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

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REGULAR  W A V E C O N D I T I O N S IN A D I R E C T I O N A L By M a r k D . Shaver B . E n g . (Mechanical) C a r l e t o n U n i v e r s i t y  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR T H E DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING  W e accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t , 1989  © M a r k D . Shaver, 1989  W A V E BASIN  In presenting this thesis i n p a r t i a l fulfilment of the requirements for a n advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree that the L i b r a r y shall make it freely available for reference a n d study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of m y department or by his or her representatives. It is understood that copying or p u b l i c a t i o n of this thesis for financial gain shall not be allowed w i t h o u t m y w r i t t e n permission.  C i v i l Engineering T h e University of B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, C a n a d a V 6 T 1W5  Date:  Abstract  T h i s thesis represents a s m a l l step toward improved generation of realistic sea states i n laboratory wave basins. E x p e r i m e n t s were conducted i n an offshore directional wave basin equipped w i t h a segmented wave generator.  Regular waves were generated for  several periods a n d propagation directions, and the resulting wave elevations were measured throughout the basin. M o t i o n s of the 60 wavemaker segments were based on the snake principle of directional wave generation. Results are summarized a n d compared w i t h predictions of a b o u n d a r y element n u m e r i c a l method. F i n d i n g s 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 a n d reflection effects w h i c h affect the wave field. T h e experiments were conducted at the H y d r a u l i c s L a b o r a t o r y of the N a t i o n a l Research C o u n c i l of C a n a d a , i n O t t a w a . A six metre square array of wave probes was used to measure wave elevations at discrete points spaced on a 2m g r i d a r o u n d the wave basin. Three wave periods were investigated, w i t h waves directed n o r m a l to the generator face, as well as i n three oblique propagation directions. For a l l but the most severe propagation direction, tests were r u n a second time w i t h a large surface piercing c y l i n d r i c a l structure positioned i n the test area.  M a x i m u m waveheight results were  plotted, a n d compared w i t h numerical model predictions at the same locations around the basin.  A linear diffraction computer p r o g r a m based on the b o u n d a r y element  m e t h o d was used to predict the wave field at the same points around the basin. B y this m e t h o d , generator segment faces a n d reflecting walls are represented b y a d i s t r i b u t i o n of discrete sources a r o u n d the basin boundaries.  ii  Measured elevation time series were analysed using a m u l t i p l e regression screening program w h i c h extracts a prescribed number of sinusoidal components f r o m the signal of interest.  T h e program was modified to accommodate h a r m o n i c analysis.  Funda-  mental a n d second harmonic components were synthesized f r o m each t i m e series. T h e second h a r m o n i c component was generally an order of magnitude smaller t h a n the fundamental component. Discrepancies between these results a n d the linear numerical model predictions are a t t r i b u t e d to nonlinear effects, a n d to basin resonance.  The  linear diffraction computer model is seen to predict the wave field to a h i g h degree of accuracy, even though imprecise boundary definition was necessary at the current level of program development.  iii  . . . 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 h i m 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, M r . 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 1  2  Introduction  1  1.1  General  1  1.2  Literature Review  3  1.3  Scope of the Project  5  Theoretical Background  7  2.1  T w o - D i m e n s i o n a l Plane Wavemaker T h e o r y  7  2.2  T h e Snake P r i n c i p l e of Wave Generation  10  2.2.1  12  2.3  3  xv  Refinement B y Use of Reflective P a r t i a l Side W a l l s  T h e Diffraction P r i n c i p l e of Wave Generation  16  2.3.1  A n a l o g y W i t h a Breakwater G a p  16  2.3.2  Point Source Representation of Generator M o t i o n s  17  Experimental Setup  21  3.1  22  T h e N R C Offshore Wave B a s i n v  4  5  6  3.2  T h e Wave Probe Array  25  3.3  T h e G r a v i t y Base Structure  28  The Test Program  28  4.1  31  Data Analysis  32  5.1  Modifications to the S C R E N P r o g r a m  35  5.2  Types of P l o t s Generated  36  Results and Discussion  39  6.1  Synopsis  40  6.2  Effect of Prescribed Waveheight  42  6.3  Repeatability Test  44  6.4  M a x i m u m Waveheight P l o t s  44  6.4.1  P l o t s F o r N o r m a l l y P r o p a g a t i n g Wave Trains  47  6.4.2  P l o t s F o r Oblique Wave Trains  49  6.5  Phase P l o t s F r o m the E x p e r i m e n t a l Results  54  6.6  N u m e r i c a l M o d e l Predictions of Wave C o n d i t i o n s  56  6.6.1  T h e L i n e a r Diffraction C o m p u t e r P r o g r a m  56  6.6.2  Waveheight P l o t s of N u m e r i c a l M o d e l Predictions  58  6.6.3  N u m e r i c a l M o d e l Phase Predictions  63  6.6.4  N u m e r i c a l M o d e l Surface E l e v a t i o n P l o t s  64  6.7  7  Synthesis of D r i v i n g Signals  F u r t h e r Considerations  65  Conclusions and Recommendations  65  7.1  69  Conclusions  vi  7.2  Recommendations  8  References  9  Tables  10 Figures  List of Tables  1  Tests C o n d u c t e d a n d Pertinent Parameters  2  Side W a l l Lengths Specified For D r i v e Signal Synthesis, a n d Resultant Generator M o t i o n s  76  77  vm  List of Figures  1  E l e v a t i o n V i e w of the Wavemaker Configuration a n d C o o r d i n a t e System  78  2  Definition of Coordinate System, P l a n V i e w  78  3  O b l i q u e Wave W i t h Interference Wave, Conventional M e t h o d of O b l i q u e Wave Generation  79  4  Superposition of Direct a n d Reflected Indirect Waves  79  5  V i e w of the N R C Offshore Wave B a s i n  80  6  Wave P r o b e A r r a y R i g i n C a l i b r a t i o n P o s i t i o n  80  7  B a s i n L a y o u t , W i t h the Four P r o b e A r r a y Positions Indicated  81  8  Use of B a s i n S y m m e t r y to Measure Wave C o n d i t i o n s Over O n l y H a l f the B a s i n  82  9  D i a g r a m of the P r o b e A r r a y Frame  83  10  V i e w of the F o u r Probes Facing the Generator  84  11  V i e w F r o m Wave Generator A s B a s i n F i l l s  84  12  Closeup of 20cm Spacer i n Place F o r C a l i b r a t i o n of P r o b e A r r a y  13  Closeup of Brass F i t t i n g H o l d i n g a Wave P r o b e  14  T h e Shortest Wave T r a i n P r o p a g a t i n g A t + 3 0 °  86  15  T h e Shortest Wave T r a i n P r o p a g a t i n g A t - 3 0 °  86  16  Intermediate L e n g t h Waves P r o p a g a t i n g A t —45°  87  17  V i e w Showing the P r o b e A r r a y In the F i n a l P o s i t i o n  87  18  Wave Generator Segment D r i v e Signals for Test 4, T = 1.25sec A t  19  . . . .  85  '. .  85  0 = +30°  88  D r i v e Signal P l o t s Illustrating Dependence of Paddle Stroke  90  ix  20  K e y O p e r a t i o n a l Steps of the S C R E N M u l t i p l e Regression  Screening  Program 21  91  Examples of P l o t s Generated For E a c h Test, For the Case of One A r r a y P o s i t i o n of Test 5, T = 1.75sec, 9 = 30°  22  H a l f Waveheight Test: Hsy  P l o t s Over H a l f the B a s i n of the T = 1.25sec,  9 = 30° Results 23  98  H a l f Waveheight Test: Phase P l o t s Over H a l f the B a s i n of the T = 1.25sec, 9 = 30° Results  24  Repeatability Test: Hsy  99  Plots For a Single A r r a y P o s i t i o n of the  T = 1.25sec, 9 = 30° Results 25  92  100  Repeatability Test: Phase P l o t s For a Single A r r a y P o s i t i o n of the T = 1.25sec, 9 = 30° Results  101  26  Au  A2 and Hsy  P l o t s For Test 7: T = 1.25sec, 9 = 45°  102  27  Ax, A2 and Hsy  P l o t s For Test 5: T = 1.75sec, 9 = 30°  103  28  Au  A2 and Hsy  P l o t s For Test 20: T = 1.75sec, 9 = 3 0 ° , C y l i n d r i c a l  G B S i n Place  104  29  Hsy  P l o t s For N o r m a l l y Propagating Wave Trains  105  30  Hsy  P l o t s 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  P l o t s For Wave Trains P r o p a g a t i n g O b l i q u e l y A t 9 = 3 0 ° ; Results  For the Three Wavelengths Investigated 32  Hsy  107  P l o t s For the Intermediate L e n g t h Wave T r a i n Directed A t the Three  O b l i q u e Angles Tested  108  33  Hsy  P l o t For Test 12, T = 2.25sec, 9 = 60°  109  34  Hsy  P l o t For Test 15, T = 2.25sec, 9 = 3 0 ° , Involving the Simple Snake  35  M e t h o d of Generation W i t h o u t Corner Reflection  110  Hsy  Ill  P l o t s For T = 1.75sec, 9 = 30°, Wave F i e l d and G B S Cases x  36  Phase P l o t For Test 4, T = 1.25sec, 0 = 30°  112  37  Phase P l o t For Test 7, T = 1.25sec, 0 = 4 5 °  113  38  Phase P l o t For Test 8, T = 1.75sec, 0 = 4 5 °  '. . 114  39  Rectangular B a s i n Configuration For Use W i t h the S E G E N L i n e a r Diffraction C o m p u t e r P r o g r a m  40  115  Use of the A r b i t r a r y B a s i n Configuration For the S E G E N L i n e a r Diffraction C o m p u t e r P r o g r a m  41  E x p e r i m e n t a l Hsy  116  a n d N u m e r i c a l M o d e l M a x i m u m Waveheight (Ai)  Results C o m p a r e d For T = 1.25sec, 0 = 0° 42  E x p e r i m e n t a l Hsy  and N u m e r i c a l M o d e l M a x i m u m Waveheight (Ay)  Results C o m p a r e d For T = 1.75sec, 0 = 0° 43  E x p e r i m e n t a l Hsy  E x p e r i m e n t a l Hsy 0 =  45  30°  120  E x p e r i m e n t a l Hay and N u m e r i c a l A\ Results C o m p a r e d For T = 1.75sec,  E x p e r i m e n t a l Hsy  121 and N u m e r i c a l Ay Results C o m p a r e d For T = 2.25sec,  0 = 30° 47  E x p e r i m e n t a l Hsy  122 and N u m e r i c a l Ai Results For the Simple Snake M e t h o d  of Generation, T = 2.25sec, 0 = 30° 48  E x p e r i m e n t a l Hsy  E x p e r i m e n t a l Hsy  123  and N u m e r i c a l A\ Results C o m p a r e d For T = 1.75sec,  0 = 45° 49  119  and N u m e r i c a l Ay Results C o m p a r e d For T = 1.25sec,  0 = 30° 46  118  and N u m e r i c a l M o d e l M a x i m u m Waveheight (Ay)  Results C o m p a r e d For T = 2.25sec, 0 = 0° 44  117  124 and N u m e r i c a l A-y Results C o m p a r e d For T = 2.25sec,  0 = 60°  125  xi  50  E x p e r i m e n t a l Hsy a n d N u m e r i c a l  Results C o m p a r e d For T = 1.25sec,  9 = 0 ° , W i t h the G B S I n Place 51  126  E x p e r i m e n t a l Hsy a n d N u m e r i c a l A\ Results C o m p a r e d For T = 1.75sec,  9 = 0 ° , W i t h the G B S In Place 52  127  E x p e r i m e n t a l Hsy a n d N u m e r i c a l A\ Results C o m p a r e d For T = 2.25sec,  9 = 0 ° , W i t h the G B S In Place 53  E x p e r i m e n t a l Hsy  128  a n d N u m e r i c a l Ai Results C o m p a r e d F o r 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° 54  129  E x p e r i m e n t a l Hsy a n d N u m e r i c a l Ax Results C o m p a r e d For T — 1.25sec,  9 = 3 0 ° , W i t h the G B S In Place 55  130  E x p e r i m e n t a l Hsy a n d N u m e r i c a l Ax Results C o m p a r e d For T = 1.75sec,  9 = 3 0 ° , W i t h the G B S In Place 56  131  E x p e r i m e n t a l Hsy a n d N u m e r i c a l Ax Results C o m p a r e d For T = 2.25sec,  9 = 3 0 ° , W i t h the G B S In Place 57  132  E x p e r i m e n t a l Hsy a n d N u m e r i c a l A\ Results C o m p a r e d For T = 1.25sec,  9 = 4 5 ° , W i t h the G B S In Place 58  133  E x p e r i m e n t a l Hsy a n d N u m e r i c a l A\ Results C o m p a r e d For T = 1.75sec,  9 = 4 5 ° , W i t h the G B S In Place 59  134  E x p e r i m e n t a l Hsy a n d N u m e r i c a l Ax Results C o m p a r e d For T = 2.25sec,  9 = 4 5 ° , W i t h the G B S In Place 60  135  N u m e r i c a l l y Derived Phase P l o t Corresponding to Test 2, T = 1.75sec, 0 = 0°  61  136  N u m e r i c a l l y Derived Phase P l o t Corresponding to Test 4, T = 1.25sec, 0 = 30°  137  xii  62  N u m e r i c a l l y Derived Phase P l o t Corresponding to Test 8, T = 1.75sec, 0 = 45°  63  138  Three-Dimensional Perspective B l o c k D i a g r a m Showing Instantaneous W a t e r Surface E l e v a t i o n F r o m the N u m e r i c a l Result Corresponding to Test 5, T = 1.75sec, 9 = 30°  64  139  A m p l i t u d e vs. Frequency L i n e Spectra F o r Selected P r o b e P e r t a i n i n g to Test 10, T = 1.25sec a n d 6 = 6 0 °  xin  Locations 140  Acknowledgment  I w i s h to thank m y supervisor, D r . M i c h a e l Isaacson, for his guidance a n d good counsel throughout the preparation of this thesis. I w i s h also to extend m y deep gratitude to the staff of the H y d r a u l i c s L a b o r a t o r y of the N a t i o n a l Research C o u n c i l i n O t t a w a , for p e r m i t t i n g me the use of their facilities during a busy period. I w o u l d 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 i n s t a l l a t i o n of apparatus. T h a n k s also to D a n Pelletier, B r i a n W i l s o n and R o n G i r a r d , who accommodated  a  large a n a l y t i c a l demand u p o n the computer resources d u r i n g a p e r i o d of t r a n s i t i o n , and to M i k e M i l e s for occasional insightful tips along the way.  xiv  Nomenclature  a  - wave amplitude  a2  - two-dimensional wave amplitude  A  - constant used i n derivation of 2-D generator transfer function  Ay  - a m p l i t u d e of the fundamental synthesized wave component  A2  - a m p l i t u d e of the second harmonic synthesized wave component  A-wave  - primary, or direct wave t r a i n  b  - generator horizontal displacement  b2  - two-dimensional generator horizontal displacement  B  - generator segment w i d t h  Bpn  - segment source strength m a t r i x coefficients  B-wave  - secondary, or indirect wave t r a i n w h i c h joins i n phase w i t h the A-wave u p o n reflection from the side w a l l  C  - constant associated w i t h evanescent wave components  n  T>  - diameter of c y l i n d r i c a l surface piercing structure  /  -  fa  - fixed fundamental frequency value supplied as i n p u t to S C R E N  f(z)  - dimensionless f o r m of generator displacement a m p l i t u d e as a function of elevation z  /(£)  - source strength d i s t r i b u t i o n function  F(kh)  - 2-D transfer function relating wave amplitude to generator displacement, as a function of the depth parameter kh  T  -  frequency  3-D transfer function relating wave amplitude to generator displacement  xv  g  -  gravitational constant  —*  G, G(x, £) -  Green's function  GBS  -  gravity base structure  GEDAP  -  Generalized E x p e r i m e n t control, D a t a acquisition a n d data A n a l y s i s Package  h  - still water depth (using the convention suggested by the International A s s o c i a t i o n for H y d r a u l i c Research (1986))  H  - wave height  HQ 1^  -  H a n k e l function of the first k i n d a n d zero order  i  -  V^T  J0  -  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  -  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  - r a d i a l distance f r o m any point to a source location:  a counter, or n o r m a l direction when w r i t t e n as dn  r  = 1^- £1  S  -  sidewall length  S  -  h o r i z o n t a l contour along all non-absorbing surfaces, at still water level  SCREN  -  m u l t i p l e regression screening program xvi  t  -  time  T  -  wave p e r i o d  u  - h o r i z o n t a l fluid particle velocity  V, V(x)  - velocity a m p l i t u d e along surface S  x  - h o r i z o n t a l coordinate direction n o r m a l to generator face  x  -  y  - h o r i z o n t a l coordinate direction along generator face  z  - v e r t i c a l coordinate direction, measured from basin b o t t o m  A  - generator source facet length  e  - h o r i z o n t a l stroke amplitude of generator paddle at the base of the paddle (for flapper mode, e = 0 )  £  -  rj  - water surface elevation  8  - wave t r a i n propagation angle, measured f r o m the x axis  v  - vertical distance between the basin b o t t o m a n d the generator base  —* —* £, £ ( x , y) r  a point (x,y)  i n the wave field  wavemaker displacement function  - l o c a t i o n of a source point i n the x , y plane - 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  xvii  1  1.1  Introduction  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. S u i t a b i l i t y of this m e t h o d is l i m i t e d by effects associated w i t h the finite size of these basins. A short-crested, or multi-directional r a n d o m sea state may be considered a superposition of many infinitely long-crested, regular wave trains of various frequencies, a n d travelling i n numerous directions. R e p r o d u c t i o n of such sea states b y this concept relies o n a n ability to generate waves w h i c h propagate obliquely away f r o m the wave generator. M a n y laboratories use segmented wavemakers which consist of a straight or curved bank of i n d i v i d u a l l y controlled, narrow wave b o a r d segments, forming one or more basin wall. O b l i q u e regular wave trains are produced using the snake principle, introduced b y Biesel i n 1954.  T h e generator m o t i o n is sinusoidal b o t h i n time a n d  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 d r a w n by diffraction into the area adjacent to the region which is i n the direct p a t h of propagat i o n . T h 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 a d d i t i o n , the snake m e t h o d is unable to account for waves reflected back into the p r i m a r y wave field f r o m 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 m e t h o d of generation (Isaacson, 1984, 1985). Referred to as the diffraction m e t h o d , the wave field created by the m o t i o n of a single generator segment moving i n isolation is considered analogous to the diffraction pattern formed by waves propagating t h r o u g h a breakwater gap of the same w i d t h as the oscillating 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 a n d diffraction effects w i t h i n the basin. W h i l e the snake principle m e t h o d has been implemented a n d involves relatively straightforward p r o g r a m m i n g , the diffraction m e t h o d is comparatively complex, a n d untried i n practice. For each frequency component to be superposed to f o r m a directional sea state, a series of large m a t r i x equations must be solved. T h e appeal of the m e t h o d is enhanced b y present trends i n computer technology, toward increasingly powerful array processing machines at d i m i n i s h i n g cost. Experiments were conducted i n the directional wave basin of the N a t i o n a l Research C o u n c i l ( N R C ) of C a n a d a i n O t t a w a . E l e v a t i o n time series of n o r m a l l y a n d obliquely propagating, long-crested regular wave trains were recorded at discrete locations w i t h i n the test area. A modified f o r m of the snake principle of generation was used to determine generator segment motions for a desired oblique wave height, p e r i o d a n d angle of propagation. T h e wave height was held constant between tests. T h e p e r i o d a n d direct i o n of propagation were varied to investigate wave conditions over ranges frequently encountered i n research endeavours.  T h e test program also i n c l u d e d a series w i t h a  surface piercing impervious cylinder placed i n the test area, to m o d e l a large gravity  1  Introduction  3  base structure. A r a n d o m wave analysis program was modified to evaluate the fundamental and harmonic wave components present i n the data.  P l o t s derived from this information  enable v i s u a l 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. E x a m i n a t i o n of the time series stripped of fundamental a n d 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 o n linearized, or small-amplitude theory, as put forth by Havelock (1929). T w o dimensional theory for a piston wavemaker is reviewed by U r s e l l , D e a n a n d Y u (1960), who affirm empirically the use of sinusoidal generator motions for wave trains of m o d erate steepness. T h e y discuss effects not predicted b y 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 m o t i o n are negligible for small amplitude waves.  Discrep-  ancies between e m p i r i c a l results a n d theory not due to the effects just mentioned are a t t r i b u t e d p r i m a r i l y to nonlinear, finite-amplitude effects. Recent efforts have been made to extend wave generation to include second order components which are b o u n d to the fundamental wave.  H i g h harmonics distort the  wave profiles, sharpening crests and flattening troughs.  L i n e a r wave generation can-  not reproduce these components, instead producing free harmonic components which d i m i n i s h the realism of generated sea states. Sand and M a n s a r d (1986) address this  1  Introduction  4  problem, a n d provide a second order theory for satisfactory r e p r o d u c t i o n of irregular sea states. T h e first treatment of the theoretical relationships between segmented generator motions a n d the resulting wave field was that of Biesel (1954). O n e aspect considered was the generation of oblique waves by a straight, segmented generator. H e developed the snake principle m e t h o d , and established a p r a c t i c a l l i m i t to the mean direction of propagation attainable without spurious secondary wave effects. Subsequent m a t u r a t i o n of two-dimensional wave generator theory enabled G i l b e r t (1976) to extend Biesel's results to relate generator displacements to wave field elevations. T h i s transfer function for a specified wave field has been rederived a n d extended by S a n d (1979). Sand a n d M y n e t t (1987) discuss the p r a c t i c a l limitations of generating directional waves i n a finite basin by this m e t h o d . T h e y examine the l i m i t a t i o n , by diffraction a n d reflection, of the area over which a homogeneous wave field m a y be produced. T h e y also indicate that local disturbances i m m e d i a t e l y i n front of the wave boards further decrease the homogeneous test area. Funke a n d M i l e s (1987) present a technique of using p a r t i a 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 t u a l extension of the generator, a n d effectively enlarge the homogeneous test area. A t the opposite end of the generator, reflections from the side wall are suppressed by holding a sector of generator segments adjacent to the w a l l motionless. A n alternative m e t h o d of relating generator displacements to the wave field was introduced by Naeser (1979). He drew an analogy between a single generator segment oscillating i n a straight wall, and a gap of the same w i d t h i n a straight breakwater, t h r o u g h w h i c h waves pass and diffract.  A directional sea state w o u l d be produced  1  Introduction  5  by superposition of a d d i t i o n a l segment a n d frequency contributions. Naeser (1981) subsequently described how this 'diffraction m e t h o d ' might be applied i n practice. For smaller ratios of segment w i d t h to wave length, B/L,  the assumptions u p o n  w h i c h Naeser developed this concept become unrealistic. Isaacson (1984) presents a more general analysis, based o n the diffraction principle, b u t unconstrained by these restrictions. T h e breakwater gap analogy is reproduced b y a point wave source representation, using p o t e n t i a l flow theory. Isaacson treats the case of a single, straight generator bank, a n d also covers L - a n d U-shaped, m u l t i p l e generator configurations not pertinent to this study. A single oscillating generator segment may be approximated b y a point wave source, for s m a l l values of B/L.  A s B/L  increases, a single segment produces wave crest  contours which are non-circular, i n agreement w i t h breakwater gap results. T h e point source representation m a y be extended for this case by d i s t r i b u t i n g a d d i t i o n a l sources along the segment face. Reflections from basin walls a n d structures situated w i t h i n the wave field m a y also be accounted for b y point source representation of these surfaces. Isaacson demonstrates the validity of this approach by n u m e r i c a l comparisons w i t h Naeser's breakwater analogue.  Subsequent comparison of wave fields predicted b y  the snake principle a n d the diffraction principle (Isaacson, 1985) indicates that the diffraction m e t h o d appears to reproduce the desired wave field realistically.  1.3  Scope of the P r o j e c t  T h e 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 n o r m a l to the generator face. T h r e e wave lengths t y p i c a l of the range encountered i n laboratory studies were  1  Introduction  6  investigated. A p r i n c i p a l goal of the project was to acquire sets of wave elevation data spanning the entire basin test area. In a d d i t i o n , a large c y l i n d r i c a 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, a n d reflection f r o m solid boundaries are qualitatively understood. Measured wave conditions are complicated further by harmonic factors, resonance effects w i t h i n the basin, a n d by nonlinear effects generally a t t r i b u t e d to wave generation aspects a n d the energy dissipation processes occurring w i t h i n the artificial beaches situated i n front of solid basin boundaries. A n existing analysis program was modified to enable the removal of synthesized fundamental a n d harmonic wave components f r o m measured wave elevation time series. T h e synthesized time series remaining after extraction of these components are composed of higher harmonics a n d other effects.  V i s u a l inspection of these residual  series provides some i n d i c a t i o n of the magnitudes a n d spatial extent of these effects for various wave conditions. Long-crested regular wave tests provide the simplest d a t a f r o m which to examine underlying phenomena. Fundamental a n d second harmonic amplitude a n d phase plots were produced to facilitate inspection of diffraction a n d 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 b y Isaacson.  A p p l i c a b i l i t y of the diffraction  m e t h o d is thereby verified, by confirming the ability of the m e t h o d to account for diffraction a n d reflection effects.  2  Theoretical Background  T o date most laboratory wave tanks are equipped w i t h some f o r m of wave m a k i n g machine along one boundary of the tank. Such facilities include long, narrow flumes for two-dimensional studies, and broader coastal and offshore basins. L i n e a r wave theory has generally proven sufficient i n predicting resultant wave m o t i o n a n d generator power requirements, although nonlinear factors are increasingly being addressed as a n a l y t i c a l methods evolve.  2.1  Two-Dimensional Plane Wavemaker Theory  Detailed development of two-dimensional wavemaker theory using a paddle is presented by D e a n and D a l r y m p l e (1984, pp. 172-178). T h e b o u n d a r y value p r o b l e m corresponds to two-dimensional waves propagating i n an incompressible, inviscid fluid, and due to a prescribed wave generator m o t i o n . A Cartesian coordinate system is defined i n F i g u r e 1, w i t h x measured horizontally orthogonal to the generator face, a n d z  measured  vertically u p w a r d from the flume b o t t o m . T h e flow is assumed to be i r r o t a t i o n a l , a n d may be described by a velocity potential <j> which satisfies the Laplace equation w i t h i n the fluid region. T h i s potential is also subject to various b o u n d a r y conditions on the flume b o t t o m , at the free surface and the generator face, a n d to a r a d i a t i o n c o n d i t i o n at large distance x f r o m the generator. T h e horizontal displacement £ of the wavemaker is given b y  ( = b2f(z)smujt where f(z)  (2.1)  is a dimensionless function of the generator displacement a m p l i t u d e 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 still water level z = h. T h e subscript signifies the two-dimensional case. H o r i z o n t a l displacement is sinusoidal i n time t, w i t h angular frequency u>. T h e displacement amplitude is assumed s m a l l w i t h respect to still water d e p t h h a n d the characteristic wave length of the resulting wave m o t i o n . Because of this, the free surface b o u n d a r y conditions may be linearized, a n d the k i n e m a t i c b o u n d a r y c o n d i t i o n at the generator face may be applied at the generator e q u i l i b r i u m location x = 0:  u = ^(0,z,t) ox  = b2f(z) cocosut  (2.2)  A solution to the b o u n d a r y value problem provides an expression of the velocity potential as  (p =  Acosh kh sin (kx — out) CO  + 53 Cne~ knX  c o s  knh cos ut  (2.3)  n=l  where A a n d Cn are constants.  T h e first t e r m represents a progressive wave t r a i n  produced b y the generator. T h e series of terms represents standing wave components whose amplitudes decay w i t h distance away f r o m the wavemaker. T h e progressive wave number k a n d standing wave numbers kn are defined b y their corresponding dispersion relations:  gktanhkh  = u?  (2.4)  and  gkn t a n knh = — u> 2 i n which g is the gravitational constant.  for  n — 1,2,3,...  (2-5)  E q u a t i o n 2.4 has a single solution for k,  whereas E q u a t i o n 2.5 has an infinite number of solutions for kn. T h e decaying standing waves are referred to as evanescent waves. These are caused  2  Theoretical  Background  9  by an energetic adjustment of the waveform f r o m its i n i t i a l profile adjacent to the waveb o a r d , w i t h its linear shape a n d m o t i o n characteristics, to the downstream nonlinear progressive wave profile. E x a m i n a t i o n of the series terms i n E q u a t i o n 2.3 reveals that the heights of these evanescent waves decay to less t h a n one percent of their i n 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. T h e 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)  T h e general two-dimensional transfer function relating this wave amplitude a 2 to the generator displacement amplitude b2 is given b y  F(kh) v  s  !  a Aksinhkh = -1 = — :—r—7T2  b2  f h ,. . , , f(z)coshkzdz w  .  /  2kh + sinh2Jfc/» Jo  y  U s i n g the n o t a t i o n presented i n F i g u r e 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 a n d the transfer function is given s i m p l y by F(kh)  =  = t a n h kh.  A s a more p r a c t i c a l case, many generator motions are i n c l u d e d i n the general representation ,of f(z)  given as  {  0  c+ (l-c)(*=£)  for z < v  f o r z ^  (2.8)  T h i s corresponds to a fixed vertical plate f r o m z — 0 to z = v a n d a linear variation of generator amplitude from z = v to z = h w i t h f(z)  = e at z =  (2.7) J  2  Theoretical  Background  10  v. F o r this generalized generator m o t i o n , the transfer function F(kh) is given as 4 sinh. kh F ( k h )  =  f  2tt+sinh2tt { £ ? (  + llz^(kh  s i n h k  h  ~s  i n h  *")  sinhfc/i —fci/sinh kh + c o s h — coshfc/i)j(2.9)  For a paddle extending to the flume b o t t o m , simplified expressions for piston, flapper and combined piston/flapper modes are presented b y Isaacson (1984, p . 11).  2.2  T h e Snake P r i n c i p l e of W a v e G e n e r a t i o n  It is possible to generate i n a laboratory basin three-dimensional waves w h i c h propagate obliquely from the generator face, using a segmented wavemaker. T h e direction of propagation depends o n the wave machine segment motions. L i n e a r i t y of the problem allows superposition of m u l t i p l e generator motions to generate realistic sea states i n a basin. A t most research facilities capable of directional wave generation, the programm i n g of segment motions has to date been based on the snake, or serpentine principle. F i r s t consider a straight, infinitely long bank of generator segments, each of i n finitesimal w i d t h .  T h e coordinate system used is defined i n F i g u r e 2.  Horizontal  displacement of each segment is sinusoidal i n time. T h e snake p r i n c i p l e 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 motion. T h i s produces a regular wave t r a i n w h i c h 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, multi-directional wave fields. T h e wavemaker displacement, sinusoidal i n b o t h time a n d 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 still water level z = h, for this three-dimensional case. T h e generator shape function f(z) water level f(h)  is as defined i n Section 2.1, such that at still  = 1. W r i t t e n i n complex n o t a t i o n , E q u a t i o n 2.10 becomes  ( = ibf(z) where i =  e  «(^-"<)  (2.11)  . T h i s type of generator m o t i o n results i n an oblique m o n o c h r o m a t i c  wave t r a i n of amplitude a, wave number k a n d 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)  T h e velocity p o t e n t i a l is  ua cosh kz <p — ——:—-—— sin (kx cos 9 -\- ky sin 9 — ut) k sinn kh =  ga cosh kz . — — sm (kx cos 9 + ky sm9 — ut) u coshfc/i  (2.13)  T h e relationship between the propagation direction 9 a n d the wave numbers k a n d  k', a n d corresponding wave lengths L and Z/, is given by  s  i  n  6  =  T  =  T>  R e c a l l i n g the two-dimensional transfer function F(kh)  ( 2  -  relating wave a m p l i t u d e a2  1 4 )  to  generator displacement amplitude 6 2 at still water level, the three-dimensional transfer function is  o  cos 9  2  Theoretical  Background  12  T h e p r o b l e m may be extended to the case of a finite generator segment w i d t h B. T h e rfi 1  segment mid-point location is given by  xn = 0  y„ = (n - 1)B  T h e instantaneous displacement of the n  („  (2.16)  segment, at z = h , is  =  bsin(k'(n  - 1)B - ut)  =  6 s i n ( ( n - l)kB sm6 - ut)  (2.17)  T h e wave t r a i n generated is as before, but w i t h a different wave amplitude. T h e wave amplitude to generator displacement amplitude transfer function is now modified to  a  F(kh) sinr  b  (2.18)  cos 6  where r = \kB sin6> = \k'B , If the segment w i d t h to wave length ratio B/L  is not sufficiently s m a l l , secondary  propagating wave components w i l l also be generated.  Biesel stipulated that i n order  to avoid these undesired components,  f  s  TTilhT«i  T h e theoretical l i m i t corresponding to 6 = 90° gives B/L  < ' 219  — 0.5, but cannot be realized;  there is then no preferred wave propagation direction. Biesel therefore proposed a more practical B/L  l i m i t as T  T h i s gives B/L  2.2.1  ^  L = 0.41 when 6 = 90° .  -T-^i  ^7  v/2 + | s i n 0 |  ( 2  V  2 0  )  '  Refinement By Use of Reflective Partial Side Walls  A m a j o r problem i n reproducing multi-directional sea states i n a finite basin concerns the s m a l l size of the test area over which a homogeneous wave s i m u l a t i o n occurs. T h i s  2  Theoretical  Background  13  aspect has been considered by Sand a n d M y n e t t (1987) a n d subsequently b y Funke a n d M i l e s (1987). Diffraction a n d reflection effects significantly influence the effective test area. T h e existence of evanescent waves which decay exponentially away f r o m the wave boards prohibits use of the region w i t h i n 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 m a i n t a i n uniform crest elevations when r u n n i n g regular, long-crested waves n o r m a l 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.  T h i s reflected energy  contaminates the wave field i n the test area by its interaction w i t h incident waves. T h e performance of side beaches is easily appreciated by n o t i n g the r a p i d dissipation of such 'cross-waves' after a test. Depending on the predominant sea conditions i n a test program, the side wall 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 h a n d , and reflections from such side walls on the other. A s i l l u s t r a t e d i n F i g u r e 3, using the simple snake m e t h o d it is not possible to generate waves directly into the region bounded by side wall ' R ' a n d the line ' A - A ' , w h i c h projects f r o m the corner at angle 8 to the side wall. T h i s b o u n d a r y represents the ideal edge of the plane waves propagating directly away f r o m the generator i n the desired direction.  T h e region adjacent to the direct wave field is referred to as the  diffraction zone, into which some wave energy is d r a w n by the process of diffraction, which involves bending of wave crests a n d waveheight attenuation due to lateral transfer of energy along wave crests. T h e result is a reduction of waveheights i n the direct wave field, thereby reducing the area i n which a homogeneous wave t r a i n is produced.  2  Theoretical  Background  14  Funke a n d M i l e s (1987) present a technique which enlarges the homogeneous test region using waves reflected from one corner, a n d by keeping generator segments adjacent to the opposite side wall motionless to reduce unwanted reflections. T h e y demonstrate the rationale for this approach over the simple snake m e t h o d . Referring to Figure 4, the 'indirect reflection' principle consists of generating a secondary wave toward the side wall so that, when reflected, it joins i n phase w i t h the direct, or primary, wave. T h i s has the effect of extending the effective test area by translating b o u n d a r y line A - A so that it projects f r o m the end of the left side w a l l . L i n e ' B - B ' , d r a w n at angle 9 to the reflecting side w a l l R , a n d extending f r o m the side w a l l outer end to the generator face, marks the boundary of the secondary wave t r a i n . T h e 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. T h i s is given s i m p l y as „ tan 9 m = S—— B  (2.21)  where S is the length of the side wall, a n d m is rounded up to the nearest integer value. For these m segments the secondary, or 'B-wave', m o t i o n is superposed on the p r i m a r y 'A-wave' m o t i o n . T h e B-wave m o t i o n mimics the m o t i o n of m v i r t u a l segments w h i c h would extend the generator to the left of the side wall. A s s u m i n g perfect reflection of the B-wave f r o m the wall, the reflected indirect wave is of the same a m p l i t u d e (and frequency) as the A-wave, and it joins i n phase w i t h the A-wave to f o r m a wider wave crest of u n i f o r m height. O n the other side of the basin, line ' C - C , d r a w n similarly to B - B , defines the region i n w h i c h the propagating A-wave w o u l d reflect f r o m the wall, heading back into the direct wave field at an angle of —9° to the side wall. Therefore, the segments w h i c h lie w i t h i n this region are held motionless, so that the only energy w h i c h reflects back into the wave field is due to diffraction of the direct wave t r a i n into this 'quiescent zone'.  2  Theoretical  Background  15  T h e phase relationship between adjacent wave boards, m o v i n g i n snakelike fashion, is given b y <P = j?  (2-22)  where M is the (non-integer) number of segments comprising one wavelength L' along the generator face:  M = -—— B sin 9  (2.23) ;  v  th  T h e phase angle of the n  segment is 1  27T  <Pn = ( n -  (2-24)  Jf  and the n ^ n wave b o a r d displacement is given by Co(n, t) = jsm(ut  + (n-±)jf)  (2.25)  where T is the transfer function given b y E q u a t i o n 2.15, a n d a and u are the desired wave a m p l i t u d e and frequency, respectively. T h e subscript D refers to the direct wave component, while V w i l l refer to the v i r t u a l , or indirect, component. T h e displacement of the n'^  v i r t u a l segment to the left of the wall w o u l d 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 t u a l segments, a wave t r a i n is generated w h i c h is directed against the side w a l l , a n d u p o n reflection produces the same indirect waves as w o u l d the v i r t u a l segments. T h i s is achieved by prescribing motions for segments 1 to m b y substituting n' = 1 — n i n E q u a t i o n 2.26. T h e n : Cv(n, t) = - sin (ut - ( „ - - ) — ) for n = 1, 2 , . . . , m  (2.27)  2  Theoretical  Background  16  C o m b i n i n g the direct a n d indirect wave motions, the first m segments have comb i n e d 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 a n d the quiescent section at the other end of the generator, E q u a t i o n 2.25 applies. T h i s equation shows that, for simple snakelike m o t i o n , a l l segments move w i t h a phase shift relative to one another. However, as shown by E q u a t i o n 2.28, the first m segments a l l have the same phase, but different displacement amplitudes. T h e first m segments have displacement amplitudes (at z = h) v a r y i n g f r o m 0 to which is twice the value of the segments responsible for generating direct waves only. T h i s is the case when, for single frequency oblique waves, the n o r m a l velocities of the direct a n d indirect waves are i n phase. T h u s , the stroke l i m i 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. T h i s problem is less severe for short-crested wave generation, b o t h because the component wave amplitudes f r o m the extreme directions are typically s m a l l , 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 t h a n for long-crested oblique waves.  2.3 2.3.1  T h e Diffraction P r i n c i p l e of W a v e G e n e r a t i o n A n a l o g y W i t h a Breakwater  Gap  T h e wave field produced by a single generator segment m o v i n g i n isolation was perceived by Naeser (1979) to be analogous to the diffracted wave p a t t e r n produced by a wave  2  Theoretical  Background  17  t r a i n passing at n o r m a l incidence through a gap of the same w i d t h i n a straight section of breakwater. T h e wave crest pattern beyond the gap generally resembles circles, depending on the breakwater gap width-to-wavelength ratio, B/L. to the Shore Protection  Manual (1984), for B/L  flattened  According  > 5 , the diffraction effects due to each  w i n g of the breakwater are nearly independent. In other words, the diffraction p a t t e r n is comparable to a m i r r o r e d 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 H = -j= cos2 (0) for x > 2B yLx  B , | cot 9 \ > — L  5  (2.29)  where  B B S = 2 ( - + 0 . 6 ) 3 w i t h — >0.2 Li  LI  (2.30)  a n d H0 is the wave height close to the paddle, the same height as w o u l d exist throughout a flume of w i d t h B. T h e cosine-power f o r m of the wave height expression requires a superposition process of the same f o r m to produce a short-crested sea state. Naeser's approach is found to be of l i m i t e d applicability i n the context of laboratory wave generation. T h e requirement that B/L  > 0.2 is a c o n d i t i o n w h i c h 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 i n t r o d u c e d 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 b y superposing other frequency components.  2  Theoretical  Background  18  Consider the wave m o t i o n throughout the basin to be represented b y a velocity potential given b y  where x = (x, y), a n d <f> (x) is the two-dimensional potential function. C o m p l e x notat i o n is used, w i t h the real portion representing the physical quantities described. T h e corresponding surface elevation is given by  n(x,t)  = — <t>{x)e- iwt 9  (2.32)  T h e three-dimensional potential $ satisfies the Laplace equation, w h i c h requires that <j> must satisfy the H e l m h o l t z equation w i t h i n the fluid:  <*•> T h e potential <p is also subject to boundary conditions along the generator face a n d all reflective surfaces, and to a r a d i a t i o n condition along energy absorbing sides of the basin. A c c o r d i n g to potential theory, <f> m a y be represented as due to point sources dist r i b u t e d along any reflective surfaces, including the wave boards:  *( g)  =  'hJs /(^  ^' dS  -  (2 34)  Here / ( i f ) is the source strength distribution function, S represents the horizontal contour along a l l non-absorbing surfaces at still water level, a n d dS denotes a differential length along S.  A l o n g this surface, the boundary condition equates the flow velocity  n o r m a l to S to the velocity of active generator segment surfaces, or to zero for quiescent segments and reflective walls. T h e potential at any point x = (x,y) due to a source at point £ = £(x,y) —* described by a Green's function G(x,£)  m a y be  w h i c h represents a concentric wave t r a i n ema-  nating f r o m the point source. T h i s function must satisfy b o t h the H e l m h o l t z equation  2  Theoretical  Background  19  a n d the r a d i a t i o n boundary c o n d i t i o n , a n d is given by  G = inH^ where HQ 1^ (kr)  (kr)  (2.35)  is the H a n k e l function of the first k i n d a n d zero order, HQ 1^ (kr)  Jo (kr) + i YQ (kr),  =  r = \x — £J, a n d J 0 a n d Y0 are zero order Bessel Functions of the  first a n d second k i n d , respectively. T h e surface b o u n d a r y c o n d i t i o n along S is given by the following line integral equation:  i n w h i c h V is the velocity at point x on S where the boundary c o n d i t i o n is applied, a n d n represents the unit direction n o r m a l to <S at point x. E q u a t i o n 2.36 is generally incorrect i n that it assumes the v a r i a t i o n of generator velocity w i t h elevation to follow that corresponding to E q u a t i o n 2.31, such that the shape function f(z) shape function f(z),  = cosh kz j cosh kh.  For the actual case of any other generator  an approximate generator transfer function expression may be  developed analogous to the two-dimensional expression of E q u a t i o n 2.7. T h e resulting expression for the velocity amplitude V(x)  along surface <S, w i t h x o n the generator  face, is  V(x) —  U  a  t a n h kh  — tob ^ ( ^ ) t a n h kh  ^2 37)  T h e value of V is set to zero for inactive sections, as well as at reflective surfaces such as side walls a n d structures i n the test area. T h e integral E q u a t i o n 2.36 for the source strength function f(£)  may be solved  numerically by discretizing the S contour into N short, straight lengths, a n d w i t h a constant source strength over each of these segments, or facets. T h e N  source facet  —* is of length A „ , w i t h a point source of strength / ( 0 n  a  * the facet mid-point l o c a t i o n .  T h e t o t a l number of sources N w i l l generally exceed the number of wave generator  2  Theoretical  Background  20  segment sources Ng . T h i 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 t h a n one source per generator segment may be required. For s m a l l 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 a d d i t i o n a l  sources along the face of the generator segment to more accurately represent an i n creasingly non-circular wave field. Consider each generator segment to be d i v i d e d into  M  equal facets each of length A = B/M.  each facet. C o m p u t a t i o n s for B/L  A point source is located at the centre of  = 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. T o offset the large number of equations required i n a model consisting of m u l t i p l e point sources per segment, Isaacson suggests the use of the asymptotic form of the H a n k e l function for large values of kr. T h i s is given as for kr >  2TT  (2.39)  T h i s a p p r o x i m a t i o n increases computational efficiency at a cost of about two percent i n accuracy.  2  Theoretical  Background  21  —* B y the discretization approach, the integral equation for / ( £ ) is approximated by a m a t r i x equation for the source strengths at each segment: N £  BpnUO  = XP  for p =  ... , N  1,2  (2.40)  n=l  Here, for p = n  1 'pn  i  o  r  p  ¥  z  (2.41)  n  and Ap = <  2ub^ anh2i  ^or Po  0  otherwise  t  n  a  g e n e r a t o r face  (2.42)  In E q u a t i o n 2.41,  MfkM an = -cos^ or = where /? is the angle between the line j o i n i n g fp at the point x*p . In E q u a t i o n 2.42,  -ZTrib  c o s ( * r )  (2.43)  to £ n a n d the o u t w a r d n o r m a l vector  denotes the displacement amplitude of the  generator segment associated w i t h the p^ 1 point source.  3  3.1  E x p e r i m e n t a l Setup  T h e N R C Offshore W a v e B a s i n  A c q u i s i t i o n a n d i n i t i a l analysis of a l l experimental d a t a took place at the H y d r a u l i c s L a b o r a t o r y of the N a t i o n a l Research C o u n c i l ( N R C ) of C a n a d a i n O t t a w a , f r o m June to A u g u s t of 1988. T h e experiments were conducted i n the Offshore E n g i n e e r i n g Wave B a s i n , which has since 1986 been equipped w i t h a segmented wave generator capable of p r o d u c i n g directional sea states. In recent years, increasing emphasis has been placed o n 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 a n d frequency d o m a i n , as well as the space domain. A p o r t i o n of the 60 segment, directional wave machine installed i n the N R C offshore basin is seen i n F i g u r e 5. E a c h of the half-metre wide wave boards is driven by a h y d r a u l i c actuator under i n d i v i d u a l computer control. T h e actuators are linked to lever arms for mechanical stroke amplification 1 . T h 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 m a y 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 m a y be accommodated i n this way, b u t conversion between modes involves the relocation of 60 linkages. T h e piston mode is used most often. T h e waveboards of the N R C segmented generator may be relocated vertically to accommodate different water For signal generation purposes, waveboard displacement £(x) at the still water level z = h is prescribed by the stroke angle of the lever arm. 1  22  3  Experimental  Setup  23  depths. O p e r a t i n g i n piston mode i n an elevated position provides a compromise over the range of waves most frequently encountered i n offshore studies. T h e 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 F i g u r e 6, behind the large t u b u l a r steel structure used to support an array of wave probes. T h e basin is three metres deep, a n d is usually operated at a water depth of about two metres. T h e configuration used throughout the experiments, a n d depicted i n F i g u r e 7, results i n a working area of 30 by 19.2 metres. T h i s corresponds to the layout most frequently used for offshore conditions. T h e end wave absorber along the wall opposite the wave machine is 3.5m i n breadth. T h e wave absorbers positioned i n front of the three passive walls of the basin are based on a design developed at the H y d r a u l i c s Laboratory. K n o w n as a vertical progressive porosity, or upright wave absorber, the design consists of vertical rows of perforated galvanized steel sheets supported rigidly on a t u b u l a r steel frame. M o d u l a r design a n d adjustable fittings allow modifications of the i n s t a l l a t i o n 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. T h i s concept enables development of compact absorbers which l i m i t reflections to below 10% consistently over a wide range of water depths a n d wave conditions encountered i n laboratory studies. F o r details, refer to Jamieson a n d M a n s a r d , 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, a n d are installed for wave generation purposes, as o u t l i n e d i n Section 2.2.1. T h e side w a l l lengths can be varied to suit particular applications. T h e 9.2m length for b o t h side walls used i n the present study was prescribed b y a separate experiment r u n concurrently i n the basin.  3  Experimental  Setup  24  T h e experimental program coincided w i t h a p e r i o d of t r a n s i t i o n w i t h i n the laboratory. A l l aspects of d a t a acquisition a n d analysis, wave synthesis a n d generation were i n the process of conversion from a H e w l e t t - P a c k a r d ( H P ) 1000 computer system to a M i c r o V A X network. A t the time of testing, the d a t a acquisition package was still 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 generator drive signals, the sampling of wave elevation (probe) signals, a n d for subsequent analysis of data.  G E D A P was developed at the N R C H y d r a u l i c s Laboratory, a n d is  an acronym for g e n e r a l i z e d E x p e r i m e n t control, D a t a acquisition a n d d a t a A n a l y s i s Package. D a t a acquisition was r u n on an H P 1 0 0 0 computer operating under software control w h i c h permits multi-user, foreground/background activities (Funke, et. a l . , 1980).  Use of an H P alphanumeric/graphic t e r m i n a l enables full control of d a t a ac-  quisition f r o m a station at the experiment site. T h e a n a l o g / d i g i t a l subsystem  consists  m a i n l y of a N E F F Series 500 serial d a t a link w i t h remote converters. T h e G E D A P syst e m 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 t e r m i n a l . A software package provides commands for various aspects of the segmented wave generator operation, a n d makes use of drive signals created b y G E D A P software for each of the 60 waveboards. G E D A P logically a n d physically organizes the a n a l o g / d i g i t a l interface into analog i n p u t / o u t p u t ports, each consisting of 16 analog input a n d two analog output channels. T h i s arrangement is implemented through a card rack of 16 slots. E a c h measurement instrument is connected to a printed circuit card which fits into one of the 16 slots. T h i s provides a pick-up for power and reference voltages a n d transmission of the analog signal. Successful experimentation depends u p o n reliable c a l i b r a t i o n of instruments.  3  Experimental  Setup  25  These are freed of linearization electronics by implementing nonlinear compensation operation t h r o u g h G E D A P ; this allows use of cheaper, more stable a n d more easily maintained measuring devices.  C a l i b r a t i o n 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 t h e n carried t h r o u g h to data analysis a n d graphic output phases.  3.2  The Wave Probe A r r a y  It was essential to place wave elevation probes at a large number of g r i d positions to measure the wave field throughout the basin. T h e g r i d spacing h a d to be s m a l l enough to provide acceptably fine resolution of wave elevation i n f o r m a t i o n , while not resulting i n a cumbersome quantity of data. T h i 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.  T h e off-  shore basin is s y m m e t r i c a l about a centreline r u n n i n g perpendicular to the wave m a chine face.  Consequently, the entire wave field was i n effect measured while m a k i n g  actual physical measurements over only one half of the basin.  T h i s concept is illus-  trated i n Figure 8. T w o similar wave trains generated at propagation angles of positive a n d negative 6 degrees f o r m a mirrored pair which yields measurements  representa-  tive of conditions throughout the entire test area, a n d corresponding to the positive 8 propagation direction. Figures 7 a n d 8 indicate that measurement over the half-basin g r i d was achieved b y locating a square array of 16 wave probes i n four positions. These probes occupied one i n p u t / o u t p u t port.  A single stationary probe was necessary to reference phase d a t a  between tests. T h i s necessitated the use of a second i n p u t / o u t p u t port.  T h e array  of probes h a d to be supported rigidly to m i n i m i z e instrument v i b r a t i o n due to wave  3  Experimental  Setup  26  action. T h e structure supporting this array h a d to be largely transparent to the waves, a n d light enough to be easily moved between positions. L a b o r a t o r y staff suggested that this large-scale frame needed to hold the 6 by 6 metre probe array be constructed using steel pipe a n d fittings which are stocked for construction of wave absorber modules. B y use of aircraft cable cross braces to stiffen the structure, a r i g i d yet transparent frame evolved. T h e probe array support frame is illustrated i n F i g u r e 9 a n d seen i n Figures 6, 10 a n d 11. C a l i b r a t i o n of wave probes is achieved by i m m e r s i n g each probe i n still water to at least three user-prescribed depths.  A G E D A P p r o g r a m for calibrating  each probe calculates a quadratic p o l y n o m i a l , fit to the voltages registered for each of these depth readings. A s m a l l second order t e r m is indicative of an acceptably linear probe c a l i b r a t i o n . 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 g r i d spacing, the support frame was designed so that the probe array could translate vertically as a u n i t , assisted by a winched cable led from an overhead pulley. T h e probes were mounted on a superstructure w h i c h could slide up and down the vertical corner members of the core structure. In p l a n view, the p r i n c i p a l structure is 4 by 4 metres square, while the probe array superstructure covers a 6 by 6 metre region. T h e probe array was designed so as to minimize the necessary number of placements to acquire measurements throughout the test area. A s seen i n Figure 7, the 6 x 6 m array meets this goal. T h e support structure was stiffened w i t h cable bracing. T h i s ensured m i n i m a l v i b r a t i o n of the large planar superstructure without resorting to heavy reinforcement i n the t h i r d dimension. For calibration purposes, support blocks of precise lengths were fabricated. These snapped i n place on the upright tubes when i n use.  T h i s arrangement is shown i n  3  Experimental  F i g u r e 12.  Setup  27  A s seen i n Figure 13, inserts were fabricated w h i c h fit inside standard  fittings to h o l d probes on the assembly. A l l probes were at least one metre long, a n d were m o u n t e d for about 40% immersion i n still water. R i g a n d probe dimensions were established i n light of requirements for the separate experiment r u n concurrently i n the basin. A s seen i n F i g u r e 6, the array r i g also incorporated components provided for another concurrently r u n experiment. These consisted of an e x t r a pipe length o n the superstructure to h o l d wave probes, a n d a current measuring assembly inside the rig, below still water level. These additions increased the weight of the structure to 300-350kg. F l o t a t i o n balloons o n each corner of the r i g were used to facilitate its movement from one measurement l o c a t i o n to the next. F o r each move, several balloons were secured to the r i g at a submerged level. These provided enough flotation that the author was able to singlehandedly move the r i g a r o u n d the basin, pulling at floor level. T h e services of only one other person were required to assist i n m o v i n g instrument cables suspended from roof girders. Wave probes were calibrated at the beginning of each of two series of tests, first without a n d 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 i g was moved i n a circular loop, so that at the end of the wave field series, the rig was adjacent to the c a l i b r a t i o n 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 i g , a n d at intervals during test sets. These readings do not entail vertical movement of probes, but update probe voltage magnitudes at the still water level. T h e basin water level was m o n i t o r e d for leakage, a n d offsets taken accordingly. Evanescent waves from the generator, described i n Section 2.1, d i m i n i s h to less t h a n 10% of their i n i t i a l height at a distance of about 4.0m away from the waveboards, a n d  3  Experimental  Setup  28  to less t h a n 1% at a distance of 6.0m. T h e wave probe array was positioned so that the closest measurements to the generator were along a line 4.0m from the generator n e u t r a l position. A s is visible i n Figures 14 a n d 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 t h a n two metres from side a n d end beaches.  3.3  T h e G r a v i t y Base Structure  Tests were conducted w i t h a surface piercing, impervious cylinder to simulate the presence of a gravity base structure of large diameter. T h e 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. T h e wave heights chosen were adequate for the measurement a n d observation of a l l pertinent wave effects, yet d i d not cause m o t i o n of the structure, nor severely tax the generator h y d r a u l i c 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. previous G B S study.  T h i s location was close to the position used i n a  It also corresponded to a corner probe location on the array,  when the array was situated i n 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 position, the cantilevered member from which this probe was supported was s i m p l y cut off, a n d the probe removed for the final tests. T h i s configuration is visible i n F i g u r e 17.  4  T h e Test P r o g r a m  T h i s study deals solely w i t h regular, long-crested wave trains.  Tests were grouped  according to two series. T h e p r i n c i p a l series was comprised of wave fields without the disturbing influence of a structure. A shorter series of tests involved wave trains f r o m the first series, but w i t h a large, surface piercing, circular cylinder i n place. T h e tests of this latter series are referred to as ' G B S ' tests, to distinguish t h e m f r o m the wave field tests. Four propagation directions were considered.  In a d d i t i o n to waves  propagating  n o r m a l to the generator face {9 = 0 ° ) , wave trains were directed at three angles to the n o r m a l direction: 3 0 ° , 45° a n d 60°. A pair of tests was r u n for each of the oblique wave conditions: for each wave t r a i n propagated at + 0 ° , a m a t c h i n g test at —9° was conducted. T h e positive sense of 9 is shown i n Figure 2. B a s i n s y m m e t r y allowed wave conditions over the complete test area to be measured by r u n n i n g tests i n such pairs, while only actually m o v i n g the probe array around half the basin.  T h i s concept is  illustrated i n Figure 7. Waves were produced w i t h the generator segments m o v i n g i n piston mode. T h e y were elevated 0.48m above the basin floor. T h i s value corresponds to parameter v of E q u a t i o n 2.8, a n d as seen i n Figure 1.  T h i s configuration has proven adequate for  deep a n d 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 c o m m o n l y encountered i n laboratory studies. These three sets are sometimes referred to b y wavelength as short, intermediate or m e d i u m , and long wave cases, corresponding to periods of  29  4  The Test  Program  30  1.25, 1.75, a n d 2.25 seconds, respectively. For the side w a l l lengths used (9.2m), the 60° wave tests are of l i m i t e d p r a c t i c a l importance. A large p r o p o r t i o n 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 w i t h i n the basin, a n d for assessing the p r e d i c t i o n capabilities of the n u m e r i c a l model. T h e G B S series excluded any tests at 6 0 ° . 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 r u n w i t h waves propagating at 30° without use of the corner reflection generation m e t h o d , for b o t h wave field measurements a n d 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).  T o 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 w i d t h to wavelength ratio B/L a value close to Isaacson's suggested l i m i t of 0.2.  — 0.205,  A n u m e r i c a l m o d e l was r u n to  appraise the effect of using two point sources per generator segment (M=2)  for this  case, compared to only one. T h e short wavelength provides a deep water condition.  T h e m e d i u m a n d long  wavelengths fall 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 i n 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 f r o m 0.027 to 0.082.  T h e upper l i m i t  corresponds to short wave conditions a n d can result i n breaking steepnesses w h e n waves superpose.  4  4.1  The Test  Program  31  Synthesis of D r i v i n g Signals  G E D A P programs are provided for the synthesis of wave generator drive signals. F o r regular waves directed n o r m a l to the generator, a l l segments move i n unison, so that only one signal file is produced. F o r oblique regular waves, i n d i v i d u a l signal files are needed for each of the 60 segments.  E a c h sinusoidal signal is synthesized from 35 or  more discrete values, calculated at a n interval of 0.1 sec. A t least one complete cycle is produced; this is repeated without t r u n c a t i o n as required during operation, thereby avoiding the need for large signal files. Synthesis is executed t h r o u g h a configuration parameter file, which prescribes paddle stroke mode, a n d other options. F o r this study, displacement control of segment motions was used, rather t h a n velocity control. Displacement signals are expressed i n terms of the lever a r m stroke angle. P l o t s of selected drive signals are included to illustrate some aspects of generator motions.  I n 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 - a n d B-wave region. T h e simple snake m e t h o d defines the m o t i o n of segments 12-46, while segments 47-60 are held motionless. A s 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.  T h e former move i n a phase-locked standing wave pattern;  segment 3 is almost quiescent at a n o d a l position. A phase shift is evident between adjacent segments i n the simple snake region of the generator. T w o plots are presented i n F i g u r e 19. P l o t 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 l o 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 E q u a t i o n 2.15.  4  T h e Test  Program  32  T h e d r i v i n g signal synthesis p r o g r a m utilizes the corner reflection technique, unless zero side w a l l lengths are specified. N o r m a l l y the program calculates the number of 'B-wave' segments using E q u a t i o n 2.21. For the purposes of this study, side wall length values w h i c h were prescribed to the synthesis program were altered f r o m the true lengths to improve the wavefield. These values are presented i n Table 2. A l s o specified are the numbers of those segments involved i n combined A - a n d 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. T h i s is reflected from the side wall back into the wave field.  T o lessen this effect,  the apparent side wall length at this end of the generator, a n d hence the number of stilled segments, was increased. A t the other end of the generator, almost a l l energy propagating toward the side wall reflects to j o i n w i t h the p r i m a r y wave t r a i n . However, some wave attenuation does occur at the edge of this B-wave t r a i n .  Increasing the  apparent side wall length slightly may send a 'wider' wave toward the side wall, reducing the degree of attenuation at the outer end of the wall. A n y 'excess' wave passes into the side beach beyond the w a l l . T h i s second change was used to a m i n i m a l extent, setting side wall lengths at 9.3m rather t h a n the actual value of 9.2m. In only one case, that of the short wavelength at 9 = 6 0 ° , d i d this i n fact result i n a larger (integer) number of segments being assigned to B-wave p r o d u c t i o n .  5  D a t a Analysis  Capacitive-wire wave probes were the only instruments used.  These provided t i m e  series elevation records at discrete locations a r o u n d the basin. Seventeen probes were used, consisting of sixteen probes w h i c h formed the movable array, a n d one reference probe on a stationary, guyed pole situated along the basin centreline. A l l necessary signal c o n d i t i o n i n g was achieved through analog cards on the electronics racks. T h e chief a n a l y t i c a l tool used was a modified version of a powerful p r o g r a m k n o w n as S C R E N . T h e name refers to its function of m u l t i p l e regression screening of a given time series. T h e program was developed i n the N R C H y d r a u l i c s L a b o r a t o r y for the analysis of irregular wave trains, a n d was modified to accommodate harmonic components of regular waves. T h e 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 p r i n c i p a l frequency components of a measured wave t r a i n 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 i n the root program are as follows. Before reading the data, the p r o g r a m requests values of certain parameters which define the desired analysis. O n e 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. T h e d a t a file is then read a n d the mean value removed, which may be reinstated if requested. T h e presence of a mean value indicates a change i n still water level since the most recent probe offset reading. T h e R M S value of the time series is then determined.  33  4  T h e Test  Program  34  Users specify the number of parameters to be o p t i m i z e d . A three parameter fit optimizes the a m p l i t u d e , phase a n d frequency of each fitted sinusoidal signal, whereas a two parameter fit optimizes the amplitude a n d phase, but not the frequency. In the c o m p u t a t i o n loop for the current frequency component, the t i m e series u n dergoes a Fast Fourier Transformation a n d 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 w h i c h this occurs.  T h i s m e t h o d is sensitive  to a sample d u r a t i o n w h i c h is not an integer m u l t i p l e of the p e r i o d of the selected wave component. Better amplitude a n d frequency approximations are next made b y quadratic interpolation. These values are then used as i n i t i a l guesses i n a nonlinear regression o p t i m i z a t i o n routine w h i c h fits a sinusoidal curve to the data. T h e regression is performed using the Gauss-Newton a l g o r i t h m , which is perhaps the most efficient a l g o r i t h m available for least squares curve f i t t i n g 2 . Corrections to starting values of parameters are computed b y iterative cycles u n t i l the change i n the error s u m of squares between successive iterations falls below a user specified tolerance. T h e phase angle is also determined using quadratic interpolation. H a v i n g established the final estimates of these values, the computed sinusoid is removed from the time series, a n d the procedure recycles for the next frequency, if requested.  T h e p r o g r a m cycles u n t i l either  the prescribed number of frequency components have been extracted, or the variance of the amplitude exceeds a l i m i t i n g percentage of the original time series R M S value, whichever comes first. T h i s l i m i t 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 a n d phase. T h e p r o g r a m outputs four files, each of w h i c h contain a l l pertinent parameter values i n a c o m m o n header. T w o time series are output. T h e synthesized time function is the 2 For 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  s u m of a l l sinusoidal functions at their computed amplitude a n d phase. A l s o output is the residual time series remaining after the mean a n d a l l o p t i m a l l y fitted sinusoids have been removed f r o m the original time series. T h e other two files contain line spectra, of the amplitudes a n d the phases of the sinusoidal time functions, each plotted as functions of frequency.  5.1  Modifications to the S C R E N  Program  T h e S C R E N p r o g r a m was modified to extract harmonic components f r o m the elevation time series. T h e flowchart presented i n F i g u r e 20 illustrates key operational steps. A n input parameter indicates the number of harmonic components to be analysed. If given a value greater t h a n the default value of one, this parameter causes a two-parameter fit (amplitude a n d phase) to be performed for the second a n d higher harmonics. O t h erwise, the p r o g r a m 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 w o u l d treat the second harm o n i c 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- T h i 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. T h 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. T h i s w o u l d cause the program to perform a fit at a frequency shifted away from the actual fundamental value. T h e 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 r u n for the reference probe channels of a l l tests of the same wavelength. T h e mode of these values was used as the fixed fundamental value  fa.  A s an illustration of this procedure for the case of the intermediate wavelength,  the desired frequency is 1/1.75 = 0.57143Hz, whereas o p t i m i z a t i o n of amplitude a n d 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 a n d 4 x 0.5714. T h e value used for fG  was 0.5712Hz.  T h e S C R E N program was modified to provide basic wave parameters of the fundamental component, which represents the generated wave t r a i n w i t h o u t its b o u n d harmonic component. T h e user specifies the still water depth. F r o m this a n d the fundamental frequency value fo,  the wave number, wave length a n d phase velocity are  obtained on the basis of linear wave theory. T h e synthesized time series consists of superposed fundamental a n d second harmonic components. In a p r o g r a m loop, m a x i m u m a n d m i n i m u m values of this nonsinusoidal waveform are stored, yielding the wave height (Hsy).  T h e p r o g r a m stores this a n d 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 p l o t t i n g routines is an i m p o r t a n t 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, f r o m a test involving the intermediate wavelength, propagating at 8 = 30°. T h e test location is indicated i n Figure 21(a), for a positive propagation angle.  5  Data  Analysis  37  For every test set, one page was devoted to the reference probe (number 17) positioned along the basin centreline, 7.37m i n front of the generator n e u t r a l position. T h i s page is presented i n Figure 21(b), and consists of two plots a n d a listing of pertinent parameters. Prescribed values of computational limits are listed along w i t h the number of parameters o p t i m i z e d (2 = a m p l i t u d e a n d phase). F u n d a m e n t a l a n d second harmonic values of frequency, amplitude a n d phase are displayed, as well as the wave parameters calculated b y the modified S C R E N program. A l s o shown are the magnitude of the mean value which was removed by the program, the R M S value of the original measured time series, a n d of the synthesized a n d residual time series.  RMS  values offer an i n i t i a l measure of the quality of fit of the synthesized series compared to the measured trace. T h e plot along the top of the page is the entire 50 second record of the time series measured by the reference probe. T h e larger plot consists of three superposed traces for the first ten seconds of the sample. T h e solid trace is that of the original measured time series.  T h e dashed line which roughly m i m i c s 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. T h e remaining dotted trace of smaller magnitude corresponds to the residual time series of higher harmonic a n d other components. Quick inspection of the residual trace affords some insight into the magnitude a n d nature of effects other t h a n 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. E x a m p l e s of these are presented i n Figure 21(c)-(f), w h i c h relate to the probe 17 plot of Figure 21(b). I n d i v i d u a l probe locations are seen i n F i g u r e 21(a). T h e 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 v i s u a l record of a l l probe wave elevation records for each test set. For any one wave t r a i n and propagation direction, d a t a sets f r o m eight tests were u n i t e d to represent wave conditions throughout the basin test area. U s i n g a spreadsheet program, these eight d a t a sets were merged, then m a n i p u l a t e d to create d a t a files of fundamental a m p l i t u d e ( A i ) , synthesized wave height (Hsy),  a n d phase p r i o r to  plotting. T h e G o l d e n G r a p h i c s System, a p l o t t i n g package for personal computers, was used to generate a l l plots of basin conditions from these d a t a files. F i r s t , a g r i d d i n g program creates a regularly spaced g r i d from a data file. For full basin plots, 90 g r i d lines along the longest side were used. A 45 line g r i d was specified for half-basin plots. T h i s g r i d size represents a six-fold increase i n resolution over the original spacing of d a t a points as set by probe locations. A n increase i n the density of the g r i d results i n improvements i n the accuracy a n d smoothness of the plot, at the expense of increased c o m p u t a t i o n a l effort. T h e smoothness of plots generated is a function of the original data, the g r i d density a n d the value assigned to a smoothing factor w h i c h ranges f r o m 0 to 1. A value of 1 corresponds to no smoothing, and yields a plot w i t h very jagged extrema.  A large  degree of smoothing has the effect of flattening peaks a n d troughs to the extent that important information is lost. T h e default value of 0.95 was applied throughout this study. Phase plots were produced by manually m a n i p u l a t i n g the phase predictions generated by the S C R E N program. Phase values for a series were related between different test sets t h r o u g h the reference probe. For each set, the phase value corresponding to the reference probe was subtracted from a l 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 all probes i n a series of eight test sets were then related. A t this p o i n t , phases ranged  5  Data  Analysis  39  f r o m 0° to 3 6 0 ° , which resulted i n problems when a contouring p r o g r a m evaluated neighbouring c r e s t / t r o u g h contours. For this reason, phase values were increased or decreased b y 180° increments i n p r o p o r t i o n to distance away f r o m the reference probe location, i n the propagation direction. T h e contouring p r o g r a m 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. T h e 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 a n d troughs. A n o t h e r p r o g r a m which uses the same gridded d a t a produces three-dimensional perspective block diagrams for surface representation. V e r t i c a l exaggeration of surface features is controlled by a h e i g h t / w i d t h ratio parameter.  6  6.1  Results and Discussion  Synopsis  E i g h t sets of test d a t a were combined to f o r m a complete d a t a file for each case of a p a r t i c u l a r wavelength a n d propagation direction.  These consist of four test sets  corresponding to each of the m a t c h i n g positive a n d negative propagation directions. For the 9 = 0° cases, basin s y m m e t r y enables evaluation over only half the basin, such that only four tests were merged. Table 1 provides a s u m m a r y 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° a n d 6 0 ° , while for the G B S tests w i t h the cylinder i n place, the 9 = 60° case was excluded. E x p e r i m e n t a l m a x i m u m waveheight plots are presented for a l l but tests 9 a n d 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 experimental waveheights a n d numerical predictions. Figures 60 to 62 consist of numerically derived phase plots, a n d presented i n F i g u r e 63 is a three-dimensional instantaneous surface elevation plot f r o m one numerical r u n . Included i n the first oblique wave m a x i m u m waveheight plot presented for each of the three wavelengths investigated is a scaled representation of the pertinent wavelength, as shown by a group of three wave crests. These are Figure 26(c) for the shortest wavelength, F i g u r e 27(c) for the intermediate length a n d Figure 31(c) for the longest wave. For a l l other waveheight plots, propagation direction is indicated b y an arrow.  40  6  Results  and  Discussion  41  T h e x — y coordinate axes are depicted i n F i g u r e 2 w i t h the origin at the m i d p o i n t of the first generator segment face. figures  T h i s corresponds to E q u a t i o n 2.16.  In a l l other  showing these axes, the origin is positioned adjacent to the side w a l l , which  simplifies p l o t t e d results a n d coordinate values prescribed for n u m e r i c a l 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 a n d i n Figure 24, concerning the repeatability test.  T h i s contour interval was  also used for the plots of Figures 29 a n d 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 p a t h of the wave t r a i n 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, w h i c h include second order waveheight plots.  Second order  a n d other nonlinear components were consistently an order of magnitude smaller t h a n the fundamental, linear wave components. T w o contour intervals are also employed i n the plots of Figures 41-43. These involve comparison of experimental a n d n u m e r i c a l waveheight results for tests at 8 — 0 ° . These plots illustrate the reduced reflection levels present i n these tests, a n d that the linear diffraction numerical m o d e l underestimates the effects found i n the basin, because no account is made of nonlinear contributions. T h i s results i n shallower waveheight gradients, and hence, sparser contours t h a n 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 n u m e r i c a l results f r o m experimental plots. However, findings would not likely warrant such effort. Second order  6  Results  and  Discussion  42  effects are responsible for most differences between experimental a n d numerical waveheights. These effects vary i n prominence around the basin, a n d f r o m one test c o n d i t i o n to the next. E r r o r plots are not expected to reveal trends not already apparent i n the waveheight plots. E n e r g y reflected f r o m the downstream side wall f r o m oblique wave trains usually produces a p a r t i a 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 = 6 0 ° , a standing wave between the side walls was produced, w h i c h 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 i m p o r t a n t to include the upstream side wall as part of the basin boundary i n numerical models. Boundaries were most precisely modelled for tests involving the simple snake m e t h o d of generation. Corresponding model predictions agree very well w i t h experimental findings. T h i s i l lustrates that to achieve o p t i m u m linear model predictions, basin boundaries should be modelled as accurately as possible. Readers m a k i n g 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. O t h e r features visible i n Figures 21 a n d 64 are considered briefly i n Section 6.7.  6.2  Effect of Prescribed Waveheight  Referring to Table 1, tests 4 a n d 13 b o t h correspond to the case of the short wavelength a n d 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  T h e measurements f r o m test 13, over half the basin, allow comparison w i t h the full waveheight result of test 4. For test 13, a drive signal span of half the value used for test 4 was specified. T h e resulting waveheights recorded were generally only about 45 percent of the full height values, i n consequence of the nonlinearity of the propagating waves. I n Figure 22, two separate synthesized wave height (H3y) Refer to Section 6.4 for a definition of Hsy. waveheight H,  H  = 20cm.  plots are presented.  These plots are normalized to the desired  corresponding to either H =  10cm or the full waveheight value of  P l o t s cover only the half of the basin over w h i c h measurements  were  actually taken for the half height test 13. In general, the half height plot bears a strong resemblance to its full height counterpart, i n d i c a t i n g that m a x i m u m waveheight findings are independent of the prescribed waveheight. T h e fact that the smaller waveheight was about ten percent lower t h a n the target value is evident when the two plots are compared. T h e plots are most not a b l y s i m i l a r i n the quiescent zone i n the top left corner of each plot. T h e only m a j o r discrepancy is found i n a h i g h waveheight region indicated by a ' D ' i n the half height Figure 22(b). T h e bracketed region ' R ' of the full height plot of F i g u r e 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 a n d troughs. T h e zero phase line intersects the basin centreline at the reference probe location. T h e half height plot is shown dotted. T h e figure shows that the wavelength of the full height wave t r a i n is greater t h a n that of the half height case.  T h e full height crests a n d troughs fall  increasingly ahead of the half height contours, i n the 'downstream' direction. T h i s is another consequence of the nonlinearity of the waves.  6  6.3  Results  and  Discussion  44  Repeatability Test  A s seen i n F i g u r e 24, a p a i r of normalized Hsy  plots corresponding to tests 4 a n d 14  ( T = 1.25sec, 8 = 3 0 ° ) was obtained i n order to assess the repeatability of results. These plots pertain to duplicate sets of tests at a single array position, as indicated by region R i n Figure 22. T h e y show consistent repeatability throughout the region. W h i l e m i n o r discrepancies on the order of ten percent result i n some alteration of contour shapes, similar trends are apparent i n b o t h plots. E x c e p t 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, a n d exhibit relatively strong second harmonic components. T h e two superposed phase plots are shown i n F i g u r e 25, a n d display excellent agreement.  T h e 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 t h a n i n Figure 23.  6.4  Maximum Waveheight Plots  A l l contour plots presented were generated f r o m grids produced w i t h a low degree of smoothing. T h i s results i n negligible loss of i n f o r m a t i o n , at the expense of unrealistically jagged peaks at the probe locations. Trends indicated by contours surrounding these peaks are fairly representative of actual levels between probes.  P l o t s may be  compared more closely t h a n if greater smoothing h a d been used. For a l l series of four or eight test sets, contour plots of fundamental amplitude A\ a n d synthesized waveheight Hsy  were produced. Values of Hsy  were normalized w i t h  respect to the desired waveheight H; Ai values were n o r m a l i z e d w i t h respect to  H/2.  6  Results  and  Discussion  45  Considering a regular wave t r a i n as having several superposed h a r m o n i c components, the elevation t i m e series at any point i n the wave field is given b y CO  r] = ^2 An cos(nut  - <f>n)  (6.1)  71=1  T h e 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)  T h e n , the synthesized waveheight Hsy is given b y  Hsy  = 2(r]Sy)ma,x  (6.3)  where ?7max signifies the m a x i m u m value of rj w i t h respect to time.  T h e residual  time series contains higher harmonic a n d other contributions not represented by the fundamental plus second harmonic waveform of height Hsy. C o m p a r i n g pairs of A\ and Hsy plots, only m i n o r differences are apparent. C o n t o u r plots of the second harmonic amplitude A2 were also generated for a number of series. H a v i n g n o r m a l i z e d the Ax results to correspond to Hsy plots, Ax = 1.0 is equivalent to the desired a m p l i t u d e H/2 = 0.1m. Hence, the A2 values were m u l t i p l i e d by a factor of ten before p l o t t i n g , i n order to be comparable w i t h the normalized Hsy a n d Ax plots. It is perhaps appropriate, at this point, to briefly outline second h a r m o n i c wave components.  A regular wave is never t r u l y linear; linearity implies that the wave  elevation time record is a pure sinusoid. I n fact, measured wave profiles are distorted, such that troughs are flatter, a n d crests steeper t h a n for linear waves. A consequence of this is that troughs are wider, and crests narrower t h a n for sinusoids, so that about 60 percent of the t o t a l waveheight lies above the still water level.  A regular wave  always binds some higher harmonic energy which propagates at the same celerity as  6  Results  and  Discussion  the fundamental wave component.  46  T h i s b o u n d second harmonic component has a  frequency w h i c h is twice the fundamental value. It is the p r i n c i p a l cause of the nonlinear distortion of the wave profile, a n d is p a r t i c u l a r l y significant i n shallow water situations. In a d d i t i o n to the b o u n d 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 a n d b o u n d second order components. T h e result is a beating oscillation of the envelope height i n the flume.  T h e free harmonic is p a r t l y due to  linear wave generation, which does not satisfy the second order b o u n d a r y conditions pertaining to the b o u n d second harmonic. T h e amount of free second order energy generated i n this way is dependent on the water depth, the wave length a n d height of the fundamental component, a n d the mode of operation of the wave generator. A n o t h e r 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 r e m a i n b o u n d to the reflected fundamental. In F i g u r e 26, contour plots of A\, A2 and H3y  are presented for test 7, for the  wave field corresponding to the short wave propagating at 4 5 ° ( T = 1.25sec, 6 = 4 5 ° ) . T h e magnitude of the A2 component i n this case is relatively significant, compared to most other tests. A h i g h A2 component is apparent i n region ' A ' of F i g u r e 26(b), one metre away f r o m the 'upstream' side wall i n the corner reflection region. T h i s may be evidence of free harmonics caused by reflection f r o m the side wall. A l t e r n a t i v e l y , the h i g h second harmonic reading could be due to a relative phase difference of about 180° between the two crossing wave trains i n this region. T h e program S C R E N may treat the indirect B-wave t r a i n as a second harmonic associated w i t h the direct A-wave train. A s i m i l a r l y h i g h A2 region is apparent i n the corner indicated by label ' B ' . T h i s  6  Results  and  Discussion  47  may be due to s m a l l reflections f r o m the m u t u a l l y perpendicular beach faces. A n o t h e r region of h i g h A2 values is indicated by label ' C i n F i g u r e 26(b).  Elsewhere i n the  basin, the normalized A2 values range from about 0.1 to 0.14, a n d likely represent the b o u n d second harmonic component of this short, but fairly steep wave t r a i n . A s presented i n Figures 27 a n d 28, another two sets of A\-A2-Hsy  plots depict  conditions w i t h o u t a n d w i t h the G B S , respectively. These correspond to tests 5 a n d 20, a n d p e r t a i n to the m e d i u m wavelength ( T = 1.75sec) propagating at 6 = 3 0 ° . Very little difference is observed between the A\ a n d Hsy  plots i n these two figures, compared  to the plots of Figure 26. T h e ubiquitous b o u n d second h a r m o n i c component of this longer, flatter wave t r a i n appears i n the A2 plot, F i g u r e 27(b), to be about 0.08 i n magnitude, compared to 0.1 or greater for Figure 26(b) of test 7. A s i n Figure 26, a region of h i g h 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 a n d 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 h i g h second harmonic component upstream of the cylinder. T h e points of highest A2 values i n Figure 28(c) are characterized by low standard error magnitudes, i n d i c a t i n g fairly accurate synthesis.  6.4.1  Plots For Normally Propagating Wave Trains  For wave trains directed normally f r o m the generator face, side walls should ideally extend as far as the end beaches. beaches w o u l d be eliminated.  A s such, attenuation of waveheights due to side  However, as discussed i n Section 2.2.1, side beaches  are installed i n a d d i t i o n 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. D u e to slight imperfections i n waveboard synchronization, cross-waves are produced, which are largely confined to the region near the generator, and which undergo reflections between the side walls. T h i s effect is p a r t i c u l a r y pronounced for side walls w h i c h extend along the entire length of the basin. Figure 29 pertains to tests 1 and 2, corresponding to 9 = 0° , a n d T = 1.25 and 1.75sec, respectively.  F i g u r e 30 shows corresponding plots for tests 3 a n d 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 u p o n waves r u n n i n g parallel to the smooth side w a l l faces may be considered negligible.  A s the waves r u n past  the side wall ends, they encounter rougher walls formed b y the outermost sheets of the vertical, progressive porosity wave absorbers. In a d d i t i o n , the region of still water inside the absorbing structures results i n diffraction into the beach of a p o r t i o n of the waves w h i c h h a d previously been r u n n i n g entirely parallel to the beach face. Wave attenuation patterns are visible i n the plots of Figures 29 a n d 30, as seen b y the wake shaped region ' A ' of Figure 29(a). T h e 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 w a l l to intersect the unit waveheight contour, which represents the border of the attenuation zone. T h i s is shown as line ' B C ' i n Figure 30(a). T h e perpendicular distance from this point to the side beach, as seen by line ' C D ' , was then measured. T h e ratio of this value to the wave length gives a non-dimensional measure of the i n t r u s i o n of the attenuation effect into the wave field. T h i s ratio ranged from about 0.35 to 0.56, i n d i c a t i n g that the extent of these attenuation zones is reasonably similar for the three wavelengths tested, when scaled according to wavelength.  The  relative effects of friction 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 frictional effects along the beach faces.  6.4.2  Plots For Oblique Wave Trains  M a x i m u m waveheight plots f r o m several oblique wave field tests are presented i n Figures 31 to 33. These are representative of Hsy wave trains.  results f r o m the other tests involving oblique  Figure 31 compares tests 4, 5 a n d 6, corresponding to 8 =  T = 1.25, 1.75 a n d 2.25sec, respectively. F i g u r e 32 presents Hsy  30° a n d  plots of tests 5, 8 a n d  11, which p e r t a i n to the m e d i u m wavelength ( T = 1.75sec), at each of 8 = 3 0 ° , 45° a n d 60°. F i g u r e 33 shows the Hsy  result of test 12, corresponding to T = 2.25sec a n d  9 = 60°. O f the three oblique propagation directions tested, 30° is the direction most likely to be used i n practice, for the length of side walls used throughout the tests. Figure 31 exhibits the 8 = 30° results at all three wavelengths under study. T h e short wavelength test 4 is the only case for which the d a t a acquisition failed. Contours are absent f r o m the region not covered b y any probe points, a n d the probe array position is indicated by a dashed line.  Neighbouring contours suggest trends by e x t r a p o l a t i o n back into  the blank region. T h e u n i t waveheight contour may be thought to r u n approximately parallel to the 0.5 to 0.9 contours, at about 3 5 - 4 0 ° to the basin side. T h i s concurs w i t h the theoretical p r e d i c t i o n that the unit waveheight contour should f o r m an angle w i t h the side about 8° greater t h a n the propagation direction. T h e u n i t contour, b y this assumption, intersects the side w a l l near x = 4 m . Despite inaccuracies, it is clearly apparent that this point of intersection lies well before the end of the side w a l l . T h i s is evidence that the indirect or B-wave propagating toward the side wall is subject to diffraction a n d attenuation by lateral energy transfer along the crest, such that it impacts the outer p o r t i o n 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. T h e measured waveheight approaches the desired magnitude well outside of the geometric quiescent zone boundary. T h e 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 t h a n the desired value H. T h i s may be due to a standing wave system i n the basin, or to energy propagating across the p a t h of the direct wave field, having been reflected f r o m the side w a l l . T h e plot of F i g u r e 31(b), for test 5 at T = 1.75sec, shows s i m i l a r trends. T h e unit waveheight contour bordering the diffraction zone forms an angle of about 35° w i t h the basin side, a n d intersects the side wall at about x — 3 m . C o m p a r e d to the less certain short wave results of test 4 i n Figure 31(a), it appears that the p r i m a r y diffraction zone is larger for this longer wave, and that attenuation of the B-wave is also more pronounced. M o r e energy enters the quiescent zone. T h e unit waveheight contour is thereby shifted closer to the geometric border of this zone. Conversely, more energy is reflected back i n t o the basin from the quiescent end side w a l l . T h e central region of the basin is characterized by contours w h i c h r u n approximately parallel to those bordering the diffraction zone. T h e pattern suggests alternating high a n d low regions of a p a r t i a l standing wave system set up i n the basin perpendicular to the propagation direction. P l o t 31(c) for the long wave case of test 6 is less informative. T h e unit waveheight contour intersects the basin side between x = 3 m and x = 4 m , at an angle of about 4 0 ° . T h e quiescent zone is less clearly apparent t h a n for the other wavelength cases. In the central region of the basin, a pattern of alternating high a n d low sections is visible. These apparent node/antinode regions are i n this case oriented w i t h their long axes perpendicular to, rather t h a n parallel to the incident wave propagation direction. L o n g waves incident u p o n the beach structures are absorbed less effectively t h a n at shorter wavelengths. E n e r g y propagated i n the incident direction is p a r t i a l l y reflected from  6  Results  and  Discussion  51  the absorbers a n d 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 t h a n for the shorter wave cases, indicating 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 3 0 ° , 45° a n d 60°. T h e 30° plot of F i g u r e 32(a) is the same case as presented i n F i g u r e 31(b). T h e 45° result i n Figure 32(b) strongly resembles the 30° plot. T h e diffraction zone boundary of unit waveheight is inclined at about 50° to the basin side, or 5° greater t h a n the propagation direction.  It intersects the side w a l l  between about x = 5 a n d x = 7m., Hence, at this increased angle, attenuation of the B-wave prior to reaching the side wall is of decreased importance. T h e quiescent zone is about the same size as i n the 6 = 30° case of plot 32(a). T h e system of nodes a n d antinodes i n the central region is more distinct and extends further toward the wave generator t h a n at 9 = 3 0 ° . Energy reflected from the quiescent zone side wall may be the p r i n c i p a l cause of this effect. Some of the B-wave p o r t i o n w h i c h propagates beyond the side w a l l end may be reflected first from the side beach, then from the end beach, so as to pass back t h r o u g h the basin central region. Waves reflected from the quiescent zone side w a l l appear to contribute to this effect i n that sector of the test area. Nodes a n d antinodes appear to range from about 0.6 to 1.4 times the desired waveheight, suggesting that reflection on the order of 40% is occurring i n the basin. R i g orous tests are yet to be conducted to appraise the performance of the vertical wave absorbers w i t h waves impinging obliquely u p o n the face of such a structure. Reflections likely increase directly w i t h propagation direction 0, due to reduced apparent porosities of absorber sheets compared to the case of waves s t r i k i n g from a n o r m a l i n cident direction. T h e 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, a n d is beyond the scope of the present work. L o o k i n g to the corner reflection region of plot 32(b), a standing wave p a t t e r n of decreasing envelope height is apparent, i n the y direction away f r o m the side wall. Evanescent waves from B-wave components produced b y generator segments near the side wall appear to be superposed w i t h waves reflected back from the far side wall. These generator segments move w i t h about twice the n o r m a l stroke to simulate the effect of v i r t u a l segments, resulting i n extreme waveheights near the corner. T h e 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 n o d a l position roughly coincides w i t h y = Om. T h i s could indicate a condition 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 wall simulate open ocean waves entering the basin at that point. V i s u a l observation of the wave m o t i o n discloses a p a r t i a l standing wave i n 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. F o r example, at  8 — 4 5 ° , diffraction into the quiescent region should not contribute m u c h 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. T h i s suggestion is supported by observations that the beating envelope amplitude is greatest i n the corner reflection region, where segment amplitudes are greatest, a n d falls markedly into the quiescent region where segments are motionless. T h i s beating carries t h r o u g h 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.  T h e plot shows an  extremely contaminated wave field, and a diminished area over w h i c h waveheights close to the desired value are found. T h e diffraction region extends over more t h a n a t h i r d of the basin. Whereas the far side beach likely performs better t h a n 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 a n d the front line of wave probes at x = 4 m . Three large regions of excessive waveheights are seen. T h e largest a n d highest of these is a region near the reflected-wave corner, due to the combined effects of the corner reflection process a n d the standing wave system. Reflections f r o m the quiescent end side w a l l back into the wave field account for the high waveheight region near that side. A n o t h e r region of high waveheights is found i n the central p o r t i o n of the basin, as at the other propagation angles. T h e 60° case for the longest wavelength is shown i n F i g u r e 33, a n d corresponds to test 12. A l m o s t a l l wave energy is seen to be trapped between the side walls as a standing wave system. T h i s was investigated further by use of an equation for resonance between two solid walls, provided i n the Shore Protection of the  Manual (1984). T h e p e r i o d  mode of oscillation is given as for n = 1,2  ...  (6.4)  where A is the basin w i d t h of 30m. For the eighth mode, this predicts a period of 2.27 seconds, w h i c h is very close to the 2.25sec period of the wave t r a i n . T h i s corresponds to a standing wave envelope wavelength of 7.5m, w h i c h is nearly a m u l t i p l e of the 2m probe spacing, a n d agrees well w i t h the contours as plotted from the d a t a points. T h e Hsy  plot presented i n F i g u r e 34 corresponds to test 15, involving the simple  6  Results  and  Discussion  54  snake m e t h o d of generation without corner reflection, for the longest wave directed at 3 0 ° .  T h i s plot relates to plot (c) of F i g u r e 31, w i t h corner reflection.  A larger  diffraction zone is evident t h a n when the corner reflection technique is employed. Near the downstream side wall, waveheights are well above the target value, p a r t i c u l a r l y near the outer end of the side wall. T h e importance of h o l d i n g generator segments adjacent to the downstream side w a l l motionless is clearly apparent, by comparison w i t h F i g u r e 31(c).  F u r t h e r into the incident wave field, reflections result i n a region of standing  wave activity. T h i s greatly reduces the effective test area i n the basin, w h i c h is seen to be l i m i t e d to a s m a l l zone near the basin centre a n d 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° w i t h the cylinder i n place. T h i 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 a n d location. T h e entire region i n front of the cylinder is far more complicated by small regions of higher or lower m a x i m u m waveheight. T h e quiescent zone is greatly reduced i n size. T h e cylinder reflects energy back toward the side walls a n d the generator face. T h i s is seen b y 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.  A n o t h e r such point lies' abeam of the cylinder w i t h respect to the incident  wave direction, a n d corresponds to a point of antinodal superposition of incident a n d perpendicularly propagating reflected waves.  6.5  P h a s e P l o t s F r o m the E x p e r i m e n t a l 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 l o c a t i o n . A pattern of alternating crests a n d troughs corresponds to a 180° contour interval. T h e shorter the wavelength, the coarser the resolution of the d a t a g r i d w i t h respect to wavelength. For the short wavelength tests of T = 1.25sec, plots show reasonably straight crests/troughs.  F i g u r e 36 presents the 30° result for test 4, w i t h one d a t a set  missing i n the corner reflection region of the basin. T h e 4 5 ° result of test 7 is found i n F i g u r e 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. E a c h plot hints at such curvature i n b o t h the p r i n c i p a l diffraction zone a n d the quiescent zone, especially the 45° case. T h e corresponding 45° plot for the m e d i u m wavelength ( T = 1.75sec) result of test 8 is presented i n F i g u r e 38. In this case, increased d a t a resolution relative to wavelength effectively magnifies m i n o r phase discrepancies between neighbouring probe locations. A s a result, c r e s t / t r o u g h contours appear less straight t h a n for the short wave. T h i s plot does, however, appear to show the curvature effect more strongly. One may visualize the diffraction b o u n d a r y at about 10 degrees greater t h a n the propagation direction. T h e corresponding plot for the G B S series closely resembles this wave field plot, i n d i c a t i n g that fundamental phase results are largely unaffected b y the presence of the cylinder, a n d are repeatable. T h i s is the extent of usefulness of the phase d a t a i n this f o r m .  T h e long wave  plots are less informative, w i t h contours having greater tendency to meander.  Small  differences i n frequency result i n large differences i n 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 b y the finite sampling rate (l/10sec) used. Some loss of information inevitably occurs, p a r t i c u l a r l y near wave elevation peaks.  6  Results  6.6  and  Discussion  56  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 University of B r i t i s h C o l u m b i a . T h e 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 comparison w i t h the experimental findings f r o m the N R C wave basin. M o d e l predictions of waveheight, phase a n d direction are produced at prescribed points i n the testing area. T h e p r o g r a m treats circular, rectangular or arbitrary basin configurations. Generator segment faces and reflecting walls are represented b y a d i s t r i b u t i o n of discrete sources w i t h i n facets around the basin boundaries, using the b o u n d a r y element m e t h o d . Users specify still water depth, desired waveheight, p e r i o d a n d propagation d i rection. Generator segments are modelled b y fixing velocities of adjacent fluid particles to appropriate values calculated at the centre of each facet of the boundary. Reflective walls a n d structures are treated as perfectly reflecting surfaces, b y fixing fluid velocities n o r m a l 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. T h e number a n d locations of points i n the test area at w h i c h the wave potential is to be evaluated are specified b y the user. For each model r u n , predictions were made at points corresponding to the same full basin, two-metre g r i d as i n the physical experiments. T h i s ensured full compatability between plots of the physical a n d numerical model results. T h e rectangular basin configuration corresponds to a full bank of active generator segments along one boundary, flanked by p a r t i a l side walls of equal length adjacent  6  Results  and  Discussion  57  to each end of the wavemaker. T h i s configuration was suitable for the 8 = 0° cases, a n d 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, a n d the number of sources a n d facet w i d t h for the generator bank a n d for the side walls. of each facet.  O n e source is located at the centre  F u r t h e r modification of the S E G E N p r o g r a m is required to include  generator motions which produce B-wave components at one end, a n d a sector of quiescent generator segments at the other end. Such modification was not warranted for the purposes of the present study. T h e a r b i t r a r y basin configuration o p t i o n was employed i n order to model wave generation w i t h corner reflection a n d quiescent sectors, a n d w i t h a reflecting structure i n place. R e c a l l i n g that the corner reflection technique is based on the concept of v i r t u a l segments extending beyond the actual generator end, this concept was u t i l i z e d b y removing the reflection-end side wall, a n d adding the appropriate number of generator segments i n the negative y direction, according to E q u a t i o n 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 a r b i t r a r y basin configuration is illustrated i n F i g u r e 40. Input files include the number a n d location of generator a n d side wall segments, a n d of test area d a t a points, as well as still water depth, desired wave height, period a n d propagation direction. A l l models used a facet length of 0.5m, except for a single case i n w h i c h this length was halved to 0.25m, to provide two sources per generator segment.  6  Results  6.6.2  and  Discussion  58  Waveheight Plots of Numerical Model Predictions  6.6.2.1  Normally Propagating Wave Trains  N u m e r i c a l runs at 6 = 0° made use of the rectangular basin configuration, except when the c y l i n d r i c a l structure was included. E x p e r i m e n t a l a n d n u m e r i c a l m a x i m u m waveheight results for the three wave field cases are compared i n Figures 41, 42 a n d 43, corresponding to tests 1, 2 a n d 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. P l o t s of full-basin numerical predictions display symmetry about the basin centreline, as expected, a n d a i d visualization of the half-basin experimental plots over the entire basin. T h e numerical models predict effects to be less pronounced t h a n were actually measured.  Note that model predictions are linear i n nature, a n d so are analogous  to experimental A\ fundamental wave magnitudes. are i n fact compared w i t h experimental Hsy measurements  However, these linear predictions  findings.  P r e d i c t i o n s agree well w i t h  i n light of this, and considering the absence of reflective boundaries  where the beaches were positioned.  A l s o , i n reality harmonic a n d other nonlinear  effects of lesser significance are present. T h e zones of waveheight attenuation along the side beaches are predicted well b y these models w h i c h impose no boundary conditions other t h a n at the generator and fully reflecting surfaces.  Such a model is equivalent to having a quiescent zone of  water beyond the side wall end. T h e attenuation effect observed is caused b y linear diffraction of wave energy into this zone. T h e degree of intrusion of the attenuation zone into the wave field, as evaluated by the location of the u n i t 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 t h a n was actually measured. T h i s is clearly evident by noting the 0.9 waveheight contour i n each of the plots of F i g u r e 42. These differences between numerical a n d measured results may be a t t r i b u t e d to friction imparted by the rough w a l l formed by the outermost surface of the wave absorbing structure.  6.6.2.2  O b l i q u e l y P r o p a g a t i n g Wave T r a i n s  T h e case of the short wave propagating at 30° (test 4, T = 1.25sec) is considered i n Figure 44.  T h e a r b i t r a r y basin configuration was used.  wavelength ratio, B/L  = 0.205.  T h e generator segment to  Hence, this wavelength perhaps warrants m u l t i p l e  facet representation of generator segments. O n e r u n of the m o d e l was made w i t h halfmetre facets, such that M = 1, a n d a second r u n corresponds to M — 2. T h e facet size set for generator segments was also used to model the side walls. Hence, the 9.2m side walls were modelled b y walls 9.0m a n d 9.25m long, for facet lengths of 0.5m ( M = 1), a n d 0.25m ( M = 2), respectively. A s seen i n Figure 44(c), doubling the number of facets m i n i m a l l y affected the result, a n d 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 w a l l . N u m e r i c a l waveheight estimates throughout the test area are seen to be about 10% lower t h a n measured findings. T h i s is a t t r i b u t e d to neglect of nonlinear effects, and to absence of a reflecting wall along the upstream side of the basin. N u m e r i c a l model predictions of diffraction a n d quiescent zone boundaries intrude less into the test area t h a n measurements indicate. T h e numerical models accurately predict a region of large waveheights adjacent to the diffraction zone boundary, as well as a s m a l l 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 t h a n the model  6  Results  predicts.  and  Discussion  60  T h i s is likely due to absence of an upstream side w a l l i n the model.  The  fact that they are modelled well w i t h only one side wall indicates that such waveheight variation may be ascribed to reflection from the downstream side w a l l , rather t h a n to a resonant effect w i t h i n the basin. Figure 45 presents experimental and numerical results for the case of the intermediate wavelength at 0 = 30° (test 5, T = 1.75sec). T h e arbitrary basin configuration was used as before.  T h i s case provides perhaps the most s t r i k i n g example of model  performance. V e r y good agreement is noted between experimental and numerical findings, for b o t h 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 F i g u r e 46.  W h i l e the nu-  merically derived plot exhibits less pronounced effects t h a n 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 t h a n is actually the case. T h i s is particularly true i n the p o r t i o n of the basin situated between the side walls, i n consequence of the absence of a corner reflection side w a l l i n the model. T h e model concurs that the diffraction and quiescent zones each intrude far into the test area. T h e 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 compared i n F i g u r e 47. U s i n g a facet length of 0.5m, nine metre walls were modelled along each side, a n d the wave generator m o t i o n m i m i c k e d reality. A s a result, model predictions m a t c h the experimental result to w i t h i n 10% throughout the basin. T h i s case  6  Results  and  Discussion  61  shows that a linear diffraction model is well suited to p r e d i c t i n g predominant characteristics 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 a n d 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 t u a l segments. G o o d comparison between experiment a n d model is observed, as seen i n plots 48(a) a n d (b), respectively. T h e standing wave pattern apparent i n the corner reflection region of the experimental plot was not predicted by the numerical model. It is expected that if the combined principal- a n d B-wave overstroke motions of generator segments at this end of the wavemaker were accurately modelled, the large waveheights near this region w o u l d be correctly predicted. T h e result of an a d d i t i o n a l numerical r u n for the test 8 case is presented i n F i g u r e 48(c).  T h i s corresponds to a model w i t h b o t h side walls i n place, a n d no v i r t u a l  generator segments, while the quiescent end segments are still modelled. A n inferior prediction results, i n d i c a t i n g that for this wave condition the absence of corner reflection outweighs the effect of resonance between two side walls. T h i s is p a r t i c u l a r l y evident by the very large diffraction region found i n the plot. T h e last wave field case considered is that of the long wave propagating at 6 0 ° , a n d is found i n F i g u r e 49. T h i s corresponds to test 12, w i t h T = 2.25sec. N u m e r i c a l models were r u n as i n the previous cases presented i n F i g u r e 48. P l o t 49(b) corresponds to the model configured to include v i r t u a l generator segments to represent the B-wave generation. C o m p a r i s o n 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 h a n d , the model w i t h no v i r t u a l segments, but w i t h b o t h side walls i n place, is seen i n plot 49(c) to predict conditions very well. T h e system of standing waves formed  6  Results  and  Discussion  62  between the side walls indicates waveheights throughout the basin only about 10% lower t h a n were measured, evidence that this phenomenon is of predominant importance at this extreme propagation angle. It is expected that a more precise m o d e l of generator motions w i t h the reflecting side w a l l i n place w o u l d more closely predict the correct standing wave pattern.  6.6.2.3  Results W i t h the C y l i n d r i c a l Structure i n Place  Tests were also r u n to evaluate the performance of linear diffraction numerical models w i t h the cylindrical structure i n place.  T h e cylinder was formed of 38 facets, each  about 0.13m i n length, or four times smaller t h a n the 0.5m facet length used to model basin boundaries. T h e 0 = 0° plots corresponding to tests 16, 17 a n d 18 are presented i n Figures 50, 51 a n d 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 configur a t i o n , a n d closely simulate generator motions a n d side walls. Hence, discrepancies i n the upstream region between the side walls, into which energy is reflected by the structure, are likely due to i n a b i l i t y 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 m a y be ascribed to absence of a downsteam reflecting boundary. T h e result of another model which closely represents upstream boundary conditions is presented i n F i g u r e 53. T h i s corresponds to test 25, involving the simple snake m e t h o d of generation for T = 2.25sec a n d 6 = 3 0 ° . A g a i n , excellent spatial p r e d i c t i o n of various effects throughout the basin is observed, a n d m a x i m u m waveheight levels are about 10% lower t h a n measured.  Improved agreement between m o d e l a n d 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 t h a n for n o r m a l l y propagating waves. T h i s suggests that nonlinear effects associated w i t h the reflection process, part i c u l a r l y the free second harmonic component generated, are of greater significance to accuracy of G B S test predictions t h a n reflections f r o m 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. T h e three cases m a t c h i n g tests 19, 20 a n d 21 at 8 = 30° are presented i n Figures 54, 55 a n d 56, respectively. Results corresponding to tests 22, 23 a n d 24 at 8 — 45° are found i n Figures 57, 58 a n d 59, respectively. B-wave generation was modelled using the a r b i t r a r y basin configuration w i t h v i r t u a l segments, as for comparable wave field cases. A s before, large waveheights measured near overstroke segments due to B-wave generation were not predicted b y these models. Locations of most noteworthy features were generally well predicted. T h r o u g h o u t 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 w a l l modelled, suggest that at oblique angles reflections between the structure a n d the upstream side wall are less important t h a n harmonic effects due to reflection a n d fundamental wave generation. Less energy is trapped between the side walls than for n o r m a l l y propagating wave trains, w i t h the structure i n place.  6.6.3  Numerical Model Phase Predictions  Three phase plots f r o m numerical models corresponding to tests 2, 4 a n d 8 are presented.  A t 8 = 0 ° , the intermediate wavelength case of test 2 is seen i n F i g u r e 60.  T h i s plot clearly indicates the reliability of phase predictions a r o u n d the basin, by the straightness of crest a n d 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 p a t h of direct wave propagation.  6  Results  and  Discussion  64  T h e phase plot for test 4, T = 1.25sec a n d 9 = 3 0 ° , is found i n F i g u r e 61. Slight bending of crest/trough contours is apparent i n b o t h the diffraction a n d quiescent zones. T h i s tendency is slightly more pronounced t h a n i n the comparable experimentally derived plot of F i g u r e 36. T h e numerically derived phase plot for the test 8 c o n d i t i o n of T = 1.75sec a n d 45° is seen i n F i g u r e 62 to have less straight contours t h a n for the shorter wavelength case of F i g u r e 61. T h i s is similar to the corresponding experiment a l plot of Figure 38, a n d is likely associated w i t h the d a t a resolution as discussed i n Section 6.5.  6.6.4  Numerical Model Surface Elevation Plots  T h e S E G E N program also generates instantaneous surface elevation predictions at the prescribed d a t a point locations. T h e case corresponding to test 5 ( T = 1.75sec, 9 = 3 0 ° ) was plotted using a three-dimensional surface p l o t t i n g program, a n d is presented i n Figure 63. T h i s p r o g r a m uses the same gridded data as for the waveheight a n d phase contour plots, a n d produces perspective block diagrams for surface representation. T h e height-to-width ratio used for the plot was 0.1.  V a r i a t i o n 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 p l o t t i n g technique, such variation was visible on the water surface d u r i n g testing. A cross-mode present i n the basin, due to reflections f r o m the quiescent end side w a l l a n d elsewhere, accounts for this corruption of the incident wave t r a i n .  In order to produce similar  plots f r o m experimental results, values could be extracted f r o m measured elevation time series at a p a r t i c u l a r reference time. A l l time series would need first to be phase related, as was done to produce phase contour plots. C o m p a r i s o n between experimentally a n d numerically derived surface plots should show that the linear models  underestimate  somewhat the level of cross-mode interference. T h e numerically generated plot of F i g u r e 63 is representative of the tests conducted, a n d depicts such effects as the exaggerated  6  Results  and  Discussion  65  waveheight i n the corner reflection region, the p r i n c i p a l diffraction zone, a n d to a lesser extent, wave attenuation into the quiescent end diffraction zone.  6.7  Further Considerations  T h e time series plots presented i n F i g u r e 21 typify results f r o m most tests. O n l y the n o r m a l l y propagating wave field tests are characterized by more strongly sinusoidal records.  T  T h e examples of F i g u r e 21 are for a n array position of oblique test 5 at  = 1.75sec a n d 8 = 3 0 ° .  A t most probe locations, fairly strong second harmonic  components were present, as seen b y regular kinks between m a j o r peaks.  A beat is  apparent i n the envelope of many plots. T h i s is a t t r i b u t e d to free harmonics caused by linear generation a n d reflections. T h e case of probe 11, i n F i g u r e 21(e), is an example of a p a r t i c u l a r l y 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 w o u l d choose the second h a r m o n i c as the fundamental component, using a three parameter fit by frequency as well as amplitude a n d phase. T h e low waveheight of position 11 is quite pronounced relative to neighbouring points. W i t h reference to F i g u r e 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 envelope t h a n most of the wave field results.  T h i s is a t t r i b u t e d to increased levels of  free harmonic energy due to reflection. In a d d i t i o n to beating, some records show a slow oscillation caused by comparatively long wave components. N o p a r t i c u l a r l y good example of this is found i n F i g u r e 21. T h e residual time 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 w o r t h l o o k i n g briefly at a few signals i n greater detail. Figure 64 presents line spectra relating to some probe measurements f r o m test 10,  T = 1.25sec a n d 6 = 6 0 ° . These spectra derive from S C R E N analyses involving threeparameter fits.  T h e p r o g r a m was r u n to select frequency as well as amplitude and  phase, for the largest ten components.  T h e results were plotted as line spectra of  amplitude versus frequency. F i g u r e 64(a) locates these probes i n the basin. T h i s is an oblique test at 6 = 6 0 ° , 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 s o given are amplitude, frequency and phase values produced by the S C R E N program. T h e results for probe number 1 are presented i n F i g u r e 64(b). Some energy is seen to have shifted away from the forcing frequency of 0.8Hz. T h e 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 a m p l i t u d e at or slightly below twice the forcing frequency. T h e time series trace does not show a second harmonic kink, as is found i n most of the F i g u r e 21 plots. T h e only clear evidence of the presence of harmonic energy is the distinctive beating pattern. T h i s case is noteworthy for a long wave component of appreciable magnitude, w i t h a p e r i o d of about 46 seconds. T h i s is evidenced b y 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). T h e line spectrum amplitude scale is half that of Figure 64(b).  T h e fundamental component is less t h a n half the  magnitude of that of the probe 1 trace, while the chief second harmonic component is twice as large, a n d hence of greater relative importance. A l s o , more energy has shifted away from p r i n c i p a l frequencies i n b o t h the fundamental and second harmonic ranges. These two effects result i n a visible second harmonic kink i n the time series trace, and  6  Results  and  Discussion  67  a more pronounced a n d distinctive beating pattern t h a n is seen i n Figure 64(b). Wave elevations at probe 7, located further i n t o the diffraction zone, are seen i n plot 64(d) to be of still smaller magnitude. F i n d i n g s are similar to those at probe 13 i n plot 64(c).  T h e relative size of second order spikes compared to the fundamental  magnitude is intermediate between those at probes 13 a n d 1. T h e absence of a k i n k i n the probe 1 time series trace is a t t r i b u t e d to less frequency shifting of second harmonic energy t h a n present at probes 13 and 7. T h e 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, a n d a comparatively confused time series trace, corresponding to probe 12. T h i s probe is located well into the diffraction zone, a n d close to the end beach. T h i 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 h i g h frequency ripple is notable on the water surface. T h e first waves progressing toward the beach are smooth i n appearance. A s waves impact the outermost absorber sheets, a ripple is generated by runup against this barrier. T h i s ripple works its way back into the wave field, roughening the surface as it progresses toward the wavemaker. T h e 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 t h a n i n previous plots. A p p a r e n t l y , the runup process results i n generation of s m a l l waves at a frequency about five percent greater t h a n the second harmonic value. Increased understanding of the frequency shifts occurring due to resonant phenomena, beach r u n u p , internal beach and other processes must await more detailed, specific investigation. C e r t a i n l y , the wave field present i n a basin subject to regular wave generation cannot on close inspection be considered t r u l y sinusoidal. U n d e r most wave conditions, a n d i n central regions of the basin, the contaminating effects noted are restricted to less t h a n ten percent of the magnitude of the fundamental wave. However,  6  Results  and  Discussion  68  there are several instances a n d locations where this is not the case. Adequate predict i o n a n d correction of these effects is crucial to i m p r o v i n g wave conditions w i t h i n the test area of such a basin.  7  7.1  Conclusions and Recommendations  Conclusions  E l e v a t i o n 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, p e r i o d a n d propagation direction.  Three wave lengths were investigated, at three oblique angles as well as  propagating n o r m a l to the generator face. A shorter series of similar tests investigated the effect of a large surface piercing cylindrical structure i n the test area.  P l o t s of  m a x i m u m waveheight a n d phase contours were produced f r o m measured data. S i m i l a r plots were obtained f r o m the predictions of a linear diffraction n u m e r i c a l m o d e l , 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. N o r m a l i z e d m a x i m u m waveheight results were found to be independent of prescribed waveheight by repeating one test series at a reduced prescribed waveheight. One waveheight value was prescribed for all other tests. C o m p a r i s o n of phase plots i n dicated that the reduced-height wavelength was shorter t h a n the full-height value. T h i s is a t t r i b u t e d to nonlinear effects inherent to the wave characteristics a n d generation. A test conducted twice demonstrated excellent repeatability of results. It was found that the effects of wave diffraction and attenuation, a n d wave energy transmission t h r o u g h the beach structures, increase w i t h wavelength. T h 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, a n d smaller regions of acceptable wave conditions. E n e r g y diffracted into the quiescent zone reflects back into the incident wave field. Generated indirect waves impact the outer p o r t i o n of the reflection side w a l l at reduced levels. For most oblique wave cases, a p a r t i a l standing wave system develops i n the basin. Reflection of energy from the downstream side wall is the predominant cause, such that the standing wave is aligned perpendicular to the incident wave direction. F o r the longest wavelength tested, at 6 = 3 0 ° , the standing wave system sets up i n line w i t h the propagation direction. T h i s comparatively long wave penetrates the downstream beaches more effectively t h a n shorter waves.  Reflections from the basin boundaries  outweigh those from the downstream side wall, i n this case. F o r the longest wavelength at 6 = 6 0 ° , almost a l l energy was trapped between the side walls. T h e p e r i o d of these waves is close to a resonant frequency, treating the p r o b l e m 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; i n t u r n , 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. T h i s results i n a standing wave system which diminishes i n strength away from the combined-wave generator sector. Diffraction of this spurious energy results i n 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 width. Generation of oblique waves by the simple snake method results i n prohibitively high levels of reflected energy throughout the basin, p a r t i c u l a r l y w i t h the side w a l l length used.  Presence of a structure i n the test area results i n 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 t h a n fundamental wave contributions. Free h a r m o n i c energy due to reflections a n d linear wave generation account for beating of the waveheight envelope i n the test area. E x p e r i m e n t a l l y derived phase plots d i d not a i d evaluation of wave crest curvature due to diffraction. C o m p a r i s o n of experimental results w i t h predictions provided by the linear diffract i o n computer program indicated that the numerical model waveheight predictions were generally about 10% lower t h a n measured.  W i t h i n this tolerance, waveheight  a n d spatial predictions were found to be i n excellent agreement w i t h  measurements.  Discrepancies are a t t r i b u t e d to second order a n d other nonlinear effects not accommodated by the program. Waveheight gradients i n plots f r o m numerical results are less steep t h a n those i n experimentally derived plots. U s i n g numerical models w i t h 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, w h i c h d i d 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 t u a l segments w i t h o u t an upstream side wall. Such a configuration usually predicted well the effects measured throughout the basin, except that the high i n 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 resonance was a predominant effect, the upstream side w a l l was required for accurate prediction of a standing wave system close to the generator.  For cases involving the  simple snake m e t h o d of generation, segment motions a n d b o t h side walls were well  7  Conclusions  modelled.  and  Recommendations  72  These results displayed excellent agreement w i t h measured findings, a n d  illustrate the importance of accurate modelling of a l l basin boundaries. W i t h a structure present i n the test area, numerical predictions resemble experimental plots to w i t h i n 20%. M o d e l s were incapable of predicting the reflected second order energy present between the side walls. Locations of standing waves near the cylinder were well modelled. Absence of boundaries downstream of the cylinder resulted i n waveheights lower t h a n measured. Findings indicate that continued development of the linear diffraction program is warranted, to include p a r t i a l l y reflecting beach boundaries a n d accurate represention of complex generator motions, w i t h b o t h side walls i n place.  Excellent agreement  between m o d e l predictions a n d experimental results advocates developing another prog r a m which involves the linear diffraction technique.  T h i s p r o g r a m w o u l d prescribe  modifications to generator segment motions, so as to compensate for diffraction a n d reflection effects w i t h i n a laboratory basin. Wave contributions due to nonlinear phenomena are a n order of magnitude smaller t h a n fundamental, linear effects.  Development  of a linear program w i l l substantially improve the quality of wave fields generated. T h 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 i n a directional wave basin. E x p e r i m e n t a l tests should be conducted o n the type of upright wave absorber used i n the basin, i n order to assess performance at oblique angles of incidence. Measurements inside the beach structures should y i e l d some understanding of frequency effects u p o n residual wave reflections. R a w a n d analysed d a t a pertaining to the present study warrant closer scrutiny a n d more extensive analysis.  7  Conclusions  and  Recommendations  73  Rigorous comparison between experimental results a n d predictions f r o m 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 time series produced by the S C R E N p r o g r a m likely contain several  clues to nonlinear effects present at discrete locations.  D e t a i l e d evaluation of line  spectra w o u l d divulge much of the story. T h e trend toward more sophisticated design aspects w i l l be well served b y such work. T h e biggest single improvement w i l l surely come f r o m linear generator m o t i o n corrections afforded by the diffraction technique under continuing development.  8  References  Biesel, F . (1954) " W a v e M a c h i n e s " , Proceedings of the 1 st Conference Hoboken, N . J . , pp. 288-304. D e a n , R . G . & D a l r y m p l e , R . A . (1984) Water Wave Mechanics tists, P r e n t i c e - H a l l , Inc., Englewood Cliffs, N . J .  on Ships and Waves,  for Engineers  and Scien-  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 Analysis, N a t i o n a l Research C o u n c i l of C a n a d a , H y d r a u l i c s L a b o r a t o r y Technical Report, LTR-HY-75, March. Funke, E . R . & M i l e s , M . D . 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Naeser, H . (1981) " N H L - Ocean B a s i n Capabilities and L i m i t a t i o n s " , Proceedings of the International Symposium on Hydrodynamics in Ocean Engineering, T r o n d h e i m , pp. 1191-1210. Sand, S . E . (1979) Three-Dimensional Structure of Ocean Waves, Institute of H y d r o d y namics and H y d r a u l i c Engineering, Technical University of D e n m a r k , thesis and series paper no. 24. Sand, S . E . & M a n s a r d , E . P . D . (1986) Description and Reproduction of Higher Harmonic Waves, N a t i o n a l Research C o u n c i l of C a n a d a , H y d r a u l i c s L a b o r a t o r y Technical Report, TR-HY-012, Jan. Sand, S . E . & M y n e t t , A . E . (1987) " D i r e c t i o n a l Wave Generation and A n a l y s i s " , Proceedings of the 22 nd Congress of the International Association for Hydraulic Research, Lausanne, S w i t z e r l a n d , pp. 209-235.  Shore Protection  Manual  (1984), U . S . A r m y Corps of Engineers, C o a s t a l Engineering  Research Center, 4 * n ed., V o l . 1. U r s e l l , F . , D e a n , R . G . & Y u , Y . S . (1960) "Forced S m a l l - A m p l i t u d e W a t e r Waves: A C o m p a r i s o n of T h e o r y and E x p e r i m e n t " , Journal of Fluid Mechanics, V o l . 7, P a r t 1, Jan.  Tables test  T  9 [deg]  L [m]  H [m]  /i  H L  B L  T> L  frac  comments  #  [sec]  1  1.25  .2  .82  .082  .205  .607  1/2  (SZ)  1.75  0 0  2.44  2  4.74  .2  .42  .042  .106  .313  1/2  (MZ)  3  2.25  0  7.40  .2  .27  .027  .068  .200  1/2  (LZ)  4  1.25  30  2.44  .2  .82  .082  .205  .607  1  (S30) one test faulty  5 6  1.75  30  4.74  .2  .42  .042  .106  .313  1  2.25  30  7.40  .2  .27  .027  .068  .200  1  (M30) (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  '(L45)  10  1.25  60  2.44  .2  .82  .082  .205  .607  1  (S60)  11 12  1.75  60  4.74  .42  .042  7.40  .27  .027  .200  1 1  (M60)  60  .106 .068  .313  2.25  .2 .2  13  1.25  30  2.44  .1  .82  .041  .205  .607  1/2  (S30) h a l f waveheight  14  1.25  30  2.44  .2  .82  .082  .205  .607  1/8  (S30) r e p e a t a b i l i t y  15  2.25  30  7.40  .2  .27  .027  .068  .200  1  16  1.25  0  2.44  .2  .82  .082  .205  .607  1/2  ( C S Z ) with cylinder  17 18  1.75 2.25  0 0  4.74  .2 .2  .42 .27  .042 .027  .106 .068  .313 .200  1/2 1/2  ( C M Z ) with cylinder ( C L Z ) with cylinder  19 20 21  1.25  2.44 4.74  .2  .82  .082  .2  .042  .313  1 1  (CS30).with cylinder  .42  .205 .106  .607  1.75  30 30  2.25  30  7.40  .2  .27  .027  .068  .200  1  (CL30) with cylinder  45 45 45  2.44 4.74  .82 .42 .27  .082 .042  7.40  .2 .2 .2  .027  .205 .106 .068  .607  23 24  1.25 1.75 2.25  .313 .200  1 1 1  ( C S 4 5 ) w i t h cylinder ( C M 4 5 ) w i t h cylinder ( C L 4 5 ) 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.  22  7.40  TABLE 1  (L60)  ( L 3 0 N O ) simple snake  ( C M 3 0 ) with cylinder  Tests C o n d u c t e d a n d P e r t i n e n t Parameters  ('frac' is the f r a c t i o n of the b a s i n measured — 8 tests comprise a f u l l test series; a b b r e v i a t e d 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 =>• c y l i n d e r i n place; N O signifies no corner reflection, simple snake m e t h o d used for a l l 60 generator segments) 76  .2  sr s, [m] H n/a n/a  0  .2  n/a  2.25  0  .2  1.25 1.75  30 30  T [sec]  [deg]  H [m]  1  1.25  0  2  1.75  3 4  test  #  5  9  mg  comments  n/a  n/a  (SZ)  n/a  n/a  n/a  (MZ)  n/a  n/a  n/a  n/a  (LZ)  .2 .2  9.3  1-11 1-11  47-60 47-60  (S30) one test f a u l t y  9.3  12 12  9.3  12  1-11  47-60  (L30)  (M30)  6  2.25  30  .2  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)  9.3  10  1-19  41-60  (L45)  9  2.25  45  .2  10 11  1.25 1.75  60 60  .2 .2  9.3 9.3  9 9  1-29 1-29  33-60 33-60  (S60) (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) h a l f waveheight  14  1.25  30  .2  9.3  12  1-11  47-60  (S30) r e p e a t a b i l i t y  15  2.25  30  .2  0  0  0  0  16 17  1.25 1.75  0 0  .2 .2  n/a n/a  n/a n/a  n/a  n/a n/a  (CSZ) with cylinder ( C M Z ) with cylinder  18  2.25  0  .2  n/a  n/a  n/a  ( C L Z ) with cylinder  19  1.25 1.75 2.25  30  .2  9.3  30 30  .2 .2  9.3 9.3  12 12  20 21  n/a n/a  ( L 3 0 N O ) simple snake  1-11  47-60  12  1-11 1-11  47-60 47-60  (CS45) with cylinder ( C M 4 5 ) with cylinder 41-60j (CL45) with cylinder  22 23 24  1.25 1.75  45 45  .2 .2  9.3 9.3  10 10  1-19 1-19  2.25  45  .2  9.3  10  1-19  25  2.25  30  .2  0  0  0  (CS30) with cylinder ( C M 3 0 ) with cylinder (CL30) with cylinder  41-60 41-60 0  ( C L 3 0 N O ) snake/cyl.  TABLE 2 Side W a l l L e n g t h s Specified F o r D r i v e S i g n a l Synthesis, Resultant Generator Motions  and  (S is the specified side wall l e n g t h , subscripts r a n d q refer to corner reflection a n d quiescent zone ends of the generator, respectively; TO.A/B signifies the generator segments i n v o l v e d i n b o t h A - a n d B-wave generation; m g pertains to the quiescent segments; segments between these two sets move a c c o r d i n g to the simple snake p r i n c i p l e ; segment 1 is at the corner reflection e n d of the generator)  Figures  FIGURE 1 System  E l e v a t i o n V i e w of the Wavemaker Configuration and Coordinate  generator segment n=i  .  ,  propagation direction FIGURE 2  D e f i n i t i o n of C o o r d i n a t e S y s t e m , P l a n V i e w  78  SEGMENTED  WAVE  GENERATOR  DIFFRACTION ZONE O F REFLECTED  WAVES  ZONE O F INTERFERENCE WITH UNDESIRABLE REFLECTIONS  • DIFFRACTION  WAVE  FIGURE 3  ZONE  ABSOROERS  O b l i q u e W a v e W i t h Interference W a v e , C o n v e n t i o n a l M e t h o d  of O b l i q u e W a v e G e n e r a t i o n (after F u n k e a n d M i l e s , 1987)  FIGURE 4  S u p e r p o s i t i o n of Direct a n d Reflected Indirect Waves (after F u n k e a n d M i l e 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.  FIGURE 7  B a s i n L a y o u t , W i t h the Four P r o b e A r r a y Positions Indicated  a of  t e s t the  a r r a y  p o s i t i o n 16  probe  frame  F I G U R E 8 Use of B a s i n S y m m e t r y to Measure Wave Conditions Over O n l y H a l f the B a s i n , W h i l e i n Effect A c q u i r i n g Readings T h r o u g h o u t the T o t a l Area  6m  FIGURE 9  D i a g r a m of the P r o b e 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 Fitting Supporting the Spacer.  F I G U R E 13 Closeup of Brass Fitting Holding a Wave Probe O n A n Array Superstructure Member Using a Standard Fitting  F I G U R E 14 The Shortest Wave Train Propagating A t + 3 0 ° . Note the Quiescent 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 A t - 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 A t - 4 5 ° Past the Surface Piercing Cylindrical Structure. Flotation Balloons Are Seen Stored Atop the Probe Array R i g .  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  1.2  0.1 Time  (a)  1.6  (seconds)  D r i v e Signals F o r Segments 1-4, C o r n e r Reflection 10.0  CT>  5.0  CT> C  0.0  -5.0  •10.0 0.0  0.4  0.8  1.2  1.6  2.0  2.1  T 1 me t s e c o n d s )  (b)  Segments 4-7, C o r n e r Reflection  10.0  O)  5.0  0.0  -5.0  •10.0 0.0  0.1  0.8 Time  (c)  1.2  1.6  2.0  2.4  (seconds )  Segments 7-10, C o r n e r Reflection  F I G U R E 18 Wave Generator Segment D r i v e Signals For Test 4, T = 1.25sec at 9 = + 3 0 ° . Segments 1-11 A r e Subject to C o m b i n e d A - and B-Wave M o t i o n , Segments 12-46 M o v e A c c o r d i n g T o the Simple Snake P r i n c i p l e , and Segments 47-60 A r e H e l d 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  Time  18(f)  (seconds)  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 Time  1.:  1.6  2.0  2.4  (seconds)  (a) D r i v e Signals F o r A l l 60 Segments, For Tests 1 ( S Z E R O ) , 2 ( M Z E R O ) a n d 3 ( L Z E R O ) A t 8 = 0 ° . These Show T h a t P a d d l e Stroke Increases D i r e c t l y W i t h W a v e l e n g t h F o r a G i v e n Waveheight.  (b) D r i v e Signals F o r the Intermediate W a v e l e n g t h ( T = 1.75sec), F o r A l l Segments a n d 8 = 0° ( M Z E R O ) , a n d F o r Segment 30 a n d 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 a n d 11, Respectively, and Show T h a t For a G i v e n W a v e h e i g h t , P a d d l e Stroke Varies Inversely W i t h P r o p a g a t i o n D i r e c t i o n .  F I G U R E 19 D r i v e S i g n a l P l o t s I l l u s t r a t i n g Dependence of P a d d l e Stroke O n W a v e l e n g t h a n d P r o p a g a t i o n D i r e c t i o n F o r a G i v e n Waveheight  91  parameters requested o n l i n e : -  still  -  number of frequency components  water depth? i s the number of harmonic  to be extracted?  components > 1 ?  -  number of harmonic components?  -  tolerance below which the error  yes  sum of squares of two successive i t e r a t i o n s must f a l l  for  optimization  terminate?  -  loop to  the  number of parameters to be optimized = 2 (ampl. & phase)  l i m i t to variance of amplitudes,  o p t i o n a l l y supply value of  as percentage of measured time  fundamental  s e r i e s RMS value?  frequency  number of parameters to be optimized? 2 = amplitude and phase read data (measured time s e r i e s )  3 = amplitude, phase and frequency  remove mean value  ••T  perform FFT on data  f i n d maximum amplitude and a s s o c i a t e d frequency  using these values perform quadratic  interpolation  to improve estimates ( a l s o get phase i n i t i a l  I  using current estimates of amplitude,  estimate)  frequency  and phase, perform n o n - l i n e a r optimization to  fit  s i n u s o i d a l curve to data  yes i s the e r r o r sum of squares between current and previous i t e r a t i o n ,  f o r each of the  parameters  being optimized, within the s p e c i f i e d tolerance? remove s i n u s o i d from time s e r i e s , and add  I  to preceding s i n u s o i d a l components (ie.,  to synthesized time s e r i e s )  have the requested number of frequency components been e x t r a c t e d ,  or the s p e c i f i e d  percentage variance of amplitude been exceeded?  yes  output the fundamental component wave parameters, the synthesized time s e r i e s , the residual time s e r i e s , the l i n e spectra of amplitude v s . frequency and of phase v s . f r e q .  F I G U R E 20 K e y O p e r a t i o n a l 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 , M o d i f i e d to E x t r a c t C o m p o n e n t s F r o m the M e a s u r e d E l e v a t i o n T i m e Series  (a) D i a g r a m I n d i c a t i n g the A r r a y P o s i t i o n W i t h I n d i v i d u a l P r o b e Locations  F I G U R E 21 E x a m p l e s of P l o t s Generated F o r E a c h Test, F o r the C a s e of O n e A r r a y P o s i t i o n of Test 5, T = 1.75sec, 6 = 3 0 ° .  93  S3ai3M  ol(h) 2  U  M e a s u r e d E l e v a t i o n T i m e Series, P l o t s of S C R E N Results a n d Pertai n t P a r a m e t e r s For Reference P r o b e 17 ( O r i g i n a l T i m e Senes - sohd Irne, Synthesized - dashed line, R e s i d u a l - dotted line)  S3M13W  S3H13W  21(c)  S3M13W  S3H13H  M e a s u r e d T i m e Series P l o t s F o r Probes 1-4  S3H13H  S3M13H  21(d)  S3M13H  S3S13H  M e a s u r e d T i m e Series P l o t s F o r Probes 5-8  21-6 sgqcuj JOJ; s^o^ sstjag stuix paJris^ajAj  METRES  METRES  METRES  (d)iz  METRES  S3di3W  (f)  S3ai3H  S3M13H  M e a s u r e d T i m e Series P l o t s F o r Probes 13-16  Side  Wall Side  Beach  XI  c  Y  (a) F u l l Waveheight P l o t , Test 4; Hsy Values N o r m a l i z e d to Desired Waveheight of 20cm. ( T h e bracketed region ( R ) locates the repeatability test. See Figures 24 a n d 25.)  F I G U R E 22  (  hei  " l  h  e  i  S  h  t  P I  °''TeSt  H a l f W a v e h f f , , P l o t s O v e r H a l f the B a s i n of t h e  T = 1.25sec, 8 = 30° Resn the F u l l a n d R e d u c e d Waveheights  - ™™ Normalized to Desired Wave-  B  t  F I G U R E 23 H a l f Waveheight Test: P h a s e P l o t s O v e r H a l f the B a s i n of the T = 1.25sec, 9 = 30° Results F o r B o t h the F u l l a n d R e d u c e d Waveheights (Tests 4 a n d 13, Respectively). A l t e r n a t i n g Crests a n d T r o u g h s A r e P l o t ted (i.e., C o n t o u r Interval = 1 8 0 ° ) . R e d u c e d Waveheight (Test 13) R e s u l t Shown Dotted.  F I G U R E 24 R e p e a t a b i l i t y Test: Hsy P l o t s For a Single A r r a y P o s i t i o n of the T = 1.25sec, 6 = 30° Results. (Refer to Figure 22(a) for location of test, as indicated b y region ' R ' ; C o n t o u r Interval = 0.05)  F I G U R E 25 R e p e a t a b i l i t y Test: Phase P l o t s F o r a Single A r r a y P o s i t i o n of the T = 1.25sec, 8 = 30° Results, Tests 4 a n d 14 (14 shown dotted). C o n t o u r Interval = 1 8 0 ° ^  102  FIGURE 26  A a , A 2 and Hsy Plots For Test 7: T = 1.25sec, 0 = 4 5 °  103  ( a)  A\  (b)  F I G U R E 27  Au  A2 ( x l O ; note different contour interval)  A2 and Hsy  P l o t s For Test 5: T = 1.75sec, 6 = 30  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 P l o t s For N o r m a l l y P r o p a g a t i n g Wave T r a i n s (6 = C o n t o u r Interval = 0.05)  0°;  106  (a)  W a v e F i e l d Case, Test 3  F I G U R E 30 Hsy P l o t s F o r T = 2.25sec, 0 = 0 ° , W i t h a n d W i t h o u t the G B S ( C o n t o u r Interval = 0.05)  (b)  G B S Case, Test 18  107  FIGURE 31 Hsy Plots For Wave Trains Propagating Obliquely At 6 = 3 0 ° ; Results For the Three Wavelengths Investigated  108  (a)  8 = 30°, Test 5  (b)  F I G U R E 32 (T =  Hsy  9 = 45°, Test 8  P l o t s For the Intermediate L e n g t h W a v e T r a i n  1.75sec) D i r e c t e d A t the T h r e e O b l i q u e Angles Tested  (c)  0  =  6 0 ° , Test 11  109  F I G U R E 33  Hay Plot For Test 12, T = 2.25sec, 9 = 60°  F I G U R E 34 Hsy P l o t F o r Test 15, T = 2.25sec, 6 = 3 0 ° , I n v o l v i n g the S i m p l e Snake M e t h o d of G e n e r a t i o n W i t h o u t C o r n e r Reflection  111  (a)  Test 5, Wave F i e l d  F I G U R E 35 Cases  (b)  Hsy  P l o t s For T = 1.75sec, 6  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 P h a s e P l o t F o r Test 4, T = 1.25sec, 6 = 3 0 ° . T h e C o n t o u r Interval U s e d Is 1 8 0 ° , R e s u l t i n g In a P a t t e r n of A l t e r n a t i n g Crests a n d Troughs  113  FIGURE 37 Phase Plot For Test 7, T = 1.25sec, 6 = 45°. Contour Interval = 180°  FIGURE 38 Phase Plot For Test 8, T = 1.75sec, 6 = 45° Contour Interval = 180°  wave g e n e r a t o r 30m  /60 0.5m\ \facets / •  H  '18 0 . 5 m facets  fully a b s o r b e n t  beach 1.2m  s h a d i n g i n d i c a t e s r e g i o n of 2-metre  spaced data  points  FIGURE 39 Rectangular Basin Configuration For Use With the SEGEN Linear Diffraction Computer Program, Consisting Of a Full Bank of Active Generator Segments, Flanked On Each End By Side Walls of Similar Lengths  wave |<f-5.5m^-  generator 7 . 0 m -=>|  23.0m I  I  I  I  I  I  I  \,imu///////i/.  side 9.0m  fei.Om  fully a b s o r b e n t  wall  for 0 . 5 m facets /length = 9.25m\ for 0 . 2 5 m \ facets j  beach  shading  indicates region  2—metre  spaced data  of  points  (a) E x a m p l e C o r r e s p o n d i n g T o Test 4, U s i n g the V i r t u a l Segment Concept T o M o d e l B - W a v e G e n e r a t i o n , and M o d e l l i n g Quiescent Zone Segments B y a Reflecting W a l l wave <=  — Li ^> Ji — i  L  side wall  1.0m  Y  —  23.0m i i i  i  1  1  Y  1  i  r  1  i  i  — 7 . 0 m  1  12m  Y// -5\  generator  ///  / /  9.0m  //  /  L.  A HY-  1.0m  J  fully a b s o r b e n t  beach  s h a d i n g i n d i c a t e s region 2-metre  spaced data  of  points  (b) E x a m p l e C o r r e s p o n d i n g T o Tests 5 a n d 12, W i t h B o t h Side W a l l s In P l a c e T o C o m p a r e W i t h the V i r t u a l Segment M o d e l s , a n d I n c l u d i n g the Cylindrical Structure  F I G U R E 40 Use of the A r b i t r a r y B a s i n C o n f i g u r a t i o n F o r the S E G E N Linear Diffraction Computer Program  117  (a)  E x p e r i m e n t a l Result, Test 1. C o n t o u r Interval =  0.05  F I G U R E 41 E x p e r i m e n t a l Hsy and N u m e r i c a l M o d e l M a x i m u m Waveheight (Ai) Results C o m p a r e d For T = 1.25sec, 9 = 0° (note different contour intervals)  (b) N u m e r i c a l Result, R e c t a n g u l a r B a s i n C o n f i g u r a t i o n . C o n t o u r Interval = 0.02  i  118  (a)  E x p e r i m e n t a l R e s u l t , Test 2. C o n t o u r Interval =  0.05  F I G U R E 42 E x p e r i m e n t a l Hsy and N u m e r i c a l M o d e l M a x i m u m Waveheight ( A i ) Results C o m p a r e d For T = 1.75sec, 9 = 0°  119  (b) Numerical Result, Rectangular Basin Configuration. Contour Interval = 0.02  120  (a)  E x p e r i m e n t a l Result, Test 4  (b)  Numerical Result, M = 1 (Arbitrary Basin Configuration)  F I G U R E 44 E x p e r i m e n t a l Hsy T = 1.25sec, d = 30°  a n d N u m e r i c a l Ax Results Compared F o r  (c)  Numerical Result, M — 2  121  (a)  E x p e r i m e n t a l R e s u l t , Test 5  F I G U R E 45 E x p e r i m e n t a l H3y T = 1.75sec, 6 = 30°  (b)  Numerical Result, A r b i t r a r y Basin Configuration  and N u m e r i c a l A\ R e s u l t s C o m p a r e d F o r  122  (a)  E x p e r i m e n t a l R e s u l t , Test 6  F I G U R E 46 E x p e r i m e n t a l Hsy T = 2.25sec, 9 = 30°  (b)  Numerical Result, A r b i t r a r y Basin Configuration  a n d N u m e r i c a l Ai Results C o m p a r e d For  123  F I G U R E 47 E x p e r i m e n t a l Hsy and N u m e r i c a l Ax Results F o r the S i m p l e Snake M e t h o d of Generation, T = 2.25sec, 6 = 30°  124  (a)  E x p e r i m e n t a l Result, Test 8  (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 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 N o V i r t u a l Segments F I G U R E 48  E x p e r i m e n t a l Hsy  T = 1.75sec, 0 = 45°  and N u m e r i c a l Ai Results C o m p a r e d For  125  (a)  E x p e r i m e n t a l Result, Test 12  (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 Configuration  (c)  N u m e r i c a l Result, W i t h Quiescent Zone A s I n (b), C o r n e r Reflection S i d e W a l l a n d N o V i r t u a l Segments  F I G U R E 49 E x p e r i m e n t a l Hsy T = 2.25sec, 6 = 60°  and N u m e r i c a l Ax  Results C o m p a r e d For  126  (b)  N u m e r i c a l Result, Rectangular Basin Configuration  (b)  Numerical Result, Rectangular Basin Configuration  128  (a)  F I G U R E 52  E x p e r i m e n t a l R e s u l t , Test IS  E x p e r i m e n t a l 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, 9 = 0 ° , W i t h the G B S In P l a c e  (b)  N u m e r i c a l Result, Rectangular Basin Configuration  129  (a)  E x p e r i m e n t a l R e s u l t , Test 25  (b)  N u m e r i c a l Result, Rectangular B a s i n Configuration  F I G U R E 53 E x p e r i m e n t a l Hsy and N u m e r i c a l Ai Results C o m p a r e d F o r the S i m p l e Snake M e t h o d of Generation W i t h the G B S In P l a c e , T = 2.25sec, 6 = 3 0 ° (relates to F i g u r e 56)  (a)  E x p e r i m e n t a l Result, Test 19  (b)  Numerical Result, A r b i t r a r y Basin Configuration  F I G U R E 54 E x p e r i m e n t a l Hsy and N u m e r i c a l Ax Results C o m p a r e d F o r T = 1.25sec, 9 = 30°, W i t h the G B S In P l a c e  E x p e r i m e n t a l Result, Test 20  (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 C o n f i g u r a t i o n  F I G U R E 55 E x p e r i m e n t a l Hsy and N u m e r i c a l Ax R e s u l t s C o m p a r e d F o r T = 1.75sec, 6 = 3 0 ° , W i t h the G B S In P l a c e  132  (a)  E x p e r i m e n t a l R e s u l t , Test 21  (b)  Numerical Result, A r b i t r a r y Basin Configurati  F I G U R E 56 E x p e r i m e n t a l Hsy and N u m e r i c a l Ax Results C o m p a r e d F o r T = 2.25sec, 8 = 30°, W i t h the G B S In P l a c e  133  (a)  E x p e r i m e n t a l Result, Test 22  (b)  Numerical Result, A r b i t r a r y Basin Configuration i  F I G U R E 57  E x p e r i m e n t a l Hsy  and N u m e r i c a l Ax R e s u l t s C o m p a r e d F o r  T = 1.25sec, 6 = 4 5 ° , W i t h the G B S In P l a c e  134  (a)  E x p e r i m e n t a l R e s u l t , Test 23  (b)  Numerical Result, A r b i t r a r y Basin Configuration  F I G U R E 58 E x p e r i m e n t a l Hsy and N u m e r i c a l Ax Results C o m p a r e d F o r T = 1.75sec, d = 4 5 ° , W i t h the G B S In P l a c e  135  (a)  E x p e r i m e n t a l Result, Test 24  (b)  Numerical Result, A r b i t r a r y Basin Configuration  F I G U R E 59 E x p e r i m e n t a l Hsy and N u m e r i c a l A\ Results C o m p a r e d F o r T = 2.25sec, 9 = 4 5 ° , W i t h the G B S In P l a c e  136  Y X  F I G U R E 60 N u m e r i c a l l y Derived Phase P l o t C o r r e s p o n d i n g T o Test 2, T = 1.75sec, 6 = 0 ° . C o n t o u r Interval = 180°  137  F I G U R E 61  N u m e r i c a l l y D e r i v e d P h a s e P l o t C o r r e s p o n d i n g T o Test 4,  T = 1.25sec, 6 = 30°. C o n t o u r Interval = 180°  138  F I G U R E 62 N u m e r i c a l l y D e r i v e d Phase P l o t C o r r e s p o n d i n g T o Test 8, T = 1.75sec, 8 = 4 5 ° . C o n t o u r Interval = 1 8 0 °  F I G U R E 63 T h r e e - D i m e n s i o n a l Perspective B l o c k D i a g r a m S h o w i n g In stantaneous W a t e r Surface E l e v a t i o n F r o m the N u m e r i c a l R e s u l t C o r r e s p o n d i n g T o Test 5, T = 1.75sec, 9 = 3 0 °  F I G U R E 64  A m p l i t u d e vs. Frequency L i n e S p e c t r a F o r Selected P r o b e  L o c a t i o n s P e r t a i n i n g T o Test 10, T = 1.25sec a n d 9 = 6 0 °  141  LO  r—  LJ  CO ' <Z CO X is LL  B  Q  O  D> N  K  N  CO 1-1  i-".  •0 N 0O CO O O 0! 0>  C LLJ  C  CM  CO  00  r>. K  o n N  H  C  N  i-i N  co co co *o co  i " f J O IN N l> b H H CO li") i - i CM N CO Ii0 i l l ilO N O CO  a  fi  LO  O O C i - » t H i - » i h O O O  LL  o  CM  ro N -o o- co co  T-i cs o- co n i-t K  o  O CO CO N ~0 CM i - " o o o  i-i o o o o c  o o o o o o o o  O ^0 CO o  O C  rH O  c © o c  tsa  >, (J  CS cu cu  LO  S3H13W  L_  o  [ui]  64(b)  apn^ijdiuy  A m p l i t u d e vs. Frequency L i n e S p e c t r u m a n d M e a s u r e d E l e v a t i o n T i m e  Series P l o t F o r P r o b e 1  142  LO |N-  CO  -0 •?  *  CO  N  in o  CO NO NO IS CO  L O  CN CO n o- o CO IS c o IN CO 0co N C0 CJ If) CM CM CM CM CM i-i  < r X u U- Q  N  <r o o CO  a u.  LU N X LL.  N iio to in Ch 7-1 CO NO K CO 0* o N 0- N r-( CO m Is co  o o  -  N T o N CO CM co <? m =? CM CM «r o CN CO CM c> Is IN. iiO NO NO N  O o o i-l  —t  L O  o o  ~  CM  • 1  o N •0 N N N T N CN o IlO CO Is IlO sT NO .7-1 o NO CM n CO co Is sO IlO NO in r-t o NT NT s ' 1-t Is -0 o m 1-1 1-1  _1 LL  X.  m o iH O  IT)  o o o o o o o o o o o o o• o• * * * * o o o o o o•o o o o  mm  •  —<  LO  >N o G CO 3  cr CO I-i  LO  fa  LO  S38I3W <Z3  LO  LO  <=>  C3>  [m]  apniijduiy  64(c) Amplitude vs. Frequency Line Spectrum and Measured Elevation Time Series Plot For Probe 13  143  L O  |n-  i.i  •  U)  <Z  •  •  CM  co  Q  o  UJ LL  K  * T-i m  en  o-  o * CO•  o  N  tH  T"i  CO  * K  CO  -0  *  CO C N S3 in CM <J o o CM CO IJO O CM C N IIO CM t-i CO CO h if) N o * • • • *  N  IX *  IlO  l>  co  CM  -0  T-i 03 Is  t-i  o o o  T-i  o  iH  m  CM tH Is o <T CM CN i-i O sO co •0 T-i CO Is S3 CO CO *0 S3 CM o T-i N . <r CO CO o o c o o sT O  *  o  o6 * • o o o  o  o  o  S3 O  <r CO CM T •0 rs 0- CO * * • *  •0  <.  o  IlO CM  tH  o o• TH  r  CN  CO N  tH 0>  IjO . - i  X  o o o • *  CM CN 0CM CM  c  o  Is CO T-t CM  1-1  o o  o o  CO CO  • • * • o o o o o o «  *  LO  *  o a  LO  LO  S3M13W  L L O  LO  LO  NO  [in]  LO CVI  9pn^i|duiy  64(d) A m p l i t u d e vs. Frequency L i n e S p e c t r u m a n d M e a s u r e d E l e v a t i o n T i m e Series P l o t F o r P r o b e 7  144 >  L O  LU  X 0.  0; Q  CO i H PJ O 1 0 Fl 0 * TI o» o B N > >0 B H N > b~ P i sT i-i PJ 0 > i n PJ r ! N ri N fj (1 f )  •C N \i~ ~£  03 CO N 0> N  CO P J n C- <T n  in c* in co in  CO © P i 'TI CO CO CO N * C  -o  PJ  c* o <r m  o - PO cc o ^ - -o *o N  n  O O O T I O O T I T I T I O  L 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 o-  «?  •  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  \  tS)  L O  O  >  cr L O  I> •  L O  <  <  > >  S3ai3H  L. CO  [uu]  apn^ijduiy  64(e) A m p l i t u d e vs. Frequency L i n e S p e c t r u m a n d M e a s u r e d E l e v a t i o n T i m e Series P l o t F o r P r o b e 12  

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