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Onshore/offshore transport mechanisms Walsh, Bruce William 1989

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ONSHORE/OFFSHORE TRANSPORT MECHANISMS  by BRUCE WILLIAM WALSH B . A . S c , The U n i v e r s i t y o f B r i t i s h Columbia, 1986  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department o f C i v i l  Engineering  We accept t h i s t h e s i s as conforming t o t h e r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1989  (c)  Bruce W i l l i a m Walsh, 1989  In  presenting  degree  this  at the  thesis  in  University of  partial  fulfilment  British Columbia,  of  the  I agree  requirements  for  an  advanced  that the Library shall make it  freely available for reference and study. I further agree that permission for extensive copying  of  department  this or  thesis by  for scholarly  his  publication of this thesis  or  her  may  representatives.  It  be is  granted  by the head  understood  that  for financial gain shall not be allowed without  permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  purposes  of  my  copying  or  my written  ABSTRACT The  o n s h o r e / o f f s h o r e t r a n s p o r t mechanisms are  C a r e f u l and f i n d any  d e t a i l e d observations  o f the s u r f zone are made t o  d i f f e r e n c e s or s i m i l a r i t i e s between the r e s u l t i n g flows  caused by d i f f e r e n t breaking s p i l l i n g and breaking  investigated.  plunging  types.  breakers.  Emphasis i s p l a c e d on  Even though the p h y s i c a l  p r o p e r t i e s are s i g n i f i c a n t l y d i f f e r e n t , the  time-  averaged p r o p e r t i e s of the s u r f zone f o r each type of wave are s i m i l a r .  the  breaking  U s i n g t h i s as a b a s i s , a model of the beach  i s developed u s i n g a c o n t r o l volume t h a t i s bounded by the beach f a c e , the time-averaged water l e v e l , and breaking  point.  the onshore and The  The  a v e r t i c a l p l a n e at  the  momentum a c t i n g on the c o n t r o l volume i n  o f f s h o r e h o r i z o n t a l d i r e c t i o n s i s balanced.  model shows t h a t the onshore/.offshore sediment  transport  i s p r i m a r i l y dependent upon the magnitude o f the wave  setup shoreward o f the b r e a k i n g the beach.  p o i n t , and  I n c r e a s i n g the p e r m e a b i l i t y  the p e r m e a b i l i t y  causes a r e d u c t i o n  of in  the o f f s h o r e net shear s t r e s s a c t i n g along the beach face which r e s u l t s i n an i n c r e a s i n g s l o p e . d i f f e r e n c e between a g r a v e l and g r a v e l beach b e i n g The  U s i n g t h i s simple model,  the  sand beach can be e x p l a i n e d ,  the  steeper.  model i s used t o c a l c u l a t e d the o f f s h o r e net  shear  s t r e s s f o r a p l a n e impermeable beach i n the l a b o r a t o r y . c a l c u l a t i o n g i v e s the r i g h t order o f magnitude (<10  The  N/m2), but  proves t o be s e n s i t i v e t o s m a l l i n a c c u r a c i e s i n the measurement of the  setup.  ii  TABLE OF CONTENTS Page Abstract  i i  Table o f Contents  i i i  L i s t of Tables  vi  L i s t of Figures  v i i  Acknowledgement  xi  CHAPTER 1  INTRODUCTION  1  CHAPTER 2  BREAKING WAVE TYPES  4  2.1  Introduction  4  2.2  P h y s i c a l C l a s s i f i c a t i o n o f Breaking Waves ....  5  2.3  Breaking I n d i c e s :  HQ/LQ,  Beach Slope, and Breaking Types ....  3  2.3.1  Comparison o f t h e Breaking I n d i c e s ....  15  2.3.2  V a r i a t i o n o f t h e Breaking I n d i c e s  19  2.3.2.1  Experimental Procedures and C a l c u l a t i o n s ....  2.3.3  19  2.3.2.2  Wave E f f e c t s  23  2.3.2.3  Breaking D e f i n i t i o n s  27  2.3.2.4  Personal Judgement  27  Breaking I n d i c e s on N a t u r a l Beaches ...  2.4  Experimental Design and Procedure  2.5  Experimental R e s u l t s  2.6  Discussion  28 32  .'  34 36  iii  Page CHAPTER 3  WAVE HEIGHT AND DEPTH AT BREAKING  3.1  Introduction  3.2  D e f i n i t i o n o f t h e Breaking 3.2.1  38 38  Height  and Depth ..  The Height-to-Depth R a t i o  39 42  3.3  Influence o f the Wave Steepness  47  3.4  Influence o f t h e Depth on Breaking  57  3.5  Influence o f the Beach Slope on Breaking  60  3.6  Discussion  65  CHAPTER 4  FLOW DYNAMICS WITHIN THE SURF ZONE  68  4.1  Introduction  68  4.2  T r a n s i t i o n t o R o t a t i o n a l Flow  70  4.3  E s t a b l i s h e d Turbulent  75  4.4  Mass Transport  76  4.5  Setup i n t h e Surf Zone  88  4.5.1  Experimental  Design and Procedure  92  4.5.2  Experimental  Results  94  4.6  CHAPTER 5  Flow  Discussion  99  DEVELOPMENT OF THE BEACH FACE CONTROL VOLUME  5.1  Introduction  5.2  Formulation  5.3 5.4  102 102  o f the Model  103  5.2.1  Assumption o f an Impermeable Beach ....  105  5.2.2  Inclusion of the E f f e c t of Permeability  115  D i f f e r e n t Beaches Subject t o the Same Wave A t t a c k Same Beaches Subject t o V a r y i n g Wave A t t a c k iv  ..  117 127  Page 5.5  Oblique Wave Attack  131  5.6  Discussion  132  CHAPTER 6  TESTING OF THE BEACH FACE CONTROL VOLUME MODEL ....  136  6.1  Introduction  13 6  6.2  Model f o r an Impermeable Beach  137  6.3  Results  139  6.4  S e n s i t i v i t y o f the Model  147  6.5  Discussion  152  CHAPTER 7  SUMMARY AND DISCUSSIONS  BIBLIOGRAPHY  155  163  APPENDICES A - S h o a l i n g C h a r a c t e r i s t i c s o f an O s c i l l a t o r y Wave .... B - Observations  168  o f P a r t i c l e Motions Under Breaking Waves  171  C - Wave Setup/Setdown Data  184  D - Wave R e f l e c t i o n Data  207  E - Laboratory  209  Study o f Breakers (Iversen 1952)  v  LIST OF TABLES Table  Title  Page  2.1  Transition points  22  3.1  Maximum height-to-depth r a t i o s f o r s o l i t a r y waves ( a f t e r G a l v i n 1972)  43  Beach s l o p e e f f e c t on breaker h e i g h t f o r o s c i l l a t o r y waves  61  Beach s l o p e e f f e c t on breaker h e i g h t f o r a 1:15 s l o p e  61  The c a l c u l a t e d shear s t r e s s u s i n g the data c o l l e c t e d f o r each wave c o n d i t i o n  140  S e n s i t i v i t y o f the impermeable beach model  149  3.2 3.3 6.1 6.2  vi  LIST OF FIGURES Figure  Title  Page  2.1  Breaking wave p r o f i l e s  6  2.2  Breaker types f o r three beach s l o p e s (Galvin 1 9 6 8 ) .  10  Breaker type as a f u n c t i o n o f the o f f s h o r e parameter ( G a l v i n 1 9 6 8 )  11  Breaker type as a f u n c t i o n o f t h e onshore parameter ( G a l v i n 1 9 6 8 )  11  2.3 2.4 2.5  V a r i a t i o n o f the breaker type with t h e beach s l o p e and H Q / T (Iversen 1 9 5 3 ) 2  2.6  2.7 2.8 2.9 2.10 2.11  12  V a r i a t i o n o f the breaker h e i g h t w i t h the deepwater wave steepness and t h e beach s l o p e (Iversen 1 9 5 3 )  13  V a r i a t i o n o f the breaker type with t h e deepwater wave steepness and t h e beach s l o p e  14  Regions f o r which s u r g i n g , p l u n g i n g , and s p i l l i n g breakers occur (Weggel 1 9 7 2 )  16  Non-dimensional depth a t b r e a k i n g v e r s u s breaker steepness (SPM 1 9 8 4 )  17  Breaker h e i g h t index versus deepwater wave steepness  17  Breaker h e i g h t index versus deepwater wave 18  steepness f o r d i f f e r e n t s l o p e s 2.12  Spill-Plunge transition  20  2.13  Plunge-Surge  21  2.14  The breaker types i n c l u d e d i n G a l v i n ' s study  transition (1968)  25  2.15  E f f e c t o f backrush on the breaker type  26  2.16  T r a n s i t i o n zones  29  2.17  Comparison o f the breaker i n d i c e s f o r n a t u r a l beaches O v e r a l l flume setup (not t o s c a l e )  30 33  2.18  vii  Figure 2.19  Title Experimental r e s u l t s f o r  Page HQ/LQ  and the breaker  type on the 1:15 s l o p e  35  3.1  V a r i a b l e s of the wave a t t h e p o i n t o f breaking ....  3.2  Breaker height index versus deepwater wave steepness Non-dimensional depth a t b r e a k i n g versus breaker steepness (SPM 1984)  3.3 3.4  41 45 45  Dependence of H^/d^ on deepwater wave steepness and beach slope  46  3.5  P l o t o f data (Iversen 1952) f o r 1:10 s l o p e  . ..  48  3.6  P l o t o f data (Iversen 1952) f o r 1:20 s l o p e  49  3.7  P l o t o f data (Iversen 1952) f o r 1:30 s l o p e  50  3.8  P l o t o f data (Iversen 1952) f o r 1:50 s l o p e  51  Comparison of experimental r e s u l t s with Miche's formula 3.10 Miche's formula and t h e s h o a l i n g equation f o r  53  3.9  selected values of  55  HQ/LQ  3.11  Miche's l i m i t  56  3.12  Breaking response o f a s o l i t a r y wave  59  3.13  H /H  Q  versus H / L  Q  62  3.14  d /H  Q  versus H / L  Q  64  5.15  Trends o f important r a t i o s f o r o s c i l l a t o r y waves and s o l i t a r y waves R e l a t i o n s h i p s between the beach s l o p e , the deepwater wave steepness, and t h e breaking conditions  3.16  b  b  0  0  4.1  Surf zone regions  4.2  Vortex chain produced breaker  4.3 4.4  64a 65a 69  by a symmetric  spilling 72  Asymmetrical p l u n g i n g breaker and t h e splash-plunge c y c l e (Longuet-Higgins T u r b u l e n t bore  1953 ....  74 77  viii  Figure  Title  Page  4.5  T y p i c a l mass t r a n s p o r t v e l o c i t y p r o f i l e s  79  4.6  E f f e c t o f i n c r e a s i n g beach roughness on the mass transport velocity  80  4.7  V e r t i c a l d i s t r i b u t i o n o f the d r i f t v e l o c i t y a t the b r e a k i n g p o i n t  82  4.8  Transport mechanism o f f s h o r e o f the n u l l p o i n t ....  84  4.9  Transport mechanism onshore o f the n u l l p o i n t  86  5.10 a) C i r c u l a t i o n c e l l s on e i t h e r s i d e o f t h e n u l l p o i n t , b) Mass t r a n s p o r t v e l o c i t y p r o f i l e s i n the s u r f zone  87  4.11  D e f i n i t i o n s k e t c h of t h e wave setup  4.12  Setup curves o f S / H  4.13  Setup p r o f i l e s f o r experimental run 2  4.14  C a l c u l a t e d t o t a l setup versus measured t o t a l setup f o r an experimental 1:15 s l o p e a) T o t a l setup versus deepwater steepness b) R e l a t i v e t o t a l setup versus deepwater steepness Components o f t h e t o t a l setup f o r experimental run 7, run 10, run 11  4.15 4.16 5.1  w  b  versus H^/gT  ..  90 93  2  95 97 98 100  Beach f a c e c o n t r o l volume and t h e f o r c e s a c t i n g upon i t  104  5.2  S i m p l i f i e d beach f a c e c o n t r o l volume  106  5.3  Forces a c t i n g upon a p a r t i c l e l a y i n g on t h e beach .  108  5.4  A s p e c u l a t i v e e x p l a n a t i o n f o r an o f f s h o r e net shear s t r e s s  110  5.5  The o f f s h o r e c o n t r o l volume used t o c a l c u l a t e M '  5.6  I n f i l t r a t i o n and e x f i l t r a t i o n on a beach face  5.7  Undamped s o l u t i o n f o r the same wave a t t a c k i n g d i f f e r e n t beaches (equation 5.22) Damped s o l u t i o n f o r the same wave a t t a c k i n g d i f f e r e n t beaches (equation 5.23) w i t h s e l e c t e d data from Dalrymple and Thompson (1976)  5.8  b  ix  .  112 119 . 123  126  Figure 5.9 5.10  6.1 6.2 6.3 6.4  Title  Page  Undamped s o l u t i o n f o r d i f f e r e n t waves a t t a c k i n g the same beach (equation 5.28) Damped s o l u t i o n f o r d i f f e r e n t waves a t t a c k i n g the same beach (equation 5.29) w i t h s e l e c t e d data from Dalrymple and Thompson (1976)  129  130  Shear s t r e s s r e q u i r e d f o r a moveable bed based upon S h i e l d s entrainment f u n c t i o n  14 3  C a l c u l a t e d shear s t r e s s v e r s u s H/gT , depth = 47.5 cm  144  C a l c u l a t e d shear s t r e s s v e r s u s H/gT , depth = 45.0 cm  145  C a l c u l a t e d shear s t r e s s v e r s u s H/gT f o r changing wave c o n d i t i o n s  148  2  2  2  x  ACKNOWLEDGEMENTS The author i s very g r a t e f u l f o r the f i n a n c i a l  support,  ouragement, and guidance g i v e n by h i s s u p e r v i s o r , C. Quick.  xi  Professor  CHAPTER  1:  INTRODUCTION  I t i s w e l l documented t h a t beaches can undergo q u i t e changes when exposed t o severe storm a t t a c k .  large  I t i s known t h a t  l a r g e q u a n t i t i e s o f m a t e r i a l can be moved o f f s h o r e , c a u s i n g reduction  a  i n beach s l o p e and p e r m i t t i n g wave a t t a c k t o cut a t  the backshore r e g i o n , o f t e n w i t h v e r y d e s t r u c t i v e r e s u l t s . the other hand, d u r i n g p e r i o d s  On  of calm weather c o n d i t i o n s ,  g e n t l e r waves can move sediment shoreward so t h a t the beach repairs either p a r t i a l l y  or completely, depending upon the  i n t e r v a l u n t i l the next storm. considerable  and  i t i s economically  p r e d i c t the n a t u r e and The  Damage d u r i n g the storm can  extent  s h a l l o w water i s w e l l known  However, as the waves s h o a l and  beach, the c h a r a c t e r i s t i c s and not v e r y w e l l understood and  break upon a  a c t i o n o f the b r e a k i n g  are d i f f i c u l t t o model.  waves g e n e r a l l y a r r i v e o b l i q u e l y t o the s h o r e l i n e and r e s o l v e d i n t o longshore and  to  of damage.  motion of waves i n deep and  and w e l l modelled.  d e s i r a b l e t o be a b l e  be  waves are Incident can  o n s h o r e / o f f s h o r e components.  1  be The  main focus o f r e s e a r c h has been on the wave-induced longshore c u r r e n t s and t h e longshore t r a n s p o r t .  L i t t l e has been done on  the o n s h o r e / o f f s h o r e components and many q u e s t i o n s r e g a r d i n g these a r i s e . 1.  Among t h e more p r e s s i n g ones a r e :  Why a r e t h e r e d i f f e r e n t types o f b r e a k i n g waves and  what a r e t h e c h a r a c t e r i s t i c s a s s o c i a t e d w i t h breaking? 2.  Does t h e breaker type i n f l u e n c e t h e o n - o f f s h o r e  c u r r e n t s and t r a n s p o r t mechanisms i n t h e s u r f zone? 3.  What a r e t h e r e l a t i o n s h i p s between t h e b r e a k i n g waves,  beach s l o p e , and sediments f o r o n s h o r e / o f f s h o r e t r a n s p o r t to 4.  occur? How does t h e o n s h o r e / o f f s h o r e t r a n s p o r t change f o r  changing wave and beach c o n d i t i o n s ? This thesis w i l l  t r y t o answer some o f these q u e s t i o n s ,  remembering t h a t t h e nearshore p r o c e s s e s a r e complex and i n t e r c o n n e c t e d , and t h a t many a r e s t i l l n o t p r o p e r l y understood. Presented i n t h i s t h e s i s i s a breakdown o f t h e events o c c u r r i n g i n t h e s u r f zone w i t h r e s p e c t t o o n l y t h e onshore/offshore d i r e c t i o n .  Chapter two d e s c r i b e s t h e p h y s i c a l  c h a r a c t e r i s t i c s o f each b r e a k e r type and t h e r e l a t i o n s h i p between t h e b r e a k e r type, deepwater wave steepness, and beach slope.  Experimental measurements a r e compared w i t h those  by o t h e r r e s e a r c h e r s .  found  Chapter t h r e e d e f i n e s t h e wave h e i g h t and  depth a t b r e a k i n g , and d i s c u s s e s t h e i n f l u e n c e o f wave steepness, b r e a k i n g depth, and t h e beach s l o p e on t h e b r e a k i n g process.  2  Chapter f o u r d e s c r i b e s t h e f l u i d motions produced w i t h i n the  s u r f zone w i t h p a r t i c u l a r emphasis on the t r a n s f o r m a t i o n o f  i r r o t a t i o n a l wave motion i n t o r o t a t i o n a l and u l t i m a t e l y t u r b u l e n t flows.  Both the mass t r a n s p o r t v e l o c i t i e s and the  water l e v e l changes, which are r e s u l t s o f t h e s e flows, are discussed.  E x p e r i m e n t a l o b s e r v a t i o n s o f f l u i d and p a r t i c l e  motions, as w e l l as c a r e f u l measurements o f water l e v e l changes, are  used. Chapter f i v e shows the development o f an o n - o f f s h o r e  t r a n s p o r t model based upon a c o n t r o l volume o f the s u r f  zone  which uses the f i n d i n g s o f t h e p r e v i o u s c h a p t e r s as a b a s i s . c h a p t e r s i x , the model i s t e s t e d u s i n g the e x p e r i m e n t a l  In  results  found f o r water l e v e l changes mentioned i n c h a p t e r f o u r . L a s t l y , c o n c l u s i o n s drawn from a l l t h e r e s u l t s a r e g i v e n i n c h a p t e r seven.  3  CHAPTER  2.1  2:  BREAKING  WAVE  TYPES  INTRODUCTION  The most v i s i b l e aspect o f a b r e a k i n g wave i s i t s shape as i t breaks.  As w i t h any complex phenomena, an important  s t e p i s t o c a t e g o r i z e the events t h a t are observed. b r e a k i n g waves, the shapes have been c l a s s i f i e d b r e a k e r t y p e s , each type dominating  For  into four  f o r c e r t a i n v a l u e s of  deepwater steepness and beach s l o p e s . b r e a k e r t y p e s are r e l a t i v e l y  first  Although the  easy t o d i s t i n g u i s h ,  distinct  the  t r a n s i t i o n s t h a t separate the breaker types are d i f f i c u l t determine.  S i n c e the flow dynamics v a r y f o r d i f f e r e n t  to  breaker  t y p e s i t i s important t o understand the c o n d i t i o n s t h a t are a s s o c i a t e d w i t h each breaker t y p e . T h i s c h a p t e r d e s c r i b e s the p h y s i c a l c h a r a c t e r i s t i c s of each b r e a k e r type and the r e l a t i o n s h i p between b r e a k e r type, deepwater steepness, and beach s l o p e .  4  2.2  PHYSICAL  CLASSIFICATIONS  OF BREAKING  WAVES  The b r e a k i n g wave can be g e n e r a l l y c l a s s i f i e d i n t o one o f f o u r groups:  s p i l l i n g ; plunging; c o l l a p s i n g ; or, surging  ( G a l v i n 1968, 1972).  The p h y s i c a l appearance  o f each type i s  shown i n f i g u r e 2.1 and t h e d e s c r i p t i o n o f each type as g i v e n by G a l v i n a r e as f o l l o w s : 1) S p i l l i n g - White t u r b u l e n t water appears a t t h e c r e s t preceded by a s m a l l j e t o f water. The t u r b u l e n c e " s p i l l s " down t h e f a c e o f t h e wave. The wave shape as a whole remains symmetric. 2) P l u n g i n g - The f r o n t o f t h e wave f a c e reaches v e r t i c a l and t h e c r e s t forms a j e t which plunges ahead of t h e wave. 3) C o l l a p s i n g - The f r o n t o f t h e wave f a c e reaches v e r t i c a l and t h e lower p o r t i o n o f t h e wave a c t s as a truncated plunging breaker. 4) S u r g i n g - The wave s t a y s r e l a t i v e l y smooth as i t moves up t h e beach, except f o r minor t u r b u l e n c e a t t h e wave-shoreline i n t e r f a c e . Turbulence i s generated by bottom boundary shear. The d e s c r i p t i v e terms a r e used t o d i s t i n g u i s h t h e breaker a t t h e p o i n t o f i n i t i a l motion. However, t h r e e problems a r i s e by the use o f t h i s c l a s s i f i c a t i o n  system  When t h e b r e a k i n g p r o f i l e i s an i n t e r m e d i a t e o f any two o f the above p r o f i l e s i t i s d i f f i c u l t t o determine how t h e breaker can b e s t be d e s c r i b e d .  T h i s i s s o l v e d by u s i n g a b r e a k e r type  index which p r e d i c t s t h e breaker type based on t h e v a l u e o f H /L 0  0  and t h e beach s l o p e .  One must remember t h a t a continuous  sequence e x i s t s between s p i l l i n g , p l u n g i n g , c o l l a p s i n g , and s u r g i n g b r e a k e r s and t r a n s i t i o n types a r e p o s s i b l e . Breakers can f a l l o u t s i d e o f t h i s continuous sequence. example o f t h i s i s a s u r g i n g wave t h a t i s f o r c e d t o a c t as a p l u n g i n g b r e a k e r due t o t h e a c t i o n o f t h e backrush.  5  For  An  -b)  Plunging  Figure 2.1  Breaking wave  6  profiles  example, a b r e a k e r index can p r e d i c t a s u r g i n g breaker, but a " f o r c e d " p l u n g i n g type breaker a c t u a l l y o c c u r s .  Obviously, i n  t h i s case t h e p e r m e a b i l i t y and s l o p e o f t h e beach p l a y important roles. The f i n a l problem motions  i s one o f s c a l e .  The i n i t i a l  breaking  o f both t h e p l u n g i n g and s p i l l i n g b r e a k e r s a r e the same  except t h e s c a l e f o r which they occur ( G a l v i n 1968, 1972; M i l l e r 1976;  Basco 1985).  Plunging breakers are g e n e r a l l y c a t e g o r i z e d  by a l a r g e - s c a l e , v i s i b l e c u r l i n g o f t h e wave c r e s t around an inner a i r core.  The f a l l i n g j e t h i t s t h e oncoming trough  suddenly t r a n s f o r m i n g t h e i r r o t a t i o n a l motion t o r o t a t i o n a l motion over a l a r g e percentage o f t h e water column.  An  i d e n t i f i a b l e time and d i s t a n c e a r e r e q u i r e d f o r t h e wave t o c u r l over b e f o r e r e a c h i n g t h e j e t impact p o i n t o r plunge p o i n t . the s p i l l i n g breaker, on t h e o t h e r hand, t h e i n i t i a l ,  For  low-level  impact o f t h e j e t g e n e r a l l y o c c u r s q u i c k l y and i s not v i s i b l e t o the eye.  The s m a l l s c a l e j e t r a p i d l y c r e a t e s t u r b u l e n c e a t the  c r e s t which s l i d e s down t h e f a c e o f t h e wave.  V o r t i c i t y and  t u r b u l e n c e a r e c r e a t e d and t h e flow g r a d u a l l y changes from i r r o t a t i o n a l t o r o t a t i o n a l motion over an i n c r e a s i n g of  percentage  t h e water column. Both s p i l l i n g and p l u n g i n g b r e a k e r s a r e i n i t i a t e d by a j e t  h i t t i n g t h e forward f a c e o f t h e wave.  The i n i t i a l behaviour o f  each type i s i d e n t i c a l , except f o r t h e d i f f e r e n c e i n s c a l e , but t h e r e s u l t i n g motions  are very d i f f e r e n t .  i d e n t i f y t h e b r e a k e r based on i t s t o t a l  7  Care must be taken t o  motion.  A s i d e from these problems,  the c l a s s i f i c a t i o n o f breakers  g i v e n by G a l v i n i s w i d e l y used as a q u a l i t a t i v e d e s c r i p t i o n ,  but  does not l e n d i t s e l f t o p r o d u c i n g q u a n t i t a t i v e v a l u e s . 2.3  BREAKING  INDICES:  HQ/LQ,  BEACH  SLOPE,  BREAKING  TYPES  There i s agreement t h a t breaker types form a spectrum  of  shapes going from s p i l l i n g t o p l u n g i n g t o c o l l a p s i n g t o s u r g i n g , and v i c e v e r s a .  The c y c l e i s complex because the beach s l o p e  depends upon the breaker type and the breaker type depends upon the beach s l o p e . General o b s e r v a t i o n s o f beaches can be made.  Steep waves  on low angle beaches tend t o form s p i l l i n g b r e a k e r s , whereas l e s s steep waves on h i g h e r angle beach s l o p e s tend t o form plunging or surging breakers. P a t r i c k and Wiegel  Results of Iversen  (1953),  (1955), G a l v i n (1968), and p e r s o n a l  o b s e r v a t i o n s c o n f i r m these statements.  Thornton  e t a l . (1976)  measured the k i n e m a t i c s o f v a r i o u s types of b r e a k i n g waves and f i n d t h a t the manner i n which waves break depends v e r y much on the c h a r a c t e r i s t i c s o f the deepwater steepness and the nearshore bottom s l o p e . B r e a k i n g waves were o r i g i n a l l y c l a s s i f i e d by G a l v i n (1968) who  found t h a t two d i m e n s i o n l e s s parameters  determine  c o u l d be used t o  the type o f b r e a k i n g wave r e s u l t i n g from o f f s h o r e or  i n s h o r e wave c h a r a c t e r i s t i c s .  The breaker types f o r waves on  impermeable beaches w i t h s l o p e m, wave p e r i o d T, deepwater wavelength  L , Q  and e i t h e r deepwater o r b r e a k e r h e i g h t , H  can be s o r t e d by two d i m e n s i o n l e s s combinations  8  of these  Q  or  H^,  variables.  The o f f s h o r e parameter i s ,  H /(L m )  (2,1)  2  0  0  and, t h e i n s h o r e parameter i s ,  H /(gmT )  (2,2)  2  b  Both parameters a r e e m p i r i c a l l y determined. H /L 0  0  and ( H / g T )  a r e p l o t t e d f o r each s l o p e  1 / / 2  b  Initially, (figure 2 . 2 ) .  The r e g u l a r i t i e s i n f i g u r e 2 . 2 suggests t h a t t h e s l o p e can be ( f i g u r e s 2 . 3 and 2 . 4 ) .  i n c l u d e d i n t h e c l a s s i f y i n g parameters  For t h e o f f s h o r e and i n s h o r e parameters, r e s p e c t i v e l y , t h e surge-plunge t r a n s i t i o n s a r e 0 . 0 9 transitions are 4 . 8  and  and  0.003  and t h e p l u n g e - s p i l l  As e i t h e r o f t h e s e parameters  0.068.  i n c r e a s e , t h e b r e a k e r type goes from s u r g i n g t o c o l l a p s i n g t o plunging t o s p i l l i n g . Iversen  (1953)  p r o v i d e s a graph o f t h e v a r i a t i o n o f t h e  b r e a k e r c h a r a c t e r i s t i c s w i t h beach s l o p e , deepwater and p e r i o d  (figure  2.5).  The parameter  the deepwater wave steepness, H / L 0  plotted i n figure 2 . 6 .  HQ/T  2  wave h e i g h t ,  i s equivalent to  because L =gT /27T, and i s 2  Q /  Q  I v e r s e n g i v e s o n l y t h e ranges o f H / T  2  Q  over which each o f t h e t h r e e b r e a k e r types a r e observed; b r e a k e r t y p e s f o r i n d i v i d u a l waves a r e n o t a v a i l a b l e .  In addition, f o r  two o f t h e t h r e e beach s l o p e s i n v e s t i g a t e d , t h e plunge-surge t r a n s i t i o n may n o t have been reached; t h e r e f o r e , t h e lower of H / L 0  0  limit  f o r which p l u n g i n g b r e a k e r s a r e observed a r e used as  the t r a n s i t i o n  points.  P a t r i c k and Wiegel  (1955)  p r o v i d e a graph o f t h e v a r i a t i o n  o f b r e a k e r c h a r a c t e r i s t i c s w i t h deep water wave steepness and o f f s h o r e beach steepness ( f i g u r e 2 . 7 ) .  9  The l a b o r a t o r y data upon  A O • • ®  Spilling Plunging Collapsing Surging Plunging affected by reflection  °- °L  •—g-oo  °- '  D—rJ-O-*—#CDO—L-c—®0-O-OO——OO  2  ,0  c—9-am 9 Jo OcL  0.05 L o «n .c u o  <= I I0"  1 4  1 1111 llJ I0"  1  1  '  I  I I0"  < I I I  5  I  Computed Deep-woter Steepness, H / L 0  to  0.20I—c&Jbo—'  O—OO-Q  I  l l tii I 10*  0  •  o.ioL_a3#U>j^-o_o-c 0.05 I  l  2  oL&Cr-  ob  o to 0.02  0.04  0.06  0.08  Breoker Sleepness, ( H / g T b  F i gure  2 ,2  2  0.10 )  0.12  l / 2  B r e a k e r "types f o r 3 b e a c h s l o p e s ( G a l v i n 1968)  10  A O • • <g)  Surging-Collapsing  10-3  10-A  I0'  2  2,3  Plunging  10"' H /(L m ) z  0  Figure  Spilling  10°  10'  0  B r e a k e r t y p e as a f u n c t i o n o f f s h o r e parameter (Galvin  Plunging  Surging-Collapsing  10'  10"  10"  Spilling Plunging Collapsing Surging Plunging affected by reflection  of the 1968)  Spilling  10'  H /{gmn b  F i gure  2,4  Breaker onshore  t y p e as a f u n c t i o n parameter (Galvin  ii  of the 1968)  10'  SPILL ING  • I-10 pL U N G I N G SURGING  Btach Slop, - 1:20  <  5PILLI NG  ginq  p  Btach S/opt • /.SO  SPI  PLU  001  0 02  0.04  OX* DEEP  gure  2 ,5  WATER  0.1 WAVE,  02  04  0.6  Q8  H /T* 0  V a r i a t i on o f t h e b r e a k e r "type w i "th b e a c h s I ope an d H o / T ( I v e r s e n - 1953) 2  12  Beach s i o 38  1:10  3D  Dlungu  Su  BeaoT Slo >e  20  Sail  Plunge  B e a c i Slo >e  5') Flu ige  O.0O1  0.01  0.1  Deepwater steepness, Ho/Lo  F i gure  2,6  Var i a t i on o f t h e b r e a k e r t y p e w i t h d e e p w a t e r wave s t e e p n e s s and t h e beach s l o p e ( I v e r s e n 1953)  13  the  u, i " 0 a 1/20  z o <  U  1  1/10  "*"  1/20 1/50  A •  Plunging  • I JTTT  II  1/10 1/20 1/50  Spilling  1  a aoi  F i gure 2 , 7  0.00B  1 1  0.01  o.o<  i  M. ,  DCEP WATER WAVE HEIGHT  L. ,  OCEP WATER WAVE L£N€TM  Surging  J 0.04  OJM  0.0*  (after  I v e r s e n 1953)  V a r i a t i o n o f the b r e a k e r type with the d e e p w a t e r wave s t e e p n e s s a n d t h e b e a c h s l o p e ( P a t r i c k and Wiege I 1955)  14  which P a t r i c k and Wiegel base t h e i r graph has n o t been published.  For two o f the t h r e e beach s l o p e s i n v e s t i g a t e d , t h e  p l u n g i n g t o s u r g i n g t r a n s i t i o n may n o t have been reached. lower l i m i t o f H  0  / L  f o r which the p l u n g i n g breaker  Q  are  The observed  are used as the t r a n s i t i o n p o i n t . Weggel terms o f  H  Q  / L  (1972)  0  ,  p l o t s the r e s u l t s o f P a t r i c k and Wiegel i n  beach s l o p e s , and breaker type.  The r e g i o n s o f  t r a n s i t i o n are shown i n f i g u r e 2 . 8 . Iversen (1972)  (1952,  Galvin  1953),  establish that  H  B  / H  Q  (1968),  (figures  (1970)  The U.S.  Army Corps o f  p r e s e n t s t h e r e s u l t s o f Weggel and Goda as two graphs 2.9  and  2.10).  of P a t r i c k and Wiegel.  Weggel's r e s u l t s are based upon those Figure 2 . 1 0  p r e s e n t s Goda's e m p i r i c a l l y  d e r i v e d r e l a t i o n s h i p between the breaker h e i g h t index, and deepwater wave steepness,  H /gT H  0  / L  Q  f o r s e v e r a l beach s l o p e s  f o r which each type o f breaker can be expected i n f o r m a t i o n i s a l s o presented i s reached  by I v e r s e n  (1952)  i n figure 2 . 1 0 .  Comparisons o f the B r e a k i n g (1953),  t o occur.  This  A similar  result  and, again, the t r a n s i t i o n p o i n t s  g i v e n by P a t r i c k e t a l . a r e p l o t t e d i n f i g u r e  Iversen  H^/HQ,  , for various slopes.  Q  P a t r i c k e t a l . p r e s e n t ranges o f  2.3.1  and Weggel  and d^E^ depend on the beach s l o p e  and the i n c i d e n t wave steepness. Engineers  Goda  Patrick et a l .  2.11.  Indices  (1955),  and G a l v i n  (1968)  p r o v i d e t h e o n l y independent data on the r e l a t i o n s h i p between beach s l o p e , deepwater wave steepness,  and breaker  types.  i s p o s s i b l e t o p r e d i c t the breaker type g i v e n the s l o p e and  15  If i t  007  0 06  v  H i = 0.01 t 0.30 m I 0  0.03 ion H  - Spilling  0.04  i n - piuo  0.03  0.02 Ol(l- « 0.01  !  f 1 Region I - Surging 0  Figure  0.01  2.8  0.02  0 03  0 04  003 0 06 Beach Slope, m  1  0.07  1  0 08  0 09  0.10  R e g i o n s f o r which s u r g i n g , p l u n g i n g , and s p i l l i n g b r e a k e r s o c c u r (Weggel 1972)  16  0  0.002  0.004  Figure'2,10  00004  0.006  0.008  0.010  0.012  0.016  B r e a k e r height index v e r s u s wave s t e e p n e s s (SPM 1984)  00006  0001  0002  0 004 HJ,  2,9  0 006  0 01  0.018  0.020  deepwater  002 0 03 (olUr Goda, 1970)  •J N o n - d i m e n s i o n aI d e p t h a t b r e a k i n g v e r s u s b r e a k e r s t e e p n e s s (SPM 1984) 1  Figure  0.014  17  Figure 2,11  B r e a k e r height index v e r s u s d e e p w a t e r wave s t e e p n e s s f o r d i f f e r e n t s l o p e s  18  H / L , then t h i s i n f o r m a t i o n c o u l d be use t o q u a n t i t a t i v e l y 0  0  understand  the breaking  process.  F i g u r e s 2.12 and 2.13 a r e t h e p l o t s o f t h e v a l u e s o f the breaker spilling et  i n d i c e s presented  i n t a b l e 2.1.  Comparison o f t h e  t o p l u n g i n g t r a n s i t i o n show agreement between P a t r i c k  a l . and I v e r s e n , w i t h G a l v i n ' s r e s u l t s p l o t t i n g w e l l below.  In f a c t , f o r v a l u e s o f low beach s l o p e , t h e d i f f e r e n c e i s o f an o r d e r o f magnitude.  For the plunging t o surging t r a n s i t i o n  t h e r e i s l i t t l e agreement between t h e r e s e a r c h e r s , w i t h G a l v i n ' s r e s u l t s d i f f e r i n g by two o r d e r s o f magnitude. 2.3.2 V a r i a t i o n s o f the Breaking  Indices  D i f f e r e n c e s o f up t o two o r d e r s o f magnitude between the r e s u l t s o f t h e t h r e e r e s e a r c h e r s suggests t h a t s i g n i f i c a n t f a c t o r s are causing the v a r i a t i o n s . from d i f f e r e n c e s i n t h e procedures, d e f i n i t i o n s used by t h e r e s e a r c h e s .  These f a c t o r s may a r i s e c a l c u l a t i o n s , and These and o t h e r s w i l l  now  be d i s c u s s e d . 2.3.2.1 Experimental  Procedures  and C a l c u l a t i o n s  G a l v i n uses a 71 f o o t wave flume and a p i s t o n - t y p e wave generator.  I v e r s e n uses a 54 f o o t wave flume and a f l a p - t y p e  wave g e n e r a t o r . of  Patrick et a l .  do not p r o v i d e any i n f o r m a t i o n  t h e i r equipment. To c a l c u l a t e t h e deepwater wave h e i g h t , H , I v e r s e n and Q  P a t r i c k e t a l . use t h e wave h e i g h t measured i n t h e c o n s t a n t depth s e c t i o n o f t h e wave flume and back c a l c u l a t e H l i n e a r wave t h e o r y .  Q  using  G a l v i n uses t h e wave h e i g h t p r e d i c t e d  t h e o r e t i c a l l y f o r a g i v e n displacement 19  of the p i s t o n (Biessel  Deepwater wave s t e e p n e s s , H o / L o 0.07 0  -i  0.060  -  0.040  -  0.030  -  0.020  -  0.010  -  o.ooo  -\  0.000  1  \  0.020  0.040  1  0.080  Slope Figure  2,12  1  0.080  1  0.100  1  0.120  ( m)  Spill-Plunge 20  transition  1—  0.140  Deepwater wave steepness, H o / L o 0.010  0.008  Plunge  0.006 I versen(i953) -}—  P a t r i c k et  al.(1955)  -?K-  Galvin(1968)  0.004  0.002  Surge  0.000 0.000  0.020  0.040  0.060  Slope Figure 2.13  0.080  0.100  ( m)  Plunge-Surge  transition  0.120  0.140  a) P a t r i c k  b)  c)  and Weggel (1955)  SLOPE  SURGING  PLUNGING  SPILLING  0.020 0,033 0,050 0.100  Ho/Lo<0.0039  0.0039<Ho/Lo<0.020  Ho/Lo>0.020  Galvin  <0.0079 <0.0095  0.0079< 0.0095<  <0.035 <0.060  >0.035 >0,060  (1968)  SLDPE  SURGING  PLUNGING  SPILLING  0,020 0.033 0.050 0,100  Ho/Lo<0.000036 <0.000100 <0.000225 <0.000900  0.000036<Ho/Lo<0.00192 0.000100< <0.00520 0.000225< <0.01200 0.00090CK 0.04800  Ho/Lo>0,00192 >0.00520 >0,01200 >0.04800  PLUNGING  SPILLING  I v e r s e n (1953) SLDPE  SURGING  0.020 0.033 0.050 0.100  Ho/Lo<0.00195 <0.00390 <0.00878  Table 2,1  0.00195<Ho/Lo<0.023 0,0039< 0.00878<  Transition  22  <0.0375 <0.0605  points  Ho/Lo>0.0233 >0.0375 >0.0605  1951)  and back c a l c u l a t e s H  Q  u s i n g l i n e a r wave theory.  This  method e f f e c t i v e l y e l i m i n a t e s t h e problem o f a t t a i n i n g a s p a t i a l l y uniform wave h e i g h t i n t h e wave flume, but G a l v i n (1964) f i n d s t h a t t h e t h e o r e t i c a l h e i g h t i s g r e a t e r than t h e measured wave h e i g h t i n t h e flume. breaker  Therefore,  f o r a given  type on a g i v e n s l o p e with i d e n t i c a l c o n d i t i o n s i n the  wave flume, G a l v i n ' s r e s u l t s should p l o t above Iversen's and P a t r i c k ' s , r e g a r d l e s s o f t h e wave generator. 2.12 and 2.13 show t h i s not t o be t h e case.  However, f i g u r e s I v e r s e n ' s and  P a t r i c k ' s r e s u l t s a r e c o n s i s t e n t l y above G a l v i n ' s which i n d i c a t e s that other f a c t o r s are responsible f o r the d i f f e r e n c e i n the r e s u l t s . 2.3.2.2 Wave E f f e c t s Wave e f f e c t s t h a t i n t e r f e r e w i t h t h e development o f t y p i c a l b r e a k i n g waves a r e secondary waves, wave r e f l e c t i o n s , and backrush.  Secondary waves a r e caused by t h e breakdown o f a  l a r g e wave i n t o a primary wave and a number o f s m a l l e r secondary waves.  Wave r e f l e c t i o n s and backrush a r e i n f l u e n c e d by beach  s l o p e and beach p e r m e a b i l i t y w i t h both i n c r e a s i n g f o r i n c r e a s i n g s l o p e and d e c r e a s i n g p e r m e a b i l i t y .  (The importance o f beach  s l o p e and beach p e r m e a b i l i t y w i l l be d e s c r i b e d i n l a t e r chapters.) G a l v i n uses an impermeable c o n c r e t e beach and s l o p e s s t e e p e r than those commonly found i n nature.  Therefore,  r e f l e c t i o n s and backrush may a f f e c t t h e b r e a k i n g wave, but w i l l not a f f e c t G a l v i n ' s c a l c u l a t i o n o f H / L Q  t h i s from t h e displacement  0  s i n c e he determines  o f t h e wave g e n e r a t o r .  23  To account  f o r v a r i a t i o n s i n t h e breaker shape G a l v i n l a b e l s b r e a k e r s a c c o r d i n g t o a l i s t o f p o s s i b l e types that the data s a t i s f y the following  (f i g u r e ,,2.14) and r e q u i r e s  restrictions:  1) A dominant breaker type e x i s t s when a t l e a s t 70% of the waves i n a s t e a d y - s t a t e sequence have t h e same b r e a k e r type. 2) The b r e a k i n g o f t h e primary wave i s not h i n d e r e d by a secondary wave and i s s a t i s f i e d when the breaker i s not g i v e n a code number o f 7 o r 8, a c c o r d i n g t o f i g u r e 2.14. These p r e c a u t i o n s reduce t h e i n f l u e n c e o f secondary waves and wave r e f l e c t i o n s . backrush  None o f these p r e c a u t i o n s , though, takes  i n t o account.  Backrush  i s capable o f f o r c i n g a  c o l l a p s i n g o r s u r g i n g breaker t o a c t as a p l u n g i n g b r e a k e r .  If  f o r c e d p l u n g i n g b r e a k e r s a r e r e c o r d e d as a p l u n g i n g b r e a k e r then the plunge-surge plunging w i l l  t r a n s i t i o n w i l l move down ( f i g u r e 2.15) s i n c e  occur over a g r e a t e r range o f H / L . 0  Iversen's c a l c u l a t i o n s of H / L 0  Q  a r e a f f e c t e d by secondary  waves and r e f l e c t i o n s s i n c e t h e v a l u e o f H  Q  i s dependent upon  wave h e i g h t s measured w i t h i n t h e wave flume. of  t h i s has a l r e a d y been mentioned.  backrush  Q  The s i g n i f i c a n c e  I v e r s e n , though, takes  i n t o account so t h a t t h e plunge-surge  t r a n s i t i o n given  by I v e r s e n s h o u l d be h i g h e r than t h e t r a n s i t i o n g i v e n by G a l v i n , which i s t h e case. Beach p e r m e a b i l i t y i s an important f a c t o r backrush.  controlling  I f a beach i s permeable t h e backrush w i l l be reduced  s i n c e t h e uprush can permeate i n t o t h e beach.  I f t h e beach i s  impermeable, t h e backrush w i l l be s t r o n g s i n c e a l l o f t h e uprush must r e t u r n as backrush.  Coda 1  Type of Breaking Spilling  Description Bubbles and turbulent watai spill down front face of wart. Tbe upper 2aT of the front fact may become vertical befort breaking. Crest curls over a large air pocket. Smooth splash-up usually follows. Crest curls less and air pocket •mailer than i n 2. Breaking occurs over lower half of wave. Minimal air pocket and usually no spLajh-up. Bubbles and foam present. Wave slides up beach witif little or no bubble production! Water surface remains almost plane except where ripples majt be produced on the beach fats during runback. Small waves reflected from the preceding wave peak up tbt breaking crest. Breaking otherwise unaffected. Primary may ride in on secondary immediately befon it, or secondary immediately behind ride* i n on primary is front. First kind difficult to distinguish from 8. Plunging secondary may break just i n front of surging primary. Difficult to distinguish from 7. Runback from previous primary carries tbe secondary ws*s offshore, where it may break out of field of view or just disappear. c  2 Well-developed plunging 3 Plunging 4  Collapsing  5 Surging  Figure  6  Plunging altered by reflected wave  7  Plunging altered by secondary wave  8  Surging altered by secondary wave  0  Secondary wave washed out  2.14  B r e a k e r "types included in Galvin's(1968) study  25  ©  Slope  (m)  (T)  S p i l l - P l u n g e transition  (2)  P l u n g e - S u r g e transition as surging breakers  where f o r c e d plungers are r e c o r d e d  (3)  P l u n g e - S u r g e transition is neglected  where the effect  Figure 2.15  of the  The effect of backrush type  26  backrush  on the  breaker  S u r g i n g breakers g e n e r a l l y occur c l o s e r t o t h e s h o r e l i n e . T h e r e f o r e , a s u r g i n g wave on a permeable beach can be f o r c e d t o plunge on an impermeable beach.  Both G a l v i n and I v e r s e n use  impermeable beaches s t e e p e r than those commonly found It  i s not s u r p r i s i n g then t o f i n d t h e surge r e g i o n t o be  r e l a t i v e l y small f o r small values of H / L Q  The  i n nature.  (see f i g u r e 2.13).  0  same cannot be d e f i n i t e l y s a i d f o r waves i n t h e s p i l l - p l u n g e  transition.  G e n e r a l l y , these waves break f u r t h e r o f f s h o r e than  s u r g i n g breakers and a r e l e s s s u s c e p t i b l e t o t h e e f f e c t o f backrush. 2 . 3 .2.3 Breaking  Definitions  Galvin defines s p i l l i n g ,  p l u n g i n g , c o l l a p s i n g , and s u r g i n g  b r e a k e r s , but f i n d s t h a t t h e c o l l a p s i n g b r e a k e r s group t o g e t h e r w i t h t h e s u r g i n g breakers and i n t h e end o n l y shows t h e transition forspilling,  p l u n g i n g , and s u r g i n g b r e a k e r s .  I v e r s e n and P a t r i c k e t a l . d e f i n e s p i l l i n g , surging breakers.  p l u n g i n g , and  They a l s o grouped t h e c o l l a p s i n g type  the s u r g i n g breaker.  with  T h e r e f o r e , t h e r e s e a r c h e r s use t h e same  d e f i n i t i o n s f o r each breaker type which does not h e l p e x p l a i n the d i f f e r e n t  results.  2.3.2.4 P e r s o n a l Judgement A b r e a k i n g wave c l o s e t o a t r a n s i t i o n s t a t e w i l l c h a r a c t e r i s t i c s o f two b r e a k i n g t y p e s .  exhibit  In t h i s s i t u a t i o n , the  p e r s o n a l judgement o f t h e r e s e a r c h e r s w i l l have a s i g n i f i c a n t i n f l u e n c e on t h e f i n a l r e s u l t s .  A t a t r a n s i t i o n , t h e breaker  d e f i n i t i o n s b e g i n t o o v e r l a p , so f o r example, what one r e s e a r c h e r d e f i n e s as a p l u n g i n g breaker, 27  another  researcher  c o u l d d e f i n e as a c o l l a p s i n g breaker.  Even though the  d e f i n i t i o n s o f the b r e a k i n g types are the same and d e f i n i t e ,  the  i n t e r p r e t a t i o n of the d e f i n i t i o n are d i f f e r e n t between i n d i v i d u a l s and are a p p l i e d at d i f f e r e n t p o i n t s i n the process.  T h i s may  e x p l a i n the c o n s i s t e n t d i f f e r e n c e found  between the b r e a k i n g  F i g u r e 2.16  transitions.  shows the range o f v a r i a t i o n between the  r e s u l t s of the r e s e a r c h e r s . breaking  types and two  Three r e g i o n s d e f i n e d e f i n i t e  r e g i o n s d e f i n e areas of o v e r l a p t h a t ,  a c c o r d i n g t o the p r e v i o u s d i s c u s s i o n s , may breaking  zones.  T h i s seems reasonable  t r a n s i t i o n breaker 2.3.3 The  form t r a n s i t i o n  s i n c e some form o f a  i s expected.  Breaking breaking  breaking  I n d i c e s on N a t u r a l Beaches  i n d i c e s have been determined by u s i n g  plane  impermeable beaches i n c o n t r o l l e d l a b o r a t o r y environments. N a t u r a l beaches, however, have bars and  steps,  continuously  changing composite s l o p e s , and v a r y i n g degrees of p e r m e a b i l i t y . I t i s not s u r p r i s i n g t o f i n d t h a t the b r e a k i n g l a b o r a t o r y c o n d i t i o n s do not adequately  i n d i c e s found f o r  predict conditions in  nature. Weishar and Byrne (1978) use the r e s u l t s of 116 f i l m e d a t V i r g i n i a Beach, Va.,  on the A t l a n t i c U.S.  f i n d t h a t the c l a s s i f i c a t i o n of G a l v i n does not d i s c r i m i n a t e between p l u n g i n g and 2.17).  Both p l u n g i n g and  range, w i t h p l u n g i n g breakers  coast.  (figure  o c c u r over the whole  t e n d i n g t o bunch a t the  28  They  significantly  s p i l l i n g breakers  s p i l l i n g breakers  waves  transition  29  a  »  n  an  jfta m~J A  n'Sm-B-ftfll. -  "  6  A  J  I . .1^8 a  A  #  T  *  %  *  a  E  LQm  *J  a  n  8^  AA  io'  ii  i  ..  6  v r  b/gmT  ^  AAA  A  |rir  A  —  |  2  A • A IO*  „  t»*  Comparison o f t h e b r e a k e r natural beaches  , ie>  lfi8h$nVBAll).A n a n  m  H  A  a T>  to»  to  T  2.17  a ,  A«aAAA&<  \o-  Figure  ft  io'  Ho/  I0"»  1  n__»  10*  •<>'  indices f o r  p o i n t s g i v e n by G a l v i n .  The t r a n s i t i o n s g i v e n by P a t r i c k e t a l .  do not f i t t h e r e s u l t s e i t h e r . the s p i l l i n g breakers  More data p o i n t s a r e needed f o r  t o determine i f t h e r e i s a g r e a t e r  tendency f o r them t o bunch. The breaker  poor r e s u l t s can be e x p l a i n e d i n two ways.  E i t h e r the  i n d i c e s do not work i n t h e n a t u r a l environment, and/or  the breaker  i n d i c e s work but r e q u i r e measurements t h a t take i n t o  account t h e v a r i a b i l i t y o f t h e environment . value of H / L 0  0  For instance, the  should r e p r e s e n t t h e s p e c t r a o f incoming v a l u e s  and t h e v a l u e o f t h e s l o p e should r e p r e s e n t t h e complex, composite s l o p e s .  Izumiya and Isobe (1986) found t h a t an  e q u i v a l e n t s l o p e can be d e f i n e d as t h e mean s l o p e i n t h e d i s t a n c e o f Sd^ o f f s h o r e o f t h e b r e a k i n g p o i n t , where d^ denotes the water depth a t t h e b r e a k i n g p o i n t . r e s u l t f o r an impermeable beach.  But, t h i s a g a i n i s a  Also, using H / L 0  independent v a r i a b l e may n o t be adequate. of H / L 0  and  0  Q  as t h e  Obtaining the value  r e q u i r e s e i t h e r deep water sensors, which a r e d i f f i c u l t  expensive,  o r back c a l c u l a t i n g from nearshore c o n d i t i o n s ,  which i n t r o d u c e s c a l c u l a t i o n e r r o r s .  Therefore,  using the  nearshore c o n d i t i o n s as t h e independent v a r i a b l e s ( i e . H^/d^, H / g T , d^/gT ) may r e p r e s e n t a b e t t e r method. 2  2  b  The most probable  cause o f t h e d i s c r e p a n c y  and p r e d i c t e d b r e a k i n g types beaches.  between observed  i s t h a t t h e i n d i c e s a r e f o r plane  Waves a r e f o r c e d t o break on a beach t h a t do n o t  respond t o t h e incoming wave c o n d i t i o n s .  Dalrymple e t al.(1976)  p r o v i d e a graph t h a t p l o t s t h e e q u i l i b r i u m s l o p e a g a i n s t t h e nondimensional f a l l v e l o c i t y o f t h e beach sediment, H / V T where Q  V  f  i s the f a l l v e l o c i t y .  Therefore,  31  f  combinations o f H / L , 0  0  beach s l o p e , and breaker types f o r plane beaches may  be  i m p o s s i b l e t o a c h i e v e on n a t u r a l beaches s i n c e the l a b o r a t o r y s l o p e s are not n e c e s s a r i l y the e q u i l i b r i u m s l o p e f o r the incoming waves. 2.4  EXPERIMENTAL  DESIGN  AND  PROCEDURES  The o b j e c t i v e o f t h i s experiment  i s t o study  the  r e l a t i o n s h i p between the breaker type, beach s l o p e , and deepwater wave steepness, and t o compare these r e s u l t s w i t h those found by o t h e r r e s e a r c h e r s . A model beach was l o n g , 0.60  b u i l t i n a wave flume t h a t i s 28 meters  meters wide, and 0.70  meters deep.  The waves were  generated by a f l a p - t y p e wave paddle capable of v a r y i n g the wave h e i g h t and p e r i o d . and s l o p e d a t 1:15.  The beach was  r i g i d , wooden, and  T h i s s l o p e was  impermeable  chosen t o p r o v i d e t h e  g r e a t e s t range o f breaker types and nearshore c o n d i t i o n s . o v e r a l l setup i s shown i n f i g u r e  The  2.18.  Wave and b r e a k i n g c o n d i t i o n s were v a r i e d by changing  the  depth of the c o n s t a n t depth s e c t i o n , the p e r i o d , and the wave height.  I n i t i a l l y , the depth and p e r i o d are h e l d c o n s t a n t w h i l e  the wave h e i g h t was  changed.  For each wave h e i g h t , 10 t o 15  minutes were g i v e n f o r the c o n d i t i o n s i n the flume t o become stable.  The wave h e i g h t was  measured 2 meters o f f s h o r e of the  j u n c t i o n between the s l o p e d beach and f l a t bed. type was  The  breaker  r e c o r d e d u s i n g the d e f i n i t i o n s found i n s e c t i o n  2.2.  V i d e o p i c t u r e s of each run were made so t h a t the b r e a k i n g p r o c e s s , which i s a v e r y q u i c k p r o c e s s , c o u l d be a n a l y z e d a t a slower  speed.  32  28n  0.6n  PLAN Wave Generator  VIEW  Vave Filter  SVL  0,7n Rigid Plywood Beach  SIDE Figure 2,18  VIEW  O v e r a l l flume s e t u p  33  (not  to  scale)  Once t h e data was recorded, t h e deepwater wave  steepness  was back c a l c u l a t e d u s i n g t h e i n t e r m e d i a t e depth c o n d i t i o n s and l i n e a r wave t h e o r y . 2.5  EXPERIMENTAL  RESULTS  R e s u l t s from experiments f o r wave b r e a k i n g on a 1:15 s l o p e are shown i n f i g u r e 2.19.  The v a l u e o f H / L Q  Q  i s back c a l c u l a t e d  from t h e i n t e r m e d i a t e depth c o n d i t i o n s u s i n g l i n e a r wave theory. The b r e a k i n g types a r e d e f i n e d as i n s e c t i o n 2.2 . There appears t o be a r e g i o n o f o v e r l a p where e i t h e r a p l u n g i n g o r s p i l l i n g breaker can occur. G a l v i n ' s l i m i t roughly corresponds H /L 0  0  t o the smallest value of  f o r which a s p i l l i n g breaker o c c u r s .  P a t r i c k ' s l i m i t s roughly corresponds H /L 0  0  From t h e r e s u l t s ,  t o the l a r g e s t value of  f o r which p l u n g i n g b r e a k e r s occur.  t r a n s i t i o n boundaries  I v e r s e n ' s and  In s e c t i o n 2.3.2, the  a r e assumed t o be t h e l i m i t s o f t h e zone.  From l a b o r a t o r y experiments,  t h i s assumption appears t o be  correct. P l u n g i n g b r e a k e r s a r e a l s o observed  t o occur f o r c o n d i t i o n s  t h a t should produce s u r g i n g breakers as p r e d i c t e d by I v e r s e n and Patrick et a l .  In f a c t , s u r g i n g breakers c o u l d n o t be produced  i n t h e s e r i e s o f experiments.  The plunge-surge  t r a n s i t i o n s of  I v e r s e n and P a t r i c k e t a l . a r e assumed t o occur a t t h e end o f the plunge r e g i o n ( f i g u r e s 2.5 and 2.7).  These r e s e a r c h e r s may  have had t h e same problem; s u r g i n g breakers a r e d i f f i c u l t t o produce on lower s l o p e d beaches. plunge-surge  I f t h i s i s t h e case, then the  t r a n s i t i o n p r e d i c t e d by them w i l l be t o o h i g h f o r  lower s l o p e d beaches.  34  Deepwater wave s t e e p n e s s , H o / L o 0.070  0.060  0.050  -  0.040  -  O S p i l l i n g breaker X P l u n g i n g breaker  0.030  -  Plunge  0.020  0.010  -  Iversen(i953) - h  Patrick(l955) Galvln(1968)  0.000 0.000  0.020  0.040  0.060  Slope Figure  2.19  0.080  0.100  0.120  0.140  ( m)  Experimental r e s u l t s f o r H o / L o and t h e b r e a k e r t y p e on t h e 1:15 slope  35  2.6  DISCUSSION  The r e s u l t s o f I v e r s e n Galvin  (1953),  P a t r i c k e t a l . (1955), and  (1968) show s i g n i f i c a n t d i f f e r e n c e s , e s p e c i a l l y f o r the  plunge-surge  transition.  Up t o two o r d e r s o f magnitude separate  the t r a n s i t i o n between t h e t h r e e breaker type groups. v a r i a b i l i t y i s expected types i s a continuous  Some  s i n c e t h e t r a n s i t i o n between b r e a k i n g  change o c c u r r i n g over a range o f H / L . 0  Q  P i n p o i n t i n g t h e change i n t h e b r e a k i n g form i s d i f f i c u l t and i s a f f e c t e d by d i f f e r e n c e s i n p e r s o n a l judgement. The  r e s e a r c h e r s o b t a i n e d two r e s u l t s f o r each s l o p e  condition:  t h e type o f b r e a k i n g wave; and, t h e v a l u e o f H / L Q  a s s o c i a t e d w i t h t h e b r e a k i n g wave. upon t h e method o f c a l c u l a t i o n . wave r e f l e c t i o n s and secondary the displacement  The v a l u e o f H / L Q  0  0  depends  G a l v i n e l i m i n a t e s t h e e f f e c t of waves by c a l c u l a t i n g H / L 0  o f t h e wave g e n e r a t o r .  0  using  I v e r s e n and P a t r i c k e t  a l . use measured wave h e i g h t s , which a r e a f f e c t e d by r e f l e c t i o n s and secondary  waves, t o c a l c u l a t e H / L . 0  Q  Values of H / L 0  c a l c u l a t e d by c o n s i d e r i n g t h e g e n e r a t o r displacement  0  are larger  than those c a l c u l a t e d by c o n s i d e r i n g measured wave h e i g h t s . However, i t i s c l e a r l y shown t h a t Iversen's r e s u l t s a r e always above G a l v i n ' s .  Therefore,  t h e d i f f e r e n c e s between t h e  r e s e a r c h e r s judgements must be t h e cause o f t h e d i f f e r e n c e s i n the  results. I v e r s e n and P a t r i c k e t a l . o n l y p r o v i d e t h e end o f t h e  p l u n g i n g r e g i o n f o r t h e beach s l o p e s 1:20 and 1:50. The b e g i n n i n g o f t h e surge r e g i o n i s not g i v e n b u t i s assumed t o occur a t t h e end o f t h e p l u n g i n g r e g i o n . reasonable  T h i s assumption i s  s i n c e i t agrees w i t h G a l v i n ' s f i n d i n g s .  36  Still,  this  assumption does add u n c e r t a i n t y t o Iversen's plunge-surge t r a n s i t i o n and may  and P a t r i c k ' s  give values of H / L 0  Q  l a r g e r than  the a c t u a l v a l u e s . Even though t h e r e are d i f f e r e n c e s , experimental showed t h a t a reasonable  p r e d i c t i o n o f the breaker  results  type can  be  made g i v e n the deepwater wave steepness and the beach s l o p e , p r o v i d e d t h a t the beach i s plane and The  breaking  impermeable.  i n d i c e s do not seem t o be u s e f u l i n p r e d i c t i n g  c o n d i t i o n s i n the n a t u r a l environment.  The  c o n d i t i o n s on a  n a t u r a l beach are complex and are e f f e c t e d by composite s l o p e s , b a r s , s t e p s , s p e c t r a of conditions.  The  H  0  / L  0  ,  p e r m e a b i l i t y , and  simple models developed i n the  non-uniform laboratory  should not be expected t o g i v e exact d e t a i l s o f n a t u r a l  events,  but r a t h e r show the g e n e r a l t r e n d s , t r e n d s t h a t are observed i n nature. The breaking  key t o the o v e r a l l problem o f d e t e r m i n i n g c o n d i t i o n s from o f f s h o r e o b s e r v a t i o n s  a s e t of parameters t h a t r e p l a c e the terms "plunging",  " c o l l a p s i n g " , and  "surging".  nearshore  and  l i e s i n devising  "spilling",  At p r e s e n t , a l l  s t u d i e s are a t b e s t s e m i - q u a n t i t a t i v e i n t h a t graphs are presented  w i t h parameters such as  s t r i c t l y d e s c r i p t i v e terms.  HQ/LQ,  m,  and  regions  of  Once the d e s c r i p t i v e terms are  s u c c e s s f u l l y r e p l a c e d by adequate q u a n t i t a t i v e parameters t h a t a r e based upon a b e t t e r understanding for  the d i f f e r e n t types o f b r e a k i n g ,  o f the p h y s i c a l reasons i t w i l l be p o s s i b l e t o  t r a n s f e r the incoming wave s p e c t r a t o the b r e a k i n g  37  conditions.  CHAPTER  3.1  3;  WAVE  HEIGHT  AND DEPTH  AT  BREAKING  INTRODUCTION  As a wave moves from deep t o s h a l l o w water, i t b e g i n s t o undergo a s e r i e s o f changes.  These changes o c c u r as t h e wave  moves i n t o depths where t h e bed b e g i n s t o i n f l u e n c e t h e wave which r e s u l t s i n wave s h o a l i n g .  The wave h e i g h t b e g i n s t o  i n c r e a s e , t h e wavelength d e c r e a s e s , and t h e wave speed decreases.  The p e r i o d , though, s t a y s c o n s t a n t .  Together, these  changes e v e n t u a l l y cause t h e wave t o become u n s t a b l e and break. At  two extremes, t h e wave can be one o f two forms, e i t h e r  an o s c i l l a t o r y wave o r a s o l i t a r y wave.  O s c i l l a t o r y waves a r e  p r i m a r i l y c o n t r o l l e d by t h e wave steepness and s o l i t a r y waves are  p r i m a r i l y c o n t r o l l e d by t h e water depth.  when t h e c o n t r o l reaches a c r i t i c a l  Breaking occurs  instability.  For  o s c i l l a t o r y waves, i n s t a b i l i t y o c c u r s when t h e wave becomes too steep such t h a t water s p i l l s  down t h e f a c e o f t h e wave.  For  s o l i t a r y waves, t h e i n s t a b i l i t y o c c u r s when t h e depth becomes  38  s m a l l enough such t h a t t h e v e l o c i t y o f t h e c r e s t o f t h e wave i s s i g n i f i c a n t l y g r e a t e r than t h e v e l o c i t y o f t h e base o f the wave c a u s i n g t h e wave t o plunge forward.  In both cases, a t b r e a k i n g  the c r e s t o f t h e wave t r a v e l s f a s t e r than t h e base o f the wave. T h i s i s t h e r e s u l t o f b r e a k i n g f o r o s c i l l a t o r y waves, but t h e cause o f b r e a k i n g f o r t h e s o l i t a r y wave. As a g e n e r a l r u l e o f thumb, t h e r a t i o o f t h e b r e a k i n g h e i g h t t o t h e b r e a k i n g depth i s assumed t o be 0.83.  There are  l a r g e amounts o f l i t e r a t u r e and v a r i o u s e m p i r i c a l c r i t e r i a which attempt t o p r e d i c t b r e a k i n g and hence t h e v a l u e of t h e r a t i o . Experimental  evidence shows t h a t t h e b r e a k i n g  height-to-depth  r a t i o v a r i e s over a l a r g e range o f v a l u e s , depending upon t h e beach s l o p e and wave c o n d i t i o n s .  I f t h e mechanisms f o r  d i f f e r e n t types o f b r e a k i n g waves a r e b e t t e r understood, p h y s i c a l l y based c r i t e r i a , would a r i s e n a t u r a l l y .  such as the h e i g h t - t o - d e p t h  then ratio,  T h i s c h a p t e r uses t h e behaviour  o f the  b r e a k i n g h e i g h t - t o - d e p t h r a t i o as a c l u e t o t h e mechanisms t h a t cause b r e a k i n g . 3.2  DEFINITION  OF  THE BREAKING  HEIGHT  AND  DEPTH  The wave h e i g h t and depth a t t h e b r e a k i n g p o i n t depend upon where t h e wave i s d e f i n e d t o break.  R e f e r r i n g to the  c l a s s i f i c a t i o n s g i v e n i n c h a p t e r two, a s p i l l i n g breaker i s l o c a t e d where t h e c r e s t f i r s t becomes d i s c o n t i n u o u s . p l u n g i n g breaker,  t h e "breaking p o i n t i s l o c a t e d where t h e wave  f a c e f i r s t becomes v e r t i c a l . of  For a  F o r a s u r g i n g breaker,  the point  b r e a k i n g i s l o c a t e d where t h e maximum drawdown o f t h e water  from t h e p r e v i o u s wave o c c u r s .  39  F i g u r e 3.1 shows t h e v a r i a b l e s t h a t a r e used t o d e s c r i b e the c o n d i t i o n s a t t h e b r e a k i n g p o s i t i o n . (H )  The b r e a k e r h e i g h t  i s t h e d i f f e r e n c e between t h e maximum and minimum water  b  s u r f a c e e l e v a t i o n d u r i n g t h e passage  o f one wave.  o s c i l l a t o r y waves t h r e e depths a r e p o s s i b l e : depth  For  the s t i l l  water  ( d ) a t t h e breaker p o s i t i o n ; t h e mean water depth s  the breaker p o s i t i o n ; and, t h e depth o f t h e t r o u g h breaker p o s i t i o n .  (d ) at m  ( d ) a t the t  Because t h e r e a r e t h r e e p o s s i b l e b r e a k i n g  depths, t h e one t h a t i s used t o d e s c r i b e t h e b r e a k i n g p r o c e s s must be c l e a r l y d e f i n e d . of  d , d , or d . s  m  t  The symbol d^ i s used t o r e p r e s e n t one  The depth d  fa  i s u s u a l l y equated w i t h d  s  and  t h i s c o n v e n t i o n w i l l be used u n l e s s otherwise d e f i n e d . If  s o l i t a r y wave t h e o r y i s used t o d e s c r i b e t h e s h o a l i n g  waves, then d^ i s equated w i t h d^..  A s o l i t a r y wave i s a type of  c n o i d a l wave t h a t has an i n f i n i t e p e r i o d and i n f i n i t e and propagates  wavelength  forward i n water o f c o n s t a n t depth.  The wave setdown a t b r e a k i n g i s the d i f f e r e n c e between the mean water depth and t h e s t i l l water depth Stewart, 1964),  (Longuet-Higgins and  _ b  S  =  d  -  m  d  s  (3.1)  The setdown r e s u l t s from t h e flow o f excess momentum produced by the shoreward  movement o f t h e waves and i s f u r t h e r d i s c u s s e d i n  chapter four. The d i f f e r e n c e between t h e mean water depth, d^,  and trough  depth i s t h e t r o u g h amplitude, a  t  = d  m  -  d  40  t  (3.2)  Figure  3.1  Variables  for  wave a t  41  breaking  point  3.2.1  The h e i q h t - t o - d e p t h r a t i o a t b r e a k i n g  As a wave s h o a l s , the trough between the c r e s t s and the wave h e i g h t changes.  flattens  The shape of the wave begins t o  approach the s o l i t a r y wave shape and,  i n t h i s case, the maximum  wave h e i g h t w i l l begin t o be c o n t r o l l e d by the depth r a t h e r than by the wavelength, which i s the case f o r o s c i l l a t o r y waves. Many r e s e a r c h e r s have used s o l i t a r y wave t h e o r i e s t o study the h e i g h t - t o - d e p t h problem and the t h e o r e t i c a l r e s u l t s are shown i n t a b l e 3.1.  The  (H /d )  symmetrical  t o the p o i n t o f b r e a k i n g .  experiments  have shown t h a t ( H ^ / d ^ ^ x v a r i e s from 0.65  b  b  m a x  v a l u e s assume t h a t the wave remains  steep waves on low s l o p e s t o about 1.25  However, a c t u a l  f o r low steepness waves  on steep s l o p e s .  Generally, f o r s p i l l i n g breakers  i n the range 0.65  - 0.85  Weggel 1972)  ( I v e r s e n 1952,  for  1953;  (H^/d^)^^ is  Galvin  1972;  and i s approximately t h a t v a l u e o b t a i n e d from  h i g h e s t s o l i t a r y wave t h e o r y .  the  The good comparison between  t h e o r e t i c a l and experimental r e s u l t s f o r s p i l l i n g b r e a k e r s i s not s u r p r i s i n g s i n c e these breakers most c l o s e l y resemble the symmetrical  s o l i t a r y waves assumed by the t h e o r y .  b r e a k e r s , H^/d^  For p l u n g i n g  i n c r e a s e s and v a l u e s as l a r g e as 1.3  p e r i o d i c waves ( I v e r s e n 1952, (Ippen and K u l i n 1955;  1953)  and 3.0  for  f o r s o l i t a r y waves  C a m f i e l d and S t r e e t 1969)  have been  measured. Weishar and Byrne (1978) s t u d i e d f i l m s o f 116 waves on a n a t u r a l beach. H /dj3 b  i s 0.78.  breaking  They note t h a t the average v a l u e of  T h i s v a l u e i s w i t h i n the range e s t i m a t e d  o t h e r r e s e a r c h e r s , but they n o t i c e t h a t l a r g e r v a l a u e s of  42  by H^/d^  RESEARCHER  DATE  Boussinesq  1871  0.73  McCowan  1891  0.75  McCowan  1894  0,78  Gwyther  1900  0.83  Davies  1952  0,83  Packman  1952  1,03  Chappelear  1959  0,87  Laitone  1960  0.73  Lenau  1966  0,83  (Hb/db) max  (after  Table  3,1  Maximum t h e o r e t i c a l ratios f o r solitary  43  G a l v i n 1972)  height-to-depth waves  occur  f o r p l u n g i n g waves than f o r non-plunging  by the p r e v i o u s f i n d i n g s , t h i s should be The  waves.  As shown  expected.  Shore P r o t e c t i o n Manual p r o v i d e s two  graphs t h a t  demonstrate the r e l a t i o n between the b r e a k e r type, H ^ / d ^ , H  Q  / L  0  ( f i g u r e s 3.2  and  3.3).  F i g u r e 3.2  shows Goda's (1970)  e m p i r i c a l l y d e r i v e d r e l a t i o n s h i p between s e v e r a l beach s l o p e s . d  B  /H  B  and H / g T  F i g u r e 3.3  H^/HQ  and  u  b  H  The  curves  1 b  b  = 43.75 ( 1 !' ,  e"  are  By combining the two  )  (3.5)  ) can be shown as a f u n c t i o n of  and the beach s l o p e  i n the f i g u r e are the t h r e e b r e a k e r  o c c u r i n the r e c o g n i z e d sequence. H^/d^  (3,4)  "19.5m.  1  graphs, H^/d^  the deepwater steepness  19n  5 6  ( 1 + e  (figure  On  3.4).  regions.  These  For each s l o p e , as H / L Q  0  i n c r e a s e s and the b r e a k e r type changes from  s p i l l i n g t o p l u n g i n g t o s u r g i n g , a g r e e i n g w i t h the s t e e p e r s l o p e s H^/d^  previous  has a g r e a t e r range of  v a l u e s and v a r i e s by l a r g e percentages plunging  3.3  (3.3)  -(aH /gT ) 2  b  b =  discussion.  for  Q  by,  a  decreases  / L  i n figure  where a and b are f u n c t i o n s of the beach s l o p e and  Included  0  J  are g i v e n by,  approximated  H  shows the r e l a t i o n s h i p between  for various slopes.  2  b  and  between p l u n g i n g and  non-  breakers  I t i s i n t e r e s t i n g t o compare these r e s u l t s w i t h the found by I v e r s e n  (1952) f o r p e r i o d i c waves.  44  F i g u r e s 3.5  results through  0.002  Figure  0.004  3,2  0.0004  Figure  3.3  0.006  0.006  0.010  0.012  0.014  0.016  B r e a k e r height index v e r s u s wave s t e e p n e s s (SPM 1984)  0 0006  0 001  0002  0 004  0 006  0 01  0.018  0.020  deepwater  0 02 0 03 (ofttr Gotfo, 1970)  Non-dimensional b r e a k e r d e p t h v e r s u s breaking s t e e p n e s s (SPM 1984) 45  Hb/db 1.35  h  1.25  1.15  h  1.05  h 0.95  Beach  slope  m = 1:50  0.85  —|— m = 1:30. m = 1:20 -B-  m = 1:10 "1  0.001  1  I I I I  i  r  0.010  I I I '  0.75  0.100  Deepwater wave s t e e p n e s s , H o / L o Figure  3.4  Dependence o f H b / d b on d e e p w a t e r s t e e p n e s s and b e a c h slope 46  wave  3.8 a r e p l o t s o f H / L Q  1:50 s l o p e s .  Q  versus H / d b  For values of H / L Q  f o r 1:10, 1:20, 1:30, and  b  l e s s than approximately 0.015,  0  these f i n d i n g s agree w i t h t h e p r e d i c t i o n s made by f i g u r e 3.4. In a l l cases, t h e v a l u e o f H^/d^ i n c r e a s e s f o r d e c r e a s i n g v a l u e s of H / L 0  0  u n t i l H /L 0  begins t o decrease  Q  approaches 0.015.  A t t h i s v a l u e , H^/d^  f o r t h e 1:10 and 1:50 s l o p e s .  Whether t h i s  o c c u r s f o r t h e 1:20 and 1:30 s l o p e s i s d i f f i c u l t t o determine due t o t h e l a r g e s c a t t e r o f the data p o i n t s f o r lower v a l u e s of H / L , however, these curves do tend t o f l a t t e n . 0  0  The b r e a k i n g mechanism f o r H / L Q  0  g r e a t e r than 0.015 must be  d i f f e r e n t than t h e mechanism o c c u r r i n g f o r H / L 0  0.015.  Q  less  than  This i s further explored i n the f o l l o w i n g three sections  by a n a l y z i n g how t h e steepness, b r e a k i n g depth,  and beach s l o p e  a f f e c t breaking. 3.3  INFLUENCE  O F T H E WAVE  STEEPNESS  ON  BREAKING  The deepwater steepness, H / L , has been shown t o be a 0  Q  f a c t o r i n d e t e r m i n i n g how a wave w i l l break on a beach o f a g i v e n s l o p e ( s e c t i o n 2.3). the presence  Deepwater waves a r e n o t a f f e c t e d by  o f t h e bed and t h e maximum wave h e i g h t i s l i m i t e d  by t h e wavelength.  Michell  (1893) determines  the l i m i t i n g  deepwater steepness t o be,  (H /L ) 0  0  1/7  n a x  (3,6)  B r e a k i n g over t h e e n t i r e range o f depths i s covered by a formula proposed  by Miche  (1944),  H/L  = 0,141 tanh(27Td/L)  T h i s formula s e r v e s as a u s e f u l e n g i n e e r i n g o f wave steepness  (3.7) approximation  i n t h a t i t p r e d i c t s t h e maximum steepness a t 47  Hb/db 1.25  h  1.20  h  1.15  h  1.10  h  1.05  1.00  0.95  h  0.90  h 0.85  h  0.80  0.75  h  "1  I—I  0.001  I  | | 0.100  Deepwater wave s t e e p n e s s , H o / L o Figure  3,5  Plot o f slope  data  ( I v e r s e n 1952)  48  for  MO  0.70  0.65  Hb/db 1.20  h  1.15  h  1.10.  h  1.05  1.00  h  0.95  0.90  h 0.85  h  0.80  0.75  0.70  h 0.85  T 0.001  I  I  I—I  I  | |  0.010  3,6  Plot o f slope  data  0.60  0.100  Deepwater wave s t e e p n e s s , H o / L o Figure  1  ( I v e r s e n 1952)  49  for-  1:20  Hb/db 1.20  1.15  h 1.1 o 1.05  h  LOO  h  0.95  h  0.90  h  0.85  0.80  h  0.75  h  0.70  h  0.65  0.60  0.001  0.100  Deepwater wave s t e e p n e s s , H o / L o Fi 9  U  r  e  3  '  7  P l  ,  o t  o  slope  f  data ( I v e r s e n 1952)  50  for  1:30  Hb/db 1.20  1.15  1.10  h  1.05  h  LOO  0.95  h  0.90  h  0.85  h  0.80  h  0.75  0.70  0.65  0.60  0.001  0.010  0.100  Deepwater wave s t e e p n e s s , H o / L o Figure  3,8  Plot o f slope  data  ( I v e r s e n 1952)  51  for  1:50  which a wave becomes u n s t a b l e and then breaks.  F i g u r e 3 . 9 shows  I v e r s e n ' s r e s u l t s p l o t t e d a g a i n s t t h e wave steepness d e s c r i b e d by Miche's formula. i n t e r m e d i a t e depth range, Because o f t h i s ,  limit  The data p o i n t s f a l l w i t h i n the  a c c o r d i n g t o l i n e a r wave t h e o r y .  i t i s c o r r e c t t o assume t h a t t h e bed i s  b e g i n n i n g t o a f f e c t t h e s h o a l i n g and b r e a k i n g  characteristics.  S i n c e b r e a k i n g i s no l o n g e r s t r i c t l y c o n t r o l l e d by steepness t h e data p o i n t s a r e expected t o show some degree o f s c a t t e r about t h e l i m i t g i v e n by Miche.  I n t h e f i g u r e , t h e data  p o i n t s r e p r e s e n t i n g b r e a k i n g on t h e 1:10 s l o p e a r e more s c a t t e r e d and g e n e r a l l y p l o t above Miche's l i m i t .  As t h e s l o p e  decreases, t h e p o i n t s become l e s s s c a t t e r e d w i t h t h e b r e a k i n g on the 1:50 s l o p e b e i n g w e l l approximated  by Miche's l i m i t .  It  seems t h a t t h e g e n t l e r s l o p e a l l o w s t h e wave t o change s l o w l y , i n r e s p e c t t o t h e s h o a l i n g p r o c e s s , and c o n t i n u e t o m a i n t a i n a p r o f i l e more c o n s i s t e n t w i t h a symmetrical  s o l i t a r y wave.  On  steep beaches, t h e wave does not have enough time t o respond t o depth changes and so an unsymmetrical  b r e a k i n g form  causes  s c a t t e r about Miche's l i m i t . The  s h o a l i n g c h a r a c t e r i s t i c s o f o s c i l l a t o r y waves can be  d e s c r i b e d by u s i n g l i n e a r wave t h e o r y .  By e q u a t i n g the wave  power p e r u n i t wave c r e s t width i n deep and s h a l l o w water, the r a t i o o f wave steepness t o deepwater wave steepness can be calculated.  The c a l c u l a t i o n s a r e shown i n Appendix A and g i v e  the s h o a l i n g e q u a t i o n as, 1/2  coth(kd)  52  H/L 1.000  n •  0.100  m = 1:10  +  m = 1:20  0  m = 1:50  -  MICHE(1944)  0.010  -  SHALLOW  INTERMEDIATE DEPTH  0.001  DEEP  i 0.01  i  i  i  0.10  1.00  d/L Figure  3,9  Comparison o f experimental Miche's formula  i i r  results  with  Three assumptions a r e made f o r the c a l c u l a t i o n :  the wave  c r e s t s a r e p a r a l l e l t o t h e beach such t h a t no r e f r a c t i o n  occurs;  t h e s l o p e i s c l o s e t o b e i n g zero such t h a t t h e p r o p e r t i e s o f the wave a t any depth a r e t h e same as t h e p r o p e r t i e s o f an i d e n t i c a l wave i n a h o r i z o n t a l channel; and, t h e l o s s i n power i s negligible.  In p r i n c i p l e , t u r b u l e n c e , which r e p r e s e n t s a power  l o s s , comes from two sources.  Those a r e t h e bottom boundary  l a y e r and t h e s u r f a c e breaker.  A t b r e a k i n g , t h e c o n t r i b u t i o n of  the bottom boundary l a y e r i s t o t a l l y outweighed by t h e d i s s i p a t i o n due t o b r e a k i n g , and i s consequently  neglected.  Up  t o b r e a k i n g , though, f r i c t i o n occurs a t t h e bed b u t w i l l have an i n s i g n i f i c a n t e f f e c t on the o v e r a l l r e s u l t s Thornton  and Guza 1983).  Turbulence  (Kamphuis 1975;  i s zero s i n c e t h e flow i s  i r r o t a t i o n a l w i t h i n t h e wave. As an o s c i l l a t o r y wave s h o a l s , t h e v a l u e o f kd decreases and t h e wave p r o p e r t i e s w i l l f o l l o w those d e s c r i b e d by t h e shoaling equation.  E v e n t u a l l y t h e wave w i l l break which i s  d e f i n e d by Miche's formula.  T h i s assumes t h a t b r e a k i n g i s  i n i t i a t e d by t h e wave a t t a i n i n g c r i t i c a l  steepness.  e q u a t i o n 3.8 i s v a l i d up t o t h e p o i n t o f b r e a k i n g .  Therefore, By p l o t t i n g  these e q u a t i o n s on a s i n g l e graph t h e c o n d i t i o n s a t b r e a k i n g f o r constant values of H / L 0  Q  can be found.  These c o n d i t i o n s a r e a t  the p o i n t s o f i n t e r s e c t i o n i n f i g u r e 3.10. U s i n g t h e c o n d i t i o n s a t each i n t e r s e c t i o n p o i n t , Miche's formula can be t r a n s f e r r e d t o f i g u r e 3.4, which g i v e s t h e v a l u e o f H^/djj as f u n c t i o n s o f beach s l o p e and deepwater wave steepness.  F i g u r e 3.11 i s t h e p l o t w i t h Miche's l i m i t i n c l u d e d .  C l e a r l y , Miche's c r i t e r i a r e p r e s e n t s t h e lowest  54  limiting  H/L  H o / L o = 1:20 H o / L o = 1:100 Ho/Lo = 1:200  0.001 ' • 001  '  1  1  1—''''' 0.10  d/L Figure  3,10  Miche's formula and t h e shoaling f o r s e l e c t e d values of H o / L o 55  equation  Hb/db  h  1.25  h  1.1 5  h  1.05  h  0.95  0.85  h  I 0.001  |  0.65  0.100  Deepwater wave s t e e p n e s s , H o / L o . F i g u r e 3,11  Miche's limit  56  0.75  c o n d i t i o n a t b r e a k i n g which means t h a t i f a wave breaks  solely  from steepness e f f e c t s t h e v a l u e o f H^/d^ w i l l be a minimum. For o s c i l l a t o r y waves, l a r g e r v a l u e s o f H / d b  because wave steepness  are obtained  b  i n c r e a s e s f a s t e r than t h e wave h e i g h t f o r  a wave t r a v e l l i n g onto a beach.  T h i s i s shown by t h e s h o a l i n g  e q u a t i o n and i m p l i e s t h a t an o s c i l l a t o r y wave t h a t s t a r t s out steep enough i n deepwater w i l l grow i n steepness t h a t steepness w i l l  f a s t enough  induce b r e a k i n g i n water deeper than where  b r e a k i n g i s s o l e l y depth c o n t r o l l e d .  This e a r l y breaking  suggests t h a t h i g h i n i t i a l steepness should be a s s o c i a t e d w i t h l a r g e r b r e a k i n g depths. decrease as H / L Q  0  I f so, then H^/d^ would be expected t o  increases f o r higher values of H /L . 0  supported by I v e r s e n f o r H / L 0  0  0  This i s  g r e a t e r than 0.015 ( f i g u r e 3.5 t o  3.8), but does not appear t o be t r u e f o r H / L 0  0  l e s s than 0.015,  as w i l l now be d i s c u s s e d . 3.4  INFLUENCE  OF T H E DEPTH  ON  BREAKING  I t i s c l e a r from I v e r s e n ' s f i n d i n g s t h a t H^/d^ i s d e c r e a s i n g , o r r e l a t i v e l y constant, f o r d e c r e a s i n g v a l u e s o f H /L 0  0  below 0.015.  Presumably, a d i f f e r e n t b r e a k i n g mechanism  i s r e s p o n s i b l e f o r t h e change. The p r e v i o u s s e c t i o n assumed o s c i l l a t o r y waves broke because o f steepness e f f e c t s .  Waves t h a t no l o n g e r s h o a l as  o s c i l l a t o r y waves b e g i n t o appear as s o l i t a r y waves which break because o f depth  effects.  57  An i d e a l s o l i t a r y wave has a wavelength both approach i n f i n i t y .  and a p e r i o d t h a t  I t i s a wave o f t r a n s l a t i o n t h a t  lies  w h o l l y above t h e s t i l l water l e v e l and i s c o n t r o l l e d by the depth.  The v e l o c i t y o f such a wave i s g i v e n by, S.—  =  1  +  f(H/d)  (3.9)  VgcT Both t h e depth and t h e h e i g h t - t o - d e p t h r a t i o a r e important. Breaking o f such a wave can be e x p l a i n e d by n o t i n g t h a t as the depth i n c r e a s e s t h e wave speed i n c r e a s e s .  The change o f wave  speed w i t h depth w i l l be g r e a t e r i n more s h a l l o w depths.  Since  t h e r e i s a s i g n i f i c a n t depth i n c r e a s e i n t h e wave i t s e l f , t h e top  p o r t i o n o f t h e wave moves f a s t e r t h a t t h e bottom  wave.  of the  When t h e s i t u a t i o n becomes u n s t a b l e t h e wave w i l l  (Longuet-Higgins  break  1980).  To become s o l i t a r y , waves w i t h i n i t i a l  low steepness must  reach depths where t h e wave i s c o n t r o l l e d by t h e depth. waves must have an i n i t i a l  These  low steepness s i n c e i t i s a l r e a d y  shown t h a t waves w i t h h i g h i n i t i a l  steepness w i l l break due t o  steepness e f f e c t s . When s o l i t a r y waves a r e c o n s i d e r e d , t h e response o f t h e wave t o changing s h o a l i n g c o n d i t i o n s i s important. response s h o u l d be immediate, i n t h e response.  I d e a l l y , the  but i n r e a l i t y t h e r e w i l l be a l a g  F i g u r e 3.12 shows what o c c u r s when a s o l i t a r y  wave responds t o depth changes.  According t o the f i g u r e , the  slow response o f t h e s o l i t a r y wave w i l l r e s u l t i n t h e wave h e i g h t a t b r e a k i n g b e i n g reached f u r t h e r shoreward depth.  i n a smaller  T h e r e f o r e , t h e t h e o r e t i c a l v a l u e o f H^/d^ w i l l be l e s s  than t h a t which i s measured.  As t h e i n i t i a l wave steepness  58  Wave Height Increase  — Theoretical  gure  3,12  Actual Position  -B&»  Water Depth Decrease  Breaking r e s p o n s e  59  {  of  a solitary  wave  i n c r e a s e s , f o r low i n i t i a l v a l u e s o f H / L , t h e e f f e c t Q  s i n c e t h e h e i g h t o f t h e incoming s o l i t a r y wave w i l l increasing H /L . Q  0  increase f o r  However, s o l i t a r y wave development r e q u i r e s  time which can o n l y be p r o v i d e d by s h o a l i n g on m i l d Therefore,  increases  0  slopes.  one would expect d e c r e a s i n g v a l u e s o f H / d f o r b  decreasing values of H / L 0  0  on m i l d s l o p e s .  b  Iversen's  results  f o r t h e 1:30 and 1:50 s l o p e s show t h i s as t h e t r e n d . How can t h e r e s u l t s f o r t h e 1:10 s l o p e be explained? the s l o p e i s steep enough t o r u l e out t h e formation s o l i t a r y wave even though H / L 0  probably  i s l e s s than 0.15.  0  b e s t e x p l a i n e d by t h e t r a n s l a t i o n o f a  of a This i s  pre-breaking  o s c i l l a t o r y wave i n t o shallow water on a steep s l o p e . response o f H / d b  b  t o changing H / L Q  0  Here  The  i s t h e same r e s u l t as t h a t  j u s t d e s c r i b e d f o r t h e s o l i t a r y wave. 3.5  INFLUENCE  OF T H E BEACH  SLOPE  ON  BREAKING  Beach s l o p e a f f e c t s wave c o n d i t i o n s by c o n t r o l l i n g t h e time and d i s t a n c e spent by t h e wave moving towards b r e a k i n g .  A  g r e a t e r p e r i o d o f time, measured i n wave p e r i o d s , o r , e q u i v a l e n t l y , a g r e a t e r d i s t a n c e , measured i n wavelength, on the beach a l l o w s f r i c t i o n and n o n l i n e a r wave shape changes t o have a greater e f f e c t .  F r i c t i o n h e l p s t o reduce wave h e i g h t  (Thornton  and Guza 1983) w h i l e n o n l i n e a r changes can e i t h e r i n c r e a s e o r decrease t h e wave h e i g h t . In Iversen's  experiment, t h e breaker  g r e a t e r e x t e n t on f l a t t e r s l o p e s  h e i g h t reduces t o a  ( t a b l e 3.2, f i g u r e 3.13).  However, t h e r e s u l t s o f Ippen and K u l i n i n d i c a t e t h a t f o r s o l i t a r y waves t h e r e v e r s e i s t r u e .  60  The e q u i v a l e n t  breaker  Ho/Lo=0,01 SLDPE  Ho/Lo=0,02 Hb/Ho  0.02  1,31  1,12  0,033  1,43  1,23  0.05  1,60  1,31  0,10  1,76  1,41 (from  Table  3,2  Slope e f f e c t on t h e b r e a k i n g f o r o s c i l l a t o r y waves  3,3  1952)  height  Ho/Lo=0,01  SLDPE  Table  Iversen  Hb/Ho  0,023  1,70  0,065  1,40'  Slope e f f e c t on t h e b r e a k i n g f o r s o l i t a r y waves  61  height  Hb/Ho  0>  0.02  0.03  0.04  0.05  0.06  0.07  0.08  Ho/Lo Figure 3,13  Hb/Ho v e r s u s H o / L o ( I v e r s e n  1952)  0.09  0.10  height  index, H /H , i n c r e a s e s w i t h d e c r e a s i n g s l o p e f o r b  Q  c o n s t a n t v a l u e s o f t h e deepwater steepness ( t a b l e 3.3). For o s c i l l a t o r y waves, t h e v a r i a t i o n  of the breaking  w i t h deepwater wave h e i g h t i s shown i n f i g u r e  3.14 and t h e r e  does n o t appear t o be a s t r o n g s l o p e i n f l u e n c e . H /L 0  Q  increases, d /H b  versus H / L Q  0  0  decreases.  depth  However, as  The combined r e s u l t s  of H /d b  has p r e v i o u s l y been shown i n s e c t i o n 3.2.  show a s t r o n g s l o p e i n f l u e n c e , which, from t h e above  b  These  results,  must o n l y come from t h e i n f l u e n c e o f t h e s l o p e on H /H . b  F i g u r e 3.15 compares t h e t r e n d s found f o r both and  s o l i t a r y waves.  Q  oscillatory  The bottom two graphs show t h a t f o r a g i v e n  s l o p e t h e behaviour o f t h e r a t i o Hj^/d^ f o r i n c r e a s i n g v a l u e s o f H /L 0  Q  i s different  ratio H /d b  s o l i t a r y waves. b  Iversen  b  This finding  e x p l a i n s t h e curves o f H / L 0  I n t h e t h r e e cases,  each r e p r e s e n t i n g t h e  fora particular  significantly at H /L 0  marks t h e t r a n s i t i o n  Q  s l o p e , t h e curve  equal t o 0.015.  between b r e a k i n g  as an o s c i l l a t o r y wave.  of breaking, for a given  0  f o r t h e 1:20, 1:30, and 1:50 s l o p e s g i v e n by  characteristics  breaking  The  decreases f o r o s c i l l a t o r y waves and i n c r e a s e s f o r  b  versus H / d  f o r o s c i l l a t o r y and s o l i t a r y waves.  the value of H / d b  b  This value of H / L 0  Q  as a s o l i t a r y wave and  A l s o , r e g a r d l e s s o f t h e type i n c r e a s e s as t h e s l o p e  H /L . 0  changes  Q  63  increases  3.00  db/Ho  2.50 H  a  • •  +  m = 1:10 m = 1:20 m = 1:50  •  m = 1:30  2.00 H  1.50 H  4-  CTl  1.00  +  •  0.50  0.00 0.00  0.02  0.04  0.06  0.08  Ho/Lo Figure 3,14  clb/Ho v e r s u s  Ho/Lo (Iversen  1952)  0.10  OSCILLATORY  SOLITARY  INCREASING SLOPE  INCREASING SLOPE  Hb/Ho  Hb/Ho  Ho/Lo  Ho/Lo  db/Ho  db/Ho  Ho/Lo  Ho/Lo  Hb/db  Hb/db  Ho/Lo  Figure 3,15  Ho/Lo  Trends f o r the ratios of oscillatory ( I v e r s e n 1952) and s o l i t a r y (Ippen and Kulin 1955) waves  64-A  3.6  DISCUSSION  The wave h e i g h t and depth a t b r e a k i n g are p o i n t s of i n t e r e s t f o r r e s e a r c h e r s and e n g i n e e r s .  The  r u l e of thumb i s  t h a t the r a t i o of wave h e i g h t - t o - d e p t h a t b r e a k i n g i s  0.83.  However, as found by many r e s e a r c h e r s , the a c t u a l r a t i o can have a l a r g e range of v a l u e s .  Values as h i g h as 1.3  f o r p l u n g i n g and s u r g i n g type b r e a k e r s . v a l u e s between 0.65  and 0.85.  t o 3.0  are  found  S p i l l i n g breakers have  The r e s u l t s f o r s p i l l i n g  breakers  are c l o s e r t o the v a l u e s p r e d i c t e d t h e o r e t i c a l l y s i n c e the s p i l l i n g breaker most c l o s e l y resembles s o l i t a r y waves a t breaking. The behaviour  of the h e i g h t - t o - d e p t h r a t i o , w i t h r e s p e c t to  deepwater steepness, steepness, of H / d b  b  depth,  can be e x p l a i n e d u s i n g the e f f e c t of wave  and beach s l o p e .  versus H /L . 0  Q  F i g u r e 3.16  shows the graph  I t i s d i v i d e d i n t o t h r e e r e g i o n s where  i n each r e g i o n b r e a k i n g i s i n i t i a t e d by d i f f e r e n t mechanism. In r e g i o n one, breaking process. critical  steepness  e q u a t i o n and  wave steepness p l a y s the major r o l e i n The wave breaks when the wave reaches The  o s c i l l a t o r y wave obeys the s h o a l i n g  i n c r e a s e s i n steepness  f a s t e r than i n h e i g h t .  This  causes the wave t o break i n water deeper than i f t h e wave were s o l e l y c o n t r o l l e d by depth.  T h i s i s not t o say t h a t the depth  does not take p a r t i n the s h o a l i n g p r o c e s s , but t h a t the b r e a k i n g i s due t o a l i m i t i n g wave steepness. H /L 0  0  g r e a t e r than 0.015, the H / d b  deepwater wave steepness In r e g i o n two,  b  For v a l u e s of  r a t i o i n c r e a s e s as the  decreases.  the beach s l o p e s are low enough t o a l l o w  s o l i t a r y waves t o develop  from incoming 65  o s c i l l a t o r y waves.  The  Hb/db  H o / L o = 0,015 Figure  3,16  Relationship between t h e b e a c h slope, d e e p w a t e r wave s t e e p n e s s , and breaking conditions  65-A  s l o p e must be low s i n c e a s o l i t a r y wave needs time and d i s t a n c e to  form from an o s c i l l a t o r y wave.  Breaking  i s i n i t i a t e d by the  d i f f e r e n c e between t h e forward  velocity  base o f t h e wave, t h e v e l o c i t y  b e i n g p r o p o r t i o n a l t o the square  r o o t o f t h e depth.  of the crest  and the  Because the s o l i t a r y wave cannot respond  immediately t o changing depths, a t b r e a k i n g t h e wave moves i n t o water t h a t i s more shallow  than i s necessary  Therefore,  i n c r e a s e s as H / L  the r a t i o H /d b  the wave h e i g h t w i l l transition H /L 0  b  0  f o r breaking. i n c r e a s e s because  Q  i n c r e a s e with i n c r e a s i n g H / L . 0  The  Q  between s o l i t a r y waves and o s c i l l a t o r y waves i s when  equals 0.015.  0  The  development o f s o l i t a r y waves i s r e s t r i c t e d  t o beaches  w i t h low s l o p e s , so as t h e s l o p e i n c r e a s e s s o l i t a r y wave development ceases.  T h i s n e c e s s i t a t e s a s l o p e above which  s o l i t a r y waves a r e no l o n g e r a b l e t o develop. value of the t r a n s i t i o n 1:25.  The approximate  s l o p e , based upon Iversen's  results, i s  In r e g i o n t h r e e , because o f t h e steep s l o p e , a p r e -  breaking  o s c i l l a t o r y wave moves i n t o shallow water.  The wave  breaks from a combination o f steepness and depth e f f e c t s . the s o l i t a r y wave case, t h e r a t i o H / d b  b  As i n  i n c r e a s e s as t h e  deepwater steepness i n c r e a s e s f o r v a l u e s o f H / L Q  0  less  than  0.015. Because o f t h e d i f f e r e n t  b r e a k i n g mechanism i n each r e g i o n ,  s p e c i f i c b r e a k i n g types predominate i n t h e t h r e e r e g i o n s .  In  r e g i o n one, o s c i l l a t o r y waves break from steepness e f f e c t s and are expected t o s p i l l o r plunge s l i g h t l y . forward velocity  velocity  o f t h e wave c r e s t  In r e g i o n two, the  i s g r e a t e r than t h e forward  o f t h e base c a u s i n g s o l i t a r y waves t o plunge.  66  Finally,  i n r e g i o n t h r e e , o s c i l l a t o r y waves t r a v e l are  expected t o plunge or surge.  67  i n t o s h a l l o w water and  4;  CHAPTER  4.1  FLOW  DYNAMICS  WITHIN  THE  SURF  ZONE  INTRODUCTION  From the p o i n t of view of f l u i d mechanics, the s u r f zone i s c h a r a c t e r i z e d by the i r r e v e r s i b l e and complete t r a n s f o r m a t i o n organized  i r r o t a t i o n a l flow i n t o motions o f d i f f e r e n t types  d i f f e r e n t s c a l e s , i n c l u d i n g v o r t i c a l motions and  of  and  turbulence.  The wave goes through a s e r i e s of changes which permit  the  s u r f zone t o be broken i n t o a number of r e g i o n s where d i s t i n c t wave shapes and  fluid  flows occur.  which make up the s u r f zone.  The  F i g u r e 4.1  shows the  outer region i s characterized  by a r a p i d t r a n s f o r m a t i o n o f the wave shape from the breaker  shape t o a ( p e r i o d i c ) bore.  The  initial  inner region i s  c h a r a c t e r i z e d by the r a t h e r slow changes of the bore and run-up r e g i o n i s c h a r a c t e r i z e d by the f a c t t h a t no roller  the  surface  exists.  The for  regions  p r e v i o u s two  chapters  the s u r f zone t o e x i s t .  d i s c u s s the c o n d i t i o n s  necessary  T h i s chapter d i s c u s s e s the  68  fluid  Breaking Point  V •LITER REGION  -*G  INNER REGION  5»—  RUN-UP REGION  ,  5*-  •*=  SVL MV/L  Figure  4,1  Surf  69  zone  regions  •  motions produced w i t h i n each r e g i o n as a r e s u l t o f wave breaking w i t h p a r t i c u l a r emphasis on t h e t r a n s f o r m a t i o n o f i r r o t a t i o n a l wave motion i n t o r o t a t i o n a l and u l t i m a t e l y t u r b u l e n t flows. Both t h e mass t r a n s p o r t v e l o c i t i e s and water l e v e l changes, which a r e r e s u l t s o f these flows, a r e d i s c u s s e d .  Experimental  o b s e r v a t i o n s o f f l u i d and p a r t i c l e motions, as w e l l as c a r e f u l measurements o f water l e v e l changes, a r e used. 4.2  TRANSITION  TO ROTATIONAL  FLOW  Two b a s i c a l l y d i f f e r e n t types o f motion can be d i s t i n g u i s h e d i n t h e e v o l u t i o n o f breakers as c o n s i d e r e d The  first  here.  i s e s s e n t i a l l y i r r o t a t i o n a l motion d u r i n g wave  steepening, o v e r t u r n i n g and j e t f o r m a t i o n .  T h i s motion occurs  o u t s i d e t h e s u r f zone and w i l l not be t r e a t e d here.  The second  type o f motion i s t h a t f o l l o w i n g j e t impingement, l e a d i n g t o the g r a d u a l development o f a t u r b u l e n t bore.  These motions occur  w i t h i n t h e o u t e r r e g i o n and a r e c o n s i d e r e d f o r both s p i l l i n g and plunging breakers.  Surging b r e a k e r s a r e i g n o r e d because they do  not e x h i b i t any r o t a t i o n a l t e n d e n c i e s except a t t h e base o f the wave f a c e where t h e wave and beach meet. The  i n i t i a l c h a r a c t e r i s t i c s o f both s p i l l i n g  and p l u n g i n g  breakers  i s t h e f o r m a t i o n o f a j e t , though a t much d i f f e r e n t  scales.  C a r e f u l o b s e r v a t i o n r e v e a l s t h a t these two b r e a k i n g  t y p e s can be f u r t h e r d i v i d e d i n an e f f o r t t o h e l p d e s c r i b e t h e r e s u l t i n g f l u i d flows.  These a r e symmetric s p i l l i n g  breakers,  symmetric p l u n g i n g breakers, and asymmetric p l u n g i n g  breakers.  The  f o l l o w i n g d e s c r i p t i o n s o f each w i l l d e s c r i b e t h e breaker  70  p r o f i l e and t h e f l u i d motions w i t h i n t h e o u t e r r e g i o n o f the s u r f zone. The symmetrical  s p i l l i n g breaker i s no d i f f e r e n t from the  t y p i c a l s p i l l i n g breaker s i n c e a l l s p i l l i n g b r e a k e r s a r e symmetrical.  The symmetry c o n t i n u e s a t and a f t e r b r e a k i n g as  the wave moves forward d i s s i p a t i n g energy. initiated  Breaking i s  when a s m a l l j e t o f water appears a t t h e c r e s t  creating turbulence.  The t u r b u l e n c e g r a d u a l l y s l i d e s down the  f a c e o f t h e wave c r e a t i n g a shear s u r f a c e between t h e t u r b u l e n t white water and t h e c l e a r water on t h e wave f a c e .  The shear  l a y e r generates a s e r i e s o f v o r t i c e s which a r e l e f t behind the c r e s t as t h e wave t r a v e l s towards t h e beach.  The t r a i n o f  v o r t i c e s move s l i g h t l y forward and e l o n g a t e o b l i q u e l y t o the direction of travel.  Generally, the v o r t i c e s are confined t o  the r e g i o n n e a r e r t h e f r e e s u r f a c e ( f i g u r e 4.2). v o r t e x can s t r e t c h t o t h e bed. each subsequent  Occasionally a  The s t r e n g t h and diameter o f  v o r t e x decreases u n t i l  no more a r e c r e a t e d .  T h i s corresponds t o t h e end o f t h e o u t e r r e g i o n o f t h e s u r f zone. The symmetrical p l u n g i n g b r e a k e r a c t s as a s p i l l i n g breaker, b u t breaks as a p l u n g i n g breaker. l a r g e enough t o plunge  The j e t i s not y e t  i n f r o n t o f t h e wave f a c e .  I n s t e a d the  j e t plunges i n t o t h e f a c e c r e a t i n g an i n i t i a l p l u n g i n g v o r t e x and white t u r b u l e n t water on t h e wave f a c e . does n o t p e n e t r a t e t o t h e bed.  The p l u n g i n g j e t  The wave symmetry i s not  a f f e c t e d and t h e breaker now l o o k s and a c t s as t h e symmetric s p i l l i n g b r e a k e r as p r e v i o u s l y d e s c r i b e d . t h a t even though t h e symmetrical  The important f a c t i s  s p i l l i n g and symmetrical  71  Figure 4,2  V o r t e x train p r o d u c e d by a symmetrical spilling b r e a k e r  72  p l u n g i n g b r e a k e r s have d i f f e r e n t i n i t i a l  physical  c h a r a c t e r i s t i c s , each has a s i m i l a r e f f e c t i n t h e c r e a t i o n o f r o t a t i o n a l flow i n t h e o u t e r r e g i o n o f t h e s u r f zone. The asymmetrical  p l u n g i n g breaker i s d e f i n e d t o break when  the wave f a c e becomes v e r t i c a l .  A j e t , l a r g e r than i n t h e  p r e v i o u s two cases, p r o g r e s s i v e l y extends  from t h e c r e s t  i t c l o s e s upon t h e forward s l o p e o f t h e wave.  until  At t h i s point, a  l a r g e p l u n g i n g v o r t e x i s formed w i t h c i r c u l a t i o n around the c a v i t y i n t h e sense o f wave advance.  O b s e r v a t i o n s show t h a t the  c a v i t y q u i c k l y c o l l a p s e s w h i l e t h e a i r entrapped w i t h t h e water.  i n i t mixes  The r e s u l t i s a r e g i o n w i t h v o r t i c a l motion and  a h i g h c o n c e n t r a t i o n o f a i r bubbles which g r a d u a l l y r i s e t o the surface.  The i n i t i a l  j e t may o r may n o t r e a c h t h e bed.  However, i n e i t h e r case, t h e shape o f t h e wave i s asymmetrical. When t h e t i p o f t h e j e t s t r i k e s t h e forward f r e e s u r f a c e , the r e s u l t i n g s p l a s h can cause a wedge shaped amount o f water t o s p l a s h up.  T h i s c u r l s forward and forms another j e t which  s t r i k e s t h e water below i t , and so on ( f i g u r e 4.3).  As many as  f i v e s u c c e s s i v e v o r t i c e s can r e s u l t from t h i s p r o c e s s 1976,  Basco 1985).  (Miller  The r e s u l t i s a sequence o f two dimensional  v o r t e x s t r u c t u r e s r o t a t i n g i n t h e same d i r e c t i o n .  S i n c e they  r o t a t e i n t h e same d i r e c t i o n t h e r e g i o n s between them have a h i g h r a t e o f d i s s i p a t i o n o f t h e o r g a n i z e d motion  and t h e wave  energy. Nadoaka (1986) has observed t h a t t h e two d i m e n s i o n a l s t r u c t u r e o f t h e h o r i z o n t a l v o r t i c e s breaks down through the f o r m a t i o n o f v o r t i c e s extending o b l i q u e l y downward.  These a r e  found t o o r i g i n a t e i n t h e areas o f maximum s t r a i n r a t e between  73  Figure  4,3  Asymmetrical plunging b r e a k e r and t h e s p l a s h - p l u n g e c y c l e (Longuet-Higgins 1953)  74  the v o r t i c e s , and t o be roughly o r i e n t e d a l o n g t h e corresponding principal axis. transformed  The i n i t i a l wave motion, o r energy, i s  t o r o t a t i o n a l flow which i s then d i s s i p a t e d by the  s h e a r i n g f o r c e s between the v o r t e x s t r u c t u r e s . The bed can be d i r e c t l y a f f e c t e d by t h e v o r t i c a l s t r u c t u r e s and t h e p l u n g i n g j e t . symmetrical  G e n e r a l l y , t h e v o r t e x t r a i n formed by the  s p i l l i n g o r t h e symmetrical  r e a c h t h e bed.  p l u n g i n g wave does not  When they do t h e r e i s a n o t i c e a b l e i n c r e a s e i n  the o f f s h o r e a c t i v i t y o f m a t e r i a l s on t h e bed. of  t h e asymmetrical  p l u n g i n g breaker can p e n e t r a t e d i r e c t l y t o  the bed and throw l a r g e amounts o f sediment i n t o The  The p l u n g i n g j e t  i n i t i a l p l u n g i n g v o r t e x and subsequent  suspension.  splash-plunge  v o r t i c e s can a l s o a c t on t h e bed and move m a t e r i a l o f f s h o r e due to 4.3  t h e i r sense o f r o t a t i o n near t h e bed. ESTABLISHED  TURBULENT  FLOW  As time e l a p s e s a f t e r t h e i n i t i a l  b r e a k i n g , the l a r g e  s c a l e , o r d e r e d v o r t e x motion i n t h e o u t e r r e g i o n o f t h e s u r f zone degenerates disorder.  i n t o s m a l l s c a l e motions w i t h i n c r e a s i n g  A t some stage t h e i d e n t i f i c a t i o n o f coherent  s t r u c t u r e s i s no l o n g e r p o s s i b l e . of  When t h i s o c c u r s , t h e s c a l e s  motion can be t r e a t e d as t u r b u l e n c e . The motions o f a b r e a k i n g wave i n t h e o u t e r r e g i o n show a  s i m i l a r decrease  i n s c a l e and i n c r e a s e i n d i s o r d e r .  I d e n t i f i a b l e v o r t i c e s a r e no l o n g e r c r e a t e d .  The i n n e r r e g i o n  s t a r t s where t h e b r e a k i n g wave has been transformed t u r b u l e n t bore.  into a  The bore i s c h a r a c t e r i z e d by a steep, t u r b u l e n t  f r o n t with' an area o f r e c i r c u l a t i n g flow between t h e c r e s t and  75  the t o e ( f i g u r e 4.4). A continuous s h e a r i n g motion occurs between t h e t o e and t h e u n d i s t u r b e d i n f l o w i n g water c r e a t i n g t u r b u l e n c e which spreads behind t h e bore and decays i n t h e wake. As a r e s u l t , t h e r e i s a d i r e c t and continuous t r a n s f o r m a t i o n o f wave motion i n t o t u r b u l e n t motion. 4.4  MASS  TRANSPORT  Before wave b r e a k i n g o c c u r s , sediment i n t h e presence o f p r o g r e s s i v e shallow water waves i s observed t o be t r a n s p o r t e d as bedload  i n t h e d i r e c t i o n o f wave p r o p a g a t i o n .  be accounted waves.  T h i s movement can  f o r by t h e n o n - l i n e a r behaviour o f shallow water  In t h e absence o f a c u r r e n t t h e waves generate a mass  t r a n s p o r t v e l o c i t y and o t h e r h i g h e r o r d e r wave v e l o c i t i e s . Stokes  (1847), assuming a p e r f e c t non-viscous  f l u i d , was t h e  f i r s t t o show t h a t i n a water wave t h e f l u i d p a r t i c l e s , from t h e i r o r b i t a l motion, have a second-order Longuet-Higgins  (1953) made an important  drift  apart  velocity.  theoretical  c o n t r i b u t i o n by e x p l a i n i n g t h e observed movement o f t h e s u r f a c e water and bedload  i n t h e d i r e c t i o n o f wave p r o p a g a t i o n .  s o l u t i o n i s based  on v e r y low waves u s i n g l a m i n a r boundary  l a y e r s a t t h e bed and s u r f a c e . and experiment Experiments  The  Good agreement between t h e o r y  i s o b t a i n e d f o r v a l u e s o f kd between 0.9 and 1.5.  f o r h i g h e r v a l u e s o f kd g e n e r a l l y y i e l d a mass  t r a n s p o r t v e l o c i t y p r o f i l e resembling b e t t e r t h e Stokes  profile.  The n o n - l i n e a r i t i e s have a l s o been e x t e n s i v e l y s t u d i e d by R u s s e l and O s o r i o  (1958), A l l e n and Gibson  (1977), and Isaacson  (1978).  (1959), L i u and Davis  In a l l cases, t h e r e s u l t a r e  76  Direction of  'S^e  4.4  travel  Turbulent  77  bore  s i m i l a r t o t h e above f i n d i n g s and a r e f o r constant  depth  sections. Data from experiments g e n e r a l l y i n d i c a t e t h a t t h e d r i f t v e l o c i t y i s i n t h e d i r e c t i o n o f wave p r o p a g a t i o n near t h e bed, but  a g a i n s t t h e wave f o r t h e s e c t i o n i n t h e middle o f t h e water  column.  A t t h e s u r f a c e , t h e v e l o c i t y can be onshore o r o f f s h o r e  ( f i g u r e 4.5). For beaches t h e mass t r a n s p o r t occurs on a s l o p i n g beach and  t h e o r e t i c a l o r experimental r e s u l t s f o r mass t r a n s p o r t  v e l o c i t i e s over g e n t l y s l o p i n g bottoms a r e v e r y et  scarce.  Bijker  al.(1974) s t u d i e d t h e mass t r a n s p o r t v e l o c i t y on a s l o p i n g  beach and found t h a t t h e o r e t i c a l c o n s i d e r a t i o n s , wave theory,  based on l i n e a r  show t h a t t h e s l o p e w i l l have t h e g r e a t e s t  i n f l u e n c e on t h e mass t r a n s p o r t v e l o c i t i e s f o r r e l a t i v e l y waves on steep impermeable s l o p e s . however, remain r e l a t i v e l y s m a l l  The numerical  long  values,  ( i n f l u e n c e s l e s s than 20%).  In  a d d i t i o n , t h e i r experiments show t h a t t h e bottom mass t r a n s p o r t v e l o c i t i e s a r e more determined by t h e l o c a l depth than by t h e magnitude o f t h e bottom s l o p e .  The bottom v e l o c i t i e s  predicted  by a h o r i z o n t a l bottom t h e o r i e s a r e t o o l a r g e f o r t h e s l o p i n g bottom case.  As t h e roughness i n c r e a s e s , t h e d r i f t  change s l i g h t l y and c o n s i d e r a b l y present  velocities  when a r i p p l e - l i k e roughness i s  ( f i g u r e 4.6).  Wang e t a l . (1982) measured t h e d r i f t v e l o c i t y p r o f i l e s a t the wave b r e a k i n g  p o i n t f o r waves on a plane impermeable beach.  Their r e s u l t s are f o r s p i l l i n g , breakers.  plunging,  and t r a n s i t i o n a l  The d i f f e r e n c e i n b r e a k e r type was i n i t i a l l y  f e l t to  i n f l u e n c e t h e v e r t i c a l d r i f t v e l o c i t y p r o f i l e s a t t h e breaking  78  Direction of  wave t r a v e l E3~  SWL  Figure  4.5  Typical mass t r a n s p o r t (Longuet-Higgins 1953)  79  velocity  profiles  profile nr.  10  11  12  13  1*  15  IS  IT  M  1*  20  profile on hor. bottom  tondrough bottom 015 m |  - —*« A* • MIS m  oxp. nr. D-Sa 03 O  h . 0 411  rippltd bottom  •ip  nr. D-Sb  h i 01] n initial  ml. v«l«ctlf scat*  w i v t : nr. 5  mtasurod  T . 1.5 i. L • 275 m  H • 016 m  krt-IOi  KbAo" 0.0*6  m.t.v.  thtor profilt  profile  L H -Ihtory  (after  Fiqure  4.6  E f f e c t o f increasing b e a c h r o u g h n e s s on t h e n a s s t r a n s p o r t velocity  B i j k e r e t al.)  p o i n t ; however, experimental r e s u l t s show t h a t , i r r e s p e c t i v e of breaker type, t h e d r i f t v e l o c i t y i s onshore near t h e bed and a t the s u r f a c e , and o f f s h o r e i n t h e mid-depth r e g i o n . p r o f i l e does tend t o a more uniform d i s t r i b u t i o n  The v e l o c i t y  ( f i g u r e 4.7).  The cause o f t h i s may be a t t r i b u t e d t o t h e i n c r e a s e d v e r t i c a l momentum t r a n s f e r by t u r b u l e n c e near t h e b r e a k i n g p o i n t . So f a r t h e mass t r a n s p o r t v e l o c i t y p r o f i l e has been d e s c r i b e d up t o t h e p o i n t o f b r e a k i n g .  Onshore o f t h e b r e a k i n g  p o i n t , t h e mass t r a n s p o r t v e l o c i t y p r o f i l e i s d i f f i c u l t t o measure because o f t h e t u r b u l e n c e c r e a t e d by b r e a k i n g .  Instead,  q u a l i t a t i v e o b s e r v a t i o n s o f t h e flow p a t t e r n can be made u s i n g a v a r i e t y o f flow i n d i c a t o r s . experiment cm),  i n the present  a r e f i n e quartz sand, pea g r a v e l ( l o n g e s t a x i s < 1.0  crushed B a k e l i t e , a c r y l i c cubes ( s i d e < 2.0 mm), and  a c r y l i c spheres of  The i n d i c a t o r s used  (6 mm d i a m e t e r ) .  The purpose o f u s i n g a range  d i f f e r e n t m a t e r i a l s was t o attempt  suspended and bedload a c t i o n .  to investigate  The experiment  both  confirmed t h e w e l l  known s i t u a t i o n t h a t o f f s h o r e o f t h e b r e a k i n g zone, bedload moves onshore,  whereas onshore o f t h e b r e a k i n g zone bedload  moves o f f s h o r e , c o n v e r g i n g towards a n u l l p o i n t (Appendix B ) . F o r a c o n s t a n t wave c o n d i t i o n t h e n u l l p o i n t remained s t a t i o n a r y and was p o s i t i o n e d onshore o f t h e b r e a k i n g p o i n t and s l i g h t l y o f f s h o r e o f t h e f i r s t major v o r t e x c r e a t e d by b r e a k i n g , r e g a r d l e s s o f t h e b r e a k i n g type. C o n d i t i o n s o f f s h o r e o f t h e n u l l p o i n t a r e f i r s t examined. Suspension point.  o f t h e a c r y l i c cubes o c c u r s p r i m a r i l y a t t h e n u l l  Once t h e cubes a r e i n suspension they a r e l i f t e d  further  by t h e upward v e l o c i t i e s under a c r e s t and a r e s l o w l y moved 81  Direction  of  wave  Breaking point  travel  82  offshore.  Eventually  , the cubes f a l l t o the bed  as bedload onshore t o the n u l l p o i n t .  The  t r a v e l l e d by the cubes i s shown i n f i g u r e  and  general  are moved  path  4.8.  Each upward movement i n d i c a t e s the presence of a c r e s t each downward movement the presence of a trough. upward movement occurs f o r the f i r s t initial  suspension.  Not  few wave p e r i o d s  some s e t t l e e a r l i e r .  s e t t l i n g u s u a l l y occurs a t the b r e a k i n g breakers.  greatest a f t e r the  a l l of the cubes f o l l o w the same path.  Some are suspended e a r l i e r and  the p l u n g i n g  The  and  The  The  point, e s p e c i a l l y for  suspension i s g r e a t e r  asymmetrical p l u n g e r s i n c e both the p l u n g i n g  f o r the  vortex  and  the j e t  promote suspension. The bedload.  movement o f the other p a r t i c l e s i s predominantly However, when these p a r t i c l e s do go  i n t o suspension  they u s u a l l y remain i n suspension f o r l e s s than one and  are moved o f f s h o r e .  as  wave p e r i o d  They r e t u r n onshore t o the n u l l  point  as bedload. The  movement o f the i n d i c a t o r s suggests t h a t a mass  t r a n s p o r t v e l o c i t y e x i s t s between the b r e a k i n g p o i n t and  i s c o n s i s t e n t w i t h the p r o f i l e a t the b r e a k i n g  an onshore v e l o c i t y component near the bed and  p o i n t and  and  a t the  null point:  surface  an o f f s h o r e component v e l o c i t y i n the middle s e c t i o n of  the  water column. Onshore o f the n u l l p o i n t , the flow f i e l d The  bedload i s t r a n s p o r t e d  is different.  o f f s h o r e t o the n u l l p o i n t  suspended p a r t i c l e s are t r a n s p o r t e d  and  onshore from the n u l l  point.  Suspension of the a c r y l i c cubes o c c u r s a t the n u l l p o i n t i n the same manner as p r e v i o u s l y d e s c r i b e d ;  83  however, i n s t e a d  of  Figure  4,8  Transport  nechanisn o f f s h o r e  i  84  of  the  null  point  t r a v e l l i n g o f f s h o r e , t h e p a r t i c l e i s l i f t e d h i g h enough such that the top h a l f of the f i r s t vortex  forces the p a r t i c l e  onshore ( f i g u r e 4.9). The p a r t i c l e i s passed from one v o r t e x t o another and i s moved onshore .  These v o r t i c e s a r e e i t h e r formed  by t h e shear s u r f a c e o f symmetrical b r e a k i n g plunge c y c l e s .  waves o r by s p l a s h -  The p a r t i c l e s f i n a l l y s e t t l e s when t h e v o r t i c e s  are no l o n g e r s t r o n g enough t o keep them suspended. particle settles,  A f t e r the  i t t r a v e l s as bedload back t o t h e n u l l  point.  Once t h e p a r t i c l e begins t o t r a v e l back t o t h e n u l l p o i n t i t i s r a r e l y resuspended. observing  The flow p a t t e r n i s e a s i l y t r a c e d by  t h e movement o f t h e a c r y l i c cubes, which a r e t h e only  p a r t i c l e s moved onshore o f t h e n u l l p o i n t .  When any o f t h e  o t h e r m a t e r i a l s a r e p l a c e d onshore o f t h e n u l l p o i n t , they q u i c k l y a r e t r a n s p o r t e d o f f s h o r e t o the n u l l p o i n t as bedload. However, i n some cases,  t h e l a r g e r p i e c e s i f pea g r a v e l remain  on t h e upper s e c t i o n o f t h e beach and a r e not t r a n s p o r t e d o f f s h o r e u n t i l t h e wave a t t a c k i s i n c r e a s e d . The  g e n e r a l movement o f t h e sediment i s shown i n f i g u r e  4.10a) which i s c o n s i s t e n t w i t h mass t r a n s p o r t v e l o c i t i e s shown i n f i g u r e 4.10b).  S l i g h t l y onshore o f t h e b r e a k i n g  p o i n t , the  mass t r a n s p o r t v e l o c i t y p r o f i l e s p r e d i c t e d by waves t h e o r i e s break down. Changing t h e c h a r a c t e r i s t i c s o f t h e s h o a l i n g waves w i l l change t h e l o c a t i o n o f t h e o f t h e n u l l p o i n t . changes from s p i l l i n g t o p l u n g i n g moves onshore.  t o surging, the n u l l  point  From t h e motion o f t h e n u l l p o i n t , t h e change i n  the beach s l o p e can be determined. observations  As t h e wave  Even though t h e n u l l  point  a r e f o r a plane 1:15 s l o p e , any forward movement o f  85  Null  Figure  4,9  point  Transport  mechanism o n s h o r e o f  86  t h e null  point  Breaking point  First  major  vortex  a)  Null  point  Breaking point  b)  Null  Figure  4,10  point  a) Circulation c e l l s on e i t h e r side o f t h e null point b) Mass t r a n s p o r t v e l o c i t y p r o f i l e s in t h e s u r f zone  87  the n u l l p o i n t i n d i c a t e s a beach b u i l d i n g p r o c e s s and any o f f s h o r e movement i n d i c a t e s a beach e r o s i o n p r o c e s s . r e s p e c t , f o r experiments performed  on an impermeable plane  beach, t h e breaker c o n d i t i o n s and sediment  responses a r e f o r c e d  t o conform t o t h e c o n d i t i o n s imposed by t h e s l o p e . a n a t u r a l beach,  In t h i s  Rather, f o r  t h e beach and breaker c o n d i t i o n s depend upon  each o t h e r which e v e n t u a l l y r e s u l t s i n t h e e s t a b l i s h m e n t of an e q u i l i b r i u m beach p r o f i l e .  T h e r e f o r e , t h e beach s l o p e w i l l  i n c r e a s e as t h e n u l l p o i n t moves up t h e beach,  and w i l l  decrease  as t h e n u l l p o i n t moves o f f s h o r e . 4.5  SETUP  IN  T H E SURF  ZONE  Wave setup i s d e f i n e d as t h e s u p e r e l e v a t i o n o f t h e mean water l e v e l  (MWL) caused by t h e wave a c t i o n a l o n e .  setup needs t o be b e t t e r understood because  The t o t a l  i t might  either  o c c u r u n i f o r m i l y over t h e whole s u r f zone o r i t may f o l l o w some o t h e r non-uniform  distribution.  I f energy i s d i s s i p a t e d  immediately a t b r e a k i n g , then u n i f o r m setup would be expected. But, i f energy i s d i s s i p a t e d u n i f o r m i l y , then a f a i r l y i n c r e a s e i n t h e setup would be expected.  linear  The k i n e t i c energy o f  the waves i s c o n v e r t e d t o q u a s i - s t e a d y p o t e n t i a l energy as the wave t r a v e l s up t h e beach.  O b s e r v a t i o n s o f waves b r e a k i n g on a  1:15 s l o p e i n t h e l a b o r a t o r y i n d i c a t e t h a t t h e r e i s a decrease i n t h e water l e v e l t o below t h e s t i l l water l e v e l  (SWL) p r i o r t o  b r e a k i n g , w i t h a maximum setdown a t approximately t h e b r e a k i n g point.  As t h e waves breaks t h e mean water l e v e l begins t o r i s e  above t h e SWL and e v e n t u a l l y i n t e r s e c t s w i t h t h e shore.  88  Similar  o b s e r v a t i o n s a r e found by S a v i l l e  (1961).  The v a r i a b l e s  d e f i n i n g wave setup a r e shown i n f i g u r e 4.11. T h e o r e t i c a l s t u d i e s o f setup caused by monochromatic waves have been c a r r i e d out by Longuet-Higgins and Stewart 1962,  (I960,  1963, 1964), Bowen, Inman, and Simmons (1968), and Goda  (1975).  The major c o n t r i b u t i o n t o t h e u n d e r s t a n d i n g o f setdown  and setup i s p r o v i d e d by Longuet-Higgins and Stewart  (1963).  They develop a new term c a l l e d t h e r a d i a t i o n s t r e s s which  i s the  a t t r i b u t e d t o t h e momentum caused by t h e forward movement o f the waves and i s w r i t t e n as,  Sxx  =  x x  )  + 4 2 y  (~~ V sinh(2kd)  E  (4-1) v  ;  where E i s t h e energy o f t h e wave and i s w r i t t e n as,  E = pgH /8  (4.2)  2  As a wave t r a v e l s from deep water t o shallow water, the r a d i a t i o n s t r e s s changes from S =E/2 t o S =3E/2. xx  xx  h e i g h t g e n e r a l l y i n c r e a s e s as t h e wave s h o a l s .  The wave  T h i s b e i n g the  case, t h e r a d i a t i o n s t r e s s w i l l l a r g e l y c o n t i n u e t o i n c r e a s e up to the point of breaking.  S i n c e t h e t o t a l momentum remains  c o n s t a n t up t o b r e a k i n g , i g n o r i n g any s m a l l l o s s e s which occur a t t h e bed, t h e water depth w i l l drop, d e c r e a s i n g t h e momentum component a s s o c i a t e d w i t h t h e water depth.  Therefore, the point  o f maximum setdown s h o u l d occur near t h e b r e a k i n g p o i n t where the flow i s no l o n g e r i r r o t a t i o n a l . The computation o f setup i s o f p r a c t i c a l importance f o r a thorough d e s i g n e f f o r t r e q u i r i n g water l e v e l e s t i m a t i o n s . In the next c h a p t e r , a model o f t h e beach environment 89  i s presented  Breaking point  AS  Sj  F i g u r e 4,11  Total setup Setup Setdown a t breaking Setdown f o r c o n s t a n t  Definition s k e t c h  90  wave  of  conditions  total  setup  which r e l i e s h e a v i l y o f t h e setdown and setup and, t h e r e f o r e , r e q u i r e s an understanding o f t h e setup p r o c e s s .  The Shore  P r o t e c t i o n Manual p r e s e n t s a s e r i e s o f e q u a t i o n s t h a t a l l o w t h e v a r i a b l e s g i v e n i n f i g u r e 4.11 t o be c a l c u l a t e d .  For setdown a t  breaking, =  S  9  1  /  2  Ho T 2  b  The setup a t t h e shore i s ,  S  w  =  AS  -  S  (4.4)  b  Equations 3.4, 3.5, and 3.6 d e f i n e d h e i g h t , p e r i o d , and beach s l o p e .  b  i n terms o f the breaker  These equations a r e repeated  here f o r convenience  b-<aH /gT ) 2  b  where a and b a r e ( a p p r o x i m a t e l y ) ,  a  = 43.75 ( 1 -  • *  e"  19n  )  -19.5m. )  ( 1 + e Longuet-Higgins Saville's  data  and Stewart  (1963) have shown from a n a l y s i s of  (1961) t h a t t h e t o t a l setup f o r shallow water  waves i s ,  AS  =  0.15 d  91  b  (4.5)  Combining equations and  3.4, 3.5, and 3.6 with equations  4.3, 4.4,  4.5 g i v e s , 1/2  2  g  t  Ho  where,  d  b  ii  =  1.56 ,  (4.7) 43.75(1 -  -19.5m  1 + e 1  gT T  F i g u r e 4.12 i s a p l o t o f e q u a t i o n H /gT b  2  for various  4.5.1 The  e~  19m  )H b  2  4.4 i n terms o f S / H w  b  versus  slopes.  Experimental  design and procedures  o b j e c t i v e o f t h i s experiment i s t o determine t h e  setdown and setup p r o f i l e f o r d i f f e r e n t wave and b r e a k i n g conditions.  The beach d e s c r i b e d i n s e c t i o n 2.4 i s m o d i f i e d t o  a l l o w t h e s u r f a c e p r o f i l e t o be recorded.  Twelve manometers a r e  a t t a c h e d t o t h e beach f a c e a t 61 cm i n t e r v a l s along t h e l e n g t h o f t h e beach.  Each manometer i s made up o f a p l e x i g l a s s p i p e 2 5  mm i n diameter which i s connected t o t h e beach f a c e by 5 mm diameter f l e x i b l e t u b i n g . mm i n diameter. pressure  The opening on t h e beach f a c e i s 1.5  The manometers a r e designed  t o damp t h e  f l u c t u a t i o n a s s o c i a t e d w i t h wave motion.  This  allows  the mean water l e v e l above each manometer opening t o be measured.  The assumptions i n v o l v e d i n t r a n s l a t i n g  this  measurement i n t o t h e mean s u r f a c e e l e v a t i o n have been i n d e t a i l by Longuet-Higgins and Stewart  92  (1962).  considered  Figure  4.12  Setup c u r v e s of (SPM 1984)  Sw/Hb  versus  Hb/gT  For s e v e r a l runs,  the depth and wave h e i g h t  depth s e c t i o n a r e h e l d constant o t h e r runs, height  i n the constant  and t h e p e r i o d i s v a r i e d .  t h e p e r i o d and depth a r e h e l d constant  i s varied.  In  and the wave  F o r each run, r e l e v a n t wave data  i s recorded  and t h e setdown/setup p r o f i l e i s determined from t h e l e v e l s i n d i c a t e d by t h e damped manometers. s u f f i c i e n t time i s allowed 4.5.2 The  Experimental  data  F o r each wave c o n d i t i o n ,  f o r t h e flume t o reach  steady s t a t e .  Results  f o r 59 d i f f e r e n t wave c o n d i t i o n s i s presented i n  Appendix C. From t h e r e s u l t s c o l l e c t e d i n t h e l a b o r a t o r y , t h e p o i n t o f maximum setdown f o r s p i l l i n g b r e a k e r s i s near t h e d e f i n e d breaking first  p o i n t which i s l o c a t e d where white t u r b u l e n t water i s  seen on t h e c r e s t .  occurs  For plunging  breakers,  s l i g h t l y onshore o f t h e d e f i n e d b r e a k i n g  offshore of the f i r s t plunging vortex. the j e t f i r s t p e n e t r a t e s  setdown  p o i n t and  T h i s i s l o c a t e d where  t h e f r e e water s u r f a c e but does not y e t  produce any s i g n i f i c a n t r o t a t i o n a l flow. it  maximum  As t h e j e t  i s a l s o moving forward so t h a t t h e p l u n g i n g  vortex  penetrates i s formed  s l i g h t l y onshore o f t h e p o i n t where t h e j e t f i r s t p e n e t r a t e s the water s u r f a c e . Bowen e t a l . (1968) f i n d t h a t t h e s l o p e o f t h e setup p r o f i l e i s a constant  p r o p o r t i o n o f t h e beach s l o p e .  This trend  i s seen i n t h e m a j o r i t y o f t h e cases observed i n t h e l a b o r a t o r y . The  setup i s approximately l i n e a r beyond t h e p o i n t o f maximum  setdown.  F i g u r e 4.13 shows t h e setup p r o f i l e f o r r u n two which  i s r e p r e s e n t a t i v e o f most o f t h e setup p r o f i l e s .  94  Figure  4,13  Setup profiles  for  experimental  run 2  Longuet-Higgins  and Stewart  a n a l y s i s o f S a v i l l e ' s data  (1963) have shown from  an  (1961) t h a t the t o t a l setup f o r pure  s h a l l o w water waves where S =3E/2 i s , xx  AS  = 0.15  d  b  From the c o l l e c t e d r e s u l t s , the t o t a l setup f o r the 1:15  beach  c o n d i t i o n s i s found t o be,  AS  = 0,31  d  b  ( 4 i 8 )  which i s t w i c e the v a l u e found by Longuet-Higgins  and  Stewart.  The d i f f e r e n c e i n d i c a t e s t h a t the b r e a k i n g waves are not s t r i c t l y s h a l l o w water waves and t h a t , as Bowen e t a l . found, the t o t a l setup i s dependent upon t h e beach s l o p e . Comparison o f the t o t a l setup f o r measured and v a l u e s , u s i n g e q u a t i o n s 4.3 4.14.  through 4.5,  calculated  i s shown i n f i g u r e  In a l l cases the measured v a l u e s of t o t a l setup are  g r e a t e r than the c a l c u l a t e d v a l u e s assuming shallow water wave behaviour. 4.7,  The v a l u e o f d  b  i s not c a l c u l a t e d u s i n g e q u a t i o n  but r a t h e r i s taken a t the p o i n t of maximum setdown from  the setup  profiles.  The s c a t t e r o f the measured t o t a l setup i s b e s t seen i n f i g u r e 4.15.  T h i s i s a p l o t of the t o t a l setup v e r s u s the  deepwater wave steepness and i t shows a d e f i n i t e t r e n d t h a t as t h e deepwater wave steepness i n c r e a s e s , the t o t a l decreases.  setup  S p i l l i n g b r e a k e r s , a s s o c i a t e d w i t h h i g h e r v a l u e s of  H / L , break f u r t h e r o f f s h o r e and tend t o break over a g r e a t e r 0  0  distance.  This releases a significant  l e n g t h of the s u r f zone as t u r b u l e n c e .  amount of energy over the The energy t h a t i s not  d i s s i p a t e d as t u r b u l e n c e i s c o n v e r t e d i n t o p o t e n t i a l energy  96  as  1 60.0  C a l c u l a t e d Total Setup,  S c (mm)  140.0  120.0 H  100.0  80.0 H  60.0 H  40.0  H  20.0 H  0.0 0.0  20.0  40.0  60.0  80.0  100.0  M e a s u r e d Total Setup, Figure  4.14  120.0  140.0  160.0  S (mm)  C a l c u l a t e d t o t a l s e t u p v e r s u s measured t o t a l s e t u p f o r an experimental M 5 slope 97  a) Total S e t u p ,  S (mm)  160.0  140.0  120.0  100.0 H  80.0  H  60.0 H  40.0  20.0  -i  0.0  0.00  0  0  0.04  2  o.oe  o.oa  Deepwater S t e e p n e s s , H o / L o  0.10  b) 1.20  Relative S e t u p ,  S/db  1.00  0.80  h  o.eo  o.4o  H  0.20  h  0.00 0.00  002  0.04  0.06  Deepwater S t e e p n e s s , H o / L o  Figure  4,15  0.08  0.10  a) T o t a l s e t u p v e r s u s d e e p w a t e r s t e e p n e s s b) Relative t o t a l s e t u p v e r s u s d e e p w a t e r steepness 98  setup.  The components o f t h e t o t a l setup f o r run 7, run 10, and  run 11 a r e shown i n f i g u r e 4.16.  F o r i n c r e a s i n g p e r i o d and  d e c r e a s i n g deepwater steepness, the t o t a l setup i n c r e a s e s . 4.6  DISCUSSION The  flow dynamics w i t h i n t h e s u r f zone a r e d r i v e n by the  conversion of i r r o t a t i o n a l  flow t o r o t a t i o n a l flow.  The  r o t a t i n g f l u i d generates t u r b u l e n c e which l e a d s t o t h e d i s s i p a t i o n o f energy,  but momentum i s conserved.  i n i t i a l k i n e t i c energy  t h a t i s not d i s s i p a t e d as t u r b u l a e n c e i s  converted i n t o p o t e n t i a l energy  Any o f the  as setup.  S p i l l i n g and p l u n g i n g breakers can be f u r t h e r d i v i d e d symmetrical  spilling,  plunging breakers.  symmetrical  into  p l u n g i n g , and asymmetrical  Each o f these has a d i f f e r e n t  initial  motion, b u t t h e n e t f l u i d motions a r e s i m i l a r when observed  over  a number o f wave p e r i o d s . R e g a r d l e s s o f t h e type o f b r e a k i n g wave, s i m i l a r flows a r e maintained  i n t h e s u r f zone.  Two c i r c u l a t i o n c e l l s a r e c r e a t e d  on e i t h e r s i d e o f a p o i n t t h a t e x h i b i t s no n e t motion, the n u l l p o i n t .  called  O f f s h o r e o f t h e n u l l p o i n t , t h e sediments  t r a v e l s onshore as bedload and o f f s h o r e as suspended l o a d . o c c u r s between t h e b r e a k i n g p o i n t and t h e n u l l p o i n t . of  This  Onshore  t h e n u l l p o i n t , t h e sediments t r a v e l onshore as suspended  l o a d and o f f s h o r e as bedload.  Changing t h e wave c o n d i t i o n s o n l y  causes t h e b r e a k i n g p o i n t and n u l l p o i n t t o move e i t h e r onshore or  o f f s h o r e depending upon whether t h e wave a t t a c k decreases o r  increases.  The two c i r c u l a t i o n c e l l s s t r e t c h and c o l l a p s e y  99  SETDOWN, SETUP, TOTAL SETUP (mm)  SETDOWN, SETUP, TOTAL SETUP (mm) M  n  SETDOWN SETUP  GTJ  TOTAL s e T u p  •  b)  Run  iai  I Run 10.2  • Run 10.8  Run  10.4  SETDOWN, SETUP, TOTAL SETUP (mm) • i  SETDOWN  S3  SETUP  E3  TOTAL S E T U P  c)  • Figure  4,16  Components o f t o t a l s e t u p f o r run 7, run 10, and run 11  100  experimental  depending  upon t h e l o c a t i o n s o f the b r e a k i n g p o i n t and n u l l  point. The setdown/setup p r o f i l e s e a s i l y determined  f o r g i v e n wave c o n d i t i o n s a r e  by u s i n g damped manometers.  steady p r o f i l e develops.  In a l l cases a  The shape o f t h e p r o f i l e s  l i t t l e between c o n d i t i o n s and a r e r e l a t i v e l y l i n e a r .  differ The t o t a l  setup does i n c r e a s e as t h e deepwater steepness decreases o r as the p e r i o d i n c r e a s e s The above o b s e r v a t i o n s a r e important because they show t h a t the c o m p l e x i t i e s o f t h e s u r f zone when averaged over many wave p e r i o d s can be reduced t o a r e l a t i v e l y steady s t a t e . f i v e , t h e importance  o f these o b s e r v a t i o n s becomes apparent when  a t r a n s p o r t model i s developed the steady  In chapter  from a c o n t r o l volume d e f i n e d by  state.  101  CHAPTER 5;  5.1  DEVELOPMENT OF THE  BEACH FACE CONTROL VOLUME MODEL  INTRODUCTION The  onshore-offshore  p r o f i l e changes of a beach under wave  a t t a c k have been d e s c r i b e d by many r e s e a r c h e r s .  Some o f the  more r e c e n t work i s by H a t t o r i and Kawamata (1980), H a t t o r i (1982), S t r i v e and B a j j t e s (1984), Leont'ev (1985), Nishimura and Sunamura (1986).  and  Komar (1983) and Horikawa  summarize o t h e r r e s e a r c h e r ' s work.  (1981)  The m a j o r i t y o f onshore-  o f f s h o r e p r o f i l e models r e l y on d i r e c t  sediment t r a n s p o r t  measurements o r the comparison o f nondimensional parameters with experimentally established values. c h a p t e r takes a new  The method presented  approach and examines the beach  in this  profile  u s i n g a beach f a c e c o n t r o l volume model (Quick 1989a, 1989b). The model r e l i e s on time-averaging  o f c o n d i t i o n s w i t h i n the s u r f  zone. In t h e model, the o n s h o r e - o f f s h o r e  sediment t r a n s p o r t and  beach s l o p e i s e x p l a i n e d and d e f i n e d i n terms o f the b a s i c  102  momentum balance and r e s u l t i n g p r e s s u r e and shear  distribution  a l o n g t h e beach f a c e , as m o d i f i e d by beach p e r m e a b i l i t y .  The  model uses t h e b r e a k i n g p o i n t , t h e mean water l e v e l , and the beach f a c e as t h e boundaries  o f t h e c o n t r o l volume.  The r e l a t i o n s h i p s between t h e main v a r i a b l e s a r e analyzed by o b s e r v i n g t h e response  o f t h e main equation t o v a r y i n g wave  attacks. 5.2 FORMULATION OF THE MODEL The point.  s t a r t i n g p o i n t o f t h i s model i s a t t h e wave b r e a k i n g I t i s here t h a t complex r o t a t i o n a l motion begins and  c o n t i n u e s up t h e beach.  The incoming  wave momentum i s conserved  a t b r e a k i n g and t h e i n i t i a l k i n e t i c energy i s e i t h e r d i s s i p a t e d as t u r b u l e n c e o r converted t o p o t e n t i a l energy which i s seen as a change i n t h e water l e v e l from t h e s t i l l water l e v e l  (SWL).  The p r e v i o u s t h r e e c h a p t e r s have d e s c r i b e d and q u a n t i f i e d many of t h e changes t h a t occur from t h e b r e a k i n g i n c i d e n t waves through  t o t h e t u r b u l e n t bore t h a t runs up t h e beach f a c e t o  e v e n t u a l l y r e t u r n as backrush.  The whole p r o c e s s i s complex and  dynamic, but, f o r t u n a t e l y , can be s i m p l i f i e d by u s i n g averaging  techniques.  Time-averaging  t h e f r e e s u r f a c e o f t h e water waves r e s u l t s  i n t h e e s t a b l i s h m e n t o f a mean water l e v e l f i g u r e 5.1, t h e MWL f a c e c o n t r o l volume. breaking.  time-  (MWL).  Shown i n  forms t h e t o p boundary, AC, o f t h e beach The boundary AB i s a t t h e p o i n t o f wave  C o n d i t i o n s a t t h e seaward boundary, AB, a r e d e f i n e d  by t h e h y d r o s t a t i c p r e s s u r e d i s t r i b u t i o n from t h e MWL  together  w i t h t h e e x t r a momentum f l u x produced by t h e waves, M^.  103  The  igure  5,1  The b e a c h f a c e c o n t r o l f o r c e s acting upon it  104  volume and  the  time-averaged h y d r o s t a t i c f o r c e and t h e time-averaged net shear s t r e s s a c t along t h e boundary BC. face modifies by r e d u c i n g  Permeability  o f t h e beach  t h e f o r c e s a c t i n g on t h e beach f a c e c o n t r o l volume  t h e t o t a l setup and r e d u c i n g  t h e volume o f backrush.  These changes a r e important and w i l l be accounted f o r l a t e r s i n c e t h e development o f t h e model i s b e t t e r understood i f t h e beach i s f i r s t c o n s i d e r e d 5.2.1  impermeable.  Assumption o f an Impermeable Beach  E q u a t i n g t h e components o f t h e time-averaged momentum i n the h o r i z o n t a l d i r e c t i o n y i e l d s an e s t i m a t e o f t h e t o t a l Experimental observations  setup.  o f setup show t h a t i t i s f a i r l y  over t h e s u r f zone (Appendix C) which agrees w i t h  linear  observations  of Bowen, Inman, and Simmons (1968) who f i n d t h a t t h e setup i s l i n e a r l y r e l a t e d t o t h e beach s l o p e .  F i g u r e 5.2 shows a  s i m p l i f i c a t i o n s o f t h e f o r c e s a c t i n g on t h e beach f a c e c o n t r o l volume assuming a u n i f o r m l y profile.  s l o p e d beach and a l i n e a r setup  The t o t a l h o r i z o n t a l momentum balance f o r an  impermeable beach becomes,  A l i n e a r beach i s assumed f o r a number o f reasons.  First,  f o r s i m p l i c i t y s i n c e a l i n e a r beach w i l l not be s u b j e c t t o any s t r o n g l o c a l a c c e l e r a t i o n s a s s o c i a t e d w i t h b a r s and r i p p l e s which c o u l d a f f e c t t h e p r e s s u r e by t h e f o r c e t r i a n g l e a t BC.  force represented  i n f i g u r e 5.2  I t can be argued t h a t even f o r  normal concave beach p r o f i l e s t h e l o c a l a c c e l e r a t i o n s w i l l 105  still  Figure  5.2  Simplified  beach f a c e  106  control  volume  be s m a l l enough t o not a f f e c t t h e h o r i z o n t a l f o r c e s .  Second,  the e x t r a momentum f l u x o f t h e waves, M^, w i l l be c a l c u l a t e d u s i n g t h e r a d i a t i o n s t r e s s theory  developed by Longuet-Higgins  and  t h e setup p r o f i l e s measured i n  Stewart (1964).  And f i n a l l y ,  c h a p t e r f o u r a r e f o r a l i n e a r impermeable beach which w i l l be used t o a n a l y z e t h e model. The  v a l u e o f t h e e x t r a momentum f l u x , M , may be a f f e c t e d b  by r e f l e c t i o n s o c c u r r i n g from t h e beach f a c e .  F o r t h e 1:15  s l o p e d beach used i n t h e l a b o r a t o r y , t e s t s i n d i c a t e t h a t r e f l e c t i o n s a r e about 2% o f t h e i n c i d e n t wave h e i g h t  and a r e  i n s i g n i f i c a n t f o r t h e c o n d i t i o n s s t u d i e d i n both t h e t o t a l experiment and t h e flow dynamic o b s e r v a t i o n s  setup  (Appendix D).  However, i f t h e r e f l e c t i o n s a r e found t o be s i g n i f i c a n t t h e r e f l e c t e d momentum must be i n c l u d e d as an e x t r a shoreward momentum f l u x a c t i n g on t h e c o n t r o l volume. decreases i n t h e beach s l o p e o r i n c r e a s e s  I n any event,  i n t h e i n c i d e n t wave  steepness r e s u l t i n a r e d u c t i o n o f r e f l e c t i o n s . The  n e t shear f o r c e a c t s along t h e beach f a c e o f t h e  c o n t r o l volume.  T h i s f o r c e depends upon t h e drag f o r c e s a c t i n g  on t h e bed d u r i n g uprush and backrush c y c l e s .  In f a c t , t h e  d i r e c t i o n o f t h e n e t shear f o r c e can be onshore, o f f s h o r e , o r even zero.  F i g u r e 5.3 shows t h e f o r c e s a c t i n g on a p a r t i c l e .  Summing over a u n i t area o f t h e beach t h e uprush and backrush drag d u r i n g one wave c y c l e y i e l d s t h e n e t shear s t r e s s .  The  h o r i z o n t a l components o f t h e l o c a l n e t shear s t r e s s e s a r e i n t e g r a t e d over t h e l e n g t h o f t h e beach f a c e , BC, t o determine the n e t h o r i z o n t a l shear f o r c e a c t i n g on t h e c o n t r o l volume. The  p a r t i c l e weight does not c o n t r i b u t e t o t h e shear f o r c e ,  107  i gure  5.3  F o r c e s a c t i n g upon a p a r i : l a y i n g on t h e b e a c h f a c e  108  however, i f t h e n e t shear s t r e s s i s zero t h e g r a v i t y f o r c e along the beach face w i l l  s t i l l produce an o f f s h o r e movement when the  shear s t r e s s f o r the uprush o r backrush a r e g r e a t e r than the critical  shear s t r e s s needed f o r t h e onset o f motion.  q u i t e p o s s i b l e t h a t t h e uprush shear s t r e s s and/or shear s t r e s s can be l e s s than t h e c r i t i c a l for  It is  backrush  shear s t r e s s needed  t h e onset o f motion; however, f o r t h i s model t h e uprush and  backrush shear s t r e s s e s a r e assumed t o be much g r e a t e r than the critical  shear s t r e s s .  The c r i t i c a l  shear s t r e s s then becomes  i n s i g n i f i c a n t w i t h r e s p e c t t o t h e n e t sediment motion. t h i s c o n d i t i o n , t h e sediment movement w i l l  Under  be o f f s h o r e f o r an  onshore n e t shear s t r e s s s m a l l e r than t h e o f f s h o r e g r a v i t y force.  F o r an onshore n e t shear s t r e s s g r e a t e r than t h e  o f f s h o r e g r a v i t y f o r c e , t h e sediment movement w i l l  be onshore.  An o f f s h o r e n e t shear s t r e s s combines w i t h t h e o f f s h o r e g r a v i t y f o r c e t o produce a s t r o n g e r  o f f s h o r e sediment movement.  V i g o r o u s wave a t t a c k i s known t o cause o f f s h o r e movement of sediments, so t h a t i t appears t h a t such wave a t t a c k i s capable of h i g h o f f s h o r e shear s t r e s s e s .  F o r such a s i t u a t i o n , t h e  onshore shear s t r e s s must be l e s s than t h e o f f s h o r e s t r e s s , which i s d i f f i c u l t  t o explain unless  can r i d e over t o p o f t h e water a l r e a d y t h a t t h e uprush shear s t r e s s i s s m a l l .  shear  t h e incoming wave  on t h e beach f a c e , so F i g u r e 5.4 shows t h i s  s i t u a t i o n and i t i s seen t h a t t h e n e t shear s t r e s s a c t i n g on the beach f a c e c o n t r o l volume i s onshore. R e f e r r i n g t o e q u a t i o n 5.1, t h e second and t h i r d  terms  r e s u l t i n t h e c a l c u l a t i o n o f t h e n e t p r e s s u r e f o r c e , which must a c t i n t h e o f f s h o r e d i r e c t i o n ( i e . t h e t h i r d term i s always 109  Region of Uprush R i d i n g Dver Backrush Flow  In i t i a I Wave B r e a k i n g  B r e a k i n g Wave Mov i n g u p B e a c h  Shoreward  Stress  Predominantly Dffshore S t r e s s o f W a t e r on S e d i m e n t  Figure  5,4  A speculative explanation for o f f s h o r e net shear s t r e s s  no  an  g r e a t e r t h a t t h e second f o r any s e t u p ) .  I f the net pressure i s  o f f s h o r e , which i m p l i e s n e t o f f s h o r e t r a n s p o r t , then t h e wave setup and t h e o f f s h o r e shear s t r e s s i n c r e a s e even f u r t h e r so as t o keep t h e e q u a t i o n i n e q u i l i b r i u m .  The f o u r t h term i s t h e  h o r i z o n t a l component o f t h e time averaged n e t shear s t r e s s i n t e g r a t e d over t h e l e n g t h o f t h e beach f a c e c o n t r o l volume. The  value  i s impossible  o f t h e e x t r a momentum a t t h e b r e a k i n g  point,  M, b  t o measure and v e r y d i f f i c u l t t o d i r e c t l y  c a l c u l a t e because no wave t h e o r i e s e x i s t t h a t can account f o r the h i g h l y asymmetrical wave shapes commonly found a t breaking. To s o l v e t h e problem, a new c o n t r o l volume, t h e o f f s h o r e c o n t r o l volume, i s d e f i n e d seaward o f t h e b r e a k i n g together The  w i t h a new momentum a t breaking,  point M'  ( f i g u r e 5.5)  (Quick 1989b).  b  reason f o r u s i n g Mb' w i l l become c l e a r e r once t h e momentum  o f t h e backrush flow i s i n c l u d e d i n t h e a n a l y s i s .  The seaward  boundary, DF, i s l o c a t e d a t t h e p o i n t where t h e setdown begins t o i n c r e a s e below t h e setdown a s s o c i a t e d w i t h constant, waves.  The shoreward boundary, AB, i s a t t h e b r e a k i n g  uniform point,  the MWL i s again t h e time-averaged s u r f a c e boundary, and t h e bed forms t h e bottom boundary, BF.  Along t h e bed,  for simplicity,  the shear s t r e s s i s assumed t o be zero which i s o n l y for  s m a l l h o r i z o n t a l v e l o c i t i e s o c c u r r i n g a t t h e bed.  observations  i n c h a p t e r f o u r show t h a t t h e r e  t r a n s p o r t offshore o f the breaking there  point.  possible However,  i s onshore bedload  This indicates that  i s a n e t shear s t r e s s component a c t i n g o f f s h o r e on t h e  o f f s h o r e c o n t r o l volume.  Neglecting  i t s e f f e c t increases the  onshore momentum f l u x a c t i n g on t h e beach f a c e c o n t r o l volume,  111  112  but does not i n t e r f e r e w i t h t h e b a s i c r e l a t i o n s h i p s between the v a r i a b l e s o f t h e beach f a c e c o n t r o l volume. The momentum f l u x a t t h e boundary presence o f t h e waves, i s Mj. i s added t o M ', b  b  =  Because o f setdown an e x t r a term  giving,  /  n D  l  DF, caused by the  M, +  pA  -  pdy FJ  b  r  d  \  B  0  (5.2)  pdy  Pb v  Mj r e p r e s e n t s t h e incoming wave momentum and i s e a s i l y c a l c u l a t e d u s i n g t h e r a d i a t i o n s t r e s s formula f i r s t proposed by Longuet-Higgins and Stewart  M, 1  =  ( 2kd \ sinh(2kd)  (1964),  A  + J_ 2 /  (5.3)  E  where E i s t h e wave energy and i s equal t o 7H /8. 2  The  c o n d i t i o n s f o r Mj a r e measured a t t h e seaward boundary, e q u a t i o n 5.3 t o be v a l i d , t h e p o s i t i o n o f t h e seaward must be such t h a t t h e MWL  DF.  For  boundary  i s c o n s t a n t and t h e wave shape i s  r e l a t i v e l y constant The e f f e c t o f backrush on t h e momentum l e a v i n g t h e o f f s h o r e c o n t r o l volume must be accounted f o r . the boundary  O b s e r v a t i o n s show t h a t a t  AB t h e forward movement o f t h e wave i s suppressed.  T h i s i s caused by t h e slowing e f f e c t o f s h o a l i n g and t h e o f f s h o r e flow o f t h e backrush.  As t h e forward edge o f t h e  o f f s h o r e c o n t r o l volume i s stopped, a new wave c r o s s e s t h e o f f s h o r e boundary, volume.  DF, and compresses  into the offshore control  The backrush, i n e f f e c t , decreases t h e momentum e x i t i n g  the o f f s h o r e c o n t r o l volume.  Therefore, the r e l a t i o n s h i p  113  between t h e momentum e x i t i n g t h e o f f s h o r e c o n t r o l volume and e n t e r i n g t h e beach <b  where M  R  =  f a c e c o n t r o l volume i s ,  -  b  M  M  (5-4)  r e p r e s e n t s t h e momentum f l u x o f t h e backrush.  T y p i c a l l y , t h e p e r i o d o f t h e backrush, t  b  , i s about h a l f o f t h e  wave p e r i o d and must be taken i n t o account i n t h e f i n a l determination of M . R  The o f f s h o r e momentum f l u x o f t h e b a c k r u s h i n g water from the beach f a c e , generated by t h e p r e v i o u s b r e a k i n g wave,is,  M  =  1/T o  pqu dt  (5.5)  b  where q i s t h e backrush flow over some p o r t i o n o f t h e b r e a k i n g depth, hdjjU^ ( f i g u r e 5.4). The backrush v e l o c i t y i s r e p r e s e n t e d by u . b  The f a c t o r 1/T can be j u s t i f i e d  by c o n s i d e r i n g t h e  impulse o f t h e momentum f l u x , MjT, a s s o c i a t e d w i t h t h e incoming wave and t h e s u b t r a c t i o n o f t h e impulse o f t h e backrush, Mpt^, which o c c u r s o n l y f o r t h e i n t e g r a l time 0 t o t ^ .  The f a c t o r T  d i v i d e s a l l t h e terms when r e v e r t i n g t o momentum f l u x . By combining  equations 5.2 through 5.5, t h e e q u a t i o n f o r  the e x t r a momentum f l u x e n t e r i n g t h e beach  f a c e c o n t r o l volume  at t h e b r e a k i n g p o i n t i s found t o be,  2kd sinh(2kd) 1/T  + J_ 2  E  +  pdy FJ  P q u dt  BJ  p  d b  y  +  J pdy (5.6)  b  114  The  t o t a l momentum balance f o r t h e beach f a c e c o n t r o l  volume o f an impermeable beach i s obtained 5.1  and equation  5.6.  1/T  +  equation  The f i n a l r e s u l t g i v e s , D  1 2  2kd sinh(2kd)  by combining  E  +  B  pdy  -  P dy h  +  pdy  pqu dt b  pdy  Pb y d  BJ  +  7~  0  0  cos Od L q  (5.7)  where t h e f i r s t two l i n e s o f terms a r e t h e momentum f l u x a t t r i b u t e d t o the breaking  waves e n t e r i n g t h e beach f a c e c o n t r o l  volume; t h e t h i r d l i n e i s t h e o f f s h o r e momentum f l u x caused by the t o t a l setup o c c u r r i n g over t h e beach f a c e c o n t r o l volume; and,  t h e f i n a l term i s t h e momentum f l u x o f t h e shear s t r e s s e s  a c t i n g along t h e bottom o f t h e c o n t r o l volume. 5.2.2  Inclusion o f the E f f e c t s o f Permeability  To model r e a l beaches, p e r m e a b i l i t y must be i n c l u d e d w i t h i n the f o r m u l a t i o n o f equation backrush, water w i l l  5.7.  During wave uprush and  i n f i l t r a t e i n t o t h e beach c a u s i n g a  r e d u c t i o n i n both t h e t o t a l setup and t h e backrush flow.  Any  r e d u c t i o n i n t h e t o t a l setup causes a r e d u c t i o n i n t h e o f f s h o r e s t r e s s a c t i n g on t h e sediments which i m p l i e s an i n c r e a s e i n the o f f s h o r e component o f t h e shear f o r c e a c t i n g on t h e beach face c o n t r o l volume.  Therefore,  as a consequence o f i n c r e a s i n g beach  115  permeability, and  the offshore transport  may be r e v e r s e d .  o f sediment i s decreased  Beach p e r m e a b i l i t y  w i l l be used t o e x p l a i n  the primary a s p e c t s o f beach s l o p e behaviour (Quick 1989a, 1989b). In e q u a t i o n 5.7, t h e f i f t h term, d e s c r i b i n g t h e backrush momentum f l u x , and seventh term, d e s c r i b i n g t h e o f f s h o r e hydrostatic  f o r c e on t h e beach face c o n t r o l volume, a r e  affected. During t h e uprush and backrush, water i n f i l t r a t e s  i n t o the  beach which reduces t h e impermeable beach backrush flow by t h e infiltration  flow, q-j-.  i s r e p r e s e n t e d by u  b I  .  momentum f l u x , m o d i f i e d  MR  The backrush v e l o c i t y a l s o changes and The e q u a t i o n d e s c r i b i n g t h e backrush by p e r m e a b i l i t y , i s ,  = 1/T O  I n f i l t r a t i o n modifies  P ( q - q , ) ub. id t J  (5.8)  Q I  t h e s i x t h term o f e q u a t i o n 5.7  because, by c o n t i n u i t y , t h e i n f i l t r a t i o n , r e s i d e n t time, t  r  i n t e g r a t e d over t h e  , d i r e c t l y reduces t h e t o t a l setup and hence  reduces t h e o f f s h o r e h y d r o s t a t i c f o r c e a c t i n g on t h e beach The  change i n t h e h y d r o s t a t i c  the  infiltrated  face.  f o r c e i s c a l c u l a t e d by i n t e g r a t i n g  volume along t h e beach l e n g t h ,  L , and i s given B  by,  pdy  > = ) I  j ' BJ  pdy ( )q  •t pgq,dtdL r  B J  0  «  (5.9)  B  where t h e s u b s c r i p t I i s f o r t h e setup w i t h i n f i l t r a t i o n s u b s c r i p t 0 i s f o r t h e setup w i t h no i n f i l t r a t i o n .  116  and the  The r e s i d e n t  time, t  r  , w i l l v a r y over t h e l e n g t h o f t h e beach and i s t h e  t o t a l d u r a t i o n o f t h e uprush and backrush a t any g i v e n p o i n t . By i n c o r p o r a t i n g equations t o t a l momentum balance  5.8 and 5.9 i n equation  5.7, the  f o r t h e beach face c o n t r o l volume o f a  permeable beach i s g i v e n as,  (* D  , \ sinh(2kd) 2  -  k  + -L- ) E + 2 / ,  d  -  P dy b  +  cB pdy  F  1/T I P ( q - q , ) u d t b|  +  •c pdy  Pb Y d  BJ  + 5.3  pdy  p gq, d t d L 0  #- c r cosOdL o  B  B  B  = 0  (5.10)  B,  DIFFERENT BEACHES SUBJECTED TO THE SAME WAVE ATTACK Consider  two beaches, one composed o f f i n e sand and t h e  o t h e r composed o f l o o s e g r a v e l .  I f both a r e s u b j e c t e d t o t h e  same wave a t t a c k , t h e i n c r e a s e d p e r m e a b i l i t y o f t h e g r a v e l beach w i l l cause t h e t o t a l setup and t h e backrush flow t o be l e s s than the v a l u e s f o r t h e sand beach.  Therefore,  as t h e p e r m e a b i l i t y  of t h e beach i s i n c r e a s e d , t h e t o t a l setup decreases and t h e infiltration  flow i n c r e a s e s .  the i n f i l t r a t i o n  In equation  5.10, an i n c r e a s e i n  flow causes an i n c r e a s e i n momentum f l u x  r e a c h i n g t h e beach and an i n c r e a s e i n t h e onshore f o r c e a c t i n g on t h e beach f a c e c o n t r o l volume. balanced  These i n c r e a s e s must be  by a decrease i n t h e f i n a l shear f o r c e term which can  o n l y be a t t a i n e d by an i n c r e a s e i n t h e beach s l o p e and a  117  decrease i n the o f f s h o r e shear s t r e s s a c t i n g on the  sediments.  S i n c e the beach s l o p e i n c r e a s e s , the o f f s h o r e g r a v i t y f o r c e a c t i n g on the  sediments i n c r e a s e s and must both compensate f o r  the l o s s of o f f s h o r e shear s t r e s s and balance the i n c r e a s e the onshore f o r c e . materials w i l l  The  of  consequence i s t h a t more permeable beach  form s t e e p e r beaches.  This i s consistent  the known behaviour of sand and g r a v e l beaches.  with  F i n e sand forms  v e r y low s l o p e d beaches whereas g r a v e l forms steep  beaches,  sometimes a t o r near the angle o f repose. In a l l cases,  i n f i l t r a t i o n has been assumed t o occur  over the whole l e n g t h o f the beach. case.  As shown i n f i g u r e 5.6,  T h i s probably  evenly  i s not  the  the water t h a t i n f i l t r a t e s  into  the upper s e c t i o n s of the beach must p e r c o l a t e through the beach and  a p o r t i o n may  e x f i l t r a t e a t the lower s e c t i o n s .  The  effect  of t h i s i s t o cause a r e d u c t i o n o f the time-averaged infiltration  f o r p o i n t s lower on the beach.  exfiltration  may  Therefore,  Time-averaged  occur over the lowest s e c t i o n of the beach.  as the i n f i l t r a t i o n decreases and  exfiltration,  the o f f s h o r e shear s t r e s s w i l l  r e d u c t i o n i n the beach s l o p e .  The  reverses  to  increase causing  e f f e c t of e x f i l t r a t i o n  i n c r e a s e the net o f f s h o r e shear s t r e s s , r e d u c i n g  the  a  i s to  slope,  because, a t e q u i l i b r i u m , t h e r e must be a decrease i n the downslope g r a v i t y f o r c e t o compensate f o r the i n c r e a s e d shear s t r e s s .  T h i s argument, t h e r e f o r e , e x p l a i n s the  tendency of beaches t o e x h i b i t concave p r o f i l e s , and beaches, which are more permeable, should,  and  offshore  general gravel  in fact  do,  e x h i b i t more c o n c a v i t y than l e s s permeable sand beaches.  118  INFILTRATION  Increasing o f f s h o r e shear s t r e s s ^  EXFILTRATIDN  Infiltration, e x f i l t r a t i o a  119  and b e a c h concav  The  r e l a t i o n s h i p between t h e beach s l o p e and beach  permeability be analyzed  f o r beaches s u b j e c t e d  t o t h e same wave a t t a c k can  u s i n g t h e c o n t r o l volume.  Permeability  o f sands and  g r a v e l s i s known t o be a f u n c t i o n o f sediment s i z e .  Hazen  (1911) proposes a formula t h a t r e l a t e s t h e p e r m e a b i l i t y  of a  sediment t o t h e square o f t h e 10% f i n e r p a r t i c l e s i z e p r e s e n t i n the m a t e r i a l , s i n c e t h e drainage t h a t occurs i n a m a t e r i a l i s c o n t r o l l e d by t h e p a r t i c l e s t h a t f i l l  the pores.  The formula  is,  K = 10 D where K i s t h e p e r m e a b i l i t y  2 1Q  ( 5  i n m i l l i m e t e r s p e r second and D  the 10 % f i n e r diameter i n m i l l i m e t e r s .  the formula t o make i t d i m e n s i o n a l l y  The  is  be i n c l u d e d i n  c o r r e c t , so t h a t ,  1/1000 D ^ g / i ,  (5.12)  beach l e n g t h , L , assuming a l i n e a r dependency on both B  the wave h e i g h t  and beach s l o p e , L o< B  The  1 0  The k i n e m a t i c  v i s c o s i t y , v , and t h e g r a v i t y f o r c e , g, should  K *  ,U)  i s g i v e n as,  H/sinQ  ( j 5  r e s i d e n t time o f t h e water on t h e beach, t  c a l c u l a t e t h e volume o f i n f i l t r a t i o n .  r  3  )  , i s required to  The r e s i d e n c e  time w i l l  depend on t h e l e n g t h o f t h e beach and t h e wave speed, which, f o r s h a l l o w water waves i s w r i t t e n as p r o p o r t i o n a l t o %/ gH, the wave h e i g h t and  should  a t breaking  i s r e l a t e d t o t h e depth a t breaking  s c a l e f o r depths h i g h e r  t  r  since  oc L / ( g H )  1 /2  on t h e beach.  Therefore,  ( i4)  B  5  120  and t h e volume o f i n f i l t r a t i o n , u n i t w i d t h o f t h e beach, V,  oc  L  B  t  is, r  K  _dh_ dL  where d h / d L  B  V j , f o r t h e beach l e n g t h , per  (5.15)  B  i s t h e h y d r a u l i c g r a d i e n t t h a t d r i v e s t h e flow and  i s e a s i l y expressed as H/L  B  which, by e q u a t i o n 5.13, i s s i n O .  S u b s t i t u t i n g t h e a p p r o p r i a t e terms i n t o e q u a t i o n 5.15,  sin©  (gh)'/  The volume o f i n f i l t r a t i o n  „  2  L  -  ( 5 B  1 5 )  r e p r e s e n t s a decrease i n t h e  t o t a l wave setup and t h e r e f o r e r e p r e s e n t s an i n c r e a s e i n t h e net onshore, u p s l o p e f o r c e a c t i n g on t h e sediment l a y e r and i s g i v e n by, 7 V,  cosO  (5.17)  T h i s f o r c e must be balanced by a s t e e p e n i n g on t h e beach such t h a t t h e downslope, o f f s h o r e g r a v i t y f o r c e , G, i n c r e a s e s , and i s g i v e n by,  G oc L  B  ( 7  -  S  7)  D  M  (  5  i  1  8  )  where Efy i s t h e mean p a r t i c l e s i z e and LgDj^ r e p r e s e n t s t h e volume o f mobile sediment on t h e beach f a c e .  V,cos0  oc L  B  ( 7  -  S  7 ) D  For equilibrium,  M  sin©  (5.19)  which can be s i m p l i f i e d t o g i v e t h e equation,  (qH)  1 / 2  D/o  =  C D  M  tanO  (5.20)  v  where t h e c o e f f i c i e n t C absorbs t h e s p e c i f i c weight terms and accounts f o r t h e p r o p o r t i o n a l i t y .  121  The mean p a r t i c l e s i z e ,  D, M  depends upon the beach m a t e r i a l g r a d i n g curve and can be w r i t t e n i n terms of D ,  using a grading c o e f f i c i e n t ,  10  D  =  M  kgD  k , g  (5.21)  1 0  For beach sand, the m a t e r i a l i s u s u a l l y w e l l graded and c o e f f i c i e n t might be on the o r d e r o f 2.  is  final  form o f equation 5.21,  grading  As t h e beach m a t e r i a l  g e t s c o a r s e r , the g r a d i n g c o e f f i c i e n t w i l l The  '  increase.  after substitution for  D, M  then, 1/2  (gH)  D  1 0  =  C kg tan©  (5.22)  v  T h i s e q u a t i o n r e p r e s e n t s a nondimensional f o r m u l a t i o n t h a t can be used t o c a l c u l a t e the change i n the beach s l o p e f o r d i f f e r e n t beaches s u b j e c t e d t o the same wave a t t a c k .  To  begin  c a l c u l a t i o n s f o r a beach, the sediment s i z e f o r D g r a d i n g c o e f f i c i e n t , kg, need t o be o b t a i n e d . c o n d i t i o n , and u s i n g the c o r r e s p o n d i n g  1 0  and  the  For a g i v e n wave  beach s l o p e ,  c o e f f i c i e n t C can be estimated by u s i n g equation  the  5.22.  Once the  v a l u e s o f C and kg are known f o r a known s l o p e , then changes i n the beach s l o p e f o r d i f f e r e n t wave c o n d i t i o n s can be c a l c u l a t e d . F i g u r e 5.7  i s the p l o t o f e q u a t i o n 5.22  v a l u e s o f kg and D . 1Q  T h i s treatment  for different  r e p r e s e n t s the  different  e q u i l i b r i u m s l o p e s t h a t d i f f e r e n t beaches a t t a i n f o r a constant wave c o n d i t i o n .  Unfortunately, the v a r i a t i o n i n f i g u r e  appears t o be too g r e a t when compared t o the behaviour beaches.  One  5.7 of r e a l  must remember t h a t t h i s f o r m u l a t i o n i s based upon  simple arguments.  The  r e a l s i t u a t i o n i s much more complex,  e s p e c i a l l y f o r the i n f i l t r a t i o n  flow.  122  There are t h r e e p r i n c i p a l  MEAN PARTICLE SIZE (millimeters)  Figure  5,7  Undamped solution f o r t h e same wave a t t a c k i n g d i f f e r e n t b e a c h e s (equation  5.22)  reasons f o r t h e o v e r e s t i m a t i o n  o f the changes i n t h e beach  slope. First, turbulent  t h e seepage flow w i l l change from l a m i n a r t o  flow as t h e sediment r e s i s t a n c e i n c r e a s e s  reduces t h e i n f i l t r a t i o n  and hence  such t h a t t h e s l o p e does not have t o  i n c r e a s e as much t o compensate f o r the i n c r e a s e  i n the net  onshore, upslope f o r c e a c t i n g on t h e sediment.  A similar  r e s i s t a n c e i n c r e a s i n g mechanism w i l l be t h e e n t r a i n i n g o f a i r w i t h i n t h e sediment m a t r i x . attacks  or during  T h i s may o c c u r d u r i n g  h i g h e r wave  t h e a c t i o n o f p l u n g i n g b r e a k e r s and s p l a s h -  plunge c y c l e s where l a r g e t u r b u l e n t v o r t i c e s c a r r y e n t r a i n e d t o t h e bed. above.  The e f f e c t o f t h e a i r i s s i m i l a r t o t h a t mentioned  The a i r i n t h e sediment causes t h e i n f i l t r a t i o n  become t u r b u l e n t  thus i n c r e a s i n g t h e flow r e s i s t a n c e .  the beach m a t e r i a l  i s not n e c e s s a r i l y u n i f o r m l y  shows a v e r t i c a l g r a d a t i o n and  air  finer material  with coarser material  flow t o Finally,  graded and o f t e n at the surface  i n c r e a s i n g w i t h depth i n t o t h e beach.  I n f i l t r a t i n g water w i l l again meet i n c r e a s e d  resistance  such  t h a t t h e s l o p e w i l l n o t have t o i n c r e a s e as much t o balance the reduced onshore, upslope f o r c e a c t i n g on t h e sediments The  i n t e r a c t i o n s o f each o f these e f f e c t s a r e complex and  nonlinear  and tend t o decrease, o r damp, t h e l a r g e change i n the  beach s l o p e p r e d i c t e d by t h e simple a n a l y s i s f o r i n c r e a s i n g sediment s i z e s .  The beach does not f l a t t e n o r steepen as much  as t h e as t h e a n a l y s i s p r e d i c t s .  To show t h e e f f e c t o f these  changes on t h e a n a l y s i s , damping must be i n t r o d u c e d  i n t o the  procedure.  gradient  In t h e o r i g i n a l a n a l y s i s , t h e h y d r a u l i c  t h a t produces t h e i n f i l t r a t i o n  flow i s r e p r e s e n t e d by d h / d L and B  124  i s simply expressed Therefore, beach.  as H/L  which, by equation  B  5.13,  i s sin© ,  a h i g h e r g r a d i e n t e x i s t s f o r p o i n t s h i g h e r up  the  To mimic damping, the g r a d i e n t , dh/dL , i s h e l d B  constant.  Although t h i s i s an o v e r s i m p l i f i c a t i o n ,  i t has  the  advantage of i l l u s t r a t i n g the e f f e c t of damping w h i l e maintaining  the dimensional  Assuming d h / d L  B  t o be constant,  (gH) v F i g u r e 5.8 v a l u e s of D  1 Q  homogeneity of the a n a l y s i s .  1/2  D  -  1 0  equation  The  2  (5.23)  f o r equation  curves  5.2 3 f o r v a r i o u s  show a marked decrease i n the  s l o p e changes f o r c o a r s e r sediments. c o n s i s t e n t with observations  becomes,  C kg sin © cos©  shows the curves  and kg.  5.22  The  g e n e r a l t r e n d i s more  of n a t u r a l beaches and  i t i s seen  t h a t g r a v e l beaches can even reach the n a t u r a l angle of a common o b s e r v a t i o n .  repose,  In a d d i t i o n , s e l e c t e d r e s u l t s g i v e n  by  Dalrymple and Thompson (1976), i n the form o f e q u i l i b r i u m s l o p e v e r s u s nondimensional f a l l v e l o c i t y , are r e c a l c u l a t e d and on the graph.  These p o i n t s are dependent upon an assumed wave  h e i g h t o f 2 meters, a p e r i o d o f 8 seconds, and  converting  s e t t l i n g v e l o c i t y t o an e q u i v a l e n t mean p a r t i c l e s i z e . important 1 0  It i s  f a c t o r s i n the  The p o i n t s are v e r y approximate, but show t h a t  trend i s correct.  the  T h i s f i g u r e i s i l l u s t r a t i v e a t t h i s time  r e q u i r e s c a r e f u l experimental equation  the  t o remember t h a t the mean p a r t i c l e s i z e i s c a l c u l a t e d  by k g D ' each o f which are the important analysis.  shown  v a l u e s t o c o n f i r m the v a l i d i t y  5.23.  125  and of  36  T  1  I  I I I I  1  1  1 I I I I  1  1  1—I  I I  34 D^Bmm,  32 h 30 28 26 24 22 20 18 Real beaches may follow this type behaviour with k increasing for coarser sediments  16 14 12 10 8  Very approximate points from Dalrymple  6 h D = O.lmm 4 l0  J  ' i t i i i •1  10 - i  K)°  1  i i iii i1  j  i  i—l  10  MEAN PARTICLE SIZE (millimeters)  Figure 5.8  Damped solution f o r t h e same wave a t t a c k i n g d i f f e r e n t b e a c h e s (equation 5,23) with s e l e c t e d d a t a f r o m Dalrymple and Thompson (1976)  i i I i  IO  2  5.4  SAME BEACH SUBJECT TO A VARYING WAVE ATTACK Now c o n s i d e r t h e e f f e c t o f a v a r y i n g wave a t t a c k on t h e  same beach.  The momentum f l u x , M I' d e l i v e r e d t o t h e beach face  c o n t r o l volume v a r i e s w i t h t h e wave h e i g h t squared, where, M,  oc  H  2  (5.24)  When a beach i s i n e q u i l i b r i u m , equation balance.  The n e t incoming momentum f l u x i s b a l a n c e d by the  offshore pressure  terms and t h e beach p e r m e a b i l i t y  enough water t o a l s o a i d i n t h e balance. i n c r e a s e s , t h e beach i n f i l t r a t i o n until  5.10 i s i n  absorbs  As t h e wave h e i g h t  cannot absorb t h e e x t r a water  t h e beach f l a t t e n s out, e f f e c t i v e l y i n c r e a s i n g t h e beach  l e n g t h and a l l o w i n g more i n f i l t r a t i o n .  Therefore,  t h e wave  setup i n c r e a s e s and produces an i n c r e a s e i n t h e n e t o f f s h o r e shear s t r e s s a c t i n g on t h e sediments, thus c a u s i n g flatten until  a new e q u i l i b r i u m i s reached.  t h e beach t o  Since the slope i s  l e s s , t h e downslope g r a v i t y f o r c e i s reduced which i n d i c a t e s t h a t t h e n e t onshore shear s t r e s s i s a l s o reduced. Consequently, t h e s l o p e reduces s l i g h t l y l e s s than i s r e q u i r e d by t h e i n c r e a s e i n t h e i n f i l t r a t i o n  f o r an i n c r e a s e d wave  attack. R e f e r r i n g t o equation  5.10, any change i n t h e incoming  momentum f l u x w i l l be accompanied by a p r o p o r t i o n a l change i n the i n f i l t r a t i o n volume.  As a f i r s t approximation, t h e momentum  f l u x , Mj, i s assumed t o be p r o p o r t i o n a l t o t h e i n f i l t r a t i o n volume, V-r, g i v e n by equation  5.16 such t h a t ,  H K  dh  NV2  dL  2  M,  <*  oc  (gH)'  127  B  (5.25)  where t h e p e r m e a b i l i t y , K, has been used f o r s i m p l i c i t y . equations situation.  Two  can be w r i t t e n assuming non-damped and damped First,  using dh/dL  t o be s i n © ,  B  1/2  (g ) K  sin©  H  and u s i n g d h / d L  B  =  (5 6 ) 2  t o be constant, (  LA / 1  (gH) K  •  2  2  in 0  =  z  S  The v a l u e o f K w i l l be constant these equations  constant  constant  (5.27)  f o r t h e same beach, t h e r e f o r e  can be reduced t o , sine  oc  1  /  (5.28)  2  H  and,  sinQ Equation  oc  H  1  /  (5.29)  4  5.28 appears t o p r e d i c t t o o much v a r i a t i o n ,  although 1 the g e n e r a l tendency i s c o r r e c t as shown i n f i g u r e 5.9. Again,  t h e more damped behaviour o f equation  This equation  5.29 i s p r e f e r r e d .  shows t h a t t h e beach s l o p e w i l l decrease under  i n c r e a s e d wave a t t a c k , but w i l l be s m a l l s i n c e i t depends upon the f o u r t h r o o t o f t h e wave h e i g h t .  F o r example, a 5^" beach  s l o p e w i l l change t o a new s l o p e o f 3..5*5 f o r a f o u r i n c r e a s e i n wave h e i g h t .  fold  P r e d i c t i o n s o f t h e change i n t h e  e q u i l i b r i u m s l o p e , u s i n g equation  5.29, a r e p l o t t e d i n f i g u r e  5.10 f o r a wave h e i g h t i n c r e a s e from 1 meter t o 2 meters f o r an 8 second wave p e r i o d .  Th e q u i l i b r i u m beach s l o p e decreases, and  d e c r e a s e s t o a l a r g e r extent  f o r c o a r s e r sediments.  Also  p l o t t e d a r e p o i n t s taken from Dalrymple and Thompson (1976) f o r  128  1  1—I—I  I I I  T  1  1  1 I I I  •  I  \p\  1—I  Beoch slope decreoses under wdve dltock  I I  ^  k=2.3  k»l.7 I  I  I  10 -1  1 M i l l 10°  '  «  1  i iii i  J  • • ' ''  10  MEAN PARTICLE SIZE (millimeters)  Figure  5.9  Undanped solution f o r d i f f e r e n t waves a t t a c k i n g t h e sane b e a c h (equation 5.28)  I0  !  T  1  1  1)11  1  1  1  1 I I I I  T  TT  1—I  17 r16  ^  -  15 14 13  is  12 h  From Dalrymple using H= lm» T=6sec. S  I I  > ^ A  10 —^k*4.5  9  _. Different sediments but same wave attack and based on 3.5° slope when D =0l.k = l 7  v  8 7  From Dalrymples graph (approximate values based on H«2m,T=8sec.)  6 5  7  lft  4 3 2 I j—-J  O 10- i  1  » • ' •'  K>°  '  1  1  i ii i I  i  •  •  I  i i l i  10  MEAN PARTICLE SIZE (millimeters) Figure  5.10  Damped solution f o r d i f f e r e n t waves a t t a c k i n g t h e same b e a c h (equation 5,29) with s e l e c t e d d a t a from Dalrymple and Thompson (1976)  10*  waves w i t h h e i g h t s and p e r i o d s o f 1 meter and 6 seconds, meters and 8 seconds.  and 2  These show good agreement w i t h  p r e d i c t i o n s made by e q u a t i o n 5.29. The c o e f f i c i e n t C used i n e q u a t i o n 5.22 and equation 5.2 3 r e l a t e s t h e i n f i l t r a t i n g volume t o t h e downslope g r a v i t y Combining e q u a t i o n 5.2 3 with  component f o r a g i v e n wave h e i g h t .  e q u a t i o n 5.29, t h e v a r i a t i o n o f C w i t h r e s p e c t t o t h e wave h e i g h t can be made.  F i r s t i n t r o d u c i n g a new c o n s t a n t C  into  e q u a t i o n 5.29,  sinQ  oc  cZpr^  4  (5.30)  and t h e e q u a t i o n f o r t h e c o n s t a n t C i s ,  C  =  ,  COS 0  , , \  2  '  1/2 g  where, f o r s m a l l beach s l o p e s , c o s 0 unity.  .  °ioH  N  (5.31)  can be assumed t o equal  T h i s shows t h a t C v a r i e s l i n e a r l y w i t h t h e wave h e i g h t .  An e x a c t l y s i m i l a r r e s u l t i s reached when e q u a t i o n 5.22 and e q u a t i o n 5.28 a r e used. 5.5  OBLIQUE  WAVE  ATTACK  The whole o f t h i s t h e s i s has been d i r e c t e d towards u n d e r s t a n d i n g t h e p r o c e s s e s waves and beaches go through f o r waves c r e s t s t r a v e l l i n g p e r p e n d i c u l a r t o t h e beach. waves o f t e n approach t h e beach o b l i q u e l y .  In r e a l i t y ,  The a n a l y s i s  p r e s e n t e d i n t h i s c h a p t e r c o u l d be used p r o v i d e d t h a t t h e momentum f l u x i s s u b d i v i d e d i n t o onshore and longshore components.  I f t h e beach i s s t r a i g h t and t h e r e a r e no sediment  131  sources o r s i n k s , then i t i s reasonable t o expect t h a t t h e longshore and onshore Treatment  components a r e independent.  o f t h e onshore  component o f momentum f l u x has  a l r e a d y been d e s c r i b e d i n t h i s chapter.  The longshore component  w i l l be s u b j e c t t o t h e same i n f i l t r a t i o n and e x f i l t r a t i o n e f f e c t s so t h a t t h e longshore shear s t r e s s e s w i l l decrease f o r h i g h e r p o i n t s on t h e beach.  The o f f s h o r e g r a v i t y f o r c e , which  p l a y s a s i g n i f i c a n t role, f o r t h e onshore  component, w i l l be  absent, b u t w i l l be r e p l a c e d by shear s t r e s s e s produced longshore c u r r e n t s .  The s t r e n g t h o f t h e longshore  by any  sediment  t r a n s p o r t w i l l depend upon t h e product o f t h e shear s t r e s s e s and c u r r e n t v e l o c i t y throughout  t h e s u r f zone, m o d i f i e d by t h e  changes i n t h e shear s t r e s s caused by t h e beach p e r m e a b i l i t y . The p e r m e a b i l i t y w i l l tend t o l i m i t longshore t r a n s p o r t on t h e upper beach, and i n c r e a s e t r a n s p o r t i n t h e e x f i l t r a t i o n zone o f the lower beach. 5.6  DISCUSSION A method has been developed t o a n a l y z e t h e changes o f a  beach.  I t i s based upon time-averaging t h e complex motions  w i t h i n t h e s u r f zone.  I f these time-averaged  processes are  e s s e n t i a l l y steady, as they appear t o be as d i s c u s s e d i n chapter 4, then they a r e r e p r e s e n t a t i v e o f t h e c o n d i t i o n s w i t h i n t h e s u r f zone and t h e a n a l y s i s should be v a l i d .  U s i n g t h e mean  water l e v e l , t h e b r e a k i n g p o i n t , and t h e beach f a c e as t h e boundaries o f t h e beach f a c e c o n t r o l volume, t h e momentum i s balanced i n t h e h o r i z o n t a l d i r e c t i o n and g i v e s t h e b a s i s o f the  132  model.  The t o t a l model i s d e s c r i b e d by e q u a t i o n 5.10  which  i n c l u d e s t h e e f f e c t s o f beach p e r m e a b i l i t y . The model shows t h a t o n s h o r e - o f f s h o r e sediment t r a n s p o r t i s p r i m a r i l y dependent upon the magnitude of the wave setup shoreward o f the b r e a k i n g p o i n t , and the p e r m e a b i l i t y o f the beach.  These two  f a c t o r s c o n t r o l the magnitude o f the net  shear  s t r e s s , which, when combined w i t h the downslope component of the sediment weight, transport.  determines  t h e magnitude and d i r e c t i o n of  The magnitude of the l o c a l sediment i s probably b e s t  r e p r e s e n t e d by the i n t e g r a t e d excess stream power, which i s the excess shear s t r e s s p l u s or minus the sediment weight component m u l t i p l i e d by the l o c a l v e l o c i t y .  The excess shear s t r e s s i s  the c a l c u l a t e d shear s t r e s s minus the c r i t i c a l  shear  stress.  Based upon t h i s model, simple r e l a t i o n s h i p s between the beach s l o p e and sediment s i z e can be d e r i v e d .  Equation  5.23  a p p l i e s t o d i f f e r e n t beaches b e i n g a t t a c k e d by i d e n t i c a l wave conditions.  F i g u r e 5.8  i s a p l o t o f the e q u a t i o n and shows t h a t  the e q u a t i o n i s i n reasonable agreement w i t h results.  The second  experimental  r e l a t i o n s h i p i s g i v e n by e q u a t i o n  5.29  which s t a t e s t h a t the beach s l o p e , s i n O , i s p r o p o r t i o n a l t o the i n v e r s e o f the f o u r t h r o o t of the b r e a k i n g wave h e i g h t . shown i n f i g u r e 5.10,  t h i s r e l a t i o n s h i p appears  r e a s o n a b l e agreement w i t h o t h e r f i n d i n g s .  t o be i n  The exact power of  the b r e a k i n g wave h e i g h t r e q u i r e s more a n a l y s i s and a c c o u n t i n g o f v a r i o u s n o n l i n e a r i t i e s of the beach S e v e r a l c o n c l u s i o n s ; based about beach o b s e r v a t i o n s . tendency  As  careful environment.  on these f i n d i n g s , can be made  The f i r s t c o n c l u s i o n i s about the  o f beaches t o e x h i b i t concave p r o f i l e s .  133  According  to  the theory,  t h e o f f s h o r e shear s t r e s s a c t i n g on t h e sediments  i n c r e a s e s and t h e beach s l o p e decreases as t h e i n f i l t r a t i o n decreases.  Therefore,  f o r beaches t h a t have i n f i l t r a t i o n  the upper s e c t i o n s and e x f i l t r a t i o n  over t h e lower s e c t i o n s , the  shear s t r e s s w i l l be g r e a t e r f o r t h e lower s e c t i o n s . shear s t r e s s i s a s s o c i a t e d w i t h f l a t t e r infiltration-exfiltration G r a v e l beaches, being  over  A larger  s l o p e s such t h a t the  argument p r e d i c t s a concave  profile.  f a r more permeable than sand beaches, a r e  p r e d i c t e d , and i n f a c t are, s t e e p e r and more concave than sand beaches. The  second c o n c l u s i o n i s on t h e behaviour o f armored  beaches. cobbles  Beaches t h a t have one o r two l a y e r s o f g r a v e l o r e x h i b i t s l o p e s more c o n s i s t e n t w i t h sand beaches.  This  i s b e s t e x p l a i n e d by c o n s i d e r i n g t h e p e r m e a b i l i t y o f t h e beach, s i n c e t h e cobble  o r g r a v e l l a y e r i s u n d e r l a i d by sand, i t w i l l  be t h e p e r m e a b i l i t y o f t h e sand t h a t c o n t r o l s t h e beach response t o t h e incoming wave c o n d i t i o n s .  As a l r e a d y mentioned, t h e  o f f s h o r e shear s t r e s s a c t i n g on t h e sediments i s l a r g e r f o r l e s s permeable m a t e r i a l s .  The sand, b e i n g  l e s s permeable than t h e  upper g r a v e l l a y e r , causes t h e whole beach t o a c t as a sand beach.  As t h e g r a v e l l a y e r i n c r e a s e s i n t h i c k n e s s ,  i n c r e a s e s and t h e o f f s h o r e shear s t r e s s decreases.  infiltration The beach  b e g i n s t o steepen. The  f i n a l c o n c l u s i o n concerns t h e d i f f e r e n c e i n t h e  b e h a v i o u r o f a w e l l graded beach m a t e r i a l and a uniform material.  The lower p e r m e a b i l i t y o f t h e graded m a t e r i a l  beach will  promote an i n c r e a s e d shear s t r e s s which moves t h e sediment o f f s h o r e and reduces t h e s l o p e .  T h i s l e a d s t o an i n t e r e s t i n g  134  observation  t h a t i f a beach i s a r t i f i c i a l l y  a m a t e r i a l t h a t i n c r e a s e s t h e grading,  p r o t e c t e d by adding  t h e beach i s now more  susceptible t o erosion since the permeability  i s decreased.  new combined m a t e r i a l w i l l be moved more a g g r e s s i v e l y and t h e b r e a k i n g  The  offshore  p o i n t may move f u r t h e r onshore i f enough  m a t e r i a l i s moved o f f s h o r e .  T h i s would be p a r t i c u l a r l y t r u e f o r  the a d d i t i o n o f a w e l l graded sand t o a g r a v e l beach. The  time-averaged method o f a n a l y s i s o f  behaviour o f beaches answers many q u e s t i o n s , others.  An important q u e s t i o n  onshore-offshore but also raises  i s t h e r e l a t i o n between t h e  d i s t r i b u t i o n and magnitude o f t h e setup and t h e volumes o f water d e l i v e r e d t o t h e beach, as a f u n c t i o n o f beach p e r m e a b i l i t y . S i n c e t h e wave setup and beach p e r m e a b i l i t y a r e found t o be t h e two  important parameters c o n t r o l l i n g t h e shear s t r e s s and beach  s l o p e , c a r e f u l experiments and measurements a r e needed t o c o n f i r m these c o n c l u s i o n s . and  A l s o , t h e i n f l u e n c e o f random waves  t h e v a l i d i t y o f time-averaging  further studied.  o f random waves needs t o be  F i n a l l y , as t h i s model i s adapted t o o b l i q u e  wave a t t a c k , t h e independence o f longshore and onshore momentum f l u x and sediment t r a n s p o r t needs t o be addressed.  135  CHAPTER 6;  TESTING OF THE BEACH FACE CONTROL VOLUME MODEL  6.1 INTRODUCTION To t e s t whether t h e models developed  i n the previous  c h a p t e r g i v e reasonable r e s u l t s , an a n a l y s i s o f a plane impermeable beach i s performed.  The shear s t r e s s i s c a l c u l a t e d  u s i n g e q u a t i o n 5.7, t h e model o f an impermeable beach.  F o r each  wave c o n d i t i o n , t h e shear s t r e s s i s t h e maximum v a l u e t h a t can be c a l c u l a t e d .  As d i s c u s s e d i n t h e p r e v i o u s chapter, t h e  o f f s h o r e shear s t r e s s i n c r e a s e s as t h e beach p e r m e a b i l i t y decreases.  S i n c e t h e beach i s impermeable, t h e c a l c u l a t e d  shear  stress i s the largest p o s s i b l e f o r the given conditions. In t h i s c h a p t e r t h e equations developed  i n chapter 5 are  used t o c a l c u l a t e t h e shear s t r e s s f o r measured setup c o n d i t i o n s , and i n v e s t i g a t e t h e s e n s i t i v i t y o f t h e model and the consequent requirements  f o r t h e accuracy o f t h e experimental  measurements.  136  6.2 MODEL FOR AN IMPERMEABLE BEACH The  e q u a t i o n t h a t models an impermeable beach i s repeated  here f o r convenience.  2kd V sinh(2kd) -  V T  +  )'  1 2  A  +  r>  Pb Y d  +  B  J pdy  f  pqu.dt  +  p dy b  c pdy  B  bJ  +  rC  Jr cosOdL o  = 0  B  Each p a r t o f t h e equation must be e v a l u a t e d  i n terms o f t h e  v a r i a b l e s t h a t d e f i n e t h e o f f s h o r e and beach f a c e c o n t r o l volumes ( f i g u r e 5.2 and f i g u r e 5.3). The  f i r s t term r e p r e s e n t s t h e momentum f l u x p a s s i n g through  the boundary DF o f t h e o f f s h o r e c o n t r o l volume. i s p r o p o r t i o n a l t o t h e wave h e i g h t squared.  The energy, E,  The wave h e i g h t , H,  i s measured a t t h e boundary and t h e v a l u e o f kd i s i t e r a t i v e l y c a l c u l a t e d u s i n g t h e equation,  kd  4TT  2 z  / . /  2> z  ( d / g T ) (1/tanh(kd)) T  (6.1)  where d i s t h e depth t o t h e SWL a t t h e boundary DF. The next t h r e e terms r e p r e s e n t t h e i n c r e a s e i n t h e momentum caused by t h e setdown t h a t occurs along t h e l e n g t h o f t h e o f f s h o r e c o n t r o l volume.  Assuming t h e setdown t o be l i n e a r , and  137  knowing t h e s l o p e t o be 1:15, t h e t h r e e i n t e g r a l s a r e e q u i v a l e n t to,  \y  (d + d - s, ) ( s - S , )  (6.2)  b  b  The f i f t h term r e p r e s e n t s t h e o f f s h o r e momentum f l u x a s s o c i a t e d w i t h t h e backrush, which i s s u b t r a c t e d from t h e momentum f l u x e n t e r i n g t h e beach f a c e c o n t r o l volume. C a l c u l a t i o n o f t h e i n t e g r a l depends upon t h e backrush p e r i o d , t , b  t h e backrush depth, and t h e backrush v e l o c i t y , u^.  I n t e r e s t i n g l y , these terms a r e found t o be a p p r o x i m a t e l y c o n s t a n t f o r t h e wave c o n d i t i o n t h a t a r e s t u d i e d .  The backrush  p e r i o d i s t y p i c a l l y about h a l f o f t h e wave p e r i o d .  The backrush  depth i s 0.85 o f t h e b r e a k i n g depth, and t h e backrush v e l o c i t y i s 0.33 m/s.  I t s h o u l d be c a u t i o n e d t h a t t h e backrush v e l o c i t y  was measured a f t e r t h e s e r i e s o f runs was completed.  Instead of  g e n e r a t i n g every c o n d i t i o n a g a i n , t h e backrush v e l o c i t y was measured f o r t h e maximum and minimum c o n d i t i o n s o f each depth. The measured v e l o c i t y d i f f e r e d v e r y l i t t l e  from 0.3 3 m/s.  T h e r e f o r e , u s i n g t h e s e c o n d i t i o n s and assumptions, t h e backrush momentum f l u x i s equal t o ,  -  j  p (0.85 d ) (0.33)  (6.3)  2  b  The n e t p r e s s u r e f o r c e a c t i n g on t h e beach f a c e c o n t r o l volume i s r e p r e s e n t e d by t h e next two i n t e g r a l s .  This  force  w i l l a c t o f f s h o r e f o r any setup, and assuming a l i n e a r setup, i s equal t o ,  -  } r d  b  A S  (6-4)  138  F i n a l l y , t h e n e t shear f o r c e a c t i n g along t h e bottom o f the beach f a c e c o n t r o l volume can be expressed as,  r  0  (d +  AS)/tanO  b  (6.5)  T h i s assumes t h a t t h e n e t shear s t r e s s a c t s onshore on t h e c o n t r o l volume, o f f s h o r e on t h e sediments, and i s c o n s t a n t  along  the l e n g t h o f t h e beach. Combining t h e f i v e terms and r e a r r a n g i n g t o s o l v e f o r the shear s t r e s s , t h e equation i s ,  ~  +  +  j y  j ± 2  6.3  (d + d - S, ) ( S b  b  p (0.85 d )  (0.33)  b  7 d  A  S  '  I  t  ) ( d  q  n  b  +  S,)  2  -  Q  (6.6)  A S )  RESULTS  Data i s c o l l e c t e d f o r 59 d i f f e r e n t wave c o n d i t i o n s and i s shown i n Appendix C . and  49.3 cm.  The water depth v a r i e s between 40.0 cm  The wave h e i g h t a t boundary DF v a r i e s between 10.5  cm and 24.0 cm, and t h e wave p e r i o d v a r i e s between 1.29 seconds and  3.44 seconds.  In runs one t o eleven,  increased f o r r e l a t i v e l y constant  t h e wave p e r i o d i s  wave h e i g h t s and water depths.  For runs twelve t o f i f t e e n , the wave h e i g h t constant  i s changed f o r a  wave p e r i o d .  Table  6.1 shows t h e v a l u e o f each term i n equation  w e l l as t h e c a l c u l a t e d n e t shear s t r e s s . 139  6.6 as  A p o s i t i v e value f o r  Table 6,1 • The c a l c u l a t e d net s h e a r s t r e s s using t h e d a t a c o l l e c t e d f o r e a c h wave condition RUN  1.3 1.1 1.2 1.4 2.4 2.3 2.2 2.1 3.1 4.5 4.4 4.3 4.2 4.1 5.1 5.2 5.3 5.4 5.5 £.1 6.2 6.3 6.4 7.1 7.2 7.3 8.1 8.2 8.3 8.4 8.5 9.1 9.2 9.3 9.4 9.5 10.1 10.2 10.3 10.4 10.5 11.4 11.3 11.2 11.1  MOMENTUM  SHEAR  TERM 1  TERM 2  TERM 3  TERM 4  TERM 5 dBR=e.8Sdb ub=0.33 i / s  N/i  N/i  N/i  N/i  N/i  -74.03 -76.91 -80.85 -83.92 -17.72 -21.72 -23.68 -26.36 -49.35  0.00 0.00 -3.41 -7.29 -6.23 -6.91 0.00 -15.06 -14.05  9.56 8.68 6.58 9.32 10.02 9.97 10.07  57.73 55.64  0.253 0.269  98.35 151.07 35.57 42.81 45.34  0.235 0.188 0.267  5.B3 7.85  34.58 68.67  -14.93 -15.59 -17.35  -5.62 -4.04 -1.86 -4.94  9.29 7.42 13.12 5.52 7.37  39.59 36.17 63.69 30.37  -18.73 -18.86 -26.70 -29.28 -33.01 -34.24 -35.71  -3.72 -8.51 -5.27 -1.75 3.70  -46.15 -47.31 -56.54 -58.09 -17.49  -0.66 -5.57 -3.49 -5.84 -4.44 -7.78 -2.94 -3.32 -5.50  -28.22 -22.08 -23.75 -27.54  -6.72 -3.06 -2.46 -8.84  -26.73 -28.48 -31.46 -36.28 -38.36 -36.55 -36.92 -45.00 -48.65 -50.50  -6.94 -10.07 -2.07 -0.69 -12.20 -1.64 -4.90 -3.28 0.00 -9.14  -52.42 -56.07 -60.34  -3.26 -7.69 -3.84  -67.91  -13.78  -38.55 -42.20 -41.74  47.79 61.06 36.86 58.09 73.52 49.66 61.69 61.27  13.01 7.37 11.19 13.09 9.28 11.07 11.12 14.88 13.02 14.86 12.96 13.01 8.11 8.10 8.15 8.20 6.17 11.93 8.05 11.90 11.89 6.08 10.10 10.05 10.10 15.79 8.03  90.68 82.79 7B.73 75.95 86.56 38.93 40.77 42.73 44.15 41.22 49.32 43.95 58.26 86.22 66.98 43.35 52.74 71.17 91.21 76.97  10.03 15.58 15.61 7.75  49.43 84.23 119.09 79.28  140  N/i 2 A  -1.78 -3.38 4.87 13.02 5.77  0.260 0.256 0.367 0.264  -0.37 3.47  0.277 0.323 0.202 0.389 0.302 0.205  7.84 7.74 11.66 4.76 9.86 7.96  0.323 0.229 0.199 0.266 0.229 0.228 0.176 8.195 0.180 0.199 0.194 0.302 0.300 0.296 0.292 0.339 0.225 0.296 0.220 0.205 0.283 0.258 0.250 0.234 0.168 0.253  3.13 7.92 11.13 6.00  0.253 0.172 0.163 0.253  6.29 8.13  6.54 6.09 10.20 8.83 6.92 5.85 7.40 7.27 6.57 7.61 7.64 3.74 6.19 3.98 8.05 12.53 6.37 3.93 5.24 7.73 9.83 6.41 8.96 6.20 11.49 1.35  Table  6,1  RUN TERM 1  TERM 2  HOHENTUH TERM 3  TERH 4  N/i  N/l  N/i  N/i  -74.83  -3.37  -77.22 -38.60 -35.34 -37.83 -67.83 -68.77 -31.23  -18.59 -17.83 -2.93 -5.86 -6.89 -11.13 -3.13  13.42 6.52 6.33 9.26 8.38 18.89 14.35 8.84  188.82 82.18 57.71 75.54 54.24  -11.88 1.66  8.84 18.64  15.1  -38.83 -59.82 -47.79  -5.26  11.25  47.78 64.87 74.49  15.2 15.3 15.4  -24.77 -68.53 -15.73  -34.82 -25.25 -28.88  9.44 15.88 8.42  12.1 12.2 12.3 12.4 13.1 13.2 14.1 14.2 14.3 14.4  76.99 76.79 44.97  52.83 89.79 46.42  141  SHEAR TERM 5 dBR=B.B5db ub=8.33 a/s N/i 8.182 8.257 8.299 8.241 8.266 8.238 8.185 8.279 8.275 8.232 8.218 8.268 8.175 0.285  N/V2 8.16 8.23 2.52 11.28 5.19 3.21 2.08 5.42 4.87 4.82 7.13 8.49 1.93 2.94  the n e t shear s t r e s s i n d i c a t e s t h a t i t a c t s onshore on t h e c o n t r o l volume and o f f s h o r e on t h e beach sediments t h r e e n e t shear s t r e s s v a l u e s  A l l except  a r e p o s i t i v e , i n d i c a t i n g t h a t the  average n e t shear s t r e s s on an impermeable beach i s g e n e r a l l y o f f s h o r e , as i s expected based upon o b s e r v a t i o n s  discussed i n  c h a p t e r 4. The magnitude o f t h e c a l c u l a t e d n e t shear s t r e s s range from 0.23 N/m  2  t o 13.02 N/m . 2  the range 1.0 N/m  2  The m a j o r i t y  t o 6.0 N/m .  of the stresses f a l l i n  F i g u r e 6.1 i s used t o determine  2  the maximum s i z e d p a r t i c l e a g i v e n shear s t r e s s can move.  This  f i g u r e i s based upon Shields(1936) entrainment f u n c t i o n which i s r e a r r a n g e d t o p l o t as shear s t r e s s v e r s u s jthe m a t e r i a l  diameter.  U s i n g t h i s as a guide, t h e c a l c u l a t e d n e t shear s t r e s s e s a r e capable o f moving m a t e r i a l s t h a t range i n s i z e from 0.01 mm t o 14 mm w i t h t h e m a j o r i t y gravel  f a l l i n g between 1.5 mm t o 7mm.  ( l o n g e s t a x i s < 10 mm)  Pea  t h a t was dropped on t h e upper  l e n g t h o f t h e p l a n e beach was u s u a l l y moved o f f s h o r e t o t h e breaking  point.  upper beach.  In a few cases,  t h e pea g r a v e l remained on t h e  The pea g r a v e l seems t o be t h e l a r g e s t m a t e r i a l  t h a t can be t r a n s p o r t e d  f o r any wave c o n d i t i o n t h a t c o u l d be  generated on t h e p l a n e 1:15 beach.  Because t h e model p r e d i c t s  p a r t i c l e s i z e s c o n s i s t e n t with t h i s observation  t h e approach the  model uses i n a n a l y z i n g o n s h o r e / o f f s h o r e t r a n s p o r t  i s supported.  F i g u r e 6.2 and f i g u r e 6.3 show t h e c a l c u l a t e d n e t shear s t r e s s p l o t t e d a g a i n s t t h e wave h e i g h t The  wave h e i g h t  f o r two d i f f e r e n t depths.  i s taken a t t h e boundary DF and i s h e l d  r e l a t i v e l y constant  f o r each run  F o r runs 4 t o 7 and 8 t o 11,  the depths a r e 45.0 cm and 47.5 cm, r e s p e c t i v e l y .  142  The p e r i o d i s  Figure  6,1  S h e a r s t r e s s r e q u i r e d f o r a moveable bed, b a s e d upon Shields e n t r a p m e n t function  Calculated Shear Stress (N/rrT2) 14  -i  •  .  .  Run 3 12 H  Run 4 Run 5  -B- Run 6 -X- Run 7  10H  Run 15  6H  2H  0.002  0.004  0.006  0.008  H/gT2 Figure  6.2  C a l c u l a t e d net s h e a r d e p t h = 47,5 cm  stress  0.01  versus  T 0.012  H/gT  0.014  2  Calculated Shear Stress (N/rrT2) Run 8 -f-  Run 9 Run 10  -B- Run 11 -X- Run 14  0.002  0.004  0.006  0.008  0.01  0.012  H/C/T2 Figure  6.3  Calculated net s h e a r d e p t h = 45,0 cm  stress  versus  H/qT  2  i n c r e a s e d f o r each wave h e i g h t .  Increasing the period  e f f e c t i v e l y i n c r e a s e s t h e wavelength  c a u s i n g t h e breaker type t o  change from s p i l l i n g t o p l u n g i n g which o c c u r s f o r run 5 through run 10.  The c a l c u l a t e d net shear s t r e s s , p l o t t e d a g a i n s t H/gT , 2  decreases as t h e wave s p i l l s , but begins t o i n c r e a s e as the wave begins t o plunge.  The c a l c u l a t e d n e t shear s t r e s s c o n t i n u e s t o  i n c r e a s e u n t i l i t peaks and then decreases a g a i n .  This  behaviour can be e x p l a i n e d by examining t h e e f f e c t t h e b r e a k i n g depth and t o t a l setup have on t h e f i v e terms i n e q u a t i o n 6.6. As t h e b r e a k i n g wave changes from s p i l l i n g t o p l u n g i n g , caused by an i n c r e a s i n g p e r i o d , t h e b r e a k i n g p o i n t moves onshore r e d u c i n g t h e b r e a k i n g depth. can be s i g n i f i c a n t . ratio H /d b  b  F o r a 1:15 s l o p e , t h i s r e d u c t i o n  R e c a l l i n g f i g u r e 3.4, t h e i n c r e a s e i n the  i s l a r g e f o r a s m a l l decrease i n H / L . 0  increase i n H /d b  b r e a k i n g depth.  b  Q  The l a r g e  can be a t t r i b u t e d t o a decrease i n t h e Any decrease i n t h e b r e a k i n g depth has a s t r o n g  e f f e c t on e q u a t i o n 6.6 by r e d u c i n g t h e v a l u e s o f t h e backrush momentum f l u x and o f f s h o r e p r e s s u r e f o r c e , term 3 and term 4 respectively.  A l s o , as t h e p e r i o d i n c r e a s e s , t h e v a l u e o f kd  decreases such t h a t sinh(kd) approaches  kd c a u s i n g t h e incoming  wave momentum f l u x t o i n c r e a s e , o r term 1 t o become more negative.  As term 1 g e t s more n e g a t i v e and terms 3 and 4 g e t  s m a l l e r t h e n e t shear s t r e s s a c t i n g onshore on t h e c o n t r o l volume must decrease t o m a i n t a i n e q u i l i b r i u m .  The n e t shear  s t r e s s i s a b l e t o decrease and s t i l l m a i n t a i n e q u i l i b r i u m because  t h e onshore momentum f l u x , which i s l e s s than t h e  o f f s h o r e momentum f l u x , i s i n c r e a s i n g as t h e o f f s h o r e momentum flux i s decreasing. 146  A f t e r the breaker has changed t o a p l u n g i n g breaker, the net  shear s t r e s s begins t o i n c r e a s e f o r i n c r e a s i n g p e r i o d .  data shows t h a t the b r e a k i n g depth i n c r e a s e s and the t o t a l begins t o r a p i d l y i n c r e a s e .  The setup  T h e r e f o r e , terms 3 and 4 begin t o  i n c r e a s e a t a r a t e f a s t e r than term 1 as i t gets more n e g a t i v e . The r e s u l t  i s t h a t the net shear s t r e s s a c t i n g onshore on the  c o n t r o l volume i n c r e a s e s t o m a i n t a i n e q u i l i b r i u m . E v e n t u a l l y , the b r e a k i n g depth begins t o decrease and,  as  p r e v i o u s l y e x p l a i n e d , t h i s causes a r e d u c t i o n i n the net shear s t r e s s a c t i n g onshore on the c o n t r o l volume . F i g u r e 6.4  shows the e f f e c t  v a l u e of the net shear s t r e s s .  of changing wave h e i g h t on the As the wave h e i g h t i n c r e a s e s ,  the incoming momentum f l u x i n c r e a s e s as the square of the wave h e i g h t and the t o t a l setup i n c r e a s e s . e q u i l i b r i u m i n e q u a t i o n 6.6, 6.4  Again, t o m a i n t a i n  the net shear s t r e s s must decrease  SENSITIVITY OF THE MODEL Many o f the measurements taken i n the experiments have an  element of u n c e r t a i n t y a s s o c i a t e d w i t h them.  In many cases, the  water l e v e l i n the manometers o s c i l l a t e d  about the  1 mm  When the measured setdown or setup i s o n l y a few itself,  the o s c i l l a t i o n can cause a s i g n i f i c a n t  f i n a l measurement.  MWL.  millimeters e r r o r i n the  T h i s i s a l s o t r u e f o r the measured wave  h e i g h t , depths, and flow v e l o c i t i e s . To c a l c u l a t e the s e n s i t i v i t y o f the model, the p a r t i a l d e r i v a t i v e o f the shear s t r e s s  (equation 6.6)  each o f the v a r i a b l e s i s r e q u i r e d . t a b l e 6.2  with respect to  U s i n g the data f o r each run,  shows t h e s e n s i t i v i t y of the shear s t r e s s f o r s m a l l  147  Table 6.2  RUN  Sensitivity  of  SHEAR  the  impermeable b e a c h model  S e n s i t i v i t y Analysis  upn-o.ojuu  ub=8.33 i / s dSHST/dH N/i 2 A  1.3  -1.70  1.1 1.2 1.4 2.4 2.3 2.2 2.1 3.1 4.5 4.4  -3.38 4.87 13.02 5.77 6.29  4.3 4.2 4.1 5.1 5.2 5.3 5.4  8.13 -0.37 3.47 7.84 7.74 11.66 4.76 9.86 7.96  -0.170 -0.188 -0.173 -0.144 -0.074 -0.088 -0.097 -0.152 -0.151 -0.076 -8.095 -0.066 -0.136 -0.109 -0.073 -0.125 -0.090  5.5 6.1  6.54  6.2 6.3 6.4  6.09 10.20  7.2 7.3 8.1  A  3.13 7.92 11.13 6.00  7.1  N/i 2/u  8.83 6.92 5.85 7.48  -0.100 -0.126 -0.108  dSHST/dSI  dSHST/dSb  dSHST/dSw  dSHST/db  dSHST/dub  dSHST/di  dSHST  N/i 2/u  N/i 2/n  N/I 2/II  N/i 2/ai  (N/i 2)/(i/s)  (N/i 2)/deg  N/i 2  0.089 0.106 0.155 0.111 0.031 0.037 0.034 0.111 0.100 0.033  0.015 0.014 0.009 0.011 0.016 0.016 0.016 0.013 0.013 0.016  -0.448 -0.890 1.279 3.422 1.517 1.654 2.139 -0.097 0.912 2.860  2.232 2.450 2.312 2.421 2.416 2.445 2.532 2.893 2.466 2.516  A  0.747 0.769 0.619 0.551 8.926 0.902  A  a  A  A  0.888 1.113 0.831 0.917 1.806  -0.788  0.217  0.752 1.135 0.938  -0.506  0.246  8.848 0.019  0.015 0.016  2.036 3.066  2.777 2.330  -0.932 -0.745 -0.50B  B.2IB 0.191  8.886 0.058  0.013 0.014  3.072 2.811 2.187  0.242 0.228 0.206 0.240  8.827  1. BIG  1.251 2.592 2.094  0.014  0.822  2.662  0.245  8.072 8.038  0.016  2.081  -0.492 •8.638  0.242 0.237  0.029 8.854  0.016  2.368 2.352  -8.550 -8.548 -0.434  0.246  0.045 0.847 0.030  2.926 1.577 1.720 1.601  1.803 0.804  -0.764  0.258 0.237  -0.559  0.734 0.876 8.798  -0.189  0.797  -0.086 -0.104 -0.085  8.686 8.723 8.701 8.733 8.717 0.927  -8.478 -8.435 -8.476 -8.478  8.248 0.250 8.244  0.015 0.015 0.815 0.016 B.B15 0.016 8.816  2.681  0.264 8.256  0.039 0.032 0.045  -8.697  0.246 8.227  0.046 0.046  0.228 8.221  0.851  B.B15 0.015  0.050  0.015  2.000  2.322 1.818 1.538 1.945 1.911  7.27  -8.112 -0.115 -0.884  8.2  6.57  -0.097  8.919  8.3  7.61  -8.185  8.906  -8.688 -8.683  8.4  7.64  -8.677  8.221  0.052  0.015  3.74  -0.112 -8.144  8.899  8.5 9.1  -8.765  6.19 3.98  -0.088 -8.114  8.973 8.781 8.987  8.283 8.263 8.235  0.096 0.038 0.066  0.013 0.016 0.014  2.010 0.983 1.629 1.846  8.05 12.53 6.37  -0.093 -8.097  8.768 8.788 8.810  -0.509 -8.488  8.251 8.228  0.033 0.048  0.016  2.117  0.015  3.294 1.676  3.93 5.24  -8.187 -0.104  8.155 0.261 0.247  0.125  7.73 9.83  -8.119  8.768  -8.543  0.223  0.859  10.4  -0.092  -0.395  0.257  0.028  18.5  6.41  -0.145  0.653 8.773  -8.579  0.191  8.896  11.4 11.3  0.96 6.20  -0.131 -0.097  11.2 11.1  11.49 1.35  -0.099 -0.170  8.827 8.663 0.627  -0.560 -8.393 -0.385 -8.554  0.266 0.268 0.241  0.065 8.033 0.036  0.202  0.120  9.2 9.3 9.4 9.5 18.1 10.2 10.3  -0.143  A  -0.484 -0.508 -0.470 -0.400 -0.663 -0.649 -0.646 -0.875 -0.619 -0.675  0.760  0.263 0.261 0.147 0.149 0.260 0.251  A  8.844 0.819  8.762  -B.516 -8.668  -0.649 -8.583 -0.570  148  8.847 0.051  0.015  0.010 0.016 0.015 0.014  1.727  1.034 1.378 2.031  0.016 0.B12  2.585  0.815 B.B16  0.252 1.630  0.015 8.B12  3.020 0.355  1.685  2.518 2.302 2.302 2.196 2.268 2.057 2.165 2.224 2.558 2.551 2.599 2.611 2.740 2.162 2.468 2.259 2.417 2.618 2.280 2.299 2.387 2.123 2.482 2.192 1.975 2.197 2.316  Table  RUN  6,2  SHEAR  Sensitivity  Analysis  dSHST/dSI  dSHST/dSb  dSHST/dSv  dSHST/db  dSHST/dub  N/IA2/M  N/iA2/n  N/IA2/M  N/IA2/M  (N/iA2)/(i/s)  uon-D.gjuu  ub--fl.33 i / s dSHST/dH ll/i*2  N/iA2/ai  dSHST/di  dSHST  (N/iA2)/deg  N/iA2  12.1  8.16  -0.122  0.614  -0.377  8.236  0.054  0.015  2.144  2.097  12.2  8.23  -8.184  0.684  -8.503  8.176  0.156  8.810  0.860  2.226  12.3  2.52  -0.152  0.794  -8.595  8.189  0.119  0.811  0.661  2.389  12.4  11.20  -0.112  0.706  -0.589  8.196  0.068  8.814  2.944  2.431  13.1  5.19  -0.130  0.780  -8.546  8.232  0.063  8.814  1.364  2.316  13.2  3.21  -0.150  0.702  -8.465  8.235  0.079  8.014  0.843  2.144  14.1  2.08  -0.111  0.689  -8.411  8.275  0.046  8.816  0.546  1.876  14.2  5.42  -0.112  0.874  -8.634  0.239  0.055  0.015  1.426  2.458  14.3  4.87  -8.121  0.871  -0.625  8.241  0.860  8.015  1.071  2.485  14.4  4.82  -0.142  8.769  -8.522  0.248  0.063  0.015  1.858  2.264  15.1  7.13  -8.119  0.767  -0.529  8.237  0.052  0.015  1.876  2.336  15.2  8.49  -0.102  8.877  -8.685  0.259  0.063  0.015  8.129  2.181  15.3  1.93  -8.108  8.686  -8.487  8.273  0.046  0.816  0.507  1.834  15.4  2.94  -0.078  0.924  -8.678  0.242  0.061  0.015  0.772  2.338  149  Calculated Shear Stress (N/rrT2) 12 -  •  1 2 4  •  10 •  •  8 -  12.1  15.1  6 -  +  14.2  +  13.1.  RUN  d(cm)  H(cm)  T(s)  12.1 12.2 12.3 1 2.4  40.00 40.00 40.00 40.00  22.1 1 21.50 15.20 15.20  2.00 2.75 2.75 2.00  1 3.1 1 3.2  40.40 40.40  15.40 20.50  2.28 2.00  14.1 1 4.2 14.3 1 4.4  45.00 45.00 45.00 45.00  23.00 15.50 14.00 19.50  1.56 1.56 2.30 2.30  15.1 15.2 15.3 15.4  47.30 47.30 47.30 47.30  1 7.50 1 2.60 24.00 1 1.60  2.30 2.30 1.43 1.43  14.3 _j_  4-  "K4.4  15.4 12.3  13.2  •  -h  2 15.2^/ 0  •  0  -  15.3 14.1  12.2  |  1  0.002  0.004  1 0.006  l  I  i  0.008  0.01  0.012  H/gT2 Figure  6,4  C a l c u l a t e d net s h e a r s t r e s s f o r chanqinq wave conditions  v e r s u s H/gT  0.014  errors  i n the measurements.  The maximum p o s s i b l e v a r i a n c e of  the c a l c u l a t e d v a l u e o f the shear s t r e s s i s shown i n the f i n a l column.  T h i s c a l c u l a t i o n i s based upon the f o l l o w i n g  accuracies:  f o r the wave h e i g h t , H, 2.5 mm;  setdown, S j , 1 mm; the  f o r the constant  f o r the setdown a t b r e a k i n g , S^, 1 mm; f o r  setup, S , 1 mm; w  f o r the depth a t b r e a k i n g , d , 2.5 mm; f o r b  the backrush v e l o c i t y , u , 0.03 m/s; b  and, f o r the beach  slope,  m, 0.2 degrees. From the r e s u l t s , the model appears t o be s e n s i t i v e t o small inaccuracies of the v a r i a b l e s  i n the measurements.  The l e a s t s i g n i f i c a n t  i s the backrush v e l o c i t y and the most  significant variable  i s the c o n s t a n t setdown.  reasonable s i n c e the c o n t r i b u t i o n i s r e l a t i v e l y small,  This  seems  o f the backrush momentum f l u x  but the c o n s t a n t setdown, S j , i s an  important v a r i a b l e c o n t r o l l i n g the momentum f l u x e n t e r i n g the beach face c o n t r o l volume. The maximum p o s s i b l e v a r i a n c e o f the shear s t r e s s t h a t a l l the e r r o r s add t o g e t h e r . f a l l s within  assumes  The a c t u a l net shear  stress  the range g i v e n by c a l c u l a t e d v a l u e p l u s or minus  the maximum v a r i a n c e .  The r e s u l t s show t h a t t h e model and the  c a l c u l a t i o n s f o r the net shear s t r e s s a r e s e n s i t i v e t o s m a l l inaccuracies  i n the experimental measurements.  variance generally  ranges between +/-2.1 N/m , 2  magnitude o f the c a l c u l a t e d net shear s t r e s s .  The t o t a l regardless  of the  There does not  appear t o be any r e l a t i o n s h i p between the magnitude o f the c a l c u l a t e d net shear s t r e s s and t h e t o t a l v a r i a n c e .  However,  l a r g e c a l c u l a t e d net shear s t r e s s v a l u e s a r e r e l a t i v e l y a f f e c t e d by s m a l l e r r o r s  i n t h e measurements. 151  less  6.5  DISCUSSION From the above f i n d i n g s , s e v e r a l g e n e r a l  be drawn.  observations  can  For the plane impermeable beach, the c a l c u l a t e d net  shear s t r e s s a c t i n g on the beach i s g e n e r a l l y o f f s h o r e . agrees w i t h the f i n d i n g s i n chapter f o u r where the c e l l onshore of the breaking  This  circulation  p o i n t r o t a t e s i n such a d i r e c t i o n  as t o cause net o f f s h o r e t r a n s p o r t along the beach f a c e . Observations i n chapter f o u r a l s o show t h a t pea  gravel  a x i s < 10 mm)  i s the l a r g e s t sediment t r a n s p o r t e d  Occasionally,  the pea  (longest  offshore.  g r a v e l remained on the upper beach.  This  suggests t h a t the upper l i m i t of the a c t u a l net shear s t r e s s a c t i n g on the beach i s a b l e t o move m a t e r i a l According  10 mm  i n diameter.  t o S h i e l d s entrainment f u n c t i o n , the shear s t r e s s  r e q u i r e d t o move such a sediment i s equal t o or g r e a t e r 12.0  N/m  .  than  The model i n many cases p r e d i c t e d net shear s t r e s s e s  i n t h i s range.  T h i s i s not s a y i n g t h a t i n every c o n d i t i o n  pea  g r a v e l i s moved o f f s h o r e , but t h a t when i t i s moved o f f s h o r e a c e r t a i n net shear s t r e s s i s r e q u i r e d which the model predict.  can  T h i s suggests t h a t the c o n t r o l volume approach used to  c r e a t e the model i s v a l i d . In chapter f o u r , the breaker type d i d not appear t o s i g n i f i c a n t l y i n f l u e n c e the g e n e r a l  p a t t e r n o f flows or  the  setdown/setup p r o f i l e s , but the small change i n the magnitude can  i n f l u e n c e the c a l c u l a t e d net shear s t r e s s e s .  depth and wave h e i g h t  and  f o r an i n c r e a s i n g p e r i o d , the  shear s t r e s s decreases f o r s p i l l i n g breakers,  net  breaker.  by the l a r g e decreases i n the b r e a k i n g 152  constant  but begins t o  i n c r e a s e as the breaker transforms t o a p l u n g i n g i s explained  For a  depth  This  occurring has  up t o the s p i l l - p l u n g e t r a n s i t i o n and  upon the model.  offshore pressure  As the breaking  the e f f e c t  depth decreases  this  the  f o r c e reduces which r e q u i r e s t h a t the  net  shear s t r e s s a c t i n g o f f s h o r e on the sediments must reduce t o maintain equilibrium.  As the p e r i o d continues  to increase,  wave breaks as a p l u n g i n g  breaker but moves c l o s e r t o  plunge-surge t r a n s i t i o n .  Because of t h i s , the t o t a l  the setup  begins t o r a p i d l y i n c r e a s e which causes the o f f s h o r e f o r c e , term 4, t o a l s o i n c r e a s e .  The  the  pressure  net shear s t r e s s must  now  increase to maintain equilibrium. The magnitude of the net shear s t r e s s a c t i n g on a beach face depends h e a v i l y upon the breaking how  depth and  t o t a l setup  these v a r i a b l e s i n f l u e n c e the o f f s h o r e p r e s s u r e  acting  and  on  the c o n t r o l volume. When the wave h e i g h t  i s increased,  the net shear s t r e s s decreases.  f o r a constant  period,  I t i s thought t h a t i f the  net  shear s t r e s s decreases the beach l e n g t h must i n c r e a s e t o keep the c o n t r o l volume i n e q u i l i b r i u m .  However, t h i s i s o n l y  found  i n about h a l f of the wave c o n d i t i o n s . The  model appears t o be s e n s i t i v e t o s m a l l changes i n the  measurements.  I t i s d i f f i c u l t to obtain highly  measurements i n the s u r f zone. changing.  Water l e v e l s are c o n t i n u a l l y  To e s t a b l i s h the mean water l e v e l , the waves are  time-averaged u s i n g damped manometers a t t a c h e d face.  and  t o the beach  Even though these are damped the water l e v e l i n the  manometer s t i l l the MWL  accurate  o s c i l l a t e a few m i l l i m e t e r s .  This requires  that  be assumed t o be a t h a l f the d i s t a n c e between the upper  lower p o i n t s of o s c i l l a t i o n , which i s not n e c e s s a r i l y the  153  case.  When t h e measurements themselves  a r e o n l y tens of  m i l l i m e t e r s i n l e n g t h t h e a c t u a l v a l u e o f t h e n e t shear  stress  can f a l l w i t h i n a r e l a t i v e l y l a r g e range about t h e c a l c u l a t e d value.  The f i n d i n g s put t h e c a l c u l a t e d v a l u e s o f t h e n e t shear  s t r e s s i n doubt, but t h e i r magnitude s t i l l  seem reasonable.  I t i s important t o remember t h a t t h e f i n d i n g s a r e f o r t h e impermeable form o f t h e model.  These f i n d i n g s may o r may not  apply t o t h e permeable form o f t h e model; however, f o r s i m i l a r wave c o n d i t i o n s , t h e o f f s h o r e n e t shear s t r e s s a c t i n g on t h e sediments f o r an impermeable beach i s expected t o be g r e a t e r than t h a t found on a permeable beach.  As p e r m e a b i l i t y i s  i n c l u d e d i n t h e model t h e beach i t s e l f  i s a b l e t o respond t o the  changing wave c o n d i t i o n s which adds s u b s t a n t i a l l y t o t h e complexity o f t h e model. simple beach a r e s t i l l  F o r t h i s study, t h e r e s u l t s from t h e  difficult  to interpret.  Some t r e n d s a r e  mentioned here b u t should o n l y be accepted t o occur on t h e s i n g l e beach s t u d i e d .  The equation m o d e l l i n g t h e impermeable  beach has f i v e terms which a r e made up o f many v a r i a b l e s .  For  t h i s reason t h e r e l a t i o n s h i p between them i s v e r y complex. T r y i n g t o extend these f i n d i n g s t o o t h e r beaches i s c a u t i o n e d and r e q u i r e s f u r t h e r e x p e r i m e n t a t i o n .  However, t h e r e s u l t s a r e  p r o m i s i n g and do l e n d v a l i d i t y t o t h e model.  154  CHAPTER 7;  SUMMARY AND CONCLUSIONS  T h i s t h e s i s has presented a d i s c u s s i o n o f the events o c c u r r i n g i n t h e s u r f zone and t h e development o f a model d e s c r i b i n g t h e o n s h o r e / o f f s h o r e sediment the f o r c e s a c t i n g on a beach.  t r a n s p o r t i n terms o f  Before a model c o u l d be  formulated, a b e t t e r understanding o f t h e s u r f zone was r e q u i r e d . The f o l l o w i n g i s a summary f o r o n s h o r e / o f f s h o r e wave conditions. The most v i s i b l e aspect o f t h e s u r f zone a r e t h e d i f f e r e n t ways i n which a wave can break.  These a r e s p i l l i n g ,  c o l l a p s i n g , and s u r g i n g b r e a k e r s .  plunging,  Using the a v a i l a b l e  l i t e r a t u r e , t h e r e l a t i o n s h i p between t h e b r e a k i n g type, deepwater wave steepness, H / L , and the beach s l o p e , m, was 0  studied.  Q  Separate f i n d i n g s were compared and d i f f e r e n c e s were  e x p l a i n e d i n terms o f t h e d i s s i m i l a r i t i e s between t h e procedures, c a l c u l a t i o n methods, and d e f i n i t i o n s .  Variations  between t h e r e s u l t s were mainly caused f o r two reasons:  each  r e s e a r c h e r had a d i f f e r e n t i n t e r p r e t a t i o n o f t h e same d e f i n i t i o n  155  of t h e b r e a k i n g types and b r e a k i n g waves e x h i b i t  natural  v a r i a b i l i t y t o c o n d i t i o n s w i t h i n t h e s u r f zone.  Even  though  t h e r e were d i f f e r e n c e s , experimental r e s u l t s showed t h a t a reasonable p r e d i c t i o n o f breaker type can be made g i v e n the deepwater wave steepness and t h e beach s l o p e , p r o v i d e d t h a t the beach  i s plane and impermeable. From t h e study o f t h e wave h e i g h t and depth a t b r e a k i n g ,  the governing c o n d i t i o n s o f wave b r e a k i n g were determined. Waves break because they reach some c r i t i c a l  instability.  o s c i l l a t o r y waves t h e i n s t a b i l i t y o c c u r s because  For  t h e wave  becomes t o o steep c a u s i n g t h e wave t o s p i l l down i t s f a c e . s o l i t a r y waves t h e i n s t a b i l i t y occurs because  For  t h e depth becomes  s m a l l enough such t h a t the v e l o c i t y o f t h e c r e s t o f the wave i s s i g n i f i c a n t l y g r e a t e r than the v e l o c i t y o f t h e base o f the wave, c a u s i n g the wave t o plunge forward.  In both cases, the c r e s t  v e l o c i t y i s g r e a t e r than t h e v e l o c i t y o f t h e base o f the b r e a k i n g wave.  T h i s i s t h e . r e s u l t o f b r e a k i n g f o r an  o s c i l l a t o r y wave, but t h e cause o f b r e a k i n g f o r a s o l i t a r y wave. The v a l u e o f t h e h e i g h t - t o - d e p t h r a t i o d e f i n e s where t h e wave breaks.  F o r waves b r e a k i n g from steepness e f f e c t s , the  v a l u e o f Hjy'djj i n c r e a s e s as deepwater steepness decreases because t h e wave steepness i n c r e a s e s f a s t e r than t h e wave h e i g h t , as shown by t h e s h o a l i n g e q u a t i o n (equation 3.8).  This  was found t o occur on most s l o p e s f o r a deepwater steepness g r e a t e r than approximately 0.015.  F o r waves b r e a k i n g from  depth  e f f e c t s , t h e o s c i l l a t o r y wave has enough time t o change i n t o a s o l i t a r y wave.  F o r t h i s s i t u a t i o n , t h e v a l u e o f H^/d^ decreases  as deepwater steepness decreases because 156  t h e s o l i t a r y wave  responds  s l o w l y t o changing depth c o n d i t i o n s and t r a v e l s  water more shallow than i s necessary f o r b r e a k i n g .  into  T h i s was  found t o occur on low s l o p e s f o r a deepwater steepness l e s s 0.015.  than  A t h i r d b r e a k i n g c o n d i t i o n was found t o occur on s l o p e s  s t e e p e r than 1:25 and f o r deepwater steepnesses  l e s s than 0.015.  In t h i s case, t h e beach s l o p e i s steep enough such t h a t an o s c i l l a t o r y wave, not b e i n g a b l e t o change i n t o a s o l i t a r y wave, t r a v e l s i n t o shallow water and i s f o r c e d t o break by some combination  o f s h o a l i n g and depth e f f e c t s .  the v a l u e o f H / d b  b  decreases as  H  / L  0  For a wave w i t h any v a l u e o f  0  For t h i s  situation,  decreases below 0.015.  HQ/LQ  the value of H / d b  b  i n c r e a s e s as t h e s l o p e g e t s t e e p e r . O b s e r v a t i o n s o f t h e flow dynamics w i t h i n t h e s u r f zone p r o v i d e d i n f o r m a t i o n f o r t h e development o f t h e t r a n s p o r t model. An important o b s e r v a t i o n was o f two c i r c u l a t i o n c e l l s t h a t developed when t h e s u r f zone was viewed over a l a r g e number o f wave p e r i o d s .  These r o t a t e d i n o p p o s i t e d i r e c t i o n s such t h a t  sediment was moved along t h e bed t o a common p o i n t between them, the n u l l p o i n t .  Regardless o f t h e type o f breaker, s p i l l i n g o r  p l u n g i n g , these c i r c u l a t i o n c e l l s always formed and always c i r c u l a t e d such t h a t both moved bed m a t e r i a l t o t h e n u l l p o i n t . The n u l l p o i n t o c c u r r e d j u s t onshore o f t h e b r e a k i n g p o i n t and j u s t o f f s h o r e o f t h e f i r s t v o r t e x caused by b r e a k i n g .  Movement  of t h e n u l l p o i n t was caused by changes i n t h e wave a t t a c k . the deepwater steepness was i n c r e a s e d both t h e b r e a k i n g p o i n t and t h e n u l l p o i n t moves o f f s h o r e .  D e c r e a s i n g t h e deepwater  steepness moved t h e p o i n t onshore.  The c i r c u l a t i o n  157  cells  As  s t r e t c h e d and c o n t r a c t e d , f o l l o w i n g t h e movement o f the n u l l point. Wave dynamics o u t s i d e and w i t h i n t h e s u r f zone cause a d i f f e r e n c e between t h e mean water l e v e l and t h e s t i l l level.  water  The setdown/setup p r o f i l e was measured f o r many  d i f f e r e n t wave and depth c o n d i t i o n s . determine  The purpose o f t h i s was t o  i f t h e r e were d i f f e r e n c e s between t h e setdown and  setup f o r s p i l l i n g and p l u n g i n g b r e a k e r s .  I n a l l cases,  o f f s h o r e of t h e b r e a k i n g p o i n t , t h e mean water l e v e l the s t i l l water l e v e l .  i s below  Onshore o f t h e b r e a k i n g p o i n t t h e mean  water l e v e l remains below t h e s t i l l p o i n t o f maximum setdown.  water l e v e l and reaches a  F o r s p i l l i n g b r e a k e r s , maximum  setdown i s a t t h e b r e a k i n g p o i n t .  For plunging breakers,  maximum setdown occurs a t t h e n u l l p o i n t .  F u r t h e r onshore, the  mean water l e v e l e v e n t u a l l y r i s e s above t h e s t i l l and reaches maximum setup a t t h e t o p o f t h e beach. t h a t as t h e deepwater wave steepness  water l e v e l I t was found  i n c r e a s e s , waves tend t o  s p i l l r a t h e r than plunge and t h e t o t a l setup  decreases.  S p i l l i n g breakers, being a s s o c i a t e d with higher values of H /L , 0  Q  broke f u r t h e r o f f s h o r e than p l u n g i n g b r e a k e r s and tended t o break over a l o n g e r d i s t a n c e .  This releases a s i g n i f i c a n t  amount o f energy as t u r b u l e n c e and t h e remaining energy i s turned i n t o p o t e n t i a l energy as setup.  T h e r e f o r e , i t was found  t h a t f o r c o n s t a n t depth and wave h e i g h t , t h e t o t a l setup was g r e a t e r f o r p l u n g i n g breakers than f o r s p i l l i n g b r e a k e r s . i n d i v i d u a l setdown/setup p r o f i l e s , w i t h r e s p e c t t o t h e i r differed l i t t l e .  158  The shape,  U s i n g t h e above o b s e r v a t i o n s , i t was concluded t h a t the s u r f zone c o u l d be time-averaged t o produce a r e l a t i v e l y p i c t u r e o f a v e r y complex environment. volume as d e f i n e d  The beach face  steady  control  by Quick (1989a), i s bounded by t h e beach  f a c e , t h e mean water l e v e l , and a v e r t i c a l p l a n e a t t h e b r e a k i n g point.  When t h e beach i s i n e q u i l i b r i u m  f o r a g i v e n wave  c o n d i t i o n t h e f o r c e s a c t i n g on t h e c o n t r o l volume a r e a l s o i n equilibrium.  S i n c e momentum i s conserved a t b r e a k i n g , t h e  momentum f l u x p a s s i n g i n t o t h e c o n t r o l volume, t h e p r e s s u r e f o r c e s a c t i n g on t h e c o n t r o l volume, and t h e shear s t r e s s on t h e beach face, as m o d i f i e d by beach p e r m e a b i l i t y ,  acting  can be  equated t o form t h e b a s i s o f t h e model. The  f o l l o w i n g were observed from t h e m o d e l l i n g r e s u l t s and  the experimental study u s i n g t h e impermeable form o f the model: a)  The magnitude o f t h e net o f f s h o r e  shear s t r e s s  acting  a l o n g t h e face o f t h e beach i s p r i m a r i l y dependent upon t h e t o t a l setup and t h e beach p e r m e a b i l i t y . permeability versa.  reduces t h e n e t o f f s h o r e  Increasing the  shear s t r e s s and v i c e  The d i f f e r e n c e between g r a v e l and sand beaches i s  explained  u s i n g t h i s argument.  F o r a g i v e n wave a t t a c k ,  gravel  beaches, being more permeable than sand beaches, w i l l have a smaller  offshore  shear s t r e s s a c t i n g on t h e f a c e .  the g r a v e l can form s t e e p e r s l o p e s  and s t i l l  Therefore,  remain s t a b l e , as  compared t o sand. Concave beach p r o f i l e s can be e x p l a i n e d  by t h e e f f e c t  i n f i l t r a t i o n and e x f i l t r a t i o n have on t h e shear s t r e s s . I n f i l t r a t i o n reduces t h e shear s t r e s s w h i l e increases  t h e shear s t r e s s .  Infiltration, 159  exfiltration associated  with  steeper s l o p e s , occurs on the upper s e c t i o n of the beach where the runup p e r c o l a t e s i n t o the beach f a c e .  T h i s water must  e x f i l t r a t e on the lower s e c t i o n of the beach which i n c r e a s e s net o f f s h o r e shear s t r e s s and 5.6).  The  reduces the s l o p e  the  (see f i g u r e  steep upper s l o p e and s m a l l e r lower s l o p e form a  concave beach p r o f i l e .  The  c o n c a v i t y o f a g r a v e l beach i s  g r e a t e r than a sand beach because of the g r a v e l s g r e a t e r permeability. b)  Based on the model, simple r e l a t i o n s h i p s between the  beach s l o p e and p a r t i c l e s i z e were c a l c u l a t e d .  Equation  5.2 3  p r e d i c t s the change i n the s l o p e f o r d i f f e r e n t beaches s u b j e c t to  the same wave a t t a c k .  shown t o be an important  Damping of the flow i n t o the beach f a c t o r i n the behaviour o f the  was  slope.  As the damping i n c r e a s e s the change i n the s l o p e decreases f o r changing wave c o n d i t i o n s .  Damping occurs when the  infiltrating  water meets g r e a t e r flow r e s i s t a n c e and c o u l d be caused by a t r a n s i t i o n from laminar t o t u r b u l e n t flow w i t h i n the m a t r i x the sediment.  T h i s suggests t h a t g r a v e l beaches  of  experience  l a r g e r s l o p e changes than sand beaches f o r a change i n the wave conditions.  The  second r e l a t i o n s h i p i s g i v e n by e q u a t i o n  which s t a t e s t h a t the beach s l o p e , s i n  , i s p r o p o r t i o n a l t o the  i n v e r s e of the f o u r t h r o o t of the b r e a k i n g wave h e i g h t . s l o p e s are expected f o r beaches a t t a c k e d by l a r g e r waves. i s reasonable  5.29  Smaller This  but the exact power r e q u i r e s more a n a l y s i s and  c a r e f u l accounting  o f the n o n - l i n e a r i t i e s of the beach  environment. c)  Experimental  r e s u l t s u s i n g the impermeable form of the  model and the setup data c o l l e c t e d i n chapter 160  f o u r suggest t h a t  the shear s t r e s s a c t i n g on the beach  f a c e i s always  offshore.  T h i s agrees w i t h the o b s e r v a t i o n s of the flow dynamics w i t h i n the s u r f zone and the r o t a t i o n of the nearshore c i r c u l a t i o n a l s o mentioned  i n chapter four.  cell  An i n c r e a s e i n the beach s l o p e  f o r s p i l l i n g or p l u n g i n g waves  r e q u i r e s t h a t the b r e a k i n g p o i n t  and the n u l l p o i n t move onshore s i n c e no onshore  bedload  t r a n s p o r t o c c u r s t o steepen the s l o p e a f t e r the b r e a k i n g p o i n t . To move the n u l l p o i n t onshore, the deepwater wave steepness must be decreased. d)  The magnitudes of the c a l c u l a t e d net shear s t r e s s e s  were reasonable. v a l u e from 0.23  The o f f s h o r e net shear s t r e s s e s ranged i n N/m  2  t o 13.02  s t r e s s i s approximately 6.50  N/m . 2  N/m . 2  The average net shear Using Shields  entrainment  f u n c t i o n , the average net shear s t r e s s i s found t o be a b l e t o move m a t e r i a l about 7.5 mm l a r g e r p i e c e s of pea g r a v e l  i n diameter.  C o n s i d e r i n g t h a t the  ( l o n g e s t a x i s < 10 mm)  were  sometimes observed t o remain on the upper s e c t i o n o f the beach, the c a l c u l a t e d shear s t r e s s e s l e n d v a l i d i t y t o the model. e)  For a c o n s t a n t wave h e i g h t and an i n c r e a s i n g wave  p e r i o d , the c a l c u l a t e d net shear s t r e s s a c t i n g o f f s h o r e on the sediments decreases f o r s p i l l i n g b r e a k e r s , but begins t o i n c r e a s e once the breaker t u r n s i n t o a p l u n g i n g b r e a k e r . i s e x p l a i n e d by the l a r g e decrease i n the b r e a k i n g depth  This that  occurs when a s p i l l i n g breaker becomes a p l u n g i n g breaker f i g u r e 3.4).  (see  As the b r e a k i n g depth decreases the p r e s s u r e  f o r c e s a c t i n g on the c o n t r o l volume change which causes the o f f s h o r e shear s t r e s s t o decrease.  161  f)  The impermeable form o f the model i s s e n s i t i v e t o small  changes i n the measurements.  Measuring the c o n d i t i o n s i n the  s u r f zone i s v e r y d i f f i c u l t .  Even though the mean water l e v e l  was measured u s i n g damped manometers, s m a l l f l u c t u a t i o n s occurred. millimeter.  still  The accuracy of the measurements were t o w i t h i n one C o n s i d e r i n g t h a t many o f t h e measurements are only  m i l l i m e t e r s i n magnitude the s e n s i t i v i t y o f the model i s not surprising.  Assuming t h a t a l l the measurements a r e i n c o r r e c t by  the maximum e r r o r , the average c a l c u l a t e d net shear s t r e s s i s i n e r r o r by p l u s o r minus 2.4  N/m . 2  The o b s e r v a t i o n s , s t u d i e s , and experiments i n t h i s  thesis  were p r i n c i p a l l y conducted on p l a n e impermeable beaches. U n f o r t u n a t e l y , the advantages gained by r e d u c i n g the complexity of the beach environment are b a l a n c e d by the d i s a d v a n t a g e s o f l o o s i n g some o f the r e a l i t y of the p r o c e s s e s .  For example,  i n c i d e n t waves change the form of the beach which i n t u r n changes the form o f the i n c i d e n t waves. p l a n e impermeable beach.  T h i s i s l o s t by u s i n g a  The incoming waves a r e f o r c e d t o  respond t o t h e f i x e d beach.  A p p l i c a t i o n o f t h e r e s u l t s t o sand  beaches should be done c a u t i o u s l y .  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On t h e h i g h e s t water waves o f permanent type, B u l l e t i n o f D i s a s t e r P r e v e n t i o n Research I n s t i t u t e , Kyoto Univ., V o l . 18, p. 1.  167  Appendix A  Shoaling C h a r a c t e r i s t i c s  o f an O s c i l l a t o r y Wave  Using L i n e a r Wave Theory  168  1)  E q u a t e wave  power,  P = P  -  Cl  t h e o v e r bar d e n o t e s e n e r g y f l u x per u n i t wave c r e s t assumes no r e f r a c t i o n  -  2)  Intermediate  and s h a l l o w water  P = nEc  conditions, 2kd  r  n = 0.511  -|  + sinh(2kd)J  § 3)  Deepwater  E q u a t i n g 2)  0.5  r1  gH  2 / 8  conditions,  Po =  4)  =  nE c 0  and  0  n  Q  E  0  =  9 o /8 H  2  3),  2kd  2 k d  1  +  L  =0.5  gH  2 / 8  c = 0.5  gH  ssiinnhh((22kkdd)) J 2kd  ,  1 +  H  2 c  =  H 2 Q  C  q  sinh(2kd)J r  H/H  Q  =  (c /c)  sinh(2kd)  L2kd +  5)  Remember  that,  c/c  ,1/:  l / ;  c  0  = L/L  0  tanh(kd)  169  sinh(2kd)J  2 0  /  8  c,  S u b s t i t u t i n g 5)  6)  into  4)  sinh(2kd> -.1/2 : L2kd + s i n h ( 2 k d ) J r  H/H  r i  H/H  C)  = (coth(kd) )  1  /  2  sinh(2kd:>  r  L / L o = L / L o ( c o t h (kd))  1  /  2  L2kd + s i n h ( 2 k d ) J H/L H /L 0  sinh<2kd>  r  = <coth<kd>)  L2kd + s i n h ( 2 k d ) J  0  7) U s i n g  the  hyperbolic i d e n t i t i e s ,  c o s h * ' = 0.5<cosh(2x) 2  sinh(2x>  8)  ,1/2  3 / 2  The f i n a l H/L  result  =  is,  2cosh kd  ,1/2  = Ho/Lo  1)  2sinh(x)cosh<x)  2  r  +  coth(kd) L2kd + s i n h ( 2 k d > J  170  Appendix B  Observations o f P a r t i c l e Motions Under b r e a k i n g Waves  171  OBSERVATIONS OF PARTICLE NOTIONS UNDER BREAKING HAVES  Nave Data: Condition  Water Depth (CI)  Have Period (s)  Nave Height (CI)  1.72 1.72 1.72  20.7 17.0 13.2  47.0  1.72  20.7  47.0  1.72  24.3  6  47.0  8  47.0 47.0  1.32 1.32  20.0  7  1.32  11.0 20.8 19.4  1 2 3  47.0 48.0 47.0  4  5  9  40.0  2.26  IB  40.0  11  40.0  2.26 2.26  172  16.6  13.7  Condition 1 -Plunging breaker; i n i t i a l j e t does not p e n e t r a t e t o the bed; subsequent splash-plunge c y c l e s (3-5) are generated. The t u r b u l e n c e and v o r t i c i t y generated from t h e s e does reach the bed eventually; d e f i n i t e side wall influence. Bedload -pea g r a v e l ( p g ) ; s a n d ( s ) ; b a k e l i t e ( b ) ; a c r y l i c s p h e r e s ( a s ) ; a c r y l i c cubes(ac) Suspended - a c r y l i c cubes(ac) -pg,s: form a s m a l l r i p p l e j u s t a f t e r the i n i t i a l plunge p o i n t and the b e f o r e the f i r s t splash-plunge p o i n t , -b: t r a v e l s back and f o r t h over the r i p p l e . -as: t r a v e l s back and f o r t h over the r i p p l e sometimes jumping o f f the r i p p l e when moving on or o f f s h o r e . -ac: t r a v e l s as both bedload ans suspended l o a d . The bedload t r a v e l s forward t o the r i p p l e , i s put i n t o suspension and i s h e l d f o r up t o 10 waves. The suspended p a r t i c l e s are s l o w l y c a r r i e d o f f s h o r e t o approximately the p o i n t of b r e a k i n g ( where t h e wave f a c e becomes v e r t i c a l ) where, by t h i s time they s e t t l e t o t r a v e l as bedload back t o the r i p p l e . O f f s h o r e of the b r e a k i n g p o i n t - a l l m a t e r i a l moves forward t o the r i p p l e as bedload o n l y , -some suspended m a t e r i a l moves f u r t h e r o f f s h o r e t o s e t t l e . Onshore of the b r e a k i n g p o i n t - a l l m a t e r i a l p l a c e d c l o s e t o the top o f the beach moves o f f s h o r e t o the r i p p l e as bedload o n l y . -any of the ac t h a t are l i f t e d i n t o the r e g i o n of the v o r t i c e s remain t h e r e and t r a v e l onshore u n t i l the v o r t i c e s can no longer keep them suspended and they t r a v e l back t o the r i p p l e as bedload.  173  Condition 2 -The wave h e i g h t has been reduced thereby d e c r e a s i n g the deepwater steepness. -Breaking moves f u r t h e r up t h e beach. -The wave i s s t i l l a p l u n g i n g breaker; the i n i t i a l j e t does not s t r i k e the bed; 2-3 splash-plunge c y c l e s ; waves a r e r e g u l a r with no secondary i n f l u e n c e s ; f u r t h e r v o r t i c e s generated by the splash-plunge c y c l e reach t h e bed. Bedload pg,s,b,as,ac Suspended ac -pg,s: form a s m a l l r i p p l e ; remains r e l a t i v e l y s t a t i o n a r y j u s t a f t e r t h e i n i t i a l plunge p o i n t and b e f o r e t h e "vortex w a l l " , -b: moves back and f o r t h a c r o s s t h e r i p p l e ; p e r i o d i c a l l y gets suspended o f t h e c r e s t o f t h e r i p p l e b u t does not go v e r y f a r . -as: moves as bedload a c r o s s t h e r i p p l e . -ac: moves both as suspended l o a d and bedload; t h e ac tend t o be suspended near t h e r i p p l e e i t h e r by t h e i n c r e a s e i n t h e v e l o c i t y near t h e bed o f by t h e reduced p r e s s u r e as t h e wave c r e s t approaches; a c i r c u l a t i o n o f ac occurs w i t h t h e forward movement o f the ac as bedload and t h e o f f s h o r e movement as suspended l o a d . Onshore o f t h e v o r t e x w a l l -bedload i s t r a n s p o r t e d o f f s h o r e t o t h e r i p p l e and suspended l o a d i s t r a n s p o r t e d onshore. -suspended l o a d i s suspended a t t h e r i p p l e and i s l i f t e d h i g h enough t o be trapped i n t h e v o r t i c e s near the s u r f a c e t o be c a r r i e d onshore. -some pg was put onshore o f t h e v o r t e x w a l l and some o f i t , the l a r g e r p i e c e s , d i d not move o f f s h o r e as bedload, but remained.  174  Condition 3 -The wave h e i g h t has been reduced thereby d e c r e a s i n g the deepwater steepness. -Breaking moves f u r t h e r up t h e beach. -The breaker i s p l u n g i n g ; t h e j e t s t i l l does not p e n e t r a t e t o the bed; t h e l a r g e v o r t e x forms t h e v o r t e x w a l l and the sediment does not go p a s t i t as bedload. Bedload pg,s,b,as,ac Suspended ac -pg,s: a g a i n a s m a l l r i p p l e i s formed; t h i s time d i r e c t l y under the p l u n g i n g p o i n t o f t h e i n i t i a l p l u n g e r . Some pg p l a c e d onshore o f t h e v o r t e x w a l l remained and d i d not t r a v e l o f f s h o r e as bedload, but o n l y f o r t h e l a r g e r p i e c e s , -b: t r a v e l s on equal s i d e s o f t h e r i p p l e as bedload. -ac: move as b e f o r e ; suspension occurs p r i m a r i l y a t t h e r i p p l e , are c a r r i e d o f f s h o r e where they s e t t l e and t r a v e l as bed l o a d back t o t h e r i p p l e ; t h e h i g h e r t h e p a r t i c l e i s suspended t h e further offshore i t travels. I f i t i s i n i t i a l l y suspended h i g h enough t h e r e i s a chance i t w i l l be c a r r i e d onshore b e i n g h e l d up by a s e r i e s o f v o r t i c e s c r e a t e d by v a r i o u s splash-plunge c y c l e s , and then t r a v e l o f f s h o r e as bedload t o t h e r i p p l e ( n u l l point).  175  D i s c u s s i o n on C o n d i t i o n s 1. 2.  and 3  As t h e wave steepness was decreased t h e b r e a k i n g p o i n t moved onshore as i s expected s i n c e , i f t h e rough r u l e of thumb i s used, Hu/d i s approximately 0.75, then t h e depth must decrease a t t h e b r e a k i n g p o i n t , hence t h e p o i n t moves onshore. b  In a l l cases, the j e t d i d not p e n e t r a t e t o t h e bed. An argument may be made t h a t i f t h e j e t does not p e n e t r a t e then the p l u n g i n g breaker i s r e a l l y a s p i l l i n g type breaker from the p o i n t o f view o f t h e sediment. T h i s i s and important d i s t i n c t i o n t o make s i n c e f o r t h e same breaker type you can get d i f f e r e n t t r a n s p o r t regimes. As t h e wave h e i g h t decreases and t h e b r e a k i n g depth decreases t h e bedload sediments were more prone t o going i n t o suspension. T h i s i s probably due t o t h e bed b e i n g under more d i r e c t i n f l u e n c e o f t h e t u r b u l e n c e generated by t h e b r e a k i n g p r o c e s s because o f the r e d u c i n g depth. The v o r t i c e s generated by t h e j e t and t h e s p l a s h plunge c y c l e s . d i d not i n f l u e n c e the o f f s h o r e t r a n s p o r t o f a l l m a t e r i a l o f f s h o r e t o t h e r i p p l e (except i n some i n s t a n c e s where a l a r g e r p i e c e o f pea g r a v e l would remain). Any o f the a c r y l i c cubes t h a t made i t p a s t t h e f i r s t v o r t e x d i d so as suspended load.  176  Condition 4 -Same as c o n d i t i o n 1. -**The c o n d i t i o n s a r e r e v e r s i b l e , going from 1 t o 2 t o 3 and then back t o 1 ( i e c o n d i t i o n 4) produce the same sediment c h a r a c t e r i s t i c s as those t h a t o r i g i n a l l y e x i s t e d i n c o n d i t i o n 1. Condition 5 - P l u n g i n g breaker; j e t does not p e n e t r a t e t o t h e bed; forms a l a r g e v o r t e x a f t e r i t plunges. -The p l u n g i n g wave i s s t r o n g e r and forms a v e r y s t r o n g i n i t i a l p l u n g e r v o r t e x ; t h i s v o r t e x reaches t h e bed and remains r e l a t i v e l y s t a t i o n a r y ; obvious splash-plunge c y c l e s a r e s e t up (-3) . Bedload pg,s,b,as,ac Suspended as,ac,b -pg,s: forms a r i p p l e j u s t o f f s h o r e o f t h e p l u n g i n g p o i n t ; a c t i v e movement o f t h e pg and s w i t h i n 24 cm wide zone; i s not spread out but remains i n one group. -b: t r a v e l s back and f o r t h over t h e r i p p l e and i s o c c a s i o n a l l y suspended. -as: g e t s c a r r i e d q u i t e h i g h i n t h e water column as i t i s suspended o f f t h e c r e s t . -ac: suspended as i n t h e o t h e r c o n d i t i o n s . - O v e r a l l t h e sediments a r e much more a c t i v e w i t h o n l y t h e pg and s remaining on t h e bed.  177  Condition 6 -Spilling  breakers  Bedload pg,s,b,as,ac Suspended ac -pg:  l a r g e r angular p i e c e s a t the break p o i n t .  Onshore of the v o r t e x v a i l -No p a r t i c l e s are t r a n s p o r t e d onshore of the  178  ripple.  Condition 7 -Small p l u n g i n g b r e a k e r ; j e t does not p e n e t r a t e splash-plunge c y c l e s a r e s e t up.  t o t h e bed;  Bedload pg,s,b,as,ac Suspended ac -pg: s m a l l r i p p l e j u s t onshore o f t h e plunge p o i n t ; i n t e r m i x e d ; some o f t h e l a r g e r p i e c e s o f pg put onshore o f t h e v o r t e x w a l l remained. -b: moves as bedload on onshore s i d e o f t h e r i p p l e ; does advance over the r i p p l e w i t h t h e advance o f t h e c r e s t , -as: Moves as bedload; o c c a s i o n a l l y being l i f t e d i n t o suspension. -ac: became suspended mainly a t t h e r i p p l e and t r a v e l back t o the b r e a k i n g p o i n t and s e t t l e ; t r a v e l as bedload back t o the ripple. -Same t r a n s p o r t as p r e v i o u s c o n d i t i o n s 1, 2, 3 f o r p o i n t s onshore and o f f s h o r e r e g i o n s .  179  Condition 8 -Small s c a l e p l u n g i n g ; j e t does not p e n e t r a t e t o the bed; splash-plunge c y c l e s (2-3) . - V o r t i c e s reach t h e bed onshore o f t h e f i r s t v o r t e x . Bedload pg,s,b,as,ac Suspended ac,as,b -pg,s: a g a i n t h e r i p p l e i s formed, b u t t h i s time i t i s formed o f f s h o r e o f the plunge p o i n t . -b: moves over t h e r i p p l e o c c a s i o n a l l y g o i n g i n t o suspension a t the r i p p l e . -as: moves over t h e r i p p l e o c c a s i o n a l l y going i n t o suspension at the r i p p l e . -ac: same as b e f o r e however appears t o be a l o t more suspension.  180  Condition 9 -Large p l u n g i n g wave; j e t p e n e t r a t e s t o t h e bed; l a r g e p l u n g i n g v o r t e x i s c r e a t e d which, a c t s on t h e bed; becomes deformed because o f the bed; v e r y c l e a r splash-plunge c y c l e (3-4). Bedload/Suspended A l l o f t h e p a r t i c l e s do both t o some degree; p a r t i c l e s t h a t are suspended t r a v e l f u r t h e r o f f s h o r e than t h e p o i n t o f suspension except when t h e plunger v o r t e x t r a p s them and c a r r i e s them forward; these ac a r e u s u a l l y h i g h i n t h e water column so the p l u n g i n g v o r t e x c a r r i e s them forward. -pg,s: t h e p a r t i c l e s a r e moving i n a c h a o t i c f a s h i o n w i t h the sand and g r a v e l b e i n g the l e a s t moveable. -b,as: both move as t h e pg,s but f u r t h e r on each s i d e o f the pg,s; e a s i l y suspended. -ac: p r e s e n t i n t h e whole breaker area equal amounts suspended and on t h e bed.  181  C o n d i t i o n 10 -The wave h e i g h t has ben reduced and a p l u n g i n g b r e a k e r s t i l l o c c u r s ; t h e j e t does not plunge as v i o l e n t l y and does not appear to impact on t h e bed; t h e s p l a s h - p l u n g e c y c l e s t i l l e x i s t s but are not as pronounced. -The sediment i s not moved as v i o l e n t l y as p r e v i o u s c o n d i t i o n ; the l i m i t s a r e unmoved; t h e b r e a k i n g p o i n t has moved onshore, as expected. -The g e n e r a l p a t t e r n o f sediment remains unchanged. -pg,s:  form a r i p p l e near t h e plunge p o i n t .  182  C o n d i t i o n 11 -Wave h e i g h t has been reduced and a p l u n g i n g breaker o c c u r s ; t h e j e t does not p e n e t r a t e t o t h e bed; t h e p l u n g e r v o r t e x i s formed but i s not s t r o n g and does not e f f e c t t h e bed. Bedload pg,s,b,as,ac Suspended ac.as.b -the 3  o v e r a l l impression are r e s u l t s  s i m i l a r t o c o n d i t i o n s l , 2,  -pg,s: form a r i p p l e j u s t onshore o f t h e plunge p o i n t . -as,b: move over t h e r i p p l e and go i n t o suspension sometimes, -ac: t r a v e l as suspended l o a d o f f s h o r e and a r e put i n t o suspension near t h e r i p p l e ; bedload moves onshore; onshore o f the f i r s t v o r t e x t h e bedload i s t r a n s p o r t e d o f f s h o r e and t h e suspended l o a d i s t r a n s p o r t e d onshore.  183  Appendix C  Wave Setdown/Setup Data  184  Beach S l o p e and  Manometers  WAVE SETDOWN/SETUP DATA FOR 1:15 IMPERMEABLE SLOPE  RUN  1.3 1.1 1.2 1.4 2.4 2.3 2.2 2.1 3.1 4.5 4.4  DEPTH  HEIGHT?  PERIOD  d  DEPTH H  T  SI  CD  CI  sec  ••  48.80 48.00 40.00 40.00 49.30 49.38 49.30 49.38  22.00 22.00 22.00 12.70 12.80 12.50 12.70  2.75 3.50 1.34 1.68 2.17 2.76  17.30 10.85 18.57  2.81 pi 1.55 pi 1.79 pi  10.67  2.16 pi 2.62 pi 3.44 pi  47.38 47.50 47.58 47.50  22.00  4.3 4.2  47.50  18.75  4.1  47.50  10.50  5.1  47.58 47.50 47.50  15.09 15.12 15.18  5.2 5.3 5.4  BREAKER CONSTANT BREAKING TYPE  2.88 pl/sp 2.24 pi  SETDOWN  SETDOWN Sb IB  4.5 4.5 7.7  SETUP Sv aa  A  TOTAL  DEPTH©  SETUP  BREAKING  S  db  aa  ca  4.5  52.5  57.B  56.0 132.8 144.0  68.5 141.8  6.8 1.5 1.7  4.5 9.0 9.0 5.5 6.5 4.5 10.0 10.5 3.2 3.0  1.3 1.7  1.8 3.4  3.8  4.2  1.57 pl/sp  2.8 2.5 2.5 3.5  4.3 4.2 3.0  sp/pl pi pi pi pi pi  1.41 sp  6.5 3.7 4.5 4.5 5.8  2B.B 34.0 38.8 46.0 72.8 37.0 43.8 44.8  153.0 33.5 40.5 42.5 56.8 82.5 48.2 46.8  2B.65 18.75 14.22 20.13 21.65 21.55 21.75 12.59 16.97 20.88  45.8  16.83 28.35  48.5  51.9  11.93  57.8  61.2  15.92  48.8  44.3 47.2 49.8  28.18 15.92 24.17  43.8 46.8  47.50  15.03  1.88 pl/sp 2.13 pi  2.5  50.5  53.0  28.28  47.58 47.50  15.88  2.43 pi  3.3  3.5  58.5  17.61  52.8  1.88 pi 2.18 pi  3.8 2.8  5.6 5.8 4.5 4.8  47.8  6.3 6.4  17.73 17.07 17.37  4.8 4.8  28.85 23.91  47.58 47.50 47.50  1.51 sp 1.68 sp/pl  47.8 47.8 53.8 56.8  57.5 68.8  7.1  47.50  1.43 pl/sp  3.8  5.8  45.8  58.8  7.2 7.3 B.l  1.80 pl/sp  47.50 45.88  19.96 20.18 19.65  4.5 3.4  5.3 4.3  50.0 58.5  55.3 62.8  2.8  28.18 17.52  45.00  8.3 8.4  45.80  3.8  41.5 43.5 46.5  45.3  8.2  3.8 4.8  47.5 49.5  17.58 17.68  1.8 5.8 2.5  58.0 36.5  58.8 63.8 39.8  17.72 13.34 25.78  5.5 6.1 6.2  47.50  12.58 12.58 12.47 12.37  2.89 pi 1.29 pl/sp 1.55 sp 1.88 pi  1.8 2.8  24.82 32.15 28.13 32.10 28.80  9.1  45.00 45.00 45.00  12.98 15.86  2.21 pi 2.64 pi 1.38 sp/pl  9.2  45.00  14.80  1.56 sp/pl  8.5 1.7  5.8  46.5  51.5  17.48  9.3 9.4  45.00  14.84  1.82 pi  2.6  3.2  9.5  15.38 15.24  2.28 pi 2.76 p i  2.7 2.7  2.9 7.8  46.2 68.4  25.71  45.08 45.00  43.0 65.5  18.1  45.80  17.61  1.38 sp  1.8  184.8 48.5  21.82  10.2  45.88  17.78  1.38 sp  1.8  1.5 2.5  97.0 39.8 47.8  49.5  21.72  18.3 10.4  45.00  17.78  1.84 pi  1.5  65.8  66.5  21.82  17.76 17.68  2.17 pi  1.5  53.0  54.5  18.5  45.88 45.00  8.5 1.5  2.61 pi  2.5  5.5  85.8  11.4 11.3  45.88 45.08  20.32 19.96  1.51 pi 1.77 pi  3.0 5.8  43.5  11.2  45.80  4.5  45.80  2.17 pi 2.89 pi  46.8 67.5  11.1  19.78 28.18  2.8 3.8 3.5  98.5 46.5 51.8  34.12 17.34  6.9  11.5  85.8  8.5  1.8 1.9  52.6  186  49.0  72.8 96.5  25.78 13.13  21.67 33.67 33.72 16.75  RUN  DEPTH  HEIGHTS  PERIOD  BREAKER CONSTANT BREAKING  DEPTH d  12.1 12.2 12.3 12.4 13.1 13.2 14.1 14.2 14.3 14.4 15.1 15.2 15.3 15.4  H  CI  CI  40.190 40.00 40.00 40.00 40.40 40.40 45.00  21.50 15.20 15.20 15.40 20.50 23.00  45.00 45.00 45.00 47.30 47.30  22.00  15.50 14.00 19.58 17.50  TYPE  SETUP  TOTAL  DEPTHS  SETDOWN  SETDOWN  T  SI  Sb  5w  A3  db  sec  •i  ••  ••  ••  CI  73.0 114.0 77.0 73.0 54.0  76.5 119.0 86.0 77.0 57.0  29.00  69.0 44.0  50.5  2.00 Pi 2.75 Pi 2.75 Pi 2.00 Pi 2.28 Pi 2.28 Pi 1.56 Pi 1.56 2.30 2.30 2.30  2.5 1.0 2.5 3.0 1.0  Pi pl Pi  47.30  12.60 24.00  Pl 2.30 Pl 1.43 Pl  47.30  11.50  1.43 Pl  3.5 5.0 9.0 4.0 3.0  SETUP BREAKING  1.0 3.5 3.0 0.0 3.0 1.0  3.0 6.5 4.0  1.0 5.0  11.5 11.5  40.5 45.0  2.5  11.5  40.5  3.5 2.5 2.5  44.0 47.5 55.0 60.0  72.0 48.0 51.0 57.5 62.5 52.0 56.5 52.0  14.08 13.68 20.00 19.40 21.80 31.00 19.10 19.18 23.00 24.30 20.40 32.40 18.28  "NOTE: 1) THE VALUE OF db IS FOR THE DEPTH TO THE MHL AT THE POINT OF MAXIMUM SETDOWN 2) THE CALCULATED TOTAL SETUP IS 8.15db ( FROM SPM ) 3) THE VALUE OF kd IS CALCULATED BY ITERATION USING THE EQUATION led = 4pi 2 * d/gT 2 » l/tanh(kd) A  A  4) THE VALUE OF Sbc=(g 0.5 • Ho 2 * T)/(64pi • db 1.5) A  A  A  5) THE VALUE OF S«c-Sc-Sbc  187  ( FROM SPM )  RUN  CALC.  d/gT 2  =2Pid/L  H/Ho  Ho  CALC.  CALC.  TOTAL  SETDOWN  SETUP  SETUP  Sbc  Swc  •9  19  A  •0  ci  db/S  db/Ho  Lo=gT 2/2 A  Ho/Lo  ci  1.3  31. B  0.81019  0.6801  0.98358  22.4  16.6  14.4  3.623  0.923  624.524  0.036  1.1  28.1  B.BB813  0.5985  1.01927  21.6  20.0  8.1  3.099  0.869  783.403  0.028  1.2  21.3  B.B8539  0.4784  1.09854  20.0  32.0  -10.7  1.009  0.710 1188.741  0.017  1.4 2.4  8.B8333  8.3706  1.21398  18.1  8.82799  1.2842  0.91390  13.9  8.01781 8.B1867  0.9501  0.92454  13.8  0.6984  0.97706  12.8  1.316 6.463 5.321 5.118  0.831  32.6  10.4 28.5 27.3 27.2  1.111 1912.685 1.558 288.349 1.557 440.664  2.2  19.8 4.0 5.0 5.5  8.009 8.050  2.3  38.2 32.5 32.3  1.700  735.285  0.017  2.1  18.9 25.5  0.00668  0.5335  1.85734  12.0  13.9  5.0.  2.248  1.048 1189.343  0.818  0.00611 0.02015  0.5115  1.07268  16.1  16.3  9.2  2.857  1.0288  8.91793  11.8  3.7  26.4  4.995  1.052 1232.826 1.699 375.105  0.032  3.1 4.5  38.1  8.013  4.4  24. B  0.01511  0.8580  0.93719  11.3  5.5  18.5  3.485  1.421  5B0.259  0.023  4.3  42.5  0.01038  0.6872  0.98099  10.9  2.6  39.9  6.190  2.606  728.445  0.015  4.2  17.9  0.00705  0.5535  1.04455  10.3  10.5  7.4  2.299  1.159 1071.746  0.010  4.1  23.9  0.00409  0.4131  1.16168  9.0  6.9  17.0  2.601  1.761 1847.592  0.005  5.1  42.2  0.02435  1.1676  0.91314  16.5  4.0  38.1  6.343  1.700  310.404  0.053  5.2  23.9  0.01964  1.0118  0.91908  16.5  10.4  13.5  3.373  0.968  384.847  0.043  5.3  36.3  0.01370  8.8086  0.94672  16.0  6.3  29.9  4.933  1.507  551.829  0.029  5.4  42.4  0.01067  0.6984  8.97706  15.4  5.2  37.2  5.336  1.838  788.351  0.022  5.5  38.1  0.0082B  0.6015  1.01774  14.7  9.2  20.9  3.970  1.360  921.938  0.016  6.1  35.9  0.02124  1.0646  0.91595  19.2  7.4  28.4  4.546  1.244  355.994  0.054  6.2  36. B  0.01716  0.9281  0.92703  19.1  8.1  27.9  4.619  1.256  440.664  0.043  6.3  48.2  0.01370  0.8086  0.94672  18.8  5.2  43.0  5.591  1.783  551.829  0.033  6.4  42.2  0.01019  0.6800  0.98362  17.7  7.1  35.1  4.688  1.593  741.997  0.024  7.1  48.2  0.02368  1.1453  0.91343  21.9  5.8  42.3  6.42B  1.469  319.272  0.068  7.2  42.B  0.01494  0.8522  0.93820  21.4  8.7  33.3  5.B63  1.307  505.864  0.042  7.3  42.2  0.01108  0.7137  0.97197  20.2  8.9  33.2  4.475  1.390  681.996  0.030  8.1  26.3  0.02757  1.2733  0.91369  13.8  5.2  21.1  3.868  1.272  259.818  0.053  8.2  26.3  8.01909  8.9933  8.92050  13.6  6.1  2B.2  3.684  1.289  375.105  0.036  8.3  26.4  0.01416  0.8248  0.94336  13.2  6.6  19.8  3.556  1.331  505.864  0.026  8.4  26.6  0.00939  0.6492  0.99574  12.4  7.1  19.5  3.488  1.426  762.559  0.016  8.5  2B.B  0.00658  0.5329  1.05774  12.3  12.7  7.3  2.117  1.087 1088.171  0.011  9.1  38.7  0.02409  1.1587  8.91324  16.5  4.5  34.2  6.610  1.563  0.055  297.336  9.2  26.1  0.B18B5  8.9852  B.92118  16.1  8.6  17.5  3.379  1.083  379.968  0.042  9.3  38.6  0.01385  8.8138  0.94562  15.7  5.4  33.2  5.565  1.638  517.168  0.030  9.4  38.6  0.00948  0.6525  0.99437  15.4  6.2  32.3  3.757  1.670  755.674  0.020  9.5  19.7  0.00602  0.5078  1.07539  14.2  18.1  1.5  1.263  8.927 1189.343  0.012  IB. 1  32.7  8.024B9  1.1588  0.91324  19.3  7.8  24.9  5.388  1.132  297.336  0.065  18.2  32.6  0.02409  1.1588  B.91324  19.4  8.0  24.6  4.388  1.121  297.336  0.065  18.3  32.7  B.01355  B.S033  0.94788  18.7  9.8  22.9  3.281  1.169  528.597  0.835  18.4  51.2  0.00974  8.6628  0.99020  17.9  5.5  45.7  6.261  1.902  735.205  0.024  IB.5  26. B  0.00673  0.5395  1.05340  16.7  15.7  10.3  1.916  1.038 1063.588  8.816  11.4  32.5  0.02012  1.0275  0.91801  22.1  11.4  21.1  4.660  0.979  355.994  0.062  11.3  5B.5  0.01464  0.8417  0.94010  21.2  6.4  44.1  6.602  1.586  489.143  8.043  11.2  58.6  0.00974  0.6628  0.99020  20.0  6.9  43.7  4.683  1.688  735.205  0.027  11.1  25.1  0.00549  0.4831  1.09465  18.4  22.3  2.8  1.736  0.909 1304.022  0.014  188  RUN  CALC.  d/gT 2  kd=2Pid/L  H/Ho  CALC.  CALC.  TOTAL  SETDOWN  SETUP  SETUP  Sbc  Sue  ••  ••  A  ••  12.1 12.2 12.3 12.4 13.1 13.2 14.1 14.2 14.3 14.4 15.1 15.2 15.3 15.4  43.5 21.1 28.5 38.8 29.1 32.7 46.5 28.7 28.7 34.5 36.5  Ho  CI  8.81819 8.88539 8.88539 B.B1819 0.08732 0.00792 0.01885 0.01885 0.00867 0.00867 8.00911  0.6800 0.4783 0.4783 0.6B8B 0.5901 0.5901 0.9853 0.9853 0.6206 0.6206  8.98362 1.89862 1.09862 0.98362 1.02365 1.B2365 0.92117 B.92117 1.00843 1.BB843 1.80045 1.00045 0.91349 0.91349  38.6  0.00911  0.6382 0.6382  48.6 27.3  0.02358 0.02358  1.1420 1.1428  22.4 19.6 13.8 15.5 15.0 20.0  10.0 31.1 16.2 8.3 9.4 14.0  25.0 16.8 13.9 19.3 17.5  8.8 8.2 8.3 12.1  12.6 26.3 12.6  6.2 8.3 4.5  189  9.2  db/S  db/Ho  Lo=gT"2/2  Ho/Lo  cc  33.5 -9.9 4.3 21.7 19.7 18.7 37.7 20.4 20.4 22.4  3.791 1.183 1.591 2.597 3.484 3.028 6.139 3.979 3.745 4.000  27.3 24.4  3.888 3.923  40.3 22.8  5.735 3.500  1.297 624.524 0.719 1188.741 0.989 1180.741 1.294 624.524 1.290 811.631 1.089 811.631 1.242 379.968 1.135 379.960 1.376 825.933 1.189 825.933 1.389 825.933 1.620 825.933 1.233 319.272 1.446  319.272  0.036 0.817 0.012 0.025 0.019 0.025 0.066 0.844 0.017 0.023 0.021 0.015 0.082 0.039  SETUP EXPERIMENT I  1  1) Hater depth at tonoteter 8: 2) Have data: Runt  48.8 c i  vave height leasured at lanoieter 8  Height  Period  Breaker  H (ci)  T (sec)  type  1.1  22.88  2.24  pi  1.2 1.3 1.4  22.88 22.88 22.88  2.75 2.88 3.58  sp/pl pl/sp pi  3) V e r t i c a l displacement of NHL froa SHL: Runt  Manometer 8  1.1 1.2 1.3 1.4  aeasureients i n • i l l i i e t e r s  1  2  3  4  5  6  7  8  9  18  -4.5 -7.7  -4.5  -1.5 -6.8  -2.7  28.8  38.8  37.8  47.8  8.5  27.8  -4.5  -4.8  -6.5  -7.5  -9.8  26.5 18.8  54.8 45.8  -5.8  17.5 8.5  38.5 34.8  87.5  -4.5  -8.8 -2.7  -1.5 -7.7  18.7  -9.8  31.8  51.8  87.5  1.8  -7.8 8.2  -7.2  1.8  190  11  SETUP EXPERIMENT t  2  1) Hater depth at lonoieter I : 2) Have data: Runt  2.1 2.2 2.3 2.4  49.5 c i  vave height leasured at lanoieter 0  Height  Period  Breaker  H (ci)  T (sec)  type  12.70 12.50 12.80 12.70  2.7G 2.17 1.68 1.34  Pi Pi Pi Pi  3) Vertical displacement of HHL fro* SHL: Runt 8 2.1 2.2 2.3 2.4  Hanoieter 1  -5.5 -3.7 -4.0 -3.7  2  3  4  5  6  7  8  9  10  11  12  -10.0 -4.5 -4.5  -8.5 -3.5 -4.5  -18.0 -4.0  -10.0 -4.5  -11.0 -4.5  -6.0 -5.0  -6.5 -5.5  -5.0 7.3 2.5  7.0 12.7 12.7  -4.5  4.5  12.3  18.5 23.0 23.8 18.7  34.8 34.5 31.5  -3.5  -5.0 -4.5  -1.5 -2.7 -6.2  -4.0  -1B.0 -3.5 -6.0 -5.0  4) Location of defined breaking: Runt  2.1 2.2 2.3 2.4  leasureients i n l i l l i i e t e r s  displacement left or right of lanoieter  Location (ci) 15.5--8 33.8-7 7--12.B 1.5-7  5) Location of f i r s t u j o r turbulance: Runt  Location (ci)  2.1 2.2 2.3 2.4  8-25.8 7-16.5 8-18.8 8-12.0  displacement l e f t of right of lanoieter  191  26.4  SETUP EXPERIMENT t  3  1) Hater depth at tonometer 8: 2) Have data:  47.3 c i  vave height leasured at manometer 8  Runt  Height H (ci)  Period T (sec)  3.1  17.38  2.B1  Breaker type pi  3) V e r t i c a l displacement of NHL f r o i SHL: Runt  Hanoieter 8  3.1  1  -6.8  2 -18.5  3  4  -8.8  -18.5  4) Location of defined breaking: Runt  3.1  3.1  5 -18.5  6  7  -18.5  -11.7  8  9 -9.5  14.5  IB  11  12  23.8  26.8  54.8  displacement l e f t or right of lanoieter  Location (ci) 2.8-7  5) Location of f i r s t l a j o r turbulance: Runt  leasureients i n l i l l i i e t e r s  displacement l e f t of right of lanoieter  Location (ci) 8.8-8  192  SETUP EXPERIMENT t  4  47.5 ci  1) Hater depth at lonoieter I : 2) Have data: Runt  wave height measured at manometer 0  Height  Period  Breaker  H (cm)  T (sec)  type  4.1  10.58  4.2 4.3 4.4 4.5  10.75 10.67 10.57  3.44 2.62 2.16 1.79 1.55  18.85  Pi Pi Pi Pi Pi  3) Vertical displacement of HHL from SUL: Runt  Manometer 8  4.1 4.2 4.3  measurements i n millimeters  1  2  3  4  5  6  7  8  9  10  11  12  -3.0 -1.7  -4.0 -2.5  -4.0  -4.2 -1.8  -3.6  -4.8  -1.3  -1.5  -1.8  -1.2  -1.6  -1.5  0.0  4.8 6.9  9.7 14.5  45.5  -2.6  -3.9 -3.4  28.8  -2.0  -4.2 -2.8  0.8  -2.5 -1.5  13.5  27.5 26.0  45.5 43.0  14.5 14.5  26.0 25.0  41.5  4.4  -1.7  -2.0  -2.5  -1.8  -2.1  -2.9  -3.8  -2.8  3.9  4.5  -1.5  -1.8  -2.5  -1.8  -2.1  -3.2  -3.1  -2.7  2.2  4) Location of defined breaking: Runt  displacement left or right of manometer  location (cm)  4.1  4.5-7  4.2  7-30.0  4.3  10.0-7  4.4 4.5  7-21.0 13.5-8  5) Location of f i r s t major turbulance: Runt  Location (cm)  4.1 4.2  8.5-8 8-20.0  4.3  8-1.0  4.4 4.5  8-8.5 8-37.5  displacement l e f t of right of manometer  193  SETUP EXPERIMENT I  5  I) Hater depth at lonoieter 2) Nave data: Runt  fl:  47.5  wave height leasered at lanoieter fl  Height  Period T (sec)  H (ci) 15.89  5.1 5.2 5.3 5.4  15.12 15.18 15.B3  1.41 4.57 1.88 2.13  5.5  15.88  2.43  Breaker type P pl/sp pl/sp S  Pl Pl  3) Vertical displacement of NHL f r o i SHL: leasureients i n l i l l i i e t e r s Runt  Hanoieter 8  5.1 5.2 5.3 5.4 5.5  1  -2.B -2.5 -2.5 -3.5 -3.3  2  3  -2.5 -3.1 -4.B -3.5 -4.2  -3.4 -3.B -3.1 -2.8 -3.1  4) Location of defined breaking: Runt  location (ci)  5.1 5.2 5.3  5-38.8 2.8-6 29.5-5  5.4 5.5  5-25.5 6-2.8  4 -4.3 -3.7 -3.8 -2.5 -3.5  Location (ci)  5.1 5.2 5.3 5.4  7--7.B 7-25.8 23.B-7 7-18.8  5.5  7--21.5  -3.6 -4.8 -3.B -2.4 -3.5  6  7  8  9  -4.1 -4.1  -3.2 -4.2  2.8 3.4  -2.7 -2.8 -3.5  -8.8 -2.8 -3.2  7.8 6.5 5.5  11.8 13.5 15.5 13.8 13.5  IB  11  12  18.5  26.3 38.5  4B.B 42.8  32.9 33.5 3B.3  44.8 46.5 44.5  21.6 23.5 22.5 21.5  displacement l e f t or right of lanoieter  5) Location of f i r s t l a j o r turbulance: Runt  5  displacement l e f t of right of lanoieter  194  SETUP EXPERIMENT I  6  1) Hater depth at monometer 0: 2) Have data: Runt  wave height measured at manometer 0  Height H (cm)  6.1 6.2 6.3 6.4  47.5 c i  Period  Breaker type  T (sec)  17.61 17.73 17.07 17.37  1.51 1.68 1.88 2.18  SP sp/pl Pi Pi  3) Vertical displacement of HHl from SHL: Runt  measurements i n millimeters  Nanometer 0  1  2  3  4  5  6  7  8  9  10  11  12  6.1  -4.0  -4.5  -5.0  -5.B  -5.6  -5.5  -4.0  7.5  17.0  26.0  33.2  44.5  6.2 6.3 6.4  -4.0 -3.0 -2.8  -4.5 -3.0 -3.5  -5.0 -4.5 -3.0  -4.5 -3.8 -4.0  -4.5 -3.8 -3.0  -3.5 -2.5 -2.0  0.0 1.5 2.5  7.6 11.8 11.3  7.6 11.8 11.3  24.8 26.5 25.0  33.5 35.0 34.5  45.0 48.0 49.0  4) Location of defined breaking: Runt  Location (cm)  6.1 6.2  2.0-5 31.5-5  6.3 6.4  5-3.0 8.0-5  displacement l e f t or right of manometer  5) Location of f i r s t major turbulance: Runt  Location (cm)  6.1 6.2 6.3 6.4  3.0-7 17.5-7 31.0-7 6-11.0  displacement left of right of manometer  195  SETUP EXPERIMENT I  7  1) Mater depth at monometer B: 2) Have data:  47.5 c i  wave height measured at lanoieter B  Runl  Height H (ci)  7.1 7.2 7.3  19.96 28.10  Period T (sec)  8reaker type  1.43 1.8B 2.B9  19.65  pl/sp pl/pl Pl  3) V e r t i c a l displacement of NHL from SHL: Runl  measurements i n millimeters  Manometer B  l  2  3  4  5  6  7  8  9  IB  11  12  7.1  -3.B  -4.6  -5.B  -4.5  -4.5  -4.3  3.B  15.8  2B.B  28.B  33.B  43.8  7.2  -4.5  -6.8  -5,B  -5.3  -4.3  -4.3  1.8  12.8  2B.B  28.8  36.5  47.8  7.3  -3.4  -4.8  -4.8  -4.3  -3.8  -3.B  5.B  11.5  21.5  31.8  39.8  51.5  4) location of defined breaking: Runl  location (cm)  7.1 7.2 7.3  4-19.8 4--6.B 4-13.8  displacement l e f t or right of manometer  5) Location of f i r s t major turbulance: Runl  displacement l e f t of right of manometer  Location (cm)  7.1 7.2 7.3  6-14.8 6--1B.B 34.8-7  196  SETUP EXPERIMENT I  8  1) Hater depth at sonometer B: 2) Have data: Runt  45.B c i  vave height measured at manometer B  Height H (cm)  Period T (sec)  Breaker type  8.1  12.58  1.29  pl/sp  8.2 8.3 8.4  12.5B 12.47 12.37  1.55 1.8B 2.21  sp pi pi  8.5  12.98  2.64  pi  3) Vertical displacement of HHL from SHL: Runt 8.1 8.2 8.3 8.4 8.5  B Nanometer 1 -2.8 -1.8 -2.8 -1.8 -1.9  2  3  4  5  6  7  8  9  18  -2.B -2.B -2.8 -1.5 -3.5  -2.8  -2.6 -3.8 -2.5 -1.5 -3.3  -2.7 -2.8 -2.8 -1.5  -3.8 -4.8 -3.8 -1.8 -5.8  -2.2  4.8 4.8 5.B  31.5  -8.5 -5.B  9.2 2.7  13.8 14.1 14.3 16.8 13.3  17.5  -2.6 -2.1  22.8 24.8 23.5 26.8  34.5 37.5 39.5 43.5  -2.8 -2.8 -1.8 -3.8  4) Location of defined breaking: Runt  Location (cm)  8.1 8.2 8.3 8.4 8.5  2B.B--6 9.8-6 24.8-6 I3--6 6-11.8  displacement left or right of manometer  5) Location of f i r s t major turbulance: Runt  Location (cm)  8.1 8.2 8.3  7-7.5 7-11.8 7-12.8  8.4 8.5  7-15.8 7-31.8  measurements i n millimeters  displacement l e f t of right of manometer  197  11  12  75.5  SETUP EXPERIMENT I  9  1) Hater depth at lonoieter 8: 2) Have data:  45.B c i  wave height leasured at lanoieter B  Runl  Height H (ci)  Period T (sec)  Breaker type  9.1 9.2 9.3 9.4 9.5  15.86  1.38  sp/pl  14.80 14.84 15.38 15.24  1.56 1.82 2.2B 2.76  sp/pl pi pi pi  3) Vertical displacement of HHL f r o i SHI: leasureients i n l i l l i i e t e r s Runl  Hanoieter 1  B 9.1 9.2 9.3 9.4 9.5  -8.5 -1.7 -2.6 -2.7 -2.7  2  3  4  5  6  7  8  9  18  11  -l.B  -2.8  -2.5  -2.5 -3.2 -1.7 -3.5  -2.5 -2.7 -3.2 -2.9 -4.5  -2.8 -3.5  -2.8 -5.8 -1.7 2.7 -6.5  2.3 -2.1  18.5 8.8  17.8 19.8  23.8 28.8  32.5 39.5  2.8 3.8 -7.B  3.1 14.5 3.5  18.5 19.5 18.B  27.5 28.5 29.8  38. B 48.5 47.5  -3.2 -2.5 -5.B  4) Location of defined breaking: Runt  Location (ci)  9.1 9.2 9.3 9.4 9.5  36.8-5 5-14.8 28.8-5 6.8-5 31.8-7  displaceient left or right of lanoieter  5) Location of f i r s t lajor turbulance: Runl  -2.7 -1.7 -5.8  displaceient l e f t of right of lanoieter  Location (ci)  9.1 9.2 9.3 9.4 9.5  28.8-7 7-18.8 33.8-7 6-13.8 7-33.8  198  12  8.8 77.8  SETUP EXPERIMENT t  18  1) Hater depth at tonometer 0: 2) Have data: Runt  45.8 c i  wave height measured at lanoaeter 0  Height  Period  Breaker  H (ci)  T (sec)  type  10.1 10.2 18.3 18.4  17.61 17.70 17.70 17.76  1.38 1.61 1.84 2.17  sp sp pl pl  10.5  17.68  2.61  pl  3) Vertical displacement of NHL f r o i SHL: Runl  8 Nanoieter 1  teasureients i n millimeters  2  3  4  5  6  7  8  9  18  11  21.8 26.8 25.5 24.5 23.5  26.8 32.5  35.8 48.5  33.8 33.5 37.8  42.5 45.B 49.5  18.1 18.2  -1.8 -1.8  -2.8 -2.0  -1.2 -2.2  -1.5 -2.5  -1.5 -2.5  B.B -1.8  6.8 8.8  13.5 17.5  18.3 18.4 18.5  -4.5 -1.5 -2.5  -2.8 -0.5 -3.5  -1.5 0.0 -3.5  -1.5 0.0 -4.5  -1.5 0.0 -5.5  3.8 1.5 -5.5  12.8 9.5 8.8  18.5 19.B 11.5  4) Location of defined breaking: Runl  Location (ci)  18.1 18.2 18.3  29.B--4 32.B--4 - 4 -  18.4 18.5  4--17.B 5-32.0  displacement l e f t or right of lanoieter  5) Location of f i r s t lajor turbulance: Runl  Location (ci)  18.1 18.2 18.3 1B.4 18.5  5--12.B 5.8-6 21.8-6 18.B--6 22.8-7  displacement l e f t of right of lanoieter  199  12  6.5 72.8  SETUP EXPERIMENT I  11  1) Water depth at monometer 0; 2) Have data:  wave height measured at manometer 0  Height  Runt  Breaker  Period T (sec)  H (cm) 20.18 19.78 19.96 20.32  11.1 11.2 11.3 11.4  45  type  2.89 2.17 1.77 1.51  Pl Pl Pl Pl  3) V e r t i c a l displacement of MHL from SHI: measurements i n millimeters Runt 0 11.1 11.2  -6.9 -3.5  11.3 11.4  -3.0 -2.0  Manometer 1 -7.0 -3.5 -4.0 -2.2  2  3  4  5  6  7  B  9  18  11  12  -8.8 -4.5  -6.5 -4.0  -8.8 -2.5  -8.8 -2.5  -9.5 18.8  7.5 19.5  25.8 25.8  39.8 34.5  53.5 44.5  74.5 67.8  -5.8 -2.5  -4.8 -3.0  -3.5 -2.5  -2.5 -3.8  -11.5 5.2 8.8 0.0  12.5 11.8  28.8 21.5  26.5 27.5  34.8 34.0  42.8 41.5  4) Location of defined breaking: Runt  Location (cm)  1.1 11.2 11.3 11.4  14.8--6 6.8-4 4-9.8 4--16.B  displacement left or right of manometer  5) Location of f i r s t major turbulance: Runt  Location (cm)  11.1 11.2 11.3 11.4  7-23.8 5-22.8 5-6.8 3.B--6  displacement l e f t of right of manometer  200  SETUP EXPERIMENT t  12  1) Hater depth at lonoieter 8: 2) Have data:  48  wave height leasured at lanoieter 8  Runt  Height H (ci)  Period T (sec)  Breaker type  12.1 12.2  22.88  2.88  Pi  21.58  12.3 12.4  15.28 15.28  2.75 2.75  Pi Pi pl  2.88  3) Vertical displaceient of NHL f r o i SHL: Runt  12.1 12.2 12.3 12.4  Hanoieter 8  1  2  3  4  5  6  7  8  9  IB  11  -2.5 B.B  -3.2 -1.8 -5.5  -2.8 -1.8  B.B -3.8 -6.0  11.8 -5.8 -8.8  21.5 12.B 7.8  48. B 43.8  79.8 51.8  B.B 78.8  -4.5  -2.B -3.8 -5.8  3B.B 31.B  -2.5  -3.4 8.8 -5.B  18.B  -3.8  -4.8  -3.5  -4.8  -2.8  B.B  18.5  21.5  26.5 32.8  48.5 19.5  66.8  -4.8  4) Location of defined breaking: Runt  Location (ci)  12.1 12.2 12.3 12.4  3--1B.3 5-38.2 6-18.3 4-61.8  displaceient l e f t or right of lanoieter  5) Location of f i r s t lajor turbulance: Runt  leasureients i n l i l l i i e t e r s  displaceient l e f t of right of lanoieter  Location (ci)  201  B.B  SETUP EXPERIMENT I  13  1) Mater d e p t h a t m o n o a e t e r 0: 2) Wave d a t a :  40.4 ca  wave h e i g h t m e a s u r e d a t a a n o s e t e r 0  Runl  Height H (ci)  Period T (sec)  13.1 13.2  15.40 20.50  2.28 2.28  Breaker type pl pl  3) V e r t i c a l d i s p l a c e m e n t o f MWL from SHL: Runl 0 13.1 13.2  8.8 8.0  Manoseter 1 -3.8 -1.5  2  3  4  5  6  7  8  9  IB  11  -1.8 -3.0  -1.0 0.0  -2,0 0.0  0.0 1.5  3.2 9.5  12.5 21.0  19.5 26.5  35.8 26.0  39.0 14.5  0.8  4) L o c a t i o n o f d e f i n e d b r e a k i n g : Runt  Location (cs)  13.1 13.2  4-18.3 3-45.5  measurements i n a i l l i a e t e r s  d i s p l a c e a e n t l e f t or r i g h t of e a n o i e t e r  5) B r e a k i n g and b a c k r u s h d a t a : Runl  Breaker height H (cn)  Breaker depth db ( c a )  Backrush velocity ub ( i / s )  Backrush depth uBR ( c a )  13.1 13.2  23.0 23.0  20.0 22.0  0.33 0.37  17.0 18.3  202  ' 12  SETUP EXPERIMENT I  14  1) H a t e r d e p t h a t a o n o i e t e r 0: 2) Have d a t a :  45.0 ce  wave h e i g h t measured a t nanometer 0  Runl  Height H (ca)  Period T (sec)  14.1 14.2 14.3 14.4  23.00 15.50 14.00 19.50  1.5b 1.56 2.30 2.30  Breaker type pi pi pi pi  3) V e r t i c a l d i s p l a c e i e n t o f NHL f r o a SWL: Runl 0 14.1 14.2 14.3 14.4  -3.8 -3.0 0.0 -2.0  Manoteter 1 -6.5 -4.8 -2.2 -4.8  aeasureaents i n a i l l i a e t e r s  2  3  4  5  6  7  8  9  10  11  -5.0 -4.5 0.0 -4.5  -6.2 -4.0 8.0 0.0  -5.0 -4.0 0.0 -2.0  -3.8 -4.0 -2.5 -0.5  4.5 -3.6 -3.0 -1.0  13.0 0.0 4.0 10.5  20.5 9.5 9.5 19.5  28.5 18.5 19.0 23.5  31.5 26.5 23.5 30.5  32.7 32.6 36.8 39.0  4) L o c a t i o n o f d e f i n e d b r e a k i n g : d i s p l a c e m e n t l e f t o r r i g h t o f a a n o a e t e r Runl  Location (ca)  14.1 14.2 14.3 14.4  2-42.0 5-36.8 5-36.0 4-42.0  5) B r e a k i n g and b a c k r u s h d a t a : Runl  Breaker height H (ca)  Breaker depth db ( c a )  Backrush velocity ub ( a / s )  Backrush depth uBR ( c a )  14.1 14.2 14.3 14.4  23.0 17.5 18.0 22.5  30.5 19.0 20.0 22.5  8.30 0.33 8.31 0.34  26.5 16.5 16.7 18.5  203  12  SETUP EXPERIMENT I  15  1) H a t e r d e p t h a t a o n o a e t e r 0: 2) Have d a t a :  47.3 cn  vave h e i g h t measured a t t a n o a e t e r 0  Runl  Height H (»)  Period T (sec)  15.1 15.2 15.3 15.4  17.50 12.60 24.00 11.50  2.30 2.30 1.43 1.43  Breaker type Pl Pl Pl Pl  3) V e r t i c a l d i s p l a c e m e n t o f MWL f r o s SWL: Runl 0 15.1 15.2 15.3 15.4  0.0 0.0 -5.0 -2.5  Hanoneter 1 1.0 -1.0 -7.5 -2.5  2  3  4  5  6  7  8  9  10  11  12  1.0 0.0 -7.5 -2.5  0.0 -0.5 -6.5 -2.5  0.0 -2.5 -6.8 -4.0  0.0 0.0 -4.0 -3.5  2.5 -3.0 0.0 -5.0  3.0 0.0 12.0 -4.0  12.8 3.5 18.0 -3.0  20.0 12.5 24.5 9.5  25.5 18.5 38.5 19.5  31.0 23.0 31.0 21.5  49.0  4) L o c a t i o n o f d e f i n e d b r e a k i n g : Runl  Location (ci)  14.1 14.2 14.3 14.4  - 5 - 6 - 3 30.0-7  measurements i n n i l l i s e t e r s  d i s p l a c e n e n t l e f t or r i g h t o f aanoaeter  5) B r e a k i n g and b a c k r u s h d a t a : Runl  Breaker height H (ca)  Breaker depth db ( c a )  Backrush velocity ub ( a / s )  Backrush depth uBR ( c a )  15.1 15.2 15.3 15.4  22.0 18.5 24.0 14.5  24.0 20.5 33.5 17.5  0.34 0.32 0.34 0.27  19.3 16.6 27.0 13.1  204  45.0  There are no pages 205 and 206.  Appendix D  Wave R e f l e c t i o n Data  207  REFLECTION TESTS FOR A PLANE, IMPERMEABLE, 1:15 SLOPED BEACH  RUN  DEPTH (c«)  PERIOD (sec)  Haax (ca)  Hiin (ca)  Hi (ca)  Lo (ca)  SURF REFLECTION REFLECTED SIMILARITY COEFFICIENT WAVEHEIGHT PARAMETER * tf Hr » * (ca)  1  2.60  47.5 47.5  2.00 1.74  47.5 47.5  1.52  19.83  5 6 7 8 9 IB  . 42.5 42.5 42.5 42.5 40.0  1.39 2.44  9.18 18.54  1.80 1.40 1.33  18.68 19.31 11.94  2.72  28.72  <•  3 4  t  47.5  IB.08 14.10 19.98  1  Surf S i a i l a r i t y Paraaeter  8.74  9.41  1055.44  0.71  0.054  0.51  13.17 18.88 17.00  13.64 19.35 18.02  624.52 472.78 368.72  0.45 0.33  0.021 0.010 0.010  0.29 0.19 0.18  8.28  8.65  17.43 16.58 17.44 11.00 15.84  17.99 17.59 18.38 11.47 18.28  301.66 929.54  0.13 0.41  8.36 0.27 0.33  0.815 0.023 0.018 0.010 0.010  0.53  0.023  505.86 306.02 276.18 1155.12  0.30 0.39 0.48  - 1.0/((cot(beach slope))(Hi/Lo))  *t Reflection coefficient obtained froa figure 2-65, p.2-118, SPM «»The height of the reflected wave i s given by (REFLECTION COEFFICIENT)*Hi  208  8.32 0.18 0.11 0.42  Appendix E  Laboratory Study o f Breakers (Iversen  209  1952)  IVERSEN,H.H. (1952). LABORATORY STUDY OF BREAKERS. NATIONAL BUREAU OF STANDARDS, CIRCULAR 521, pp.9-32.  SUMMARY OF DATA - BEACH SLOPE : = r r r = r r ::  DEEP RUN HATER PERIOD HAVE HAVE T HEIGHT* STEEPHi NESS! Ho/Lo  25 5 4 17  8.8797 8.8774 8.8774  1:11  ::::::: ::::::::: ::::::::: :::::::::  seconds 1.88  feet 8.488 8.391 8.391 8.200  STILL HATER DEPTH* (at Hi) di  ::::::::::  CREST  feet 2.33 2.38 2.38 2.23  feet 8.35 8.48  8.26 8.22  8.33 8.22  feet 8.45 8.41 8.41  feet 8.72 8.75  8.8614  18 8.8581  8.92  8.258  16  1.11 1.51  0.168 8.228  2.23 2.23 2.23  8.38  1.27 1.73 1.26  0.129 8.231 0.114  2.17 2.25 2.17  8.37 8.22 8.36 8.19  0.39 8.56  8.18 8.32 8.16  8.38 8.68 8.32  1.45 1.26 1.98 1.98 2.18 2.58  8.123 8.885 8.148 8.131 8.113 0.111  2.17 2.15 2.24 2.23 2.22 2.23  8.28 0.16 8.31 8.29 8.23 8.24  8.18 8.14 8.38 8.25 8.28 8.24  8.32 8.27 8.51  22 18 8.8167 145 8.8165 7 8.8158 8 8.8125 27 8.8112 2 8.8876 23 8.8871 28 8.8854 24 8.0838  =  STAGNA- BACKHASH  BREAKER DEPTH HEI6HT DEPTH TION HEIGHT AT AT AT LOCATION Hb BREAKING BREAKIN6 BREAKING Is db Yb dBH  1.88 1.88 8.80  8.8288 8.8286  BACKHASH  8.48 8.21  8.75 8.44  8.21  8.52  0.46 8.47 0.38  feet 8.27  feet 8.53  -  210  WAVE-  VELOCITY Hb/db AT  ft/sec  ft/sec 3.55  8.976 8.976 1.000  2.74  8.18 8.18  -  8.778  8.18  8.46  8.148  -  -  1.233  -  -  8.12  8.33  8.14  0.88  8.12  -  8.26  LENGTH AT  BREAKING BREAKIN6 Vc lb=Vc»T  VBH  -  t Lo f r o i 5.12T 2; Ho f r o i Hi and di/Lo using s t a l l amplitude theory. + Constant depth portion of the channel A  CREST  r : : : : : :  8.150  -  0.133  -  8.788 1.000 1.222 1.125 1.188 1.111 1.143 1.833 1.168 8.821 1.888  -  3.18 3.48 4.65  -• 3.55  -  3.45  3.558 NA NA 2.192 2.852 3.774 7.822 NA NA NA NA NA NA 7.829 NA 8.625  SUMMARY OF DATA - BEACH SLOPE 1:21 :zzz  IUN  -.zzzzzzz:  zzzzzzzzz  DEEP WATER NAVE  STILL PERIOD HAVE HATER T HEIGHT* DEPTH* STEEPHi (at Hi) NESSi di Ho/Lo seconds  36  feet  feet  8.0767 46 0.0730 45 8.8488 37 0.8488 31 8.8368 40 8.8368  8.74 1.84 1.15 0.93 1.40  8.214  1.03  8.185  1.55 1.75 1.68 1.58 l.BB 1.58  8.8358  1.26  8.268  1.57  8.8298  1.33 1.50 1.12 1.41 1.17 1.59  8.238 8.298  1.68  8.165 8.282 8.145 8.256  1.89 1.67 1.34 1.55 1.93 2.24  8.225 8.178 8.118 8.894 8.144 8.193  42 44 34 39 43 38 3  8.8288 0.8278 B.B22B 8.8228 B.B218 8.8138 0.8138  29 47 35 8.8138 41 8.8883 48 8.8879 28 B.BB76  : : : : ===:::::: : : : : : : : :  8.383 8.385 8.286 8.33B  :::::::::  : : r : : : : : :  : = = = = = r = r ::  :::::::::  : = = = z z : tzzzzzzz  : : : : : : : :  CREST HEIGHT AT  BACKHASH STAGNA- BACKHASH CREST WAVEBREAKER DEPTH DEPTH TION VELOCITY LENGTH AT HEIGHT AT LOCATION Hb/db AT AT Hb BREAKING BREAKIN6 BREAKIN6 Xs BREAKIN6 BREAKIN6 db Yb dBH VBK Vc Lb=Vc»T  feet 8.198 8.358 8.318 8.218 8.42B 8.188 8.338  feet  feet  feet  feet  ft/sec  ft/sec  0.29 0.54 0.39 8.27 0.53 8.25  0.42 0.79 0.61 0.44 0.86 0.38  B.2B 8.35 B.23 8.19 8.33  0.33  -  -  B.648 8.795 8.778 8.792  0.18  0.32  0.60  8.728  3.28  0.34  0.59 0.56  8.24  8.51  8.758  B.971  3.13  3.944  0.24  0.48  0.65  8.882 0.87B 8.826  2.68  3.458 NA 2.888  B.818 8.952 B.833  3.28  8.34  1.68 1.58 1.56 1.58 1.6B  8.388 B.48B 8.198 B.27B 8.288 B.400  1.57 1.51 1.5B 1.47 1.49 1.57  8.388 0.270 8.148 0.150 8.258 0.360  8.44 0.29 8.16 0.18 8.25 0.39  0.46 8.23 0.33 8.21 0.48  0.76 B.38 0.53 B.36 8.77 8.72 8.49 8.27 8.29 8.45 8.65  -  0.640  -  -  -  0.28 0.47  0.648  -  -  0.38  8.74  0.22 0.12 0.30 0.28 0.18  0.12 8.16 0.22  • Lo f r o i 5.12T 2; Ho f r o i H i and di/Lo using s i a l l amplitude theory. * Constant depth portion of the channel  211  0.33  0.16  : : : : : : : : : : : : : : : : : :: : : : : : : : : : : : : : : : : : :: : : : : : : : : :  A  -  0.60 0.79  8.58  -  -  -  0.25 0.41  8.64 8.68  -  -  8.655  8.864 8.931 8.875 8.833 1.888 8.923  3.48 4.35  -  3.17  -  2.58  3.88  3.B3 2.9B  IZZZZZZZZZ  2.516 4.524 NA 2.948 NA 3.296  4.512 NA NA NA 5.018 NA 4.696 5.597 NA :::::::::  SUMMARY OF DATA - BEACH SLOPE 1:33 tzz:: z z z z z z z :: z z z z z z z :: : s z z s z z :: z s z : z : x z z : z z z s z s z z ii z z z s z z z z z z x z z z x z s r s :s z z z z z x z z : z z x x x z z  tUN  DEEP HATER HAVE  PERIOD T  STEEPNESS!  STILL HAVE HATER HEIGHT* DEPTH* Hi (at Hi)  BREAKER HEI6HT  DEPTH AT  Hb  BREAKIN6 db  feet  feet  di  Hb/db  Hb/Ho  db/Ho  Ho/Lo seconds 8 9 4 IB 3  B.B665 B.B3S3 8.8214 8.8138  1.85 1.24 1.46 1.49 1.87 1.68  feet 8.356 8.255 8.214 8.144 8.169  feet  8.358  2.83  8.112 8.173  8.253  0.225  8.8888  2.37  8.238  1.49 1.64  0.959 0.753 0.814 0.833 0.702 0.673 1.124  8.415  8.518  8.814  1.882  1.268 2.215  12 8.8874 14 B.8BS2  1.79 2.18  8.115 8.115  1.48 1.44  B.18B 8.215  0.260 8.275  0.692 0.782  1.481 1.829  2.140 2.340  6 8.BB43  1.846 2.838 1.756  2.355  8.BB99 11 B.BB93 5 8.8884 7  1.65 1.58 1.55 1.44 1.58 1.48  8.275 8.285 8.225 8.262 8.175  8.365 8.365 8.350 8.278 0.373 8.268  2.67  8.164  1.52  8.298  0.370  0.784  B.N42 8.8835 8.8827  2.29 2.52 2.52  8.116 8.117 8.893  1.43 1.43 1.42  8.238 8.288 8.198  B.28B 0.265 8.230  0.821 0.755 0.826  1 8.8825  2.65  8.897  1.41  8.188  0.244  0.738  2 16 IS  0.932 0.989 1.219 1.433 1.477 1.434 1.426  2.162  2.481 2.326 2.618  2.BB1  2.712  * Lo f r o i 5.12T"2; Ho f r o i H i and di/Lo using s i a l l a i p l i t u d e theory. • Constant depth portion of the channel *  212  8.971 1.312 1.497 1.728 2.182 2.131  SUMMARY OF DATA - BEACH SLOPE 1:50 zssz s s s s s s s :  RUN  DEEP HATER HAVE  PERIOD T  STEEPNESS* Ho/Lo  62 8.8987 58 8.0718 61 8.8786 74 8.8584 59 8.8474 78 0.8465  ::::::::::  ========= ========= =========•========= s r s r r r r z s  HAVE HEIGHT* Hi  seconds 8.81 1.88 8.98 8.95 1.8B 1.13  feet 8.381 8.348 8.279 8.222 8.238  STILL HATER DEPTH* (at Hi) di  feet 1.54 1.54 1.54 1.54 1.54 1.54  : : : r : r : : r s r r s s = r i  CREST BACKHASH STAGNA- BACKHASH BREAKER " DEPTH HEI6HT DEPTH TION HEI6HT AT AT AT LOCATION Hb BREAKING BREAKIN6 BREAKING Xs db Yb dBH VBH  feet 8.258 8.383 8.222 8.191 B.218  feet  feet  8.483 8.326 8.228  8.350  feet  8.555 8.633 8.497 8.394 8.455  feet  8.292 8.322 8.261 8.183 8.236 8.292 8.274  63 8.8376 68 8.0376  1.17  0.282 8.243  1.54  8.297 B.274  8.321  8.585 8.554  1.88  0.182  1.54  8.185  -  -  -  73 8.8385 77 0.0223 68 0.8198  1.38 1.35 1.62  8.241 0.198 8.228  1.54 1.54 1.54  8.248 8.199 8.268  8.538 8.486 8.522  8.262 8.191 8.251  66 71 78 83 81 88 82  0.0130 0.0092  1.74 2.88  1.54 1.54  8.283 8.288  8.565 8.486  8.262 8.175  0.0074 0.8874 0.0065 0.0065 0.0849  2.43  8.198 8.188 8.232  8.328 8.231 8.386 8.334 B.222  1.54  8.353  -  -  -  1.98 2.65 2.25 2.65  8.129 8.243 8.168 8.186  1.54 1.54 1.54 1.54  8.181 8.398 8.217 8.328  8.212  8.362 0.848  8.173 8.436 8.276 8.368  8.513  8.422  8.517 8.691  * Lo f r o i 5.12T 2| Ho f r o i Hi and di/Lo using s t a l l amplitude theory. * Constant depth portion of the channel A  213  CREST VELOCITY Hb/db AT  8.398  -  ft/sec  ERR - 8.752 8.18  8.16  0.490  8.11  8.468  • 8.688  8.578  8.681  - 8.838 ERR  8.578  -  WAVELENGTH AT  BREAKIN6 BREAKIN6 Vc Lb=Vc»T  ft/sec  -  :::::::::  3.88 3.98 2.75  -  8.849 8.854  2.88 3.58  ERR - 8.756 - B.861 - 8.876  3a 58 3.88 3.58  B.847 8.937  2.85 3.75  8.19  ERR - 8.854 8.11  8.776 ERR 8.070 8.758  -  2.38 3.58  -  NA 3.888 3.518 2.613 NA 3.164 4.895 NA 4.558 4.858 5.678 4.959 7.588 NA NA 6.895 7.875 NA  

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