s seen i n Figure 28, the presence of. the cylinder notably increases waveheights i n the region between the side walls, the generator and the structure, due to energy reflected back into this region by the structure. These plots exhibit trends similar to the wave field results of Figure 27, except for a relatively high second harmonic component upstream of the cylinder. The points of highest A2 values i n Figure 28(c) are characterized by low standard error magnitudes, indicat ing fairly accurate synthesis. 6.4.1 Plots For Normally Propagating Wave Trains For wave trains directed normally from the generator face, side walls should ideally extend as far as the end beaches. As such, attenuation of waveheights due to side beaches would be eliminated. However, as discussed in Section 2.2.1, side beaches are installed i n addit ion to the side walls, i n order to accommodate the generation of oblique waves. For regular waves propagating at 8 = 0 ° , a l l generator paddles 6 Results and Discussion 48 move i n unison. Due to slight imperfections in waveboard synchronization, cross-waves are produced, which are largely confined to the region near the generator, and which undergo reflections between the side walls. Th i s effect is part iculary pronounced for side walls which extend along the entire length of the basin. Figure 29 pertains to tests 1 and 2, corresponding to 9 = 0° , and T = 1.25 and 1.75sec, respectively. Figure 30 shows corresponding plots for tests 3 and 18, corre-sponding to the long wavelength ( T = 2.25sec) wave field and G B S cases, respectively. For wave trains propagating at 9 = 0° , frictional effects upon waves running parallel to the smooth side wal l faces may be considered negligible. A s the waves run past the side wal l ends, they encounter rougher walls formed by the outermost sheets of the vertical , progressive porosity wave absorbers. In addit ion, the region of s t i l l water inside the absorbing structures results i n diffraction into the beach of a port ion of the waves which had previously been running entirely parallel to the beach face. Wave attenuation patterns are visible i n the plots of Figures 29 and 30, as seen by the wake shaped region ' A ' of Figure 29(a). The extent of attenuation for each of these cases was compared by non-dimensionalizing the w i d t h of these patterns w i t h respect to wavelength. In each case, an arc of length equal to the pertinent wavelength was drawn from the end of the side wal l to intersect the unit waveheight contour, which represents the border of the attenuation zone. Th i s is shown as line ' B C ' i n Figure 30(a). The perpendicular distance from this point to the side beach, as seen by line ' C D ' , was then measured. The ratio of this value to the wave length gives a non-dimensional measure of the intrusion of the attenuation effect into the wave field. Th i s ratio ranged from about 0.35 to 0.56, indicating that the extent of these attenuation zones is reasonably s imilar for the three wavelengths tested, when scaled according to wavelength. The relative effects of fr ict ion and of waveheight attenuation due to diffraction into the beach structures w i l l be assessed i n a later section by comparing experimental results 6 Results and Discussion 49 w i t h the predictions of a numerical model which takes no account of fr ict ional effects along the beach faces. 6.4.2 Plots For Oblique Wave Trains M a x i m u m waveheight plots from several oblique wave field tests are presented i n Figures 31 to 33. These are representative of Hsy results from the other tests involving oblique wave trains. Figure 31 compares tests 4, 5 and 6, corresponding to 8 = 30° and T = 1.25, 1.75 and 2.25sec, respectively. Figure 32 presents Hsy plots of tests 5, 8 and 11, which pertain to the medium wavelength (T = 1.75sec), at each of 8 = 30 ° , 45° and 60°. Figure 33 shows the Hsy result of test 12, corresponding to T = 2.25sec and 9 = 60° . O f the three oblique propagation directions tested, 30° is the direction most l ikely to be used i n practice, for the length of side walls used throughout the tests. Figure 31 exhibits the 8 = 30° results at al l three wavelengths under study. The short wavelength test 4 is the only case for which the data acquisition failed. Contours are absent from the region not covered by any probe points, and the probe array position is indicated by a dashed line. Neighbouring contours suggest trends by extrapolation back into the blank region. The unit waveheight contour may be thought to run approximately parallel to the 0.5 to 0.9 contours, at about 35-40° to the basin side. Th i s concurs w i t h the theoretical prediction that the unit waveheight contour should form an angle wi th the side about 8° greater than the propagation direction. The unit contour, by this assumption, intersects the side wal l near x = 4m. Despite inaccuracies, it is clearly apparent that this point of intersection lies well before the end of the side wal l . Thi s is evidence that the indirect or B-wave propagating toward the side wal l is subject to diffraction and attenuation by lateral energy transfer along the crest, such that it impacts the outer port ion of the side wall at a fraction of the desired waveheight H. 6 Results and Discussion 50 A t the other end of the generator, some energy is seen to intrude into the quiescent region. The measured waveheight approaches the desired magnitude well outside of the geometric quiescent zone boundary. The spur shape of contours near the side wall outer end suggests reflection of moderately high waves i n that region. Contours i n the central region of the basin indicate waveheights greater than the desired value H. Th i s may be due to a standing wave system i n the basin, or to energy propagating across the path of the direct wave field, having been reflected from the side wal l . The plot of Figure 31(b), for test 5 at T = 1.75sec, shows s imilar trends. The unit waveheight contour bordering the diffraction zone forms an angle of about 35° w i t h the basin side, and intersects the side wal l at about x — 3m. Compared to the less certain short wave results of test 4 i n Figure 31(a), it appears that the pr imary diffraction zone is larger for this longer wave, and that attenuation of the B-wave is also more pronounced. More energy enters the quiescent zone. The unit waveheight contour is thereby shifted closer to the geometric border of this zone. Conversely, more energy is reflected back into the basin from the quiescent end side wal l . The central region of the basin is characterized by contours which run approximately parallel to those bordering the diffraction zone. The pattern suggests alternating high and low regions of a part ia l standing wave system set up i n the basin perpendicular to the propagation direction. P lot 31(c) for the long wave case of test 6 is less informative. The unit waveheight contour intersects the basin side between x = 3m and x = 4m, at an angle of about 40° . The quiescent zone is less clearly apparent than for the other wavelength cases. In the central region of the basin, a pattern of alternating high and low sections is visible. These apparent node/antinode regions are in this case oriented w i t h their long axes perpendicular to, rather than parallel to the incident wave propagation direction. Long waves incident upon the beach structures are absorbed less effectively than at shorter wavelengths. Energy propagated i n the incident direction is part ial ly reflected from 6 Results and Discussion 51 the absorbers and the basin boundaries behind the absorbers, resulting i n a standing wave system aligned i n approximately the incident wave direction. Waveheights i n the central test area are generally greater than for the shorter wave cases, indicat ing increased reflection. In order to illustrate the effect of increased angle of propagation on the wave field, Figure 32 presents plots of the intermediate length wave field tests at the three oblique angles tested of 30 ° , 45° and 60°. The 30° plot of Figure 32(a) is the same case as presented i n Figure 31(b). The 45° result i n Figure 32(b) strongly resembles the 30° plot. The diffraction zone boundary of unit waveheight is incl ined at about 50° to the basin side, or 5° greater than the propagation direction. It intersects the side wal l between about x = 5 and x = 7m., Hence, at this increased angle, attenuation of the B-wave prior to reaching the side wal l is of decreased importance. The quiescent zone is about the same size as i n the 6 = 30° case of plot 32(a). The system of nodes and antinodes i n the central region is more distinct and extends further toward the wave generator than at 9 = 30° . Energy reflected from the quiescent zone side wal l may be the pr incipal cause of this effect. Some of the B-wave port ion which propagates beyond the side wal l end may be reflected first from the side beach, then from the end beach, so as to pass back through the basin central region. Waves reflected from the quiescent zone side wa l l appear to contribute to this effect in that sector of the test area. Nodes and antinodes appear to range from about 0.6 to 1.4 times the desired wave-height, suggesting that reflection on the order of 40% is occurring i n the basin. R ig -orous tests are yet to be conducted to appraise the performance of the vertical wave absorbers w i t h waves impinging obliquely upon the face of such a structure. Reflec-tions likely increase directly wi th propagation direction 0, due to reduced apparent porosities of absorber sheets compared to the case of waves s tr iking from a normal in -cident direction. The precise nature of any standing wave patterns due to resonance i n 6 Results and Discussion 52 this three-dimensional basin is further complicated by the complex reflection processes occurring w i t h i n the absorbing structures, and is beyond the scope of the present work. Looking to the corner reflection region of plot 32(b), a standing wave pattern of decreasing envelope height is apparent, in the y direction away from the side wal l . Evanescent waves from B-wave components produced by generator segments near the side wal l appear to be superposed w i t h waves reflected back from the far side wal l . These generator segments move wi th about twice the normal stroke to simulate the effect of v i r tua l segments, resulting in extreme waveheights near the corner. The probes under consideration are close to this corner, and likely register some of this effect before the wave field has developed to the desired condition. A nodal posit ion roughly coincides wi th y = Om. This could indicate a condit ion of open basin resonance, treating the region between the side walls as a two-dimensional basin. Considered so, waves coming from the corner reflection side wal l simulate open ocean waves entering the basin at that point. V i s u a l observation of the wave motion discloses a part ia l standing wave in the vicinity of the wavemaker, beyond the region of B-wave generation. A t increased propagation angles for the longer wavelengths, an envelope indicative of some beating is apparent by the wetted waterline along the generator face. The degree of beating diminishes away from the corner reflection region. For example, at 8 — 45 ° , diffraction into the quiescent region should not contribute much to the system of energy caught between the side walls. Some of this energy seemingly stems from spurious wave components from the sides of each generator segment, which propagate parallel to the generator face. Thi s suggestion is supported by observations that the beating envelope amplitude is greatest in the corner reflection region, where segment amplitudes are greatest, and falls markedly into the quiescent region where segments are motionless. Th i s beating carries through the quiescent region more strongly when 6 Results and Discussion 53 the propagation direction is increased to 6 = 60° . Figure 32(c) shows the results for the 60° case of test 11. The plot shows an extremely contaminated wave field, and a diminished area over which waveheights close to the desired value are found. The diffraction region extends over more than a t h i r d of the basin. Whereas the far side beach likely performs better than for the other angles of incidence, a greater degree of reflection between the side walls results at this severe angle of propagation. A strong standing wave pattern was observed between the generator face and the front line of wave probes at x = 4m. Three large regions of excessive waveheights are seen. The largest and highest of these is a region near the reflected-wave corner, due to the combined effects of the corner reflection process and the standing wave system. Reflections from the quiescent end side wal l back into the wave field account for the high waveheight region near that side. Another region of high waveheights is found i n the central port ion of the basin, as at the other propagation angles. The 60° case for the longest wavelength is shown i n Figure 33, and corresponds to test 12. Almost a l l wave energy is seen to be trapped between the side walls as a standing wave system. Thi s was investigated further by use of an equation for resonance between two solid walls, provided in the Shore Protection Manual (1984). The period of the mode of oscillation is given as where A is the basin w i d t h of 30m. For the eighth mode, this predicts a period of 2.27 seconds, which is very close to the 2.25sec period of the wave tra in . Th i s corresponds to a standing wave envelope wavelength of 7.5m, which is nearly a mult iple of the 2m probe spacing, and agrees well wi th the contours as plotted from the data points. The Hsy plot presented i n Figure 34 corresponds to test 15, involving the simple for n = 1,2 . . . (6.4) 6 Results and Discussion 54 snake method of generation without corner reflection, for the longest wave directed at 30° . T h i s plot relates to plot (c) of Figure 31, w i t h corner reflection. A larger diffraction zone is evident than when the corner reflection technique is employed. Near the downstream side wal l , waveheights are well above the target value, part icularly near the outer end of the side wal l . The importance of holding generator segments adjacent to the downstream side wal l motionless is clearly apparent, by comparison w i t h Figure 31(c). Further into the incident wave field, reflections result i n a region of standing wave activity. Th i s greatly reduces the effective test area i n the basin, which is seen to be l imi ted to a small zone near the basin centre and about six to ten metres i n front of the generator face. A plot which typifies those of the G B S series of tests is that of test 20, for T = 1.75sec, 9 = 30° wi th the cylinder i n place. Thi s is shown i n Figure 35(b). Compared to the wave field test 5 result of plot 35(a), the diffraction zone is seen to be similar i n shape and location. The entire region i n front of the cylinder is far more complicated by small regions of higher or lower max imum waveheight. The quiescent zone is greatly reduced i n size. The cylinder reflects energy back toward the side walls and the gen-erator face. Th i s is seen by a strong standing wave directly upstream of the cylinder; one probe clearly corresponds to a point at which 1.7 times the desired waveheight occurs. Another such point lies' abeam of the cylinder w i t h respect to the incident wave direction, and corresponds to a point of antinodal superposition of incident and perpendicularly propagating reflected waves. 6.5 Phase Plots From the Experimental Results A s discussed i n Section 5.2, phase plots were produced w i t h zero phase at the reference probe location. A contour interval of 360° is considered to depict consecutive wave 6 Results and Discussion 55 crests, w i t h the 'zero-phase' crest passing the reference probe location. A pattern of alternating crests and troughs corresponds to a 180° contour interval . The shorter the wavelength, the coarser the resolution of the data gr id w i t h respect to wavelength. For the short wavelength tests of T = 1.25sec, plots show reasonably straight crests/troughs. Figure 36 presents the 30° result for test 4, w i t h one data set missing in the corner reflection region of the basin. The 45° result of test 7 is found i n Figure 37. These results are of visual interest, but are inconclusive regarding the extent of wave crest curvature associated w i t h the diffraction process. Each plot hints at such curvature i n both the principal diffraction zone and the quiescent zone, especially the 45° case. The corresponding 45° plot for the medium wavelength (T = 1.75sec) result of test 8 is presented i n Figure 38. In this case, increased data resolution relative to wavelength effectively magnifies minor phase discrepancies between neighbouring probe locations. As a result, crest/trough contours appear less straight than for the short wave. Th i s plot does, however, appear to show the curvature effect more strongly. One may visualize the diffraction boundary at about 10 degrees greater than the propagation direction. The corresponding plot for the G B S series closely resembles this wave field plot, indicat ing that fundamental phase results are largely unaffected by the presence of the cylinder, and are repeatable. Thi s is the extent of usefulness of the phase data i n this form. The long wave plots are less informative, w i t h contours having greater tendency to meander. Smal l differences i n frequency result i n large differences in phase, so that phase plots are very sensitive to the frequency values calculated by the S C R E N program. These values are i n t u r n affected by the finite sampling rate (l/10sec) used. Some loss of information inevitably occurs, part icularly near wave elevation peaks. 6 Results and Discussion 56 6.6 Numerical Model Predictions of Wave Conditions 6.6.1 The Linear Diffraction Computer Program A computer program referred to as S E G E N is under continuing development at the Univers i ty of B r i t i s h Columbia . The program uses linear diffraction theory to predict the wave field produced i n a laboratory basin equipped w i t h a segmented generator, as described i n Section 2.3.2. Performance of the S E G E N program was evaluated by com-parison w i t h the experimental findings from the N R C wave basin. M o d e l predictions of waveheight, phase and direction are produced at prescribed points i n the testing area. The program treats circular, rectangular or arbitrary basin configurations. Generator segment faces and reflecting walls are represented by a dis tr ibution of discrete sources w i t h i n facets around the basin boundaries, using the boundary element method. Users specify st i l l water depth, desired waveheight, period and propagation di-rection. Generator segments are modelled by fixing velocities of adjacent fluid particles to appropriate values calculated at the centre of each facet of the boundary. Reflective walls and structures are treated as perfectly reflecting surfaces, by f ixing fluid velocities normal to such surfaces to zero. For the purposes of this study, the state of program development was such that absorbing beaches were not modelled. Absence of boundary definition i n beach regions is equivalent to specifying fully absorbent beaches. The number and locations of points i n the test area at which the wave potential is to be evaluated are specified by the user. For each model r u n , predictions were made at points corresponding to the same full basin, two-metre gr id as i n the physical exper-iments. Thi s ensured full compatability between plots of the physical and numerical model results. The rectangular basin configuration corresponds to a ful l bank of active generator segments along one boundary, flanked by part ia l side walls of equal length adjacent 6 Results and Discussion 57 to each end of the wavemaker. Th i s configuration was suitable for the 8 = 0° cases, and for oblique wave generation by the simple snake principle. A sample layout of the rectangular configuration model is presented i n Figure 39. Input files include basin dimensions, the length of the side walls, and the number of sources and facet w i d t h for the generator bank and for the side walls. One source is located at the centre of each facet. Further modification of the S E G E N program is required to include generator motions which produce B-wave components at one end, and a sector of quiescent generator segments at the other end. Such modification was not warranted for the purposes of the present study. The arbitrary basin configuration option was employed i n order to model wave generation w i t h corner reflection and quiescent sectors, and w i t h a reflecting structure i n place. Recal l ing that the corner reflection technique is based on the concept of v i r tua l segments extending beyond the actual generator end, this concept was ut i l ized by removing the reflection-end side wal l , and adding the appropriate number of generator segments i n the negative y direction, according to Equat ion 2.21. A t the quiescent end of the generator, a reflecting wall prescribed along the y axis represents the sector of inactive generator segments. Use of the arbitrary basin configuration is i l lustrated i n Figure 40. Input files include the number and location of generator and side wall segments, and of test area data points, as well as s t i l l water depth, desired wave height, period and propagation direction. A l l models used a facet length of 0.5m, except for a single case i n which this length was halved to 0.25m, to provide two sources per generator segment. 6 Results and Discussion 58 6.6.2 Waveheight Plots of Numerical Model Predictions 6.6.2.1 Normally Propagating Wave Trains Numerica l runs at 6 = 0° made use of the rectangular basin configuration, except when the cyl indrical structure was included. Exper imenta l and numerical m a x i m u m wave-height results for the three wave field cases are compared in Figures 41, 42 and 43, corresponding to tests 1, 2 and 3 respectively. A smaller contour interval was used for the numerical results, i n order to depict trends i n sufficient detail for comparison w i t h experimentally derived plots. For each of the three wavelengths considered, the pairs of plots show similar features throughout the wave field. Plots of full-basin nu-merical predictions display symmetry about the basin centreline, as expected, and aid visualization of the half-basin experimental plots over the entire basin. The numerical models predict effects to be less pronounced than were actually measured. Note that model predictions are linear i n nature, and so are analogous to experimental A\ fundamental wave magnitudes. However, these linear predictions are i n fact compared w i t h experimental Hsy findings. Predictions agree well w i t h measurements i n light of this, and considering the absence of reflective boundaries where the beaches were positioned. Also , in reality harmonic and other nonlinear effects of lesser significance are present. The zones of waveheight attenuation along the side beaches are predicted well by these models which impose no boundary conditions other than at the generator and fully reflecting surfaces. Such a model is equivalent to having a quiescent zone of water beyond the side wal l end. The attenuation effect observed is caused by linear diffraction of wave energy into this zone. The degree of intrusion of the attenuation zone into the wave field, as evaluated by the location of the unit waveheight contour, agrees very closely w i t h experimental findings. Inspection of contour values w i t h i n these 6 Results and Discussion 59 zones reveals that the numerical models predict substantially less attenuation than was actually measured. Th i s is clearly evident by noting the 0.9 waveheight contour i n each of the plots of Figure 42. These differences between numerical and measured results may be attr ibuted to fr ict ion imparted by the rough wal l formed by the outermost surface of the wave absorbing structure. 6.6.2.2 Obliquely Propagating Wave Trains The case of the short wave propagating at 30° (test 4, T = 1.25sec) is considered in Figure 44. The arbitrary basin configuration was used. The generator segment to wavelength ratio, B/L = 0.205. Hence, this wavelength perhaps warrants mult iple facet representation of generator segments. One run of the model was made w i t h half-metre facets, such that M = 1, and a second run corresponds to M — 2. The facet size set for generator segments was also used to model the side walls. Hence, the 9.2m side walls were modelled by walls 9.0m and 9.25m long, for facet lengths of 0.5m ( M = 1), and 0.25m ( M = 2), respectively. As seen i n Figure 44(c), doubling the number of facets minimal ly affected the result, and was not warranted i n this case. B o t h numerical predictions compare well w i t h the experimentally derived plot. For example, note the spur shape of contours near the outer end of the quiescent zone side wal l . Numerica l waveheight estimates throughout the test area are seen to be about 10% lower than measured findings. Th i s is attr ibuted to neglect of nonlinear effects, and to absence of a reflecting wal l along the upstream side of the basin. Numerica l model predictions of diffraction and quiescent zone bound-aries intrude less into the test area than measurements indicate. The numerical models accurately predict a region of large waveheights adjacent to the diffraction zone bound-ary, as well as a small low waveheight region downstream of the basin centreline. A c t u a l locations of these regions are closer to the downstream side of the basin than the model 6 Results and Discussion 60 predicts. Th i s is l ikely due to absence of an upstream side wa l l in the model. The fact that they are modelled well wi th only one side wal l indicates that such waveheight variat ion may be ascribed to reflection from the downstream side wal l , rather than to a resonant effect w i t h i n the basin. Figure 45 presents experimental and numerical results for the case of the interme-diate wavelength at 0 = 30° (test 5, T = 1.75sec). The arbitrary basin configuration was used as before. Th i s case provides perhaps the most s tr iking example of model performance. Very good agreement is noted between experimental and numerical find-ings, for both m a x i m u m waveheight magnitudes and spatial predictions. A noteworthy exception is that the strong evanescent wave component i n the corner reflection region is absent from the numerical prediction, so that the unit waveheight contour is not deflected away from the plot corner as was found experimentally. Results for T = 2.25sec and 0 = 30° are presented i n Figure 46. W h i l e the nu-merically derived plot exhibits less pronounced effects than the experimental result, contour shapes are modelled fairly well on a regional basis. Shallower contour gra-dients of the numerical result indicate less severe variation of m a x i m u m waveheights around the basin than is actually the case. Thi s is particularly true i n the port ion of the basin situated between the side walls, in consequence of the absence of a corner reflection side wal l i n the model. The model concurs that the diffraction and quiescent zones each intrude far into the test area. The long wave test 15 using the simple snake principle of generation for T = 2.25sec and 0 = 30° was modelled w i t h the rectangular basin configuration. Results are com-pared i n Figure 47. Us ing a facet length of 0.5m, nine metre walls were modelled along each side, and the wave generator motion mimicked reality. As a result, model pre-dictions match the experimental result to w i t h i n 10% throughout the basin. Th i s case 6 Results and Discussion 61 shows that a linear diffraction model is well suited to predicting predominant char-acteristics of a wave field w i t h i n a basin, provided that a l l boundaries are accurately modelled. Investigating next the 8 — 45° case, the results of test 8 and corresponding S E G E N predictions for the intermediate wavelength ( T = 1.75sec) are presented i n Figure 48. B-wave generation was again modelled by use of v i r tua l segments. G o o d comparison between experiment and model is observed, as seen i n plots 48(a) and (b), respectively. The standing wave pattern apparent in the corner reflection region of the experimental plot was not predicted by the numerical model. It is expected that i f the combined principal- and B-wave overstroke motions of generator segments at this end of the wavemaker were accurately modelled, the large waveheights near this region would be correctly predicted. The result of an addit ional numerical run for the test 8 case is presented i n Figure 48(c). T h i s corresponds to a model w i t h both side walls i n place, and no v i r tua l generator segments, while the quiescent end segments are s t i l l modelled. A n inferior prediction results, indicat ing that for this wave condition the absence of corner reflection outweighs the effect of resonance between two side walls. Th i s is part icularly evident by the very large diffraction region found in the plot. The last wave field case considered is that of the long wave propagating at 60° , and is found i n Figure 49. Th i s corresponds to test 12, w i t h T = 2.25sec. Numerica l models were run as i n the previous cases presented in Figure 48. P lo t 49(b) corresponds to the model configured to include v i r tua l generator segments to represent the B-wave generation. Comparison w i t h the experimental result of Figure 49(a) shows that this model inadequately predicts wave conditions measured throughout the basin. O n the other hand, the model w i t h no v i r tua l segments, but w i t h both side walls i n place, is seen i n plot 49(c) to predict conditions very well. The system of standing waves formed 6 Results and Discussion 62 between the side walls indicates waveheights throughout the basin only about 10% lower than were measured, evidence that this phenomenon is of predominant importance at this extreme propagation angle. It is expected that a more precise model of generator motions w i t h the reflecting side wal l i n place would more closely predict the correct standing wave pattern. 6.6.2.3 Results W i t h the Cyl indrica l Structure in Place Tests were also run to evaluate the performance of linear diffraction numerical models w i t h the cyl indrical structure i n place. The cylinder was formed of 38 facets, each about 0.13m i n length, or four times smaller than the 0.5m facet length used to model basin boundaries. The 0 = 0° plots corresponding to tests 16, 17 and 18 are presented i n Figures 50, 51 and 52, respectively. Wave conditions are seen to be modelled generally to w i t h i n 10 or 20% of measured waveheights. These models employ the rectangular basin configu-rat ion, and closely simulate generator motions and side walls. Hence, discrepancies i n the upstream region between the side walls, into which energy is reflected by the struc-ture, are l ikely due to inabi l i ty of these linear models to predict nonlinear components associated w i t h the reflection process. Downstream of the cylinder, lower numeri-cal predictions of waveheights may be ascribed to absence of a downsteam reflecting boundary. The result of another model which closely represents upstream boundary condi-tions is presented i n Figure 53. Thi s corresponds to test 25, involving the simple snake method of generation for T = 2.25sec and 6 = 30° . A g a i n , excellent spatial prediction of various effects throughout the basin is observed, and m a x i m u m waveheight levels are about 10% lower than measured. Improved agreement between model and mea-surements is apparent for this oblique wave case, compared to the 9 = 0° cases. Less 6 Results and Discussion 63 reflected energy becomes caught between the side walls than for normally propagating waves. Th i s suggests that nonlinear effects associated w i t h the reflection process, par-t icular ly the free second harmonic component generated, are of greater significance to accuracy of G B S test predictions than reflections from beaches downstream of the side walls. A l l oblique wave tests measured w i t h the G B S i n place were modelled numerically. The three cases matching tests 19, 20 and 21 at 8 = 30° are presented i n Figures 54, 55 and 56, respectively. Results corresponding to tests 22, 23 and 24 at 8 — 45° are found i n Figures 57, 58 and 59, respectively. B-wave generation was modelled using the arbitrary basin configuration w i t h v i r tua l segments, as for comparable wave field cases. As before, large waveheights measured near overstroke segments due to B-wave generation were not predicted by these models. Locations of most noteworthy features were generally well predicted. Throughout these plots, model waveheight predictions often agree to w i t h i n 10% of measurements. These results, w i t h only one side wal l modelled, suggest that at oblique angles reflections between the structure and the upstream side wall are less important than harmonic effects due to reflection and fundamental wave generation. Less energy is trapped between the side walls than for normally propagating wave trains, w i t h the structure i n place. 6.6.3 Numerical Model Phase Predictions Three phase plots from numerical models corresponding to tests 2, 4 and 8 are pre-sented. A t 8 = 0 ° , the intermediate wavelength case of test 2 is seen i n Figure 60. Th i s plot clearly indicates the reliabil ity of phase predictions around the basin, by the straightness of crest and trough contours. Contours also show that negligible wave crest curvature is associated w i t h the attenuation process along the side beach faces. Apparently, such curvature is not significant w i t h i n the path of direct wave propagation. 6 Results and Discussion 64 The phase plot for test 4, T = 1.25sec and 9 = 30° , is found i n Figure 61. Slight bending of crest/trough contours is apparent i n both the diffraction and quiescent zones. Th i s tendency is slightly more pronounced than i n the comparable experimentally de-rived plot of Figure 36. The numerically derived phase plot for the test 8 condit ion of T = 1.75sec and 45° is seen in Figure 62 to have less straight contours than for the shorter wavelength case of Figure 61. Thi s is s imilar to the corresponding experimen-ta l plot of Figure 38, and is l ikely associated w i t h the data resolution as discussed i n Section 6.5. 6.6.4 Numerical Model Surface Elevation Plots The S E G E N program also generates instantaneous surface elevation predictions at the prescribed data point locations. The case corresponding to test 5 (T = 1.75sec, 9 = 30° ) was plotted using a three-dimensional surface plott ing program, and is presented i n Figure 63. Th i s program uses the same gridded data as for the waveheight and phase contour plots, and produces perspective block diagrams for surface representation. The height-to-width ratio used for the plot was 0.1. Var ia t ion of waveheight along any crest is readily visible. W h i l e some of this effect may be associated w i t h the plott ing technique, such variation was visible on the water surface during testing. A cross-mode present i n the basin, due to reflections from the quiescent end side wal l and elsewhere, accounts for this corruption of the incident wave tra in . In order to produce similar plots from experimental results, values could be extracted from measured elevation time series at a part icular reference time. A l l time series would need first to be phase related, as was done to produce phase contour plots. Comparison between experimentally and numerically derived surface plots should show that the linear models underestimate somewhat the level of cross-mode interference. The numerically generated plot of Figure 63 is representative of the tests conducted, and depicts such effects as the exaggerated 6 Results and Discussion 65 waveheight i n the corner reflection region, the principal diffraction zone, and to a lesser extent, wave attenuation into the quiescent end diffraction zone. 6.7 Further Considerations T h e time series plots presented in Figure 21 typify results from most tests. O n l y the normally propagating wave field tests are characterized by more strongly sinusoidal records. The examples of Figure 21 are for an array position of oblique test 5 at T = 1.75sec and 8 = 30° . A t most probe locations, fairly strong second harmonic components were present, as seen by regular kinks between major peaks. A beat is apparent i n the envelope of many plots. Thi s is attr ibuted to free harmonics caused by linear generation and reflections. The case of probe 11, i n Figure 21(e), is an example of a part icularly strong second harmonic, relative to a weak fundamental component of only about 0.02m height. In this instance, the S C R E N program would choose the second harmonic as the funda-mental component, using a three parameter fit by frequency as well as amplitude and phase. The low waveheight of posit ion 11 is quite pronounced relative to neighbouring points. W i t h reference to Figure 31(b), this probe location seems to correspond to a nodal region of a standing wave system oriented perpendicular to the incident wave direction. Some tests w i t h the G B S present exhibit greater beating of the waveheight en-velope than most of the wave field results. Thi s is attr ibuted to increased levels of free harmonic energy due to reflection. In addit ion to beating, some records show a slow oscillation caused by comparatively long wave components. No part icularly good example of this is found i n Figure 21. The residual t ime series traces for some tests indicate a strong fourth harmonic component. W h i l e extensive study of these more 6 Results and Discussion 66 subtle effects is beyond the scope of the present study, it is worth looking briefly at a few signals i n greater detail . Figure 64 presents line spectra relating to some probe measurements from test 10, T = 1.25sec and 6 = 60° . These spectra derive from S C R E N analyses involving three-parameter fits. The program was run to select frequency as well as amplitude and phase, for the largest ten components. The results were plotted as line spectra of amplitude versus frequency. Figure 64(a) locates these probes i n the basin. Th i s is an oblique test at 6 = 60 ° , so that some of the generated energy does not reach certain probes. Measured elevation time series are presented w i t h the line spectra. A l so given are amplitude, frequency and phase values produced by the S C R E N program. The results for probe number 1 are presented i n Figure 64(b). Some energy is seen to have shifted away from the forcing frequency of 0.8Hz. The resultant beating of the waveheight envelope is quite pronounced i n the accompanying time series plot. The line spectrum shows several components of about 0.01m amplitude at or slightly below twice the forcing frequency. The time series trace does not show a second harmonic kink, as is found i n most of the Figure 21 plots. The only clear evidence of the presence of harmonic energy is the distinctive beating pattern. Th i s case is noteworthy for a long wave component of appreciable magnitude, w i t h a period of about 46 seconds. Thi s is evidenced by vertical shifts of the beat envelope, as seen i n the wave elevation trace. Results for probe 13 are presented i n Figure 64(c). The line spectrum amplitude scale is half that of Figure 64(b). The fundamental component is less than half the magnitude of that of the probe 1 trace, while the chief second harmonic component is twice as large, and hence of greater relative importance. Also , more energy has shifted away from pr incipal frequencies i n both the fundamental and second harmonic ranges. These two effects result i n a visible second harmonic kink i n the t ime series trace, and 6 Results and Discussion 67 a more pronounced and distinctive beating pattern than is seen i n Figure 64(b). Wave elevations at probe 7, located further into the diffraction zone, are seen i n plot 64(d) to be of s t i l l smaller magnitude. Findings are similar to those at probe 13 i n plot 64(c). The relative size of second order spikes compared to the fundamental magnitude is intermediate between those at probes 13 and 1. The absence of a kink i n the probe 1 time series trace is attr ibuted to less frequency shifting of second harmonic energy than present at probes 13 and 7. The largest second order spike for probe 1 is at a frequency almost exactly double the fundamental, forcing frequency. Figure 64(e) provides an example of diminished wave activity, and a comparatively confused time series trace, corresponding to probe 12. Thi s probe is located well into the diffraction zone, and close to the end beach. Thi s provides an example of a performance characteristic of the type of wave absorber used i n the basin. Observed at the start of a test, a high frequency ripple is notable on the water surface. The first waves progressing toward the beach are smooth i n appearance. As waves impact the outermost absorber sheets, a ripple is generated by runup against this barrier. Th i s ripple works its way back into the wave field, roughening the surface as it progresses toward the wavemaker. The line spectrum of plot 64(e) shows significant energy shifted well above the second harmonic value (1.6Hz). Note that the fundamental spike is an order of magnitude smaller than i n previous plots. Apparently, the runup process results in generation of small waves at a frequency about five percent greater than the second harmonic value. Increased understanding of the frequency shifts occurring due to resonant phenom-ena, beach runup, internal beach and other processes must await more detailed, specific investigation. Certainly, the wave field present in a basin subject to regular wave gen-eration cannot on close inspection be considered t ru ly sinusoidal. Under most wave conditions, and i n central regions of the basin, the contaminating effects noted are re-stricted to less than ten percent of the magnitude of the fundamental wave. However, 6 Results and Discussion 68 there are several instances and locations where this is not the case. Adequate predic-t ion and correction of these effects is crucial to improving wave conditions w i t h i n the test area of such a basin. 7 Conclusions and Recommendations 7.1 Conclusions Elevation time series of long-crested regular wave trains were measured at discrete locations throughout the test area of an offshore wave basin equipped w i t h a segmented wave generator. A modified form of the snake principle of generation was used to determine generator segment motions for a desired waveheight, period and propagation direction. Three wave lengths were investigated, at three oblique angles as well as propagating normal to the generator face. A shorter series of similar tests investigated the effect of a large surface piercing cyl indrical structure in the test area. Plots of m a x i m u m waveheight and phase contours were produced from measured data. Similar plots were obtained from the predictions of a linear diffraction numerical model , which uses a point source representation of basin boundaries. These results were compared w i t h the experimental findings to assess the performance of this computer model. Normal ized m a x i m u m waveheight results were found to be independent of pre-scribed waveheight by repeating one test series at a reduced prescribed waveheight. One waveheight value was prescribed for al l other tests. Comparison of phase plots in -dicated that the reduced-height wavelength was shorter than the full-height value. Th i s is at tr ibuted to nonlinear effects inherent to the wave characteristics and generation. A test conducted twice demonstrated excellent repeatability of results. It was found that the effects of wave diffraction and attenuation, and wave energy transmission through the beach structures, increase w i t h wavelength. Th i s results i n higher levels of undesired wave energy throughout most of the test area at longer 69 7 Conclusions and Recommendations 70 wavelengths, and smaller regions of acceptable wave conditions. Energy diffracted into the quiescent zone reflects back into the incident wave field. Generated indirect waves impact the outer port ion of the reflection side wal l at reduced levels. For most oblique wave cases, a part ia l standing wave system develops i n the basin. Reflection of energy from the downstream side wal l is the predominant cause, such that the standing wave is aligned perpendicular to the incident wave direction. For the longest wavelength tested, at 6 = 30° , the standing wave system sets up i n line w i t h the propagation direction. Thi s comparatively long wave penetrates the downstream beaches more effectively than shorter waves. Reflections from the basin boundaries outweigh those from the downstream side wal l , in this case. For the longest wavelength at 6 = 60° , almost a l l energy was trapped between the side walls. The period of these waves is close to a resonant frequency, treating the problem as one of two-dimensional basin resonance between the side walls. Observation of the wetted surface on the generator face suggests that considerable spurious energy is generated from the sides of generator segments, even for the shortest wavelength at 6 = 30° . Paddle stroke varies directly w i t h wavelength; in turn , a segment of finite w i d t h w i l l generate a stronger ring-shaped wave as stroke increases. Some of this energy propagates almost parallel to the generator face. Th i s results i n a standing wave system which diminishes in strength away from the combined-wave generator sector. Diffraction of this spurious energy results in a lower envelope height w i t h i n the quiescent zone adjacent to motionless generator segments. A standing wave due to reflections between the side walls would persist more strongly across the basin wid th . Generation of oblique waves by the simple snake method results i n prohibit ively high levels of reflected energy throughout the basin, part icularly w i t h the side wal l length used. Presence of a structure in the test area results in increased levels of reflected 7 Conclusions and Recommendations 71 energy between the side walls, but does not affect fundamental wave phase results, compared to wave field findings. Second order components were consistently an order of magnitude smaller than fundamental wave contributions. Free harmonic energy due to reflections and linear wave generation account for beating of the waveheight envelope i n the test area. Experimental ly derived phase plots d id not aid evaluation of wave crest curvature due to diffraction. Comparison of experimental results wi th predictions provided by the linear diffrac-t ion computer program indicated that the numerical model waveheight predictions were generally about 10% lower than measured. W i t h i n this tolerance, waveheight and spatial predictions were found to be in excellent agreement w i t h measurements. Discrepancies are attr ibuted to second order and other nonlinear effects not accommo-dated by the program. Waveheight gradients i n plots from numerical results are less steep than those i n experimentally derived plots. Us ing numerical models wi th no boundary definition along beach faces, tests at 9 = 0° resulted i n wave attenuation zones adjacent to side beaches of about the same size as measurements indicated. However, the extent of waveheight attenuation w i t h i n these zones was markedly underestimated by the models, which d id not account for frictional effects along the rough absorber outer sheets. For oblique wave tests, the corner reflection technique was modelled using v i r tua l segments without an upstream side wal l . Such a configuration usually predicted well the effects measured throughout the basin, except that the high in i t i a l waveheights present i n the corner reflection region were underestimated, having not modelled the overstroke segments at that end of the generator. However, i n cases where basin res-onance was a predominant effect, the upstream side wal l was required for accurate prediction of a standing wave system close to the generator. For cases involving the simple snake method of generation, segment motions and both side walls were well 7 Conclusions and Recommendations 72 modelled. These results displayed excellent agreement w i t h measured findings, and illustrate the importance of accurate modell ing of a l l basin boundaries. W i t h a structure present i n the test area, numerical predictions resemble experi-mental plots to w i t h i n 20%. Models were incapable of predicting the reflected second order energy present between the side walls. Locations of standing waves near the cyl in-der were well modelled. Absence of boundaries downstream of the cylinder resulted i n waveheights lower than measured. Findings indicate that continued development of the linear diffraction program is warranted, to include part ia l ly reflecting beach boundaries and accurate represention of complex generator motions, w i t h both side walls in place. Excellent agreement between model predictions and experimental results advocates developing another pro-gram which involves the linear diffraction technique. Thi s program would prescribe modifications to generator segment motions, so as to compensate for diffraction and re-flection effects w i t h i n a laboratory basin. Wave contributions due to nonlinear phenom-ena are an order of magnitude smaller than fundamental, linear effects. Development of a linear program w i l l substantially improve the quality of wave fields generated. Th i s could then be extended by use of economical second order numerical approximations. 7.2 Recommendations Further investigation w i l l enhance understanding of the phenomena occurring in a directional wave basin. Experimental tests should be conducted on the type of upright wave absorber used i n the basin, in order to assess performance at oblique angles of incidence. Measurements inside the beach structures should yie ld some understanding of frequency effects upon residual wave reflections. Raw and analysed data pertaining to the present study warrant closer scrutiny and more extensive analysis. 7 Conclusions and Recommendations 73 Rigorous comparison between experimental results and predictions from numerical models incorporating improved boundary definition w i l l help to unravel further the intricacies of the effects which contaminate the wave field generated i n a laboratory basin. Residual t ime series produced by the S C R E N program likely contain several clues to nonlinear effects present at discrete locations. Detai led evaluation of line spectra would divulge much of the story. The trend toward more sophisticated design aspects w i l l be well served by such work. The biggest single improvement w i l l surely come from linear generator motion corrections afforded by the diffraction technique under continuing development. 8 References Biesel, F . (1954) "Wave Machines" , Proceedings of the 1 st Conference on Ships and Waves, Hoboken, N . J . , pp. 288-304. Dean, R . G . & Dalrymple , R . A . (1984) Water Wave Mechanics for Engineers and Scien-tists, Prentice-Hal l , Inc., Englewood Cliffs, N . J . Funke, E . R . , Crookshank, N . L . & W i n g h a m , M . (1980) An Introduction to GEDAP: An Integrated Software System For Experimental Control, Data Acquisition and Data Anal-ysis, Nat iona l Research Counc i l of Canada, Hydraul ics Laboratory Technical Report , L T R - H Y - 7 5 , M a r c h . Funke, E . R . & Mi les , M . D . (1987) Multi-Directional Wave Generation With Corner Re-flectors, Nat iona l Research Counci l of Canada, Hydraul ics Laboratory Technical Re-port, T R - H Y - 0 2 1 , July. Gi lber t , G . (1976) Generation of Oblique Waves, Hydraul ics Research Station, Wal l ing-ford, G . B . , Notes 18, pp. 3-4. Havelock, T . H . (1929) "Forced Surface Waves on W a t e r " , Philosophical Magazine, No . 7, V o l . 8. International Associat ion for Hydraul ic Research (1986) List of Sea State Parameters, Supplement to Bu l l e t in No. 52, Permanent International Association of Navigat ion Congresses, Brussels, Jan . Isaacson, M . (1984) Theoretical Analysis of Directional Sea State Synthesis, a report to the Nat ional Research Counc i l of Canada, by Seaconsult Mar ine Research, L t d . , Vancouver, Dec. Isaacson, M . (1985) "Laboratory Measurement and Generation of Direct ional Surface Waves" , Proceedings of the 21 st Congress of the International Association for Hydraulic Research, Melbourne, V o l . 5, pp. 116-121. 74 7 References 75 Jamieson, W . W . & Mansard , E . P . D . (1987) " A n Efficient Upright Wave Absorber" , Proceedings of the ASCE Specialty Conference on Coastal Hydrodynamics, Newark, Delaware. Mansard , E . P . D . & Funke, E . R . (1980) "The Measurement of Incident and Reflected Spec-t ra Us ing a Least Squares M e t h o d " , Proceedings of the 17 th International Conference on Coastal Engineering, Sydney, Austra l ia . Naeser, H . (1979) "Generat ion of Uni form Direct ional Spectra i n a Wave Bas in Us ing the Natura l Diffraction of Waves" , Proceedings of the Conference on Port and Ocean Engineering Under Arctic Conditions (POAC), V o l . 1, pp. 621-632. Naeser, H . (1981) " N H L - Ocean Bas in Capabilities and L imi ta t ions " , Proceedings of the International Symposium on Hydrodynamics in Ocean Engineering, Trondheim, pp. 1191-1210. Sand, S .E. (1979) Three-Dimensional Structure of Ocean Waves, Institute of Hydrody-namics and Hydraul ic Engineering, Technical University of Denmark, thesis and series paper no. 24. Sand, S .E. & Mansard , E . P . D . (1986) Description and Reproduction of Higher Harmonic Waves, Nat ional Research Counc i l of Canada, Hydraulics Laboratory Technical Report , T R - H Y - 0 1 2 , Jan . Sand, S .E. & M y n e t t , A . E . (1987) "Direct ional Wave Generation and Ana lys i s " , Pro-ceedings of the 22nd Congress of the International Association for Hydraulic Research, Lausanne, Switzerland, pp. 209-235. Shore Protection Manual (1984), U.S . A r m y Corps of Engineers, Coastal Engineering Research Center, 4 * n ed., V o l . 1. Ursel l , F . , Dean, R . G . & Y u , Y . S . (1960) "Forced Smal l -Ampl i tude Water Waves: A Comparison of Theory and Exper iment" , Journal of Fluid Mechanics, V o l . 7, Part 1, Jan. Tables test T 9 L H /i H L B L T> L frac comments # [sec] [deg] [m] [m] 1 1.25 0 2.44 .2 .82 .082 .205 .607 1/2 (SZ) 2 1.75 0 4.74 .2 .42 .042 .106 .313 1/2 ( M Z ) 3 2.25 0 7.40 .2 .27 .027 .068 .200 1/2 ( L Z ) 4 1.25 30 2.44 .2 .82 .082 .205 .607 1 (S30) one test faulty 5 1.75 30 4.74 .2 .42 .042 .106 .313 1 (M30) 6 2.25 30 7.40 .2 .27 .027 .068 .200 1 (L30) 7 1.25 45 2.44 .2 .82 .082 .205 .607 1 (S45) 8 1.75 45 4.74 .2 .42 .042 .106 .313 1 (M45) 9 2.25 45 7.40 .2 .27 .027 .068 .200 1 ' ( L 4 5 ) 10 1.25 60 2.44 .2 .82 .082 .205 .607 1 (S60) 11 1.75 60 4.74 .2 .42 .042 .106 .313 1 (M60) 12 2.25 60 7.40 .2 .27 .027 .068 .200 1 (L60) 13 1.25 30 2.44 .1 .82 .041 .205 .607 1/2 (S30) half waveheight 14 1.25 30 2.44 .2 .82 .082 .205 .607 1/8 (S30) repeatabil i ty 15 2.25 30 7.40 .2 .27 .027 .068 .200 1 ( L 3 0 N O ) simple snake 16 1.25 0 2.44 .2 .82 .082 .205 .607 1/2 ( C S Z ) w i t h cylinder 17 1.75 0 4.74 .2 .42 .042 .106 .313 1/2 ( C M Z ) w i t h cylinder 18 2.25 0 7.40 .2 .27 .027 .068 .200 1/2 ( C L Z ) w i t h cylinder 19 1.25 30 2.44 .2 .82 .082 .205 .607 1 (CS30) .wi th cylinder 20 1.75 30 4.74 .2 .42 .042 .106 .313 1 ( C M 3 0 ) w i t h cylinder 21 2.25 30 7.40 .2 .27 .027 .068 .200 1 (CL30) w i t h cylinder 22 1.25 45 2.44 .2 .82 .082 .205 .607 1 (CS45) w i t h cylinder 23 1.75 45 4.74 .2 .42 .042 .106 .313 1 ( C M 4 5 ) w i t h cylinder 24 2.25 45 7.40 .2 .27 .027 .068 .200 1 (CL45) w i t h cylinder 25 2.25 30 7.40 .2 .27 .027 .068 .200 1 ( C L 3 0 N O ) snake/cyl . T A B L E 1 Tests Conducted and Pert inent Parameters ('frac' is the fract ion of the basin measured — 8 tests comprise a fu l l test series; abbreviated test names given i n brackets: S, M , L refer to Short , M e d i u m , L o n g wavelengths, respectively; Z =>• 9 = 0 ° ; C =>• cyl inder i n place; N O signifies no corner reflection, simple snake method used for a l l 60 generator segments) 76 test T 9 H sr s, m g comments # [sec] [deg] [m] [m] H 1 1.25 0 .2 n / a n / a n / a n / a (SZ) 2 1.75 0 .2 n / a n / a n / a n / a ( M Z ) 3 2.25 0 .2 n / a n / a n / a n / a ( L Z ) 4 1.25 30 .2 9.3 12 1-11 47-60 (S30) one test faulty 5 1.75 30 .2 9.3 12 1-11 47-60 (M30) 6 2.25 30 .2 9.3 12 1-11 47-60 (L30) 7 1.25 45 .2 9.3 10 1-19 41-60 (S45) 8 1.75 45 .2 9.3 10 1-19 41-60 (M45) 9 2.25 45 .2 9.3 10 1-19 41-60 (L45) 10 1.25 60 .2 9.3 9 1-29 33-60 (S60) 11 1.75 60 .2 9.3 9 1-29 33-60 (M60) 12 2.25 60 .2 3 12 1-10 19-60 (L60) 13 1.25 30 .1 9.3 12 1-11 47-60 (S30) half waveheight 14 1.25 30 .2 9.3 12 1-11 47-60 (S30) repeatabi l i ty 15 2.25 30 .2 0 0 0 0 ( L 3 0 N O ) simple snake 16 1.25 0 .2 n / a n / a n / a n / a ( C S Z ) w i t h cyl inder 17 1.75 0 .2 n / a n / a n / a n / a ( C M Z ) w i t h cyl inder 18 2.25 0 .2 n / a n / a n / a n / a ( C L Z ) w i t h cyl inder 19 1.25 30 .2 9.3 12 1-11 47-60 (CS30) w i t h cyl inder 20 1.75 30 .2 9.3 12 1-11 47-60 ( C M 3 0 ) w i t h cyl inder 21 2.25 30 .2 9.3 12 1-11 47-60 (CL30) w i t h cyl inder 22 1.25 45 .2 9.3 10 1-19 41-60 (CS45) w i t h cyl inder 23 1.75 45 .2 9.3 10 1-19 41-60 ( C M 4 5 ) w i t h cyl inder 24 2.25 45 .2 9.3 10 1-19 41-60j (CL45) w i t h cyl inder 25 2.25 30 .2 0 0 0 0 ( C L 3 0 N O ) snake/cyl . T A B L E 2 Side W a l l Lengths Specified For Drive Signal Synthesis, and Resultant Generator Mot ions (S is the specified side wall length, subscripts r and q refer to corner reflection and quiescent zone ends of the generator, respectively; TO.A/B signifies the generator segments involved i n bo th A - and B-wave generation; m g pertains to the quiescent segments; segments between these two sets move according to the simple snake principle ; segment 1 is at the corner reflection end of the generator) Figures F I G U R E 1 Elevation V i e w of the Wavemaker Configuration and Coordinate System generator segment n = i . , propagation direction F I G U R E 2 Def ini t ion of Coordinate System, P l a n V i e w 78 S E G M E N T E D WAVE G E N E R A T O R • DIFFRACTION Z O N E DIFFRACTION ZONE OF REFLECTED WAVES ZONE OF INTERFERENCE WITH UNDESIRABLE REFLECTIONS WAVE ABSOROERS F I G U R E 3 Obl ique Wave W i t h Interference Wave , Convent ional M e t h o d of Obl ique Wave Generat ion (after Funke and Mi le s , 1987) F I G U R E 4 Superposi t ion of Direct and Reflected Indirect Waves (after Funke and Mi le s , 1987) F I G U R E 5 View of the N R C Offshore Wave Basin, Showing Wave Generator Segments Raised 0.48m Above Floor. Also Note 9.2m Long Side Wall . F I G U R E 6 Wave Probe Array Rig in Calibration Position. The 4x4m Frame Supports the 6x6m Superstructure. Note the Side and End Wave Absorbers In the Background. View From Wave Generator. F I G U R E 7 Bas in Layout , W i t h the Four Probe A r r a y Positions Indicated a t e s t p o s i t i o n o f t h e 16 p r o b e a r r a y f r a m e F I G U R E 8 Use of Bas in Symmetry to Measure Wave Conditions Over O n l y Hal f the Bas in , W h i l e in Effect Acquir ing Readings Throughout the Tota l A r e a 6 m F I G U R E 9 Diagram of the Probe A r r a y Frame F I G U R E 10 View of the Four Probes Facing the Generator, and the Author W i t h Upstretched Hand Indicating Still Water Level F I G U R E 11 View From Wave Generator As Basin Fills , Showing the Fixed Reference Probe and Cylinder Situated Along the Centreline, and the Probe Array in the First of Four Positions. Note Floor Markings. F I G U R E 12 Closeup of 20cm Spacer in Place For Calibration of Probe Array. In Its Operating Position, the Superstructure Rests On the Fitt ing Supporting the Spacer. F I G U R E 13 Closeup of Brass Fitting Holding a Wave Probe On A n Array Superstructure Member Using a Standard Fitt ing F I G U R E 14 The Shortest Wave Train Propagating At +30 ° . Note the Qui-escent Zone In the Corner. The Left Edge of the 6x6m Frame is Positioned Along the Basin Centreline. F I G U R E 15 The Shortest Wave Train Propagating At - 3 0 ° . Note the Short-Crested Pattern Due to Crossing of the Indirect and Direct Waves in the Corner Region. 87 F I G U R E 16 Intermediate Length Waves Propagating At - 4 5 ° Past the Surface Piercing Cylindrical Structure. Flotation Balloons Are Seen Stored Atop the Probe Array Rig . F I G U R E 17 View Showing the Probe Array In the Final Position. The Corner Probe Has Been Removed and the Support Member Cut to Position the Array W i t h the Structure Present. 10.0 -10.0 0.0 0.1 1.2 1.6 T i m e ( s e c o n d s ) (a) Dr ive Signals For Segments 1-4, Corner Reflection 10.0 CT> 5.0 CT> C 0.0 -5.0 •10.0 0.0 0.4 0.8 1.2 1.6 T 1 me t s e c o n d s ) 2.0 2.1 (b) Segments 4-7, Corner Reflection 10.0 O) 5.0 0.0 -5.0 •10.0 0.0 0.1 0.8 1 . 2 1.6 2.0 T i m e ( s e c o n d s ) 2.4 (c) Segments 7-10, Corner Reflection F I G U R E 18 Wave Generator Segment Drive Signals For Test 4, T = 1.25sec at 9 = + 3 0 ° . Segments 1-11 Are Subject to Combined A - and B-Wave M o t i o n , Segments 12-46 Move According To the Simple Snake Pr inciple , and Segments 47-60 A r e Held Motionless. 89 18(d) Drive Signals For Segments 10-13, Corner Reflection and Simple Snake 18(e) Drive Signals For Segments 13-16, Simple Snake 10.0 r o T i m e ( s e c o n d s ) 18(f) Drive Signals For Segments 44-46, Simple Snake, and 47-60, Quiescent 10.0 5.0 0.0 -5.0 -10.0 0.0 0.4 0.8 1 . : 1.6 T ime ( s e c o n d s ) 2.0 2.4 (a) Dr ive Signals For A l l 60 Segments, For Tests 1 ( S Z E R O ) , 2 ( M Z E R O ) and 3 ( L Z E R O ) A t 8 = 0 ° . These Show T h a t Paddle Stroke Increases Di rec t ly W i t h Wavelength For a G i v e n Waveheight. (b) D r i v e Signals For the Intermediate Wavelength ( T = 1.75sec), For A l l Segments and 8 = 0° ( M Z E R O ) , and For Segment 30 and 8 = + 3 0 ° ( M 3 0 P ) , 9 = + 4 5 ° ( M 4 5 P ) and 9 = + 6 0 ° ( M 6 0 P ) . These Correspond T o Tests 2, 5, 8 and 11, Respectively, and Show T h a t For a G i v e n Waveheight, Paddle Stroke Varies Inversely W i t h Propagat ion Direc t ion . F I G U R E 19 Dr ive Signal P lot s I l lustrat ing Dependence of Paddle Stroke O n Wavelength and Propagat ion Direct ion For a G i v e n Waveheight parameters requested onl ine: - s t i l l water depth? - number of frequency components to be extracted? - number of harmonic components? - tolerance below which the error sum of squares of two successive i terat ions must f a l l for the optimization loop to terminate? - l imi t to variance of amplitudes, as percentage of measured time ser ies RMS value? 91 read data (measured time ser ies) remove mean value is the number of harmonic components > 1 ? yes number of parameters to be optimized = 2 (ampl. & phase) opt ional ly supply value of fundamental frequency number of parameters to be optimized? 2 = amplitude and phase 3 = amplitude, phase and frequency ••T perform FFT on data f ind maximum amplitude and associated frequency using these values perform quadratic interpolat ion to improve estimates (also get phase i n i t i a l estimate) I using current estimates of amplitude, frequency and phase, perform non-l inear optimization to f i t s inusoidal curve to data is the error sum of squares between current and previous i t e ra t ion , for each of the parameters being optimized, within the speci f ied tolerance? yes I have the requested number of frequency components been extracted, or the speci f ied percentage variance of amplitude been exceeded? remove sinusoid from time s e r i e s , and add to preceding sinusoidal components ( i e . , to synthesized time ser ies) yes output the fundamental component wave para-meters, the synthesized time s e r i e s , the residual time s e r i e s , the l ine spectra of amplitude vs . frequency and of phase vs. f req . F I G U R E 20 K e y Operat ional Steps of the S C R E N M u l t i p l e Regression Screening P r o g r a m , Modi f i ed to Extrac t Components F r o m the Measured E levat ion T i m e Series (a) D i a g r a m Indicat ing the A r r a y Pos i t ion W i t h Ind iv idua l Probe Locat ions F I G U R E 21 Examples of Plots Generated For E a c h Test, For the Case of One A r r a y Pos i t ion of Test 5, T = 1.75sec, 6 = 3 0 ° . 93 S 3 a i 3 M ol(h) Measured Elevat ion T i m e Series, P lot s of S C R E N Results and Perta-2 U i n t Parameters For Reference Probe 17 (Or ig ina l T i m e Senes - sohd Irne, Synthesized - dashed l ine, Res idual - dotted line) S3M13W S3H13W S3M13W S3H13H 21(c) Measured T i m e Series Plots For Probes 1-4 S3H13H S3M13H S3M13H S3S13H 21(d) Measured T i m e Series Plots For Probes 5-8 21-6 sgqcuj JOJ; s^o^ sstjag stuix paJris^ajAj (d)iz METRES METRES METRES METRES S3di3W S 3 a i 3 H S3M13H (f) Measured T i m e Series Plots For Probes 13-16 S i d e W a l l S i d e B e a c h XI c Y (a) F u l l Waveheight P lo t , Test 4; Hsy Values Normal ized to Desired Wave-height of 20cm. (The bracketed region (R) locates the repeatability test. See Figures 24 and 25.) ( hei " l h e i S h t P I ° ' ' T e S t  B- ™™ Normalized to Desired Wave-F I G U R E 22 Hal f Wavehff , , Plots Over H a l f the B a s i n of the T = 1.25sec, 8 = 30° Resn the F u l l and Reduced Waveheights t F I G U R E 23 Ha l f Waveheight Test: Phase Plots Over H a l f the B a s i n of the T = 1.25sec, 9 = 30° Results For B o t h the F u l l and Reduced Waveheights (Tests 4 and 13, Respectively) . A l ternat ing Crests and Troughs A r e P lo t -ted (i.e., Contour Interval = 180° ) . Reduced Waveheight (Test 13) Result Shown Dot ted . F I G U R E 24 Repeatabil i ty Test: Hsy Plots For a Single A r r a y Pos i t ion of the T = 1.25sec, 6 = 30° Results. (Refer to Figure 22(a) for location of test, as indicated by region ' R ' ; Contour Interval = 0.05) F I G U R E 25 Repeatabi l i ty Test: Phase P lot s For a Single A r r a y Pos i t ion of the T = 1.25sec, 8 = 30° Results, Tests 4 and 14 (14 shown dotted). Contour Interval = 180° ^ 102 FIGURE 26 A a , A 2 and Hsy Plots For Test 7: T = 1.25sec, 0 = 45° ( a) A\ (b) A2 (x lO; note different contour interval) F I G U R E 27 Au A2 and Hsy Plots For Test 5: T = 1.75sec, 6 = 30 103 FIGURE 28 Au A2 and Hsy Plots For Test 20: T = 1.75sec, 9 = 30°, Cylindrical GBS in Place 105 F I G U R E 29 Hsy Plots For Normal ly Propagat ing Wave Trains (6 Contour Interval = 0.05) = 0 ° ; 106 (a) Wave F i e l d Case, Test 3 (b) G B S Case, Test 18 F I G U R E 30 Hsy Plots For T = 2.25sec, 0 = 0 ° , W i t h and W i t h o u t the G B S (Contour Interval = 0.05) 107 FIGURE 31 Hsy Plots For Wave Trains Propagating Obliquely At 6 Results For the Three Wavelengths Investigated = 30° ; 108 (a) 8 = 30°, Test 5 (b) 9 = 45°, Test 8 ( c ) 0 = 6 0 ° , Test 11 F I G U R E 32 Hsy P lots For the Intermediate Length Wave T r a i n ( T = 1.75sec) Directed A t the Three Obl ique Angles Tested 109 FIGURE 33 Hay Plot For Test 12, T = 2.25sec, 9 = 60° F I G U R E 34 Hsy P l o t For Test 15, T = 2.25sec, 6 = 30° , Involving the S imple Snake M e t h o d of Generat ion W i t h o u t Corner Reflect ion (a) Test 5, Wave F i e l d F I G U R E 35 Hsy P lot s For T = 1.75sec, 6 Cases 111 (b) Test 20, G B S Present 30 ° , Wave F i e l d and G B S 112 F I G U R E 36 Phase P lo t For Test 4, T = 1.25sec, 6 = 30° . T h e Contour Interval Used Is 180° , Result ing In a Pa t te rn of A l t e r n a t i n g Crests and Troughs 113 FIGURE 37 Phase Plot For Test 7, T = 1.25sec, 6 = Contour Interval = 180° 45°. FIGURE 38 Phase Plot For Test 8, T = 1.75sec, 6 = Contour Interval = 180° 45° '18 0 .5m face ts wave gene ra to r / 6 0 0 . 5 m \ \ f a c e t s / 3 0 m • H fully a b s o r b e n t beach 1.2m s h a d i n g i nd i ca tes region of 2 - m e t r e s p a c e d d a t a po in t s FIGURE 39 Rectangular Basin Configuration For Use With the SEGEN Linear Diffraction Computer Program, Consisting Of a Full Bank of Ac-tive Generator Segments, Flanked On Each End By Side Walls of Similar Lengths wave g e n e r a t o r |<f-5.5m^- 2 3 . 0 m 7 . 0 m -=>| I I I I I I I \,imu///////i/. ful ly a b s o r b e n t b e a c h s h a d i n g i n d i c a t e s reg ion of 2 — m e t r e s p a c e d d a t a p o i n t s s ide wall 9 . 0 m for 0 . 5m f a c e t s / l e n g t h = 9 . 2 5 m \ for 0 . 2 5 m fei .Om \ f a c e t s j (a) E x a m p l e Corresponding To Test 4, Us ing the V i r t u a l Segment Concept To M o d e l B-Wave Generat ion, and Mode l l ing Quiescent Zone Segments B y a Reflecting W a l l wave genera to r <= i ^ i i — 2 3 . 0 m i i i i i i i — 7 . 0 m — L > J — L Y 1 1 1 r 1 1 side 12m wall Y// /// 1.0m -5 \ Y / / // / A H 9 .0m Y- 1.0m L. fully a b s o r b e n t b e a c h J shad ing ind ica tes region of 2 - m e t r e s p a c e d d a t a po in ts (b) E x a m p l e Corresponding To Tests 5 and 12, W i t h B o t h Side Wal l s In P lace To Compare W i t h the V i r t u a l Segment Models , and Including the C y l i n d r i c a l Structure F I G U R E 40 Use of the A r b i t r a r y Bas in Configurat ion For the S E G E N L inear Dif fract ion Computer P r o g r a m (a) Exper imenta l Result , Test 1. Contour Interval = 0.05 F I G U R E 41 Exper imenta l Hsy and Numer ica l M o d e l M a x i m u m Waveheight (Ai) Results Compared For T = 1.25sec, 9 = 0° (note different contour intervals) 117 (b) N u m e r i c a l Result , Rectangular Bas in Configurat ion. Contour Interval = 0.02 i (a) Exper imenta l Result , Test 2. Contour Interval = 0.05 F I G U R E 42 Exper imenta l Hsy and Numer ica l M o d e l M a x i m u m Waveheight ( A i ) Results Compared For T = 1.75sec, 9 = 0° 118 119 (b) Numerical Result, Rectangular Basin Configuration. Contour Interval = 0.02 120 (a) Exper imenta l Result , Test 4 (b) Numerica l Result , M = 1 ( A r b i t r a r y B a s i n Conf igurat ion) (c) N u m e r i c a l Resu l t , M — 2 F I G U R E 44 Exper imenta l Hsy and N u m e r i c a l Ax Results Compared For T = 1.25sec, d = 30° 121 (a) Exper imenta l Result , Test 5 (b) N u m e r i c a l Result , A r b i t r a r y B a s i n Conf igurat ion F I G U R E 45 Exper imenta l H3y and Numerica l A\ Results C o m p a r e d For T = 1.75sec, 6 = 30° 122 (a) Exper imenta l Result , Test 6 (b) N u m e r i c a l Result , A r b i t r a r y Bas in Conf igurat ion F I G U R E 46 Exper imenta l Hsy and Numer ica l Ai Results C o m p a r e d For T = 2.25sec, 9 = 30° 123 F I G U R E 47 Exper imenta l Hsy and Numerica l Ax Results For the S imple Snake M e t h o d of Generation, T = 2.25sec, 6 = 30° 124 (a) Exper imenta l Result , Test 8 (b) Numerica l Result , A r b i t r a r y B a s i n Configuration (c) N u m e r i c a l Result , W i t h Quiescent Zone As In (b), C o r n e r Reflection Side W a l l a n d No V i r t u a l Segments F I G U R E 48 Exper imental Hsy and N u m e r i c a l Ai Results C o m p a r e d For T = 1.75sec, 0 = 45° 125 (a) Exper imenta l Result , Test 12 (b) Numerica l Result , A r b i t r a r y B a s i n Configuration (c) N u m e r i c a l Result , W i t h Quiescent Zone As In (b), C o r n e r Reflection S ide W a l l and N o V i r t u a l Segments F I G U R E 49 Exper imental Hsy and N u m e r i c a l Ax Results C o m p a r e d For T = 2.25sec, 6 = 60° 126 (b) N u m e r i c a l Result , Rectangular Bas in Conf igurat ion (b) N u m e r i c a l Result , Rectangular Bas in Conf igurat ion (a) Exper imenta l Result , Test IS F I G U R E 52 Exper imenta l Hsy and Numer ica l Ax Results Compared For T = 2.25sec, 9 = 0 ° , W i t h the G B S In Place 128 (b) N u m e r i c a l Result , Rectangular Bas in Conf igura t ion 129 (a) Exper imenta l Result , Test 25 ( b ) N u m e r i c a l Result , Rectangular B a s i n Conf igurat ion F I G U R E 53 Exper imenta l Hsy and Numer ica l Ai Results Compared For the S imple Snake M e t h o d of Generation W i t h the G B S In Place , T = 2.25sec, 6 = 30° (relates to Figure 56) (a) Exper imenta l Result , Test 19 (b) N u m e r i c a l Result , A r b i t r a r y B a s i n Configurat ion F I G U R E 54 Exper imenta l Hsy and Numer ica l Ax Results C o m p a r e d For T = 1.25sec, 9 = 30° , W i t h the G B S In Place Exper imenta l Result , Test 20 (b) N u m e r i c a l Result , A r b i t r a r y Bas in Configurat ion F I G U R E 55 Exper imenta l Hsy and Numer ica l Ax Results C o m p a r e d For T = 1.75sec, 6 = 30° , W i t h the G B S In Place 132 (a) Exper imenta l Result , Test 21 (b) N u m e r i c a l Result , A r b i t r a r y B a s i n Configurat i F I G U R E 56 Exper imenta l Hsy and Numer ica l Ax Results C o m p a r e d For T = 2.25sec, 8 = 30°, W i t h the G B S In Place 133 (a) Exper imenta l Result , Test 22 (b) N u m e r i c a l Result , A r b i t r a r y Bas in Conf igurat ion i F I G U R E 57 Exper imenta l Hsy and Numer ica l Ax Results C o m p a r e d For T = 1.25sec, 6 = 45 ° , W i t h the G B S In Place 134 (a) Exper imenta l Result , Test 23 (b) N u m e r i c a l Result , A r b i t r a r y B a s i n Configurat ion F I G U R E 58 Exper imenta l Hsy and Numer ica l Ax Results C o m p a r e d For T = 1.75sec, d = 45° , W i t h the G B S In Place 135 (a) Exper imenta l Result , Test 24 (b) N u m e r i c a l Result , A r b i t r a r y B a s i n Conf igurat ion F I G U R E 59 Exper imenta l Hsy and Numer ica l A\ Results C o m p a r e d For T = 2.25sec, 9 = 45 ° , W i t h the G B S In Place 136 Y X F I G U R E 60 Numer ica l ly Derived Phase P l o t Corresponding To Test 2, T = 1.75sec, 6 = 0 ° . Contour Interval = 180° 137 FIGURE 61 Numer ica l ly Derived Phase P l o t Corresponding To Test 4, T = 1.25sec, 6 = 30°. Contour Interval = 180° 138 F I G U R E 62 Numer ica l ly Derived Phase P l o t Corresponding To Test 8, T = 1.75sec, 8 = 45 ° . Contour Interval = 180° F I G U R E 63 Three-Dimensional Perspective B l o c k D i a g r a m Showing In stantaneous Water Surface Elevat ion F r o m the N u m e r i c a l Resul t Corre sponding To Test 5, T = 1.75sec, 9 = 30° F I G U R E 64 A m p l i t u d e vs. Frequency L ine Spectra For Selected P r o b e Locat ions Per ta in ing To Test 10, T = 1.25sec and 9 = 60° 141 L J CO ' <Z CO X is L L Q C LLJ L L B O D> N K o n H C N N CO 1-1 i-". N i-i N •0 N 0-O CO O O 0! 0> C CM r>. CO 0 0 K co co co *o co i " f J O IN N l> b a H H fi CO li") i- i CM N CO Ii0 i l l ilO N O CO O O C i - » t H i - » i h O O O ro N co co o CM -o o-T-i cs o- co n i-t K O CO CO N ~0 o CM i - " o o o i- i o o o o c O ^0 CO o O rH C O c © o c o o o o o o o o LO r— LO tsa >, (J CS cu cu LO S3H13W L_ o [ui] apn^ijdiuy 64(b) A m p l i t u d e vs. Frequency Line Spectrum and Measured E leva t ion T i m e Series P l o t For Probe 1 142 -0 •? N in o CO NO NO IS CO CO * L O <r CO CN CO n o- o CO IS co -X u co N C0 CJ If) IN CO 0- —t U- Q CM i-i CM CM CM CM N N iio to in N T o N CO <r Ch 7-1 CO NO CM co <? m =? a o K CO 0* o «r CM CM o LU N o N 0- N r-( CN CO CM c> Is u. X CO CO m Is co IN. iiO NO NO N L O LL. CM o o O o o i-l o o ~ • 1 o N •0 N N NT N CN o IlO CO Is IlO sT NO .7-1 o NO CM n _1 CO co Is sO IlO NO in r-t o L L 1-t Is -0 o m IT) NT NT s ' X. m iH 1-1 1-1 o o o o o o o O o o o o o o o o • * * * • * • • mm o o o o o o o o o o —< S38I3W <Z3 LO LO |N-LO >N o G CO 3 cr CO I -i fa LO LO LO C3> <=> [m] apniijduiy 64(c) Amplitude vs. Frequency Line Spectrum and Measured Elevation Time Series Plot For Probe 13 143 i . i K o CO -0 o o o IX • U ) • • * * • * * • * * <Z en T-i m o CO K CM co CN CM o- N T"i CO CN 0- IlO l> Q co tH CM CM CO N tH 0> CO CN S3 in IlO CM o CM IjO . - i CM <J o m S3 O o tH CM CO IJO o -0 T-i 03 Is U J N O CM CN IIO CM t-i <r CO CM T X CO CO h if) N o •0 rs 0- CO L L * • • • • * * * • * o o TH T-i o i H t-i o o o •0 CM tH Is o <T CM CN Is CO co •0 T-i i-i O sO T-t CO Is S3 CO CO *0 S3 CM CM 1-1 o T-i N . <r CO CO CO CO r sT 6 o o c o o o o <. O o o o o o c o o o * • * « * • • * • * o o o o o o o o o L O |n-LO o a LO LO S3M13W L L O LO NO LO LO CVI [in] 9pn^i|duiy 64(d) A m p l i t u d e vs. Frequency L ine Spectrum and Measured E levat ion T i m e Series P l o t For Probe 7 144 > \ > I • > < < > > L U X 0 ; 0 . Q CO i H PJ O 1 0 Fl 0 * TI o» o • C N B N > >0 B H N > \i~ ~£ b~ Pi sT i-i PJ 0> i n PJ r ! N ri N f j (1 f ) 0 3 CO P J n CO C- <T n N CO © P i 0> 'TI CO CO N CO N * C i n -o P J n c * c * o- PO i n o cc o co <r ^ - -o i n m *o N O O O T I O O T I T I T I O o- P) O 0 - O CO O TH PJ CO CO T N 0 - 0 - ro 0 - PJ PJ « ? 0 - b~ CO O S3 CO PJ PJ s i • H O C O O O O O O O o c o o o c o o o o o o o o o o o o o o S3ai3H L. L O L O L O tS) O cr L O L O C O [uu] apn^ijduiy 64(e) A m p l i t u d e vs. Frequency Line Spectrum and Measured E levat ion T i m e Series P l o t For Probe 12 

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