UBC Theses and Dissertations

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UBC Theses and Dissertations

A study of sedimentary bedwaves Lee Diaz, Song 1979

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A STUDY OF SEDIMENTARY BEDWAVES by SONG LEE DIAZ B . S c , Taiwan, 1969 M.E., Iowa S t a t e U n i v e r s i t y , 1974 THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (THE DEPARTMENT OF C I V I L ENGINEERING) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r , 1979 © Song Lee D i a z , 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ClViL. £A/6r/N^£R/A/& The University of Brit ish Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date /£, &CT j / 9 7 ? ABSTRACT Se d i m e n t b e d l o a d t r a n s p o r t and b e d f o r m f o r m a t i o n a r e c o n s i d e r e d f o r b o t h s t e a d y and o s c i l l a t o r y f l o w o f w a t e r . Some new t h e o r e t i c a l r e s u l t s a r e q u o t e d and e x p e r i m e n t a l r e s u l t s a r e r e p o r t e d w h i c h t e n d t o s u b s t a n t i a t e t h e t h e o r y . F o r s t e a d y f l o w , c r i t e r i a b a s e d on F r o u d e number and mean v e l o c i t y a r e d e v e l o p e d f o r p r e d i c t i n g t h e t y p e s and d imen-s i o n s o f b e d f o r m s w h i c h o c c u r u n d e r v a r i o u s f l o w c o n d i t i o n s and d i f f e r e n t s e d i m e n t p r o p e r t i e s . The f l o w d e p t h i s f o u n d t o be an i m p o r t a n t v a r i a b l e l i n k i n g F r o u d e number w i t h t h e mean v e l o c i t y . F o r o s c i l l a t o r y f l o w , g e n e r a l agreement i s f o u n d w i t h t h e work o f M o g r i d g e and Kamphuis and t h e e a r l i e r work o f B a g n o l d . The f r e e s u r f a c e wave p a r a m e t e r s and t h e w a t e r f l o w d e p t h a r e c o n f i r m e d t o be d o m i n a n t f a c t o r s i n d e t e r m i n i n g r i p p l e c h a r a c t e r i s t i c s . The e x p e r i m e n t a l r e s u l t s g i v e some c o n f i r m a t i o n o f a new k i n e m a t i c model p r o p o s e d t o e x p l a i n s e d i m e n t toipple b e h a v i o r . i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF FIGURES v i NOTATIONS -X ACKNOWLEDGEMENTS x i i i PART 1: RIPPLES AND DUNES GENERATED BY STEADY FLOW .. 1 CHAPTER 1.1 INTRODUCTION 1 CHAPTER 1.2 LITERATURE REVIEW 8 CHAPTER 1.3 MECHANISM OF SEDIMENT TRANSPORT . .. BY RIPPLES AND DUNES 18 KINEMATIC REQUIREMENTS 19 BRIEF DESCRIPTION OF EXNER' S MODEL • 19 PURE KINEMATIC MODEL 20 DYNAMIC REQUIREMENTS 22 CHAPTER 1.4- EXPERIMENTAL TECHNIQUE AND ANALYSIS EXPERIMENTAL RESULTS 26 EXPERIMENTAL TECHNIQUE 26 DIMENSIONAL ANALYSIS 28 DISCUSSION OF EXPERIMENTAL RESULTS 31 RELATIONSHIP BETWEEN N^ AND R_,_ 33 THE RELATIONSHIPS BETWEEN Y/L AND N,, OR R . . 35 K I I D THE RELATIONSHIPS BETWEEN BEDFORM WAVELENGTHS .. AND V OR °B 37 THE RELATIONSHIPS BETWEEN Y/H DAND N,. OR R_ ... 41 i v TABLE OF CONTENTS CONT... PAGE THE RELATIONSHIPS BETWEEN BEDFORM STEEPNESS AND N f OR R__ 43 r CB THE RELATIONSHIP BETWEEN SEDIMENT AND WATER FLOW TRANSPORT RATE 43 CHAPTER 1.5 CONCLUSIONS 47 CHAPTER 1.6 BIBLIOGRAPHY 53 APPENDIX I EXPERIMENTAL RESULTS AND FIGURES 55 (1) EXPERIMENTAL RESULTS OF PORTAGE S I L I C A SEDIMENT BEDFORMS 55 (2) EXPERIMENTAL RESULTS OF FRASER RIVER SAND BEDFORMS 55 (3) EXPERIMENTAL RESULTS OF BAKELITE SEDIMENT BEDFORMS . 56 (4) FIGURES 57 PART I I : RIPPLES GENERATED BY OSCILLATORY FLOW 76 CHAPTER 2.1 INTRODUCTION 76 CHAPTER 2.2 LITERATURE REVIEW 79 CHAPTER 2.3 EXPERIMENTAL TECHNIQUE 84 CHAPTER 2.4 ANALYSIS OF EXPERIMENTAL RESULTS 87 DIMENSIONAL ANALYSIS 87 DISCUSSION OF EXPERIMENTAL RESULTS 90 RELATIONSHIPS BETWEEN BEDFORM LENGTHS AND MAXIMUM HORIZONTAL WATER PARTICLE MOTION VELOCITIES OR DISPLACEMENTS FROM ITS MEAN POSITION IN THE VIC I N I T Y OF THE SEDIMENT CHANNEL BED 90 V TABLE OF CONTENTS CONT... PAGE RELATIONSHIPS BETWEEN BEDFORM LENGTHS AND MODIFIED FROUDE NUMBERS OR MODIFIED REYNOLD'S NUMBERS 93 RELATIONSHIPS BETWEEN BEDFORM HEIGHTS AND MODIFIED FROUDE NUMBERS, MAXIMUM HORIZONTAL WATER PARTICLE VELOCITY U AND AMPLITUDE 95 m RELATIONSHIPS BETWEEN BEDFORM STEEPNESS S„ AND MAXIMUM HORIZONTAL WATER PARTICLE DISPLACEMENT FROM ITS MEAN POSITION IN THE VICI N I T Y OF THE SEDIMENT CHANNEL BED 98 EXPERIMENTAL RESULTS AT SLOPING CHANNEL BED .... 99 SUMMARY 100 CHAPTER 2.5 COMPARISON STUDY 102 CHAPTER 2.6 CONCLUSIONS 110 CHAPTER 2.7 BIBLIOGRAPHY 115 APPENDIX I I EXPERIMENTAL RESULTS AND FIGURES 116 (1) PORTAGE S I L I C A SEDIMENT RIPPLES 116' (2) FRASER RIVER SEDIMENT RIPPLES 117 (3) BAKELITE SEDIMENT RIPPLES 117 (4) FIGURES: 117 APPENDIX I I I CALIBRATION PROCEDURE OF WATER WAVE MEASURING EQUIPMENT 131 APPENDIX IV RELATIONSHIP BETWEEN THE MAXIMUM HORIZONTAL WATER PARTICLE MOTION DISPLACEMENT AND MASS- TRANSPORT •DISTANCE ....... 133 v i L I S T OF FIGURES FI G NO. 1.1 w a t e r f l o w r e s i s t a n c e due t o t h e e x i s t a n c e o f s e d i m e n t b e d f o r m s . 1.2 c o r r e c t i o n t e r m s u s e d i n e q u a t i o n s . ( A f t e r R i c h a r d s o n e t a l 1 9 6 7 ) . 1.3 same as F i g . 2 1.4 s t a b l e t r i a n g u l a r shape bedwave i n s u b - c r i t i c a l w a t e r f l o w . 1.5 p r o f i l e s of; mean v e l o c i t y , l o n g i t u d i n a l and t r a n s v e r s e components o f w a t e r f l o w v e l o c i t y f l u c t u a t i o n s . 1.6 s t a b l e shape;: bedwave f r o m p r e s e n t e x p e r i m e n t s . 1.7 b a s i c d e s i g n o f e x p e r i m e n t a l a p p a r a t u s . 1.8 g r a i n d i a m e t e r d i s t r i b u t i o n o f P o r t a g e S i l i c a s a n d . 1.9 g r a i n d i a m e t e r d i s t r i b u t i o n o f F r a s e r R i v e r s a n d . 1.10 g r a i n d i a m e t e r d i s t r i b u t i o n o f B a k e l i t e s e d i m e n t p a r t i c l e s . 1.11 r e l a t i o n s h i p b etween R C B and N f o f P o r t a g e S i l i c a s a n d and F r a s e r R i v e r Sand b e d f o r m s . 1.12 r e l a t i o n s h i p between R^ B and o f B a k e l i t e s e d i m e n t b e d f o r m s . 1.13 r e l a t i o n s h i p between V / L p and N^ ., A n d e r s o n ' s e q u a t i o n . 1.14- r e l a t i o n s h i p between Y/L_ and R^ _,, M o d i f i e d e q u a t i o n . 1.15 argument o f i n v e r s i o n o f wave p i c t u r e s . 1.16 r e l a t i o n s h i p between L and C,.,. 1.17 r e l a t i o n s h i p b etween L and V. v i i L I S T OF FIGURES c o n t . 1.18 s i m p l i f i e d p l o t o f r e l a t i o n s h i p b etween L and V. 1.19 s i m p l i f i e d p l o t o f r e l a t i o n s h i p b etween Y/L and N f . 1.20 r e l a t i o n s h i p between s q u a r e r o o t s o f L and V. R 1.21 r e l a t i o n s h i p between s q a u r e r o o t s o f L R and pv/ps . 1.22 r e l a t i o n s h i p between Y / H D a n d N.. •n i 1.23 r e l a t i o n s h i p between H and N . R I 1.24- r e l a t i o n s h i p b etween H R/Y and . 1.25 r e l a t i o n s h i p b etween H and R_,„ . R LB 1.26 r e l a t i o n s h i p b etween H /Y- and R . 1.27 r e l a t i o n s h i p between C and V. B 1.28 r e l a t i o n s h i p between S D and N~. 1.29 r e l a t i o n s h i p b etween S n and R „ _ . 1.30 r e l a t i o n s h i p b etween s e d i m e n t t r a n s p o r t r a t e and w a t e r f l o w v e l o c i t y V. 2.1 e x p e r i m e n t a l equipment s e t up and w a t e r wave p r o b e a d j u s t m e n t 2.2 r e l a t i o n s h i p between r i p p l e l e n g t h and £ 2.3 r e l a t i o n s h i p b etween r i p p l e l e n g t h and'U • JL 2.4 r e l a t i o n s h i p between r i p p l e l e n g t h and (pUm/( ( P S _ P Js^g)) 2.5 r e l a t i o n s h i p s b etween r i p p l e l e n g t h s and (pU m/((p s-p)gY)) 0- 5 f o r e a c h t y p e o f s e d i m e n t c h a n n e l b e d f o r m s . 2.6 r e l a t i o n s h i p s between r i p p l e l e n g t h s and m o d i f i e d R e y n o l d ' s number U d 5 Q / v f o r e a c h t y p e o f s e d i e n t c h a n n e l b rms . 2.7 r e l a t i o n s h i p between r i p p l e l e n g t h and m o d i f i e d R e y n o l d ' s number U 'Y-/v v i i i 2.8 =» r e l a t i o n s h i p between r i p p l e h e i g h t and - ( p U m V ( ( P s - P ) g d 5 0 ) ) ° . 5 2.9 r e l a t i o n s h i p between r i p p l e h e i g h t and (pU m 2/((p s-p)gY)f 0- 5 2.10 r e l a t i o n s h i p s between r i p p l e h e i g h t s and m o d i f i e d R e y n o l d ' s number. U^CJJ-Q/V f o r e a c h t y p e o f s e d i m e n t c h a n n e l b e d f o r m s . 2.11 r e l a t i o n s h i p between r i p p l e h e i g h t and m o d i f i e d R e y n o l d ' s number U Y/v m 2.12 r e l a t i o n s h i p between r i p p l e h e i g h t and U . 2.13 r e l a t i o n s h i p between r i p p l e h e i g h t and £ 2.14 r e l a t i o n s h i p between r i p p l e s t e e p n e s s and E, 2.15 e x p e r i m e n t a l r e s u l t s f r o m s l o p i n g c h a n n e l b e d 2.16 r e l a t i o n s h i p between L R / d 5 Q and 2E, / d 5 Q , compare w i t h M o g r i d g e and Kamphuis' d e s i g n c u r v e s . 2.17 Same as F i g . 16. 2.18 r e l a t i o n s h i p between H /d and 2^/6.^, compare w i t h M o g r i d g e and K a m p n u i s - d e s i g n c u r v e s . 2.19 Same as F i g . 18 2.20 R e l a t i o n s h i p between L /d and 2 ^ / d 5 0 , compare w i t h M o g r i d g e and Kampnuis" d e s i g n c u r v e s . 2.21 r e l a t i o n s h i p b etween H„/d and 2 g / d 5 Q , compare w i t h M o g r i d g e and Kamphuis^ d e s i g n c u r v e s . 2.22 r e l a t i o n s h i p between r i p p l e s t e e p n e s s and H R / £ , compare w i t h H o r i k a w a and W a t a n a b e 1 s s t u d y . 2.23 r e l a t i o n s h i p between ? / L R and Um^/v, compare w i t h H o r i k a w a and Watanabe's s t u d y . 2.24 r e l a t i o n s h i p between t h e maximum h o r i z o n t a l w a t e r p a r t i c l e m o t i o n d i s p l a c e m e n t . a n d mass • t r a ' n s p o j r t j d i s t a n c e . i x The f o l l o w i n g symbols and l i n e s a r e u s e d i n above f i g u r e s t o r e p r e s e n t p r e s e n t e x p e r i m e n t a l r e s u l t s : o — — — — P o r t a g e S i l i c a s a n d b e d f o r m d i m e n s i o n s , x — . — . — F r a s e r R i v e r s a n d b e d f o r m d i m e n s i o n s . A — . . — . . — . . B a k e l i t e s a n d b e d f o r m d i m e n s i o n s . NOTATIONS: The f o l l o w i n g n o t a t i o n s and symb o l s a r e u s e d i n p a r t o f t h i s t h e s i s : x = w a t e r f l o w d i r e c t i o n Y = mean w a t e r f l o w v e l o c i t y t = t i m e B = c o n s t a n t w a t e r c h a n n e l w i d t h h = w a t e r f l o w d e p t h above a common datum Y = w a t e r f l o w d e p t h above c r e s t p o i n t s o f b e d f o r m s p = w a t e r d e n s i t y P = s e d i m e n t p a r t i c l e d e n s i t y s J d.__ = mean s e d i m e n t g r a i n d i a m e t e r 50 3 d D C.= 85% f i n e r s e d i m e n t g r a i n d i a m e t e r v = k i n e m a t i c v i s c o s i t y g = g r a v i t a t i o n a l a c c e l e r a t i o n C = w a t e r wave s p e e d S = s l o p e o f e n e r g y g r a d e l i n e f = f r i c t i o n f a c t o r U = s h e a r v e l o c i t y = s h e a r v e l o c i t y due t o t h e g r a i n r o u g h n e s s = s h e a r v e l o c i t y due t o t h e s e d i m e n t b e d f o r m s Y = s p e c i f i c w e i g h t o f w a t e r = s p e c i f i c w e i g h t o f s e d i m e n t p a r t i c l e s Q = w a t e r d i s c h a r g e r a t e q = s e d i m e n t t r a n s p o r t r a t e s L R s e d i m e n t b e d f o r m l e n g t h H. R s e d i m e n t b e d f o r m h e i g h t C B s e d i m e n t b e d f o r m m i g r a t i o n v e l o c i t y x i R = m o d i f i e d R e y n o l d ' s number i n t e r m s o f b e d f o r m m i g r a t i o n v e l o c i t y C B = C ^ / v = w a t e r f l o w Froude number a t c r e s t p o i n t s o f b e d f o r m s •n = b e d f o r m h e i g h t above a common datum a t a d i s t a n c e x ^ f r o m t r o u g h o f t h e b e d f o r m u' = t u r b u l e n t f l u c t u a t i n g v e l o c i t y components i n x d i r e c t i o n v ' = t u r b u l e n t f l u c t u a t i n g v e l o c i t y components i n y d i r e c t i o n w' = t u r b u l e n t f l u c t u a t i n g v e l o c i t y components i n z d i r e c t i o n k = a r b i t r a r y c o n s t a n t t o be d e t e r m i n e d by e x p e r i m e n t a l r e s u l t s L = w a t e r s u r f a c e wave l e n g t h w 3 H = w a t e r s u r f a c e wave h e i g h t w S_, = b e d f o r m s t e e p n e s s ri R h = n y ° ' r a u l : ' - c r a d i u s due t o g r a i n r o u g h n e s s R" = h y d r a u l i c r a d i u s due t o t h e b e d f o r m s TQ - t o t a l d r a g TQ = s u r f a c e d r a g T" = f o r m d r a g o — = mean v a l u e = E x n e r ' s e r o s i o n c o e f f i c i e n t T = c r i t i c a l s h e a r s t r e s s c T C / ( ( Y S - Y ) C I 50 = S h i e l d s ' e n t r a i n m e n t f u n c t i o n x i i I n a d d i t i o n t o t h o s e symbols l i s t e d above, t h e f o l l o w i n g s ymbols a r e a l s o u s e d i n P a r t 2 o f t h i s t h e s i s : C = w a t e r s u r f a c e wave s p e e d i n cm/sec T = w a t e r s u r f a c e wave p e r i o d i n s e c . L = s e d i m e n t b e d f o r m ( r i p p l e ) l e n g t h i n cm. R H = s e d i m e n t b e d f o r m ( r i p p l e ) h e i g h t i n cm. R U = t h e maximum h o r i z o n t a l w a t e r p a r t i c l e v e l o c i t y i n t h e m v i c i n i t y o f s e d i m e n t c h a n n e l b e d i n cm/sec. E, = maximum h o r i z o n t a l w a t e r p a r t i c l e d i s p l a c e m e n t i n t h e v i c i n i t y o f s e d i m e n t c h a n n e l b e d i n cm. W o = s e t t l i n g v e l o c i t y u s e d by H o r i k a w a and W a t a n a b e 1 s s t u d y 6 = n e t mass t r a n s p o r t d i s t a n c e moved b y t h e f l u i d i n a wave p e r i o d . i m m e d i a t e l y o u t s i d e t h e c h a n n e l bed b o u n d a r y l a y e r , t h e v a l u e o f i s much s m a l l e r t h a n t h e v a l u e o f £ K = r i p p l e wave number, w h i c h i s A = r i p p l e wave a m p l i t u d e , w h i c h i s H^/2 w = w a t e r s u r f a c e wave f r e q u e n c y , w h i c h i s 2 t r / T U = h o r i z o n t a l w a t e r p a r t i c l e v e l o c i t y Y = mean w a t e r d e p t h i n cm. L = w a t e r s u r f a c e wave l e n g t h i n cm, w h i c h i s t h e p r o d u c t W o f C and T. H = w w a t e r s u r f a c e wave h e i g h t i n cm. x i i i ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o e x p r e s s h i s d e e p e s t g r a t i t u d e t o h i s s u p e r v i s o r , Dr. M.C. Q u i c k , f o r t h e amount and q u a l i t y o f a d v i c e and g u i d a n c e t h r o u g h o u t a l l p h a s e s o f t h i s s t u d y . A d d i t i o n a l t h a n k s a r e due t o Dr. S.O. R u s s e l l , and Dr. M. I s a a c s o n f o r t h e i n v a l u a b l e s u g g e s t i o n s and a s s i s t a n c e . The a u t h o r w o u l d a l s o l i k e t o t h a n k t h e t e c h n i c i a n i n t h e H y d r a u l i c E n g i n e e r i n g L a b o r a t o r y , Mr. K. N i e l s o n , f o r h i s h e l p d u r i n g o p e r a t i o n o f t h e e x p e r i m e n t s . T h i s s t u d y was f i n a n c i a l l y s u p p o r t e d by E n v i r o n m e n t Canada and F r a s e r R i v e r D i s a s t e r Fund. 1 PART I : RIPPLES AND DUNES GENERATED BY STEADY FLOW  CHAPTER 1.1 INTRODUCTION R i v e r s i n f l o o d u n d e r g o s i g n i f i c a n t m o d i f i c a t i o n i n b e h a v i o r w h i c h c a n have i m p o r t a n t e n g i n e e r i n g and economic i m p l i c a t i o n s . F o r example, t h e maximum f l o o d l e v e l o f a r i v e r c a n be a l t e r e d by t h e s c o u r and t h e s u b s e q u e n t down-s t r e a m d e p o s i t i o n o f s a n d , o r t h e shape o f an a l l u v i a l open c h a n n e l may change w i t h d i f f e r e n t w a t e r f l o w c o n d i t i o n s . T h ese c h a n g e s c a n o c c u r i n a s s o c i a t i o n w i t h c h a n g e s i n c h a n n e l bed s u r f a c e r o u g h n e s s r e s u l t i n g f r o m t h e f o r m d r a g c a u s e d by m o d i f i c a t i o n s i n b e d f o r m f e a t u r e s . I n o t h e r s i t u a t i o n s s u c h as r e s e r v o i r s and h a r b o u r s , t h e d e p o s i t i o n o f s e d i m e n t s may r e d u c e t h e s t o r a g e c a p a c i t y o f r e s e r v o i r s and n a v i g a b i l i t y o f h a r b o u r s , w h i c h t h e n r e q u i r e c o s t l y d r e d g i n g and r e m o v a l o f d e p o s i t e d - s e d i m e n t s . Many r i v e r s ' a n d c h a n n e l s a r e o f t e n u s e d f o r n a v i g a t i o n and t h e y t o o r e q u i r e c o s t l y o p e r a t i o n s t o m a i n t a i n n a v i g a b l e waterways, e s p e c i a l l y i f t h e w a t e r f l o w i s t o c a r r y t h e s e d i m e n t l o a d s u c c e s s f u l l y . Y e t a n o t h e r p r o b l e m i s l o c a l s c o u r i n g o f s e d i m e n t n e a r p i e r s o f b r i d g e s o r s i m i l a r . s t r u c t u r e s and p o s s i b l e s c o u r i n g s h o u l d be e s t i m a t e d when d e s i g n i n g t h e s t r u c t u r e s . The total s e d i m e n t l o a d c a n be d i v i d e d i n t o two p a r t s , t h e b e d l o a d and t h e s u s p e n d e d l o a d . The b e d l o a d t r a n s p o r t i s a s s o c i a t e d w i t h t h e m i g r a t i o n o f b e d f o r m f e a t u r e s , and t h e f o r m a t i o n o f t h e s e b e d f o r m f e a t u r e s i n v o l v e s an i n t e r a c t i o n between open c h a n n e l h y d r o d y n a m i c s and t h e d y n a m i c s o f 2 s e d i m e n t m o t i o n . I n t h i s t h e s i s , we s h a l l c o n s i d e r some o f t h e mechanisms and r e l a t i o n s h i p s i n v o l v e d i n t h e f o r m a t i o n o f t h e s e b e d f o r m f e a t u r e s . The mechanism o f f o r m a t i o n o f b e d f o r m s depends upon v a r i o u s c o n d i t i o n s o f w a t e r f l o w , s e d i m e n t c h a r a c t e r i s t i c s and l o c a l i r r e g u l a r i t i e s . Many f e a t u r e s o f s e d i m e n t t r a n s p o r t and b e d f o r m f o r m a t i o n a r e s t i l l i m p e r f e c t l y u n d e r s t o o d due t o i t s c o m p l e x i t y , and t h e r e f o r e , t h e r e a r e no c o m p l e t e l y s a t i s f a c t o r y t h e o r i e s a v a i l a b l e w h i c h c a n e x p l a i n a l l a s p e c t s o f t h e e x i s t a n c e and t h e f o r m a t i o n o f s e d i m e n t b e d f o r m f e a t u r e s . The p r e s e n t s t u d y i s d i v i d e d i n t o two p a r t s . The f i r s t p a r t i n v e s t i g a t e s t h e r e l a t i o n s h i p s between t h e b e d f o r m d i m e n s i o n s as a f u n c t i o n o f t h e u n i d i r e c t i o n a l s u b - c r i t i c a l w a t e r f l o w and t h e s e d i m e n t p a r t i c l e c h a r a c t e r i s t i c s . The s e c o n d p a r t i n v e s t i g a t e s t h e r e l a t i o n s h i p s f o r t h e b e d f o r m d i m e n s i o n s p r o d u c e d by o s c i l l a t o r y w a t e r waves, and f o r a r a n g e o f s e d i m e n t p a r t i c l e c h a r a c t e r i s t i c s . F o r u n i d i r e c t i o n a l w a t e r f l o w , Simons e t . a l . ( 1 9 6 1 ) , p r e s e n t e d a c l a s s i f i c a t i o n o f b e d f o r m s w h i c h has b e e n w i d e l y a c c e p t e d . The t y p i c a l b e d f o r m s a r e r i p p l e s , dunes and a n t i -d u n e s. R i p p l e s a r e u n d e r s t o o d t o be s m a l l b e d f o r m s , whereas dunes a r e l a r g e r ones and a r e o u t o f p h a s e w i t h t h e w a t e r s u r f a c e . R i p p l e s may be s u p e r - i m p o s e d upon t h e u p s t r e a m f a c e o f t h e dunes. A n t i d u n e s , g e n e r a l l y o c c u r i n t h e s i t u a t i o n s o f h i g h w a t e r f l o w v e l o c i t i e s , may r e m a i n s t a t i o n a r y o r move e i t h e r u p s t r e a m o r downstream. We now c o n s i d e r b e d f o r m f o r m a t i o n due t o t h e a c t i o n o f t h e u n i d i r e c t i o n a l w a t e r f l o w , and l e a v e t h e b e d f o r m f o r m a t i o n 3 due t o t h e w a t e r wave a c t i o n t o t h e s e c o n d p a r t . T h e r e a r e two m a j o r a s p e c t s g o v e r n i n g t h e s e d i m e n t m o t i o n : t h e w a t e r f l o w v e l o c i t y V and t h e F r o u d e number o f w a t e r f l o w . The mean w a t e r f l o w d e p t h Y i s a v a r i a b l e l i n k i n g t h e s e two f a c t o r s . I f t h e w a t e r f l o w d e p t h i s v e r y l a r g e , t h e n t h e v e l o c i t y c a n be examined i n d e p e n d e n t l y o f t h e F r o u d e number, and t h e f o l l o w i n g o b s e r v a t i o n s m i g h t be t y p i c a l f o r a g i v e n s i z e o f s e d i m e n t p a r t i c l e s . A l o w e r c r i t i c a l v e l o c i t y w i l l e x i s t b e l o w w h i c h no s e d i m e n t p a r t i c l e s c a n move. On i n c r e a s i n g t h e v e l o c i t y , f l a t b e d s e d i m e n t m o t i o n w i l l e x i s t . A f u r t h e r i n c r e a s e t o a b o u t 1.5 t i m e s t h e l o w e r c r i t i c a l v e l o c i t y w i l l c a u s e r i p p l e s t o f o r m . As t h e v e l o c i t y i s i n c r e a s e d more, t h e s e r i p p l e s w i l l become l a r g e r and s e d -iment m o t i o n w i l l be more i n t e n s e . A t h i g h e r v e l o c i t y r a n g e s , t h e r i p p l e s w i l l become v e r y a c t i v e and c a n p r o d u c e l a r g e s e c o n d a r y f e a t u r e s w h i c h w i l l be c a l l e d r i p p l e - d u n e s (M.C. Q u i c k , 1978-1979). B e c a u s e t h e w a t e r i s deep, t h e F r o u d e number w i l l s t i l l be low, and i n p a r t i c u l a r , must be b e l o w 0.5. When t h e F r o u d e number e x c e e d s 0.5 b u t s t i l l b e low a b o u t 0.7, b e d f o r m s w i l l become p u r e dunes, and t h e s e f e a t u r e s w i l l be c a l l e d F r o u d e - d u n e s . A f u r t h e r i n c r e a s e i n v e l o c i t y w o u l d b e g i n t o p r o d u c e enough t u r b u l e n c e t o s u s p e n d t h e m a t e r i a l and e v e n t u a l l y s u s p e n s i o n o f m a t e r i a l w o u l d wipe o u t t h e s e d i m e n t bedwaves, w h i c h a r e as d e p e n d e n t on s e d i m e n t d e p o s i t i o n as t h e y a r e on s e d i m e n t e r o s i o n . Such e x p e r i m e n t s a r e h a r d t o p e r f o r m b e c a u s e o f t h e enormous w a t e r f l o w q u a n t i t i e s r e q u i r e d a t h i g h w a t e r f l o w d e p t h s . F i n e s a n d s c a n be u s e d t o d e m o n s t r a t e t h i s r a n g e o f v e l o c i t i e s a t low F r o u d e numbers. I f t h e d e p t h o f w a t e r f l o w i s a l l o w e d t o t a k e l o w e r v a l u e s , t h e n t h e F r o u d e number o f t h e w a t e r f l o w w i l l a c t as a m o d i f i e r . F o r example, e v e n a t v e r y low v e l o c i t i e s c l o s e t o t h e l o w e r c r i t i c a l v a l u e , a h i g h F r o u d e number w i l l i n t e n s i f y t h e bed s h e a r s t r e s s a t c e r t a i n l o c a t i o n s and w i l l enhance t h e s e d i m e n t p a r t i c l e m o t i o n . The f o l l o w i n g s e q u e n c e o f e v e n t s i s o b s e r v e d as t h e F r o u d e number i s s t e a d i l y i n c r e a s e d . A t v e r y low w a t e r f l o w v e l o c i t y and v e r y s m a l l F r o u d e number, t h e s e d i m e n t on t h e c h a n n e l b e d doe s n o t move a t a l l . (1) . I f t h e w a t e r f l o w v e l o c i t y i s i n c r e a s e d u n t i l t h e F r o u d e number i s l a r g e enough, sa n d on t h e c h a n n e l bed w i l l b e g i n t o move. The r e q u i r e d minimum F r o u d e number t o i n i t i a t e t h e s e d i m e n t m o t i o n v a r i e s w i t h v a r i o u s w a t e r f l o w and s e d i m e n t c h a r a c t e r i s t i c s and c a n n o t be d e f i n e d , b e c a u s e s a n d on t h e c h a n n e l b e d s t a r t s t o move when a c e r t a i n c r i t i c a l s h e a r s t r e s s i s r e a c h e d , and t h i s c r i t i c a l s h e a r s t r e s s c a n be a s s o c i a t e d w i t h s h e a r v e l o c i t y , w h i c h c a n be a s s o c i a t e d w i t h . c h a n n e l r o u g h n e s s , w a t e r f l o w d e p t h and mean g r a i n d i a m e t e r . F o r a f u r t h e r i n c r e a s e o f w a t e r f l o w v e l o c i t y and c o r r e s p o n d i n g i n c r e a s e i n F r o u d e number, t h e san d c h a n n e l bed may become r i p p l e d , i f t h e F r o u d e number i s l e s s t h a n a b o u t 0.5. (2) . As t h e w a t e r f l o w v e l o c i t y and F r o u d e number i n c r e a s e , f u r t h e r , i f t h e F r o u d e number a p p r o a c h e s , b u t i s l e s s t h a n 0.7 5 s a n d b e d f o r m f e a t u r e s change t o dunes. When dunes f i r s t b e g i n t o a p p e a r , t h e r i p p l e s may be s u p e r i m p o s e d upon t h e u p s t r e a m f a c e o f them, and t h e s e dunes a r e r i p p l e - d u n e s as m e n t i o n e d e a r l i e r . F u r t h e r i n c r e a s e o f w a t e r f l o w v e l o c i t y , c a u s e s t h e s e r i p p l e s t o d i s a p p e a r and o n l y p u r e dunes r e m a i n and t h e s e f e a t u r e s a r e F r o u d e - d u n e s . Sand p a r t i c l e s a r e moved up o v e r t h e b a c k o f t h e dunes and f a l l down t h e s t e e p down-s t r e a m f a c e . T h i s p r o c e s s o f u p s t r e a m s c o u r and downstream d e p o s i t i o n c a u s e s t h e F r o u d e - d u n e s t o m i g r a t e downstream. R i p p l e s and dunes a r e e s s e n t i a l l y s i m i l a r i n n a t u r e a l t h o u g h t h e y w i l l be shown t o d i f f e r i n some s i g n i f i c a n t r e s p e c t s . I n t h e r a n g e o f r i p p l e s , t h e r e i s no i n t e r a c t i o n between t h e w a t e r s u r f a c e and c h a n n e l bed r i p p l e s , and t h e F r o u d e number p l a y s no r o l e i n r i p p l e f o r m a t i o n e x c e p t t h a t i t must be low, s u c h as l e s s t h a n 0.5. F o r t h e r a n g e o f F r o u d e - d u n e s , t h e r e i s i n t e r a c t i o n between t h e w a t e r s u r f a c e and t h e c h a n n e l bed dunes, so t h a t t h e F r o u d e number i s s i g n i f i c a n t and w i l l be i n t h e r a n g e 0.5 t o 0.7. (3) . W i t h f u r t h e r i n c r e a s e i n w a t e r f l o w v e l o c i t y as w e l l as F r o u d e number, b u t w i t h t h e F r o u d e number s t i l l l e s s t h a n one, b e d f o r m s w i l l be washed away and t h e wavy c h a n n e l b e d f o r m f e a t u r e s w i l l become f l a t . (4) . When t h e F r o u d e number a p p r o a c h e s and e x c e e d s u n i t y , a n t i - d u n e s w i l l be f o r m e d and o c c u r i n a s s o c i a t i o n w i t h i n - p h a s e , s t a n d i n g w a t e r s u r f a c e waves. I n t h i s s u p e r - c r i t i c a l 6 f l o w r a n g e , t h e a m p l i t u d e o f t h e w a t e r s u r f a c e wave w i l l i n c r e a s e so t h a t i t i s g r e a t e r t h a n t h e a m p l i t u d e o f t h e s a n d waves. Under t h i s s i t u a t i o n , i t i s p o s s i b l e f o r t h e s c o u r o f t h e s a n d t o o c c u r f r o m downstream s i d e o f t h e b e d -f o r m f e a t u r e s and d e p o s i t i o n on t h e u p s t r e a m s i d e . A t l o w e r F r o u d e numbers, a n t i d u n e s a p p e a r as s t a n d i n g s a n d and w a t e r waves. F u r t h e r i n c r e a s e i n w a t e r f l o w v e l o c i t y o r a t h i g h e r F r o u d e numbers, t h e w a t e r s u r f a c e waves may grow, become u n s t a b l e , and b r e a k i n t h e u p s t r e a m d i r e c t i o n . I f s a n d wave b r e a k i n g o c c u r s , t h e a n t i d u n e s a r e d e s t r o y e d , t h e c h a n n e l bed becomes f l a t , and t h e f o r m a t i o n o f a n t i d u n e s s t a r t s a l l o v e r a g a i n . I n t h e b r e a k i n g o f t h e w a t e r s u r f a c e waves, t u r b u l e n c e i s c r e a t e d , and a b i g amount o f s a n d w i l l be c a r r i e d i n t o s u s p e n s i o n by e d d i e s . The b e d f o r m s g e n e r a t e d by w i n d g e n e r a t e d , o s c i l l a t o r y w a t e r wave a c t i o n a r e l i m i t e d t o r i p p l e s , w h i c h w i l l be d i s -c u s s e d i n t h e s e c o n d p a r t o f t h i s s t u d y . I n t h e f i r s t p a r t o f t h i s s t u d y , a number o f w a t e r f l u m e e x p e r i m e n t s were c a r r i e d o u t u n d e r s u b - c r i t i c a l u n i d i r e c t i o n a l w a t e r f l o w c o n d i t i o n . T h r e e t y p e s o f s e d i m e n t p a r t i c l e s , P o r t a g e S i l i c a s a n d , F r a s e r r i v e r s a n d , and B a k e l i t e s e d i m e n t p a r t i c l e s were u s e d t o s t u d y t h e d i f f e r e n c e i n v a r i o u s s e d i m e n t b e d f o r m b e h a v i o r s . E x p e r i m e n t a l r e s u l t s were examined and compared w i t h t h e p r e v i o u s t h e o r e t i c a l and e x p e r i m e n t a l s t u d i e s done by o t h e r i n v e s t i g a t o r s . 7 The mechanism i n v o l v e d i n g e n e r a t i n g t h e v a r i o u s t y p e s o f s e d i m e n t b e d f o r m s i s a l s o c o n s i d e r e d . The m a i n aim i s t o p r e d i c t t h e t y p e o f b e d f o r m w h i c h w i l l o c c u r f o r a p a r t i c u l a r s e d i m e n t and w a t e r f l o w c o n d i t i o n s . The p r e s e n t s t u d y f i r s t u s e s t h e F r o u d e number t o p r e d i c t what t y p e s o f b e d f o r m a r e p o s s i b l e and t h e n u s e s t h e mean w a t e r f l o w v e l o c i t y f o r a g i v e n s e d i m e n t s i z e t o p r e d i c t b e d f o r m t y p e and s i z e more e x a c t l y . A s e c o n d a r y aim o f t h e t h e s i s i s t o compare t h e b e h a v i o r o f l i g h t e r w e i g h t s e d i m e n t ( s u c h as B a k e l i t e s e d i m e n t p a r t i c l e s ) b e d f o r m s t o t h e b e h a v i o r s o f o t h e r r e g u l a r s e d i m e n t b e d f o r m s . As f o r f u t u r e s t u d i e s , one o f t h e m a j o r aims s h o u l d be c e n t e r e d upon c o m p u t i n g t o t a l c h a n n e l r e s i s t a n c e w h i c h does depend on p r e d i c t i n g t h e c o r r e c t b e d f o r m b e h a v i o r . 8 CHAPTER 1.2 LITERATURE REVIEW H e l m h o l t z ( 1 8 8 8 ) , u s i n g a m a t h e m a t i c a l a p p r o a c h showed t h a t a b o u n d a r y between two f l u i d s o f d i f f e r e n t d e n s i t y m o ving w i t h d i f f e r e n t v e l o c i t i e s i s s u b j e c t t o waves. B a s c h i n (1899) c o n t e s t e d t h a t b e d f o r m s i n s t r e a m s c a n be e x p l a i n e d w i t h H e l m h o l t z ' s model, and as s u c h c o n s i d e r e d t h e l o o s e s a n d t o a c t a l m o s t l i k e a f l u i d . L i u ( 1 9 5 7 ) , s u g g e s t e d t h a t , " R i p p l e s a r e c a u s e d p r i m a r i l y by t h e i n s t a b i l i t y o f t h e zone o f h i g h v e l o c i t y g r a d i e n t a t t h e s u r f a c e o f t h e s e d i m e n t - l a d e n b o t t o m " . By s p e c u l a t i n g t h a t t h e s e d i m e n t bed act-s as a f l u i d o f unknown v i s c o s i t y , L i u q u a l i t a t i v e l y compared w a t e r f l o w o v e r a s a n d bed t o t h e i n s t a b i l i t y p r o b l e m a t t h e i n t e r f a c e o f two s u p e r -imposed f l u i d s o f d i f f e r e n t d e n s i t i e s as d e s c r i b e d f i r s t by H e l m h o l t z . The l a m i n a r s u b l a y e r a t t h e s a n d bed i s compared t o t h e l a m i n a r i n t e r f a c e band i n t h e l a t t e r c a s e . The f o r m a t i o n o f r i p p l e s i s compared t o t h e f o r m a t i o n o f t h e wavy i n t e r f a c e s u r f a c e . I n h i s summary, L i u s t a t e d , " s e d i m e n t r i p p l e s a r e e x p l a i n e d as c a u s e d by t h e i n s t a b i l i t y o f t h e i n t e r f a c e , w h i c h i s a t r a n s i t i o n l a y e r between t h e f l u i d and t h e bed i t i s shown t h a t o t h e r f a c t o r s s u c h as t u r b u l e n c e , s u r f a c e waves and s m a l l i r r e g u l a r i t i e s o f t h e bed a r e n o t t h e p r i m a r y c a u s e s o f t h e s e d i m e n t r i p p l e f o r m a t i o n " . W i l l i a m s and Kemp ( 1 9 7 1 ) , commenting on L i u ' s c o n c l u s i o n p o i n t e d o u t t h a t , " i t i s o v e r s i m p l i f i c a t i o n t o assume t h a t t h e l a m i n a r s u b l a y e r i s a c o m p l e t e l y s t e a d y two d i m e n s i o n a l l a m i n a r 9 r e g i o n t h a t becomes u n s t a b l e i n s i m i l a r manner t o l a m i n a r f l o w s , n e i t h e r does t h i s a n a l o g y e x p l a i n t h a t f a c t t h a t r i p p l e f o r m as an a m p l i f y i n g and p r o p a g a t i n g d i s t u r b a n c e , and n o t s i m u l t a n e o u s l y o v e r t h e whole b e d " . V a n o n i and B r o o k s ( 1 9 5 8 ) , commenting on L i u ' s c o n c l u s i o n , p o i n t e d o u t t h a t t h e i n t e r f a c e i n s t a b i l i t y a n a l o g y w i t h r i p p l e f o r m a t i o n was d i f f i c u l t t o a c c e p t b e c a u s e a s a n d bed does n o t beha v e l i k e a d e n s e f l u i d b e c a u s e o n l y t h e s u r f a c e s a n d g r a i n s a r e f r e e t o move. A n o t h e r p o i n t o f c o n t e n t i o n was t h e f o r m a t i o n o f r i p p l e s by n a t u r a l t u r b u -l e n c e and s m a l l i r r e g u l a r i t i e s , t h e y m e n t i o n e d t h a t t u r b u l e n t e d d i e s were o b s e r v e d t o p e n e t r a t e t h e l a m i n a r s u b l a y e r . "When an eddy sweeps down t o t h e bed, p a r t i c l e s on t h e bed a r e d i s t u r b e d . When t h e p a r t i c l e s a r e p i l e d a few g r a i n s h i g h e r t h a n t h e i n i t i a l s u r f a c e a s m a l l wake d e v e l o p s w h i c h makes t h e p a r t i c l e s i m m e d i a t e l y downstream more l i k e l y t o be e n t r a i n e d " , and " w i t h a r e g u l a r r i p p l e p a t t e r n d e v e l o p i n g f r o m a random t u r b u l e n t m o t i o n as d e s c r i b e d by I n g l i s " . I n g l i s ( 1 9 4 9 ) , d e s c r i b e d t h e mechanism o f r i p p l e f o r m a t i o n as one i n w h i c h " i n t u r b u l e n t f l o w , any random c o l l e c t i o n o f p a r t i c l e s w i l l o c c u r on t h e bed, , t h i s w i l l c a u s e a s m a l l eddy t o f o r m downstream, and, due t o t h e f o r m a t i o n o f t h i s eddy, m a t e r i a l w i l l be s c o u r e d f r o m t h e t r o u g h and t h e s c o u r e d m a t e r i a l w i l l d e p o s i t a t t h e p o i n t f u r t h e r downstream t o f o r m a n o t h e r r i p p l e c r e s t , t h e p r o c e s s t h u s becomes c o n t i n u o u s " . 10 C o n s i d e r i n g t h e w a t e r - s e d i m e n t i n t e r f a c e as a p r o b l e m o f s t a b i l i t y , E x n e r ( 1 9 2 5 ) , p r o p o s e d a number o f m a t h e m a t i c a l m o d els o f t h e w a t e r l o o s e b ed i n t e r f a c e and compared t h e r e s u l t s q u a l i t a t i v e l y w i t h some l a b o r a t o r y e x p e r i m e n t s and f i e l d o b s e r v a t i o n s . A l l t h e m o dels c a n o n l y d e s c r i b e t h e g e o m e t r y and movement o f t h e b e d f o r m s b u t c a n n o t g i v e an answer t o t h e c a u s e o f t h e b e d f o r m s . Due t o t h e m a t h e m a t i c a l c o m p l e x i t y , i t i s v e r y d i f f i c u l t t o s o l v e t h e e n t i r e p r o b l e m . T h e r e f o r e , E x n e r s t a r t e d by p r o p o s i n g an i d e a l i z e d ( f r i c t i o n l e s s ) math-e m a t i c a l model and t h e n p r o g r e s s e d s l o w l y by b u i l d i n g more and more c o m p l i c a t e d m o d e l s . The b a s i c E x n e r ' s model i s d i s c u s s e d i n C h a p t e r 3 and compared t o o u r p u r e k i n e m a t i c r e q u i r e m e n t s . B a g n o l d ( 1 9 5 7 ) , c o n s i d e r e d t h e c o n d i t i o n s f o r s t a b i l i t y o f a g r a i n s u r f a c e u n d e r c e r t a i n c o n d i t i o n s . A n a l y t i c a l l y he showed t h a t f o r l i q u i d f l o w s a l l t h e s u r f a c e g r a i n s move when S h i e l d s ' e n t r a i n m e n t f u n c t i o n i s g r e a t e r t h a n a b o u t 0.4-. A t t h i s s t a g e , g r a i n s a r e i n s u s p e n s i o n and t h e m e c h a n i c a l p r o c e s s e s a r e s i m i l a r t o t h o s e f o r a i r f l o w . However, b e c a u s e o f t u r b u l e n c e and n o n u n i f o r m g r a i n s i z e d i s t r i b u t i o n , g r a i n s w i l l move on t h e s a n d bed a t l o w e r v a l u e s t h a n 0.4. B a g n o l d a l s o c o n s i d e r s t h e e q u i l i b r i u m o f s h e a r f o r c e s on t h e bed f o r f l o w s s u f f i c i e n t t o c a u s e g r a i n movement b u t i n s u f f i c i e n t t o c a u s e s u s p e n s i o n , and by making c e r t a i n a s s u m p t i o n s c o n c e r n i n g t h e n a t u r e o f t h e f l u i d and r e s i s t a n c e s t r e s s e s , he o b t a i n s an e x p r e s s i o n f o r t h e s h e a r r e s i s t a n c e d e f i c i t o f t h e b e d . I f 11 t h e e x p r e s s i o n is" p o s i t i v e , he a r g u e s t h a t t h e p l a n e bed i s u n a b l e t o r e s i s t t h e a p p l i e d s h e a r s t r e s s e s and must d e f o r m i n s u c h a way t h a t t h e s h e a r r e s i s t a n c e d e f i c i t i s made up by an a d d i t i o n a l f o r m d r a g . Hence, so c a l l e d p r i m a r y r i p p l e f e a t u r e s w i t h f l a t u p s t r e a m f a c e a r e f o r m e d i n t h i s manner. However, f o r i n c r e a s i n g f l o w c o n d i t i o n s , t h e s e p r i m a r y r i p p l e s become u n s t a b l e and e r o d e s c o u r h o l e s downstream t h a t d e p o s i t more g r a i n s on t o t h e r i p p l e s t h u s b u i l d i n g them up i n t o s e c o n d a r y r i p p l e f e a t u r e s t h a t have a much l a r g e r f o r m d r a g t h a n t h e s h e a r r e s i s t a n c e d e f i c i t . However, W i l l i a m s and Kemp ( 1 9 7 1 ) , a r g u e d t h a t B a g n o l d ' s e x p l a n a t i o n o f t h e f o r m -a t i o n o f p r i m a r y and s e c o n d a r y r i p p l e f e a t u r e s i s n o t o b s e r v e d f o r s a n d bed u n d e r w a t e r f l o w s . I n f a c t , r i p p l e s f o r m r a n d o m l y o v e r t h e bed and p r o p a g a t e downstream as t h e y grow. The s m a l l r i p p l e s f i r s t o b s e r v e d t o f o r m a r e c o m p l e t e l y t r a n s i t o r y , and t h e y r e a c h a f i n a l e q u i l i b r i u m g e o m e t r y w i t h t h e f l o w . K o n d r a t ' e v ( 1 9 6 2 ) , s u g g e s t e d t h a t a t c e r t a i n f l o w v e l o c i t i e s t h e s e d i m e n t bed o f a w a t e r f l o w c h a n n e l s i m u l t a n -e o u s l y shows f u r r o w s l y i n g a t r i g h t a n g l e s t o t h e w a t e r f l o w . The f u r r o w s a r e t h e r e s u l t s o f an a c t i v e l i f t mechanism. The s a n d w h i c h i s . c a r r i e d away f r o m t h e s e f u r r o w s i s d e p o s i t e d downstream. From t h e e x p e r i m e n t a l r e s u l t s o b t a i n e d by S o v i e t e x p e r i m e n t a l r e s e a r c h e r s G o n c h a r o v and L a p s i n , K o n d r a t ' e v ( 1 9 6 2 ) , r e p r e s e n t s t h e d a t a i n t e r m s o f F r o u d e number. I t was shown t h a t F r o u d e number c a n be u s e d as a b e d f o r m c r i t e r i o n . S p e c i f i c r a n g e s o f F r o u d e number f o r i n i t i a t i o n o f g r a i n m o t i o n , o f b e d f o r m f o r m a t i o n and t h e washaway o f b e d f o r m s c a n be d e t e r m i n e d , i f t h e s e d i m e n t g r a i n d i a m e t e r i s known. A l o g a r i t h m i c r e l a t i o n s h i p b etween F r o u d e number and R e y n o l d ' s number i n t e r m s o f b e d f o r m m i g r a t i o n v e l o c i t y i s a l s o p r e s e n t e d . S i m i l a r r e l a t i o n s h i p s have been p r e s e n t e d i n F i g . 10 and 11 f r o m t h e p r e s e n t e x p e r i m e n t a l r e s u l t s , and d i s c u s s e d i n C h a p t e r 1.5. B a s e d on e x t e n s i v e d a t a f r o m w a t e r f l u m e and s e v e r a l r i v e r s , Simons and R i c h a r d s o n ( 1 9 6 6 ) , d e v e l o p e d a c h a r t f o r p r e d i c t i n g b e d f o r m s f o r g i v e n s t r e a m power ( p r o d u c t o f bed s h e a r s t r e s s and mean w a t e r f l o w v e l o c i t y ) , and m e d i a n f a l l d i a m e t e r o f bed s e d i m e n t p a r t i c l e s . The p r e d i c t i o n o f b e d f o r m s by t h e c h a r t i s i n good > agreement w i t h t h o s e o b s e r v e d by N o r d i n ( 1 9 6 4 ) , i n t h e R i o Grande i n w h i c h t h e d e p t h s o f t h e w a t e r f l o w were l e s s t h a n 5 f e e t and t h e v e l o c i t i e s r e l a t i v e l y h i g h . However, i n t h e c a s e o f t h e M i s s i s s i p p i R i v e r , where t h e d e p t h s were up t o 50 f e e t w i t h v e l o c i t i e s i n a b o u t t h e same r a n g e as t h o s e f o r t h e R i o Grande, t h e Simon'sand R i c h a r d s o n ' s c h a r t d o e s n o t a g r e e w i t h t h e d a t a o b s e r v e d by J o r d a n ( 1 9 6 5 ) . The c h a r t p r e d i c t s f l a t b e d i n some c a s e s where e v i d e n c e shows t h a t t h e beds a r e dune c o v e r e d . R a u d k i v i ( 1 9 6 3 ) , e x p l a i n e d t h e mechanism o f r i p p l e f o r m a t i o n as f o l l o w s , "when t h e t h r e s h o l d c o n d i t i o n s o f s e d i m e n t t r a n s p o r t a r e e x c e e d e d , a d i s t u r b a n c e i n t h e p l a n e 13 s u r f a c e i s c r e a t e d b y a c h a n c e p i l i n g - u p . T h i s s u r f a c e d i s t u r b a n c e e s t a b l i s h e s an i n t e r f a c e o r s u r f a c e o f d i s c o n -t i n u i t y i n t h e f l o w s i m i l a r t o t h a t w i t h f l o w p a s t a n e g a t i v e s t e p . I n t h e s h e a r f l o w o f t h e i n t e r f a c e zone, t h e t u r b u l e n c e i n t e n s i t y i s h i g h and t h e g r a i n s a r e s t i r r e d up by t h e t u r b u l e n t a g i t a t i o n where t h e i n t e r f a c e r e a t t a c h e s i t s e l f t o t h e g r a i n b o u n d a r y . From t h e r e a t t a c h m e n t p o i n t down-s t r e a m t h e t u r b u l e n t a g i t a t i o n d e c r e a s e s and a l s o a b o u n d a r y l a y e r d e v e l o p s . Where t h e c o r e o f t h e i n t e r f a c e o f t h e d i s -c o n t i n u i t y meets t h e s a n d b o u n d a r y i t e x c a v a t e s more m a t e r i a l b e c a u s e i t ' s i n c r e a s e d t u r b u l e n t a g i t a t i o n i n t h e i n t e r f a c e b etween t h e wake and t h e main s t r e a m . T h i s e x t r a m a t e r i a l e n t r a i n e d c a n n o t be s u p p o r t e d by t h e f l o w o v e r a p l a n e b o u n d a r y . The a d d i t i o n a l e n t r a i n e d m a t e r i a l s e t t l e s o u t l e a d i n g t o a new r i p p l e f a c e " , and " i t i s n o t a p r o c e s s o f a m p l i f i c a t i o n o f a bed d i s t u r b a n c e o f a g i v e n w a v e l e n g t h , b u t a g r a d u a l g r o w t h u n t i l a s t e a d y s t a t e c o n d i t i o n i s e s t a b l i s h e d " . However, none o f L i u ' s , B a g n o l d ' s , o r R a u d k i v i ' s e x p l a n a t i o n s e t c . t a k e t h e k i n e m a t i c r e q u i r e m e n t f o r b e d f o r m f o r m a t i o n i n t o c o n s i d e r a t i o n , namely t h a t t h e s e d i m e n t t r a n s p o r t r a t e must i n c r e a s e w i t h b e d f o r m h e i g h t . I n t h e p r e s e n t s t u d y , i t w i l l be a r g u e d t h a t t h i s k i n e m a t i c r e q u i r e m e n t i s t h e k e y t o t h e b e d f o r m b e h a v i o r . A l t h o u g h R a u d k i v i 1 s work p r o p o s e s t h e most d e t a i l e d d e s c r i p t i o n o f t h e m e c h a n i c s o f t h e r i p p l e f o r m a t i o n , t h e r e s t i l l r e m a i n e d some q u e s t i o n s unanswered: 14 (1) What a r e t h e mechanisms t h a t c a u s e s a c h a n c e p i l i n g up t o o c c u r ? (2) How c a n s u c h s m a l l g r o u p i n g o f i n d i v i d u a l g r a i n s have s u c h a d r a s t i c e f f e c t on t h e w a t e r f l o w ? I n o r d e r t o be a b l e t o answer t h e s e q u e s t i o n s , many i n v e s t i g a t o r s s u c h as E i n s t e i n , H.A.; S a y r e , W.W.; Shen, H.W.; T o d o r o v i c , P.; C r i c k m o r e , M.J.; and L e a n , G.H.; have d e v e l o p e d v a r i o u s s t a t i s t i c a l o r s t o c h a s t i c p r o c e s s e s and m o d e l s . G e n e r a l l y , t h e movement o f s e d i m e n t p a r t i c l e on an a l l u v i a l bed i s t h e c o m b i n a t i o n o f t h e a c t i o n o f r o l l i n g , s l i d i n g , j u m p i n g and sometimes s u s p e n s i o n . When a s e d i m e n t p a r t i c l e i s d e p o s i t e d , i t i s u s u a l l y c o v e r e d by o t h e r p a r t i c l e s t h a t a r e d e p o s i t e d l a t e r . Hence, t h e m i g r a t i o n p r o c e s s c o n s i s t s o f s e r i e s o f s t e p l e n g t h and r e s t p e r i o d s i n t h e l o n g i t u d i n a l d i r e c t i o n . T h e s e two phenomena i . e . s t e p l e n g t h s and r e s t p e r i o d s a r e g o v e r n e d by a number o f f a c t o r s s u c h as w a t e r f l o w c o n d i t i o n s , t h e f l u i d and s e d i m e n t p a r t i c l e p r o p e r t i e s a t p a r t i c u l a r p o s i t i o n and t i m e . However, due t o t h e c o m p l e x i o f t h e f l o w c o n d i t i o n and t h e randomness o f l o c a l c o n d i t i o n s s u c h as bed p r o f i l e and c h a n n e l g e o m e t r y , t h e r e s t p e r i o d and s t e p l e n g t h f o r a g i v e n f l o w c o n d i t i o n a r e n o t c o n s t a n t and c a n n o t be p r e d i c t e d d e t e r m i n i s t i c a l l y . Hence, i t i s n e c e s s a r y t o c o n s i d e r b o t h t h e s t e p l e n g t h and r e s t p e r i o d as random v a r i a b l e s g o v e r n e d by c e r t a i n p r o b a b i l i s t i c l a w s . By p r o p e r c o n s i d e r a t i o n o f t h e randomness i n t h e mechanism, 15 t h e s t o c h a s t i c m o d e ls a r e c l a i m e d t o be i m p o r t a n t i n s o l v i n g t h e s e d i m e n t t r a n s p o r t p r o b l e m . As f o r a p l a n e bed, t h e r e s i s t a n c e t o t h e w a t e r f l o w i s m a i n l y f r o m t h e g r a i n r o u g h n e s s , and i s o f t e n r e f e r r e d t o as t h e s u r f a c e d r a g TQ . I f t h e b e d f o r m s e x i s t , a d d i t i o n a l s h e a r s t r e s s i s due t o t h e wavy b e d f o r m s and i s o f t e n r e f e r r e d t o as t h e f o r m d r a g TO' . As l o n g as s u r f a c e d r a g and f o r m d r a g do n o t e x e r t a m u t u a l i n f l u e n c e on one a n o t h e r , t h e t o t a l r e s i s t a n c e c a n be e x p r e s s e d as t h e sum o f t h e s u r f a c e d r a g and t h e f o r m d r a g . T h i s h a s b een d i s c u s s e d by E i n s t e i n , e t a l . ( 1 9 5 8 ) . M a t h e m a t i c a l l y , t h i s means: T 0 = T ; . + T ; - = YS + (i.D where and R^ ' a r e t h e h y d r a u l i c r a d i i due t o t h e g r a i n r o u g h n e s s and b e d f o r m s . F u r t h e r m o r e , t h e s h e a r v e l o c i t y c a n be e x p r e s s e d by: u3,= i i ; 2 + u ; 2 = ^ + ^ = ^- U.2) * * * p- P p I n o r d e r t o i n v e s t i g a t e how t h e b e d f o r m f e a t u r e s c hange t h e w a t e r f l o w r e s i s t a n c e , R a u d k i v i ( 1 9 6 7 ) , p r e s e n t e d demon-s t r a t i o n as t o how t h e t o t a l s h e a r s t r e s s i g v a r i e s w i t h i n c r e a s e s i n t h e mean w a t e r f l o w v e l o c i t y V. The s u r f a c e d r a g TQ , c o r r e s p o n d i n g t o t h e p l a n e bed r e s i s t a n c e , i s shown w i t h a d a s h e d l i n e i n F i g . 1, and t h e d i f f e r e n c e TQ-TQ r e p r e s e n t s t h e f o r m d r a g "t" . The d a t a comes f r o m t h e w a t e r f l u m e e x p e r i m e n t s w i t h s a n d o f mean d i a m e t e r 1.0 mm. The v a r i a t i o n o f f r i c t i o n f a c t o r f w i t h r e s p e c t t o t h e v a r i a t i o n o f mean w a t e r f l o w v e l o c i t y V i s a l s o shown. B o t h s h e a r s t r e s s p l o t 16 and f r i c t i o n f a c t o r p l o t show t h a t s e v e r a l v a l u e s o f mean w a t e r f l o w v e l o c i t i e s a r e p o s s i b l e f o r a g i v e n v a l u e o f t o t a l s h e a r s t r e s s o r f r i c t i o n f a c t o r . T h i s i n d i c a t e s t h a t t h e t o t a l s h e a r s t r e s s TQ a l o n e o r f r i c t i o n f a c t o r f a l o n e c a n n o t d e s c r i b e t h e f l o w r e s i s t a n c e a d e q u a t e l y i f t h e b e d f o r m s a r e p r e s e n t . A r e l a t i o n t h a t c o r r e l a t e d t h e a v a i l a b l e d a t a r a t h e r w e l l was g i v e n by Simons e t a l ( 1 9 6 1 ) , a s : f . „ (, (!!^o + b ) V ^ ^ ^ where a,b,c and e a r e c o n s t a n t s t o be d e t e r m i n e d e x p e r i m e n t a l l y , and 0 means "a funCfe.io.xiof" . R i c h a r d s o n e t a l ( 1 9 6 7 ) , u s e d t h e v e l o c i t y d i s t r i b u t i o n law and d e t e r m i n e d t h e e m p i r i c a l v a l u e s w i t h l a b o r a t o r y w a t e r f l u m e and f i e l d d a t a . By C h e z y e q u a t i o n and D a r c y ' s f o r m u l a : 8^77= C f f i = V / ^ ( 1 . 4 ) where C i s t h e C h e z y c o e f f i c i e n t and t h e f o l l o w i n g w a t e r f l o w r e s i s t a n c e e q u a t i o n s a r e s u g g e s t e d : ( 1 ) F o r a p l a n e bed w i t h l i t t l e o r no s e d i m e n t t r a n s p o r t , c/fi= 5.9 log10 (Y/d85) + 5.44 ( 1 . 5 ) ( 2 ) F o r p l a n e bed w i t h a p p r e c i a b l e s e d i m e n t t r a n s p o r t , C//9 = 7 , 4 1 O g 1 0 W d 8 5 J ( 1 - 6 ) (3) F o r r i p p l e s ( i n E n g l i s h u n i t s ) , 0.3 0.13  C / J ¥ = (7.66 - U* ) l o g 1 0 Y + U* + 11 (1.7) ( 4 - ) F o r dunes and a n t i d u n e s , c//g = 7.4 log { * \ 5 )'•(! - M ) ^ __ ( 1 . 8 ) 17 where A R h S i s t h e i n c r e a s e o f R hS due t o t h e f o r m r o u g h n e s s , and i s g i v e n by F i g . 2 and F i g . 3. E q u a t i o n s ( 1 . 5 ) , ( 1 . 6 ) , ( 1 . 7 ) , and (1.8) show t h a t t h e w a t e r f l o w d e p t h Y p l a y s an i m p o r t a n t r o l e i n d e t e r m i n i n g t h e w a t e r f l o w r o u g h n e s s . Q u e s t i o n s r e m a i n i n g t o be answered f r o m R a u d k i v i ' s d e m o n s t r a t i o n in^ F i g . 1, and f r o m R i c h a r d s o n ' s e q u a t i o n s a r e : (1) What i s t h e e f f e c t o f w a t e r d e p t h on w a t e r f l o w r e s i s t a n c e ? and, (2) How c a n t h e s h e a r s t r e s s be measured a c c u r a t e l y ? P r e s t o n t u b e measurements o f s h e a r s t r e s s a r e d i f f i c u l t and p e r h a p s n o t f u l l y j u s t i f i e d f o r r o u g h b o u n d a r i e s . S h e a r s t r e s s e v a l u a t e d f r o m w a t e r f l o w c h a n n e l s l o p e i s d i f f i c u l t t o measure i n t h e l a b o r a t o r y w a t e r f l u m e s , b e c a u s e t h e l e n g t h o f e x p e r i m e n t a l w a t e r f l u m e i s l i m i t e d . F i e l d v a l u e s o f s l o p e r e p r e s e n t n o t j u s t b e d f o r m e f f e c t s , b u t a l s o v a r i a t i o n s i n c h a n n e l w i d t h and d e p t h , w h i c h c a n be c o n s i d e r a b l e i n m e a n d e r i n g s a n d c h a n n e l s . N e i t h e r t h e mean w a t e r f l o w v e l o c i t y V, n o r t h e s h e a r s t r e s s T q o r s h e a r v e l o c i t y U^ ., i s s u f f i c i e n t t o d e s c r i b e t h e w a t e r f l o w c o m p l e t e l y . Thus, t h e e f f e c t s o f w a t e r d e p t h and s e d i m e n t p r o p e r t i e s s h o u l d be t a k e n i n t o c o n s i d e r a t i o n . T h i s r e v i e w o f t h e l i t e r a t u r e i n d i c a t e s t h a t t h e t r u e n a t u r e o f b e d f o r m g r o w t h i s s t i l l n o t u n d e r s t o o d , d e s p i t e t h e e x t e n s i v e e x p e r i m e n t a l d a t a a v a i l a b l e . The n e x t c h a p t e r w i l l e x p l o r e a p o s s i b l e p h y s i c a l e x p l a n a t i o n o f b e d f o r m d e v e l o p m e n t . 18 CHAPTER 1.5 MECHANISM OF SEDIMENT TRANSPORT BY RIPPLES  AND DUNES The b e d f o r m f e a t u r e s c a n be c o n s i d e r e d as t h e i n t e r -a c t i o n between t h e w a t e r f l o w and t h e i n d u c e d s a n d m o t i o n . R i p p l e s a r e b e l i e v e d t o be a l m o s t i n d e p e n d e n t o f F r o u d e number and t h u s t h e r e i s no i n t e r a c t i o n between sa n d r i p p l e s and w a t e r s u r f a c e waves. On t h e o t h e r hand, F r o u d e - d u n e s a r e F r o u d e number d e p e n d e n t and t h e r e e x i s t s i n t e r a c t i o n between t h i s t y p e o f s a n d dune and t h e w a t e r s u r f a c e waves. T h i s v i e w o f r i p p l e and dune b e h a v i o r h a s b e e n d e v e l o p e d by M.C. Q u i c k and t h e f o l l o w i n g s e c t i o n s have b e e n d e r i v e d f r o m d i s c u s s i o n s and d r a f t s o f p a p e r s w h i c h a r e now i n p r e s s ( Q u i c k , 1976-1979). The mechanism o f s e d i m e n t t r a n s p o r t by r i p p l e s and dunes c a n be s u b - d i v i d e d i n t o dynamic and k i n e m a t i c r e q u i r e m e n t s . The dynamic r e q u i r e m e n t s a r e t h e demands o f t h e d y n a m i c s w h i c h p r o v i d e t h e n e c e s s a r y f o r c e s t o r e l a t e t h e w a t e r s u r f a c e wave m o t i o n s t o t h e r e q u i r e d s a n d wave k i n e m a t i c s . The k i n e m a t i c r e q u i r e m e n t s a r i s e p u r e l y f r o m t h e e q u a t i o n o f s e d i m e n t c o n t i n u i t y and t h u s e x p l a i n t h e demands f o r a s a n d wave f o r m a t i o n and m i g r a t i o n . The f o r c e a c t i n g on i n d i v i d u a l s a n d p a r t i c l e s o r e v e n t h e a v e r a g e f o r c e a c t i n g o v e r a r e g i o n o f s a n d bed i s n o t e a s i l y m e a s u r a b l e , t h u s , t h e dynamic r e q u i r e m e n t s a r e v e r y complex due t o t h e c o m p l e x i t y o f mechanisms w h i c h l i n k w i t h t h e d i s t r i b u t i o n o f i n t e n s i t y o f s h e a r s t r e s s , c o n v e c t i v e a c c e l e r a t i o n and d i s s i p a t i o n w i t h i n t h e b o u n d a r y l a y e r . I n c o n t r a s t , k i n e m a t i c s c a n be p r e c i s e l y s p e c i f i e d f o r two d i m e n s i o n a l w a t e r f l o w c o n d i t i o n , and e v e n f o r t h r e e d i m e n s i o n a l w a t e r f l o w c o n d i t i o n , i t p r o v i d e s some i n f o r m a t i o n . F u r t h e r m o r e , dynamic r e q u i r e m e n t s f o r s e d i m e n t t r a n s p o r t by s e d i m e n t b e d f o r m s and t h e s p a t i a l v a r i a t i o n o f d y n a m i c s o v e r a s a n d wave c a n be i n f e r r e d f r o m t h e k i n e m a t i c s . KINEMATIC REQUIREMENTS.' "'_[' BRIEF DESCRIPTION OF EXNER'S MODEL: E x n e r , P.F. ( 1 9 2 5 ) , assumed t h a t t h e c a p a c i t y o f t h e w a t e r f l o w t o t r a n s p o r t s e d i m e n t depends on i t s v e l o c i t y . E r o s i o n o c c u r s i f t h e f l o w v e l o c i t y i n c r e a s e s and d e p o s i t i o n o c c u r s i f i t d e c r e a s e s i n t h e downstream d i r e c t i o n , t h u s , 3 V 3\% • b o t h s c o u r and d e p o s i t i o n depend on /•£ x , and i f ' /9t r e p r e s e n t s t h e r a t e o f s c o u r , w h i c h l e a d s t o t h e E x n e r 1 s e r o s i o n e q u a t i o n : - . 9v n q) where a E i s t h e E x n e r e r o s i o n c o e f f i c i e n t , ft i s t h e b e d f o r m h e i g h t above a common-datum. C o n t i n u i t y g i v e s Q = BV (h - ft) (1.10) where Q i s w a t e r f l o w r a t e , B i s t h e c h a n n e l w i d t h and h i s w a t e r d e p t h above common datum. F u r t h e r m o r e , assume t h e i n i t i a l b e d f o r m i s g i v e n by a c o s i n e f u n c t i o n , \ = A Q + A ] C o s ( 2 T T X / L r ) (1.11) 20 L A A where R i s b e d f o r m w a v e l e n g t h and 1 i s i t s a m p l i t u d e , 0 i s an a r b i t r a r y c o n s t a n t . C o m b i n i n g t h e s e t h r e e e q u a t i o n s ( 1 . 9 ) , ( 1 . 1 0 ) , and ( 1 . 1 1 ) , t h e g e n e r a l i z e d e q u a t i o n o f b e d f o r m f o r any g i v e n t i m e t c a n be o b t a i n e d : ^ = A o + A l C 6 S t <X " BTnTT~ ( 1- 1 2) K PURE KINEMATIC MODEL E x n e r ' s model i s n o t a t r u l y k i n e m a t i c model, he r e l a t e d b e d f o r m t o w a t e r v e l o c i t y v a r i a t i o n . Q u i c k ( 1 9 7 8 ) , s u g g e s t s a p u r e l y k i n e m a t i c model, and t h i s k i n e m a t i c b e h a v i o r d i c t a t e s t h e n e c e s s a r y t r a n s p o r t r a t e s on v a r i o u s p a r t s o f a r i p p l e o r dune and t h e dynamic b e h a v i o r must t h e n s a t i s f y t h e demands o f t h e k i n e m a t i c r e q u i r m e n t s . The f o l l o w i n g s e c t i o n i s a d a p t e d f r o m p r e l i m i n a r y d r a f t s o f a p a p e r by Q u i c k ( 1 9 7 8 ) . I t i s assumed t h a t s e d i m e n t t r a n s p o r t r a t e ^ s has t h e f o l l o w i n g r e l a t i o n s h i p w i t h b e d f o r m h e i g h t ^ above common datum, t h a t i s , qS = - k 1 n I* — — — fe (1.13) D i f f e r e n t i a t i n g and s u b s t i t u t i n g e q u a t i o n (1.5) i n t o e q u a t i o n o f s e d i m e n t c o n t i n u i t y 21 The f o l l o w i n g e q u a t i o n c a n be o b t a i n e d : nk-f" 1 :jl + |4 .„ 5 ax 9t v ' T h i s y i e l d s , dx _ n |, h n- l \= CONSTANT, ALONG dt i- (1.16) E q u a t i o n (1.16) l e a d s t h r e e p o s s i b l e c o n d i t i o n s . (1) S t e a d y k i n e m a t i c c o n d i t i o n : When n = l , b o t h d x / d t and b e d f o r m wave h e i g h t ^ a r e c o n s t a n t s due t o s t e a d y b e h a v i o r , a l l p o r t i o n s o f sand bedwave w i l l p r o p a g a t e a t c o n s t a n t s p e e d a l o n g t h e c h a n n e l and t h e b e d f o r m shape w i l l n o t be m o d i f i e d . (2) U n s t e a d y k i n e m a t i c c o n d i t i o n : When n > l , t h e k i n e m a t i c wave s p e e d d x / d t o f t h e s a n d bedwave i n c r e a s e s w i t h i n c r e a s i n g s a n d wave h e i g h t . T h i s i n d i c a t e s s e d i m e n t t r a n s p o r t r a t e i s h i g h e r a t t h e c r e s t s t h a n a t t h e t r o u g h s o f t h e b e d f o r m waves. T h e r e f o r e s a n d wave c r e s t s move t h e f a s t e s t and s a n d waves t e n d t o f l a t t e n o u t . T h i s r e s u l t does n o t a g r e e w i t h E x n e r 1 s argument. When n < l , t h e k i n e m a t i c s a n d wave s p e e d d x / d t d e c r e a s e s w i t h i n c r e a s i n g s a n d h e i g h t . I n t h i s c a s e , t h e s e d i m e n t b e d f o r m wave c r e s t s t r a v e l much s l o w e r , t h u s t h e s a n d wave s t e e p e n s and grows. (3) When n = 0, e q u a t i o n s (1.13) and ( 1 . 1 6 ) , g i v e s e d i m e n t t r a n s p o r t r a t e ^ s e q u a l s t o t h e v a l u e o f k, and t h e k i n e m a t i c wave s p e e d d x / d t e q u a l s t o z e r o . T h a t i n d i c a t e s s e d i m e n t r a n s p o r t r a t e ^ s w i l l keep c o n s t a n t o v e r e v e r y s e c t i o n o f t h e b e d f o r m s and s t a n d i n g a n t i d u n e s e x i s t . 22 D u r i n g t h e f o r m a t i o n o f a s a n d wave, t h e v a l u e o f n i s s m a l l e r t h a n one, and w i l l g r a d u a l l y a p p r o a c h one as t h e sand wave s t e e p e n s t o i t s f u l l y d e v e l o p e d c o n d i t i o n . When F r o u d e number i s l a r g e r t h a n a b o u t 0.7, t h e b e d f o r m s become f l a t a g a i n b e c a u s e t h e v a l u e o f n becomes g r e a t e r t h a n one. When t h e v a l u e o f n e q u a l s one, t h e n t h e r e e x i s t s a c o n s t a n t shape f o r s e d i m e n t b e d f o r m s . The k i n e m a t i c r e q u i r e m e n t s must e x i s t i f bedwaves a r e t o form; i t now r e m a i n s t o be s e e n w h e t h e r t h e f l u i d - s e d i m e n t d y n a m i c s c a n p r o v i d e t h e s e k i n e m a t i c r e q u i r e m e n t s . DYNAMIC REQUIREMENTS The s e d i m e n t b e d l o a d t r a n s p o r t i n r i p p l e s a n d / o r dunes i s shown t o depend m a i n l y on t h e c o n v e c t i v e a c c e l e r a t i o n o f t h e f l o w and t h e i n t e n s i t y o f s h e a r s t r e s s n e a r t h e b o u n d a r y . I n s t e a d y u n i f o r m f l o w , t h e b o u n d a r y l a y e r i s t h i c k , t h i s t h i c k and l e s s e n e r g e t i c b o u n d a r y l a y e r i s n o t v e r y C o n d u c i v e t o s e d i m e n t t r a n s p o r t . I f t h e w a t e r f l o w c a n be made t o a c c e l e r a t e , t h e h i g h e n e r g y w a t e r f l o w w i l l be mixed deep i n t o t h e b o u n d a r y l a y e r , and t h e n t h e b o u n d a r y l a y e r w i l l become t h i n and e n e r g e t i c , w h i c h w i l l p r o d u c e much more i n t e n s e s h e a r s t r e s s e s and p r o v i d e t h e n e c e s s a r y dynamic r e q u i r e m e n t s f o r t h e s e d i m e n t t r a n s p o r t . I n F i g . A , t h e dune i s shown as a s i m p l i f i e d c a s e o f a t r i a n g u l a r wave f o r m t r a n s l a t i n g i n a w a t e r c h a n n e l , t h e i n i t i a l p o s i t i o n a f t e r t r a n s l a t i n g t h r o u g h a d i s t a n c e 6x downstream a r e a l s o shown i n t h i s f i g u r e , t h e s e d i m e n t f r o m t h e u p s t r e a m s l o p e h a s b e en e r o d e d and d e p o s i t e d i n t h e down-s t r e a m t o e r e g i o n . The u p s t r e a m s l o p e i s marked o f f i n f i v e e q u a l d i s t a n c e i n c r e m e n t s numbered one t o f i v e . I t w i l l be r e a l i z e d t h a t t h e s e d i m e n t f l o w r a t e p a s t e a c h s e c t i o n must i n c r e a s e , so t h a t f i v e t i m e s t h e s e d i m e n t w i l l p a s s t h e s e c t i o n f i v e . The i n c r e a s e o f s e d i m e n t t r a n s p o r t has b e en e x p r e s s e d by E x n e r and o t h e r a u t h o r s . The q u e s t i o n r e m a i n s t o be a nswered i s how c a n t r a n s p o r t r a t e on t h e u p s t r e a m s l o p e i n c r e a s e c o n t i n u o u s l y f r o m a b o u t z e r o a t t r o u g h t o some q u i t e h i g h v a l u e a t t h e c r e s t o f dune. Q u i c k , M.C. s u g g e s t e d i n h i s s t u d y t h a t t h e mechanism f o r i n c r e a s i n g t h e s e d i m e n t t r a n s p o r t r a t e ^ s on t h e u p s t r e a m f a c e o f a F r o u d e - d u n e i s t h e c o n v e c t i v e 7 a c c e l e r a t i o n o f t h e w a t e r f l o w w h i c h i s i n d u c e d by dunes o r r i p p l e s . I n o t h e r words, t h e dunes and r i p p l e s p r o v i d e t h e c o n d i t i o n s f o r r e - e n e r g i z i n g t h e b o u n d a r y l a y e r . The i n d u c e d c o n v e c t i v e a c c e l e r a t i o n i n t h e b o u n d a r y l a y e r between w a t e r f l o w and s a n d bed i s t h e b a s i c mechanism f o r p r o d u c i n g t h i n , e n e r g e t i c b o u n d a r y l a y e r s w h i c h a r e n e c e s s a r y t o p r o d u c e h i g h b o u n d a r y s h e a r s t r e s s e s . T h e s e h i g h s h e a r s t r e s s e s a r e r e s p o n s i b l e f o r t h e h i g h t r a n s p o r t r a t e on t h e u p s t r e a m f a c e i , f.of t h e dune. S i n c e t h e s h e a r s t r e s s n e a r t h e b o u n d a r y l a y e r i s p r o -p o r t i o n a l t o t h e c o n v e c t i v e a c c e l e r a t i o n , and t h u s s e d i m e n t t r a n s p o r t r a t e , i t i s a r g u e d t h a t s h e a r s t r e s s n e a r t h e b o u n d a r y l a y e r must a l s o i n c r e a s e w i t h i n c r e a s i n g b e d f o r m h e i g h t t o g i v e * 24 i n c r e a s i n g s e d i m e n t t r a n s p o r t r a t e a t t h e u p s t r e a m f a c e o f du n e s . Sheen ( 1 9 6 4 ) , c a r r i e d o u t t h e measurements o f w a t e r f l o w v e l o c i t y d i s t r i b u t i o n and d i s t r i b u t i o n s o f t u r b u l e n t f l u c t u a t i n g v e l o c i t y components u', v ' , and w' i n t h e w a t e r f l o w o v e r r i p p l e d s a n d b e d . F i g . 5A and 5B show t h e i r r e s u l t s a t v a r i o u s s t a t i o n s a l o n g t h e r i p p l e . From t h e d i s t r i b u t i o n o f w a t e r f l o w v e l o c i t y n e a r t h e b o u n d a r y , i t i s shown t h a t c o n v e c t i v e a c c e l e r a t i o n i n c r e a s e s w i t h t h e i n c r e a s e o f t h e b e d f o r m h e i g h t \ . R a u d k i v i ( 1 9 6 3 ) , t o o k measurements o f t e m p o r a l mean s h e a r s t r e s s o v e r a sandwave b y P r e s t o n ' s s h e a r s t r e s s measurement t e c h n i q u e . I t i s f o u n d t h a t maximum s h e a r s t r e s s n e a r t h e s a n d bedwave o c c u r s a t t h e c r e s t p o i n t . Thus, t h e s e e a r l y e x i s t i n g e x p e r i m e n t a l r e s u l t s a r e i n v e r y good agreement w i t h t h e p r e s e n t argument. The b e d f o r m p r o f i l e was measured f r o m t h e p l e x i g l a s s w a l l o f l a b o r a t o r y f l u m e w h i l e c a r r y i n g o u t t h e e x p e r i m e n t s . F o r example, F i g . 6 shows one o f p o r t a g e s i l i c a s a n d dune f r o m p r e s e n t e x p e r i m e n t a l r e s u l t s , w h i c h m i g r a t e s downstream w i t h a s t a b l e f o r m . I t i s a l s o w o r t h w h i l e t o make a b r i e f d e s c r i p t i o n h e r e a b o u t P r e s t o n ' s s h e a r s t r e s s measurement t e c h n i q u e , w h i c h c o n s i s t s o f p l a c i n g a p i t o t t u b e i n c o n t a c t w i t h a w a l l and i n t e r p r e t i n g t h e dynamic p r e s s u r e r e a d i n g o b t a i n e d as a measure o f t h e l o c a l s h e a r on t h e w a l l . The r a t i o n a l e o f t h e t e c h n i q u e depends on wh e t h e r i t c a n be assumed t h a t t h e v e l o c i t y d i s t r i b u t i o n n e a r t h e r o u g h b o u n d a r y depends o n l y on t h e v i s c o s i t y and d e n s i t y o f t h e f l u i d , t h e r o u g h n e s s o f t h e bounda and t h e s h e a r s t r e s s a t t h e w a l l . Under t h i s c o n s i d e r a t i o n , 25 K a r m a n - P r a n d t l v e l o c i t y d i s t r i b u t i o n i s c o n s i d e r e d t o be a s a t i s f a c t o r y a p p r o x i m a t i o n f o r p i p e s o r c h a n n e l s , and an a n a l y s i s was p e r f o r m e d f o r t h e f u l l y r o u g h t f l o w r e g i m e t h a t g i v e s t h e r a t i o o f t h e dynamic p r e s s u r e r e a d i n g t o t h e w a l l s h e a r as a f u n c t i o n o f t h e s i z e o f t h e t u b e and o f t h e r o u g h -n e s s e l e m e n t . The dynamic p r e s s u r e r e a d i n g i n t h e t u b e i s d e t e r m i n e d by t h e v e l o c i t y d i s t r i b u t i o n and t u b e s i z e . However, t h i s a n a l y s i s assumes t h a t t h e f l o w i s i n t h e c o m p l e t e l y r o u g h r e g i m e , i n h y d r a u l i c a l l y smooth and t r a n s -i t i o n a l r e g i m e s , ,a i c o r r e c t i o n f a c t o r f o r t h i s a n a l y s i s c a n be o b t a i n e d and p r e s e n t e d as an e m p i r i c a l f o r m u l a t i o n b a s e d on t h e e x p e r i m e n t a l d a t a . A l t h o u g h some s h o r t c o m i n g s r e m a i n , s u c h as i n a d e q u a t e knowledge o f f l o w n e a r r o u g h b o u n d a r i e s and c o r r e c t f o r m o f t h e v e l o c i t y d i s t r i b u t i o n , r e s u l t s were o b t a i n e d w h i c h a t l e a s t s a t i s f i e d R a u d k i v i t h a t P r e s t o n ' s t e c h n i q u e c o u l d be u s e d s u c c e s s f u l l y f o r t h e r o u g h b o u n d a r i e s . F u r t h e r work i s r e q u i r e d t o i n v e s t i g a t e t h i s measurement o f s h e a r s t r e s s a t a r o u g h :'. w a l l and t o c h e c k t h e u s e o f a P r e s t o n t u b e u n d e r t h e s e c o n d i t i o n s . 26 CHAPTER 1.4 EXPERIMENTAL TECHNIQUE AND ANALYSIS OF  EXPERIMENTAL RESULTS EXPERIMENTAL . •TECHNIQUE, The e x p e r i m e n t s o f p r e s e n t s t u d y were c a r r i e d o u t a t t h e H y d r a u l i c E n g i n e e r i n g L a b o r a t o r y , C i v i l E n g i n e e r i n g D e p a r t m e n t o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , w i t h a u n i d i r e c t i o n a l w a t e r f l u m e o f 920 cm l o n g , 16 cm wide and 45 cm deep. F i g 7 r e p r e s e n t s t h e b a s i c d e s i g n o f t h e e n t i r e e x p e r i m e n t a l a p p a r a t u s . Water i s r e c i r c u l a t e d f r o m t h e w a t e r t a n k t o t h e w a t e r f l u m e by two s e p a r a t e d pumps and d e l i v e r y p i p e l i n e s . The c o n t r o l g a t e o p e n i n g a t t h e e x i t o f t h e w a t e r f l u m e and t h e s e t t i n g s f o r t h e c o n t r o l v a l v e s o f t h e two pumps on t h e w a t e r d e l i v e r y p i p e l i n e s a r e a d j u s t e d m a n u a l l y . Thus, t h e w a t e r f l o w r a t e o r t h e w a t e r f l o w d e p t h and v e l o c i t y c a n be a d j u s t e d b y s i m u l t a n e o u s l y a l t e r i n g t h e s i z e s o f t h e o p e n i n g o f t h e c o n t r o l g a t e a t t h e e x i t o f t h e w a t e r f l u m e and t h e s e t t i n g s f o r t h e c o n t r o l v a l v e s on t h e w a t e r d e l i v e r y p i p e l i n e s . S e d i m e n t p a r t i c l e s a r e a l s o r e c i r c u l a t e d f r o m t h e w a t e r t a n k t o t h e w a t e r f l u m e b y pump number 2, t h e i n t a k e o f w h i c h i h a s a s e d i m e n t t r a p and w e i r w h i c h d i r e c t s s e d i m e n t p a r t i c l e s t o t h e pump. A l a r g e s a n d t r a p was p l a c e d immed-i a t e l y b e l o w t h e c o n t r o l g a t e o f t h e w a t e r f l u m e and, t h u s , t h e number o f s e d i m e n t p a r t i c l e s w h i c h would be a b l e t o g e t i n t o t h e w a t e r pump number 1 i s l i m i t e d . T h e r e f o r e , t h e w a t e r pump number 1 c a n o p e r a t e s u c c e s s f u l l y w i t h o u t b e i n g damaged by s e d i m e n t , f o r w h i c h i t i s n o t d e s i g n e d . 27 One o f t h e b a s i c r e q u i r e m e n t s t o o b t a i n s i g n i f i c a n t e x p e r i m e n t a l r e s u l t s i s t h a t t h e w a t e r f l o w o v e r t h e s e d i m e n t c h a n n e l b e d must be n e a r l y u n i f o r m and s t e a d y , t h u s any a b r u p t i n c r e a s i n g o r d e c r e a s i n g w a t e r f l o w r a t e must be a v o i d e d . I n a d d i t i o n , a t w a t e r f l u m e e n t r a n c e , a w a t e r f l u m e e n t r y l e n g t h o f a b o u t two m e t e r s i n l e n g t h w i l l be r e q u i r e d , where' t h e w a t e r f l o w c a n be r e g u l a t e d i n t o a s t e a d y u n i f o r m f l o w . I t i s n o t i c e d t h a t t h e r e q u i r e d f l u m e e n t r y l e n g t h i n c r e a s e s w i t h t h e i n c r e a s i n g o f t h e w a t e r f l o w r a t e . I n t h e p r e s e n t s t u d y , a honeycomb was p l a c e d i n t h e w a t e r f l u m e e n t r a n c e t o a i d i n r e g u l a t i n g t h e i n f l o w w a t e r f l o w t o be s t e a d y and u n i f o r m . The honeycomb removes s p i n f r o m t h e w a t e r and dampens l a r g e s c a l e t u r b u l e n c e . S e d i m ent b e d f o r m d i m e n s i o n s c a n be m easured e i t h e r d i r e c t l y f r o m s e d i m e n t c h a n n e l b e d o r t h r o u g h t h e p l e x i - g l a s s w a l l s o f t h e w a t e r f l u m e . F o r e a c h r u n o f t h e e x p e r i m e n t , s e v e r a l measurements were t a k e n and t h e a v e r a g e v a l u e were u s e d as t h e e x p e r i m e n t a l r e s u l t i f , and o n l y i f , t h e d i f f e r e n c e s b etween t h e s e t h r e e o r f o u r measurements a r e a p p r o x i m a t e l y w i t h i n 10%. Water f l o w v e l o c i t i e s i n t h e v i c i n i t y o f s e d i m e n t b e d f o r m c r e s t s were measured by an 0TT-C2 m e t e r w i t h a m i n i a t u r e p r o p e l l e r . The r o t a t i n g s p e e d o f t h e m i n i a t u r e p r o p e l l e r i s p r o p o r t i o n a l t o t h e w a t e r f l o w v e l o c i t y . A s t o p w a t c h was u s e d i n d e t e r m i n i n g t h e t i m e r e q u i r e d f o r t h e m i g r a t i o n downstream o f t h e s e d i m e n t b e d f o r m o v e r a c e r t a i n d i s t a n c e , t h u s t h e b e d f o r m m i g r a t i o n v e l o c i t y c a n be e v a l u a t e d . 28 T h r e e t y p e s o f s e d i m e n t p a r t i c l e s were u s e d t o c a r r y o u t t h e p r e s e n t e x p e r i m e n t s . They a r e (1) P o r t a g e S i l i c a s a n d o f w h i c h P s=2.65 and d 5 Q=0.2mm; (2) F r a s e r r i v e r s a n d o f w h i c h p =2.66 and d._ =0.33 mm; and (3) B a k e l i t e s a n d o f s 50 w h i c h p =1.33 and d,_ =0.89 mm. S i e v i n g was u s e d t o d e t e r m i n e s 50 a t h e g r a i n s i z e d i s t r i b u t i o n s . The s e d i m e n t p a r t i c l e sample was p a s s e d t h r o u g h a s e r i e s o f U.S. s t a n d a r d t e s t s i e v e s h a v i n g s u c c e s s i v e l y s m a l l e r mesh s i z e s . The w e i g h t o f san d r e t a i n e d i n e a c h s i e v e i s d e t e r m i n e d and t h e c u m u l a t i v e p e r c e n t a g e by w e i g h t p a s s i n g e a c h s i e v e i s c a l c u l a t e d . F i g . 8, 9 and 10, show t h e g r a i n s i z e d i s t r i b u t i o n s o f t h e s e t h r e e t y p e s o f s e d i m e n t p a r t i c l e s u s e d i n p r e s e n t s t u d y . A l l p r e s e n t e x p e r i m e n t s a r e r e s t r i c t e d i n s u b - c r i t i c a l f l o w r a n g e and t h e r e s u l t s a r e t a b u l a t e d i n A p p e n d i x I . DIMENSIONAL.. ANALYSIS D i m e n s i o n a l a n a l y s i s i s a w i d e l y u s e d t e c h n i q u e i n d i s c u s s i n g t h e p r o b l e m o f b e d f o r m s f o r m a t i o n and s e d i m e n t t r a n s p o r t . I t c a n be u s e d t o s y s t e m a t i c a l l y r e a r r a n g e and r e o r g a n i z e t h e v a r i a b l e s w h i c h a r e measured d u r i n g t h e e x p e r -i m e n t s and o b t a i n some c o n v e n i e n t , i n d e p e n d e n t and n o n d i m e n s i o n a l p a r a m e t e r s f o r t h e p u r p o s e o f s t u d y i n g e x p e r i m e n t a l r e s u l t s . The a d v a n t a g e o f t h e n o n d i m e n s i o n a l p a r a m e t e r s d e v e l o p e d f r o m d i m e n s i o n a l a n a l y s i s i s t h a t i t i s p o s s i b l e t o i n d i c a t e t h e v a r i a t i o n o f b e d f o r m d i m e n s i o n s w i t h r e s p e c t t o t h e v a r i a t i o n o f one o f t h e s e d i m e n s i o n l e s s p a r a m e t e r s and keep 29 t h e r e s t o f them c o n s t a n t . On t h e o t h e r hand, t h e d i s a d v a n t a g e o f t h e s e d i m e n s i o n l e s s p a r a m e t e r s i s t h a t i t m i g h t n o t be v e r y e a s y t o i n d i c a t e t h e e f f e c t s o f e a c h i n d i v i d u a l v a r i a b l e , s u c h a s w a t e r d e p t h Y, w a t e r f l o w v e l o c i t y V, on t h e b e d f o r m d i m e n s i o n s b e c a u s e most o f t h e s e d i m e n s i o n l e s s p a r a m e t e r s f r o m d i m e n s i o n a l a n a l y s i s a r e c o m b i n a t i o n s o f t h e s e i n d i v i d u a l v a r i a b l e s . Thus, i n t h e p r e s e n t s t u d y , n o t o n l y t h e d i m e n s i o n l e s s p a r a m e t e r s d e v e l o p e d f r o m d i m e n s i o n a l a n a l y s i s , b u t a l s o many i n d i v i d u a l d i m e n s i o n a l v a r i a b l e s a r e u s e d t o f i n d o u t t h e i r own e f f e c t s on t h e b e d f o r m d i m e n s i o n s . S i n c e b e d f o r m f o r m a t i o n and s e d i m e n t t r a n s p o r t c a n be i l l u s t r a t e d by t h e p r o p e r t i e s o f w a t e r f l o w , s e d i m e n t , b e d f o r m m i g r a t i o n v e l o c i t y , and shape o f t h e open c h a n n e l , many v a r i a b l e s s h o u l d be c o n s i d e r e d t o c h a r a c t e r i z e t h e phenomena o f b e d f o r m f o r m a t i o n and s e d i m e n t t r a n s p o r t . V a r i a b l e s n e e d ed t o c h a r a c t e r i z e t h e w a t e r f l o w s a r e w a t e r f l o w v e l o c i t y V and w a t e r d e p t h Y. The v a r i a b l e s n e e d e d t o c h a r a c t e r i z e t h e f l u i d a r e i t s t e m p e r a t u r e , d e n s i t y , k i n e m a t i c v i s c o s i t y v , and g r a v i t a t i o n a l a c c e l e r a t i o n g. The v a r i a b l e s n e e d e d t o c h a r a c t e r i z e t h e s e d i m e n t and t h e t r a n s p o r t a t i o n o f b e d f o r m waves a r e s e d i m e n t p r o p e r t i e s s u c h as p , 50, and C b e d f o r m m i g r a t i o n v e l o c i t y B. I n t h e p r e s e n t s t u d y , c h a n n e l w i d t h B, and t h u s t h e shape o f open c h a n n e l , and w a t e r t e m p e r a t u r e a r e assumed t o be c o n s t a n t s . Thus, q u a n t i t a t i v e p r o p e r t y o f t h e phenomena o f 30 b e d f o r m wave f o r m a t i o n and t r a n s p o r t a t i o n may be d e s c r i b e d by, Q = f (v, P , P s , d 5 Q , V, Y, C B , g) ( i.i7) S i n c e t h r e e p h y s i c a l d i m e n s i o n s , mass, l e n g t h , and t i m e a r e i n v o l v e d and hence f o u r IT - numbers c a n be o b t a i n e d w h i c h a r e i n d e p e n d e n t and d i m e n s i o n l e s s g r o u p s o f v a r i a b l e s . D i m e n s i o n a l a n a l y s i s y i e l d s , 2 C K d r n pV 0.5 Q = f • (*-• » • (- ) ) ( U8) " ^ » t v . : < , d K where Ps'/p d e n o t e s t h e s e d i m e n t d e n s i t y , v ^ g y d e n o t e s F r o u d e number N r a t t h e c r e s t p o i n t o f dunes, 0_. d,.^ . , and v d._,_ . f ^ B 50/v 50/ v a r e m o d i f i e d p a r t i c l e R e y n o l d ' s numbers i n t e r m s o f b e d f o r m V 2 ° - 5 m i g r a t i o n v e l o c i t y and w a t e r f l o w v e l o c i t y , (-, 2 - r — - j — ) ( V p ) g d 5 0 d e f i n e s m o d i f i e d F r o u d e number. Among t h e s e f i v e d i m e n s i o n -l e s s p a r a m e t e r , s i n c e o n l y a s m a l l r a n g e o f w a t e r d e p t h s a r e t e s t e d d u r i n g c a r r y i n g o u t e x p e r i m e n t s o f p r e s e n t s t u d y , 2 0 5 v/JgT, V d 5 Q / v , and (pv / ( p s - p ) g d 5 Q ) ' a r e l i n e a r l y r e l a t e d w i t h e a c h o t h e r , t h u s , t h e v a r i a t i o n s o f b e d f o r m d i m e n s i o n s a r e s t u d i e d o n l y w i t h r e s p e c t t o t h e v a r i a t i o n s o f v/JgT and C„d^^/v , w h i c h a r e d e f i n e d as N,. and R_,D i n p r e s e n t s t u d y . B 50 r Ot> I n a d d i t i o n , s i n c e u s i n g F r o u d e number and R e y n o l d ' s number c a n be m i s l e a d i n g and c o n f u s i n g u n l e s s a l a r g e r a n g e o f w a t e r d e p t h c a n be t e s t e d w h i l e c a r r y i n g o u t t h e e x p e r i m e n t s , t h e s i g n i f i c a n t v a r i a b l e s c o u l d be d i m e n s i o n a l v a r i a b l e s s u c h as w a t e r f l o w v e l o c i t y V, w a t e r d e p t h Y, r a t h e r t h a n F r o u d e number o r R e y n o l d ' s number. Thus, i n p r e s e n t s t u d y t h e v a r i a t i o n s o f t h e b e d f o r m d i m e n s i o n s w i t h r e s p e c t t o t h e v a r i a t i o n s o f some d i m e n s i o n a l v a r i a b l e s a r e a l s o s t u d i e d . 31 A number o f e x p e r i m e n t s h a ve b e e n c a r r i e d o u t t o i n v e s t i g a t e t h e r e l a t i o n s h i p s b etween t h e b e d f o r m d i m e n s i o n s and e i t h e r d i m e n s i o n l e s s p a r a m e t e r f r o m d i m e n s i o n a l a n a l y s i s o r d i m e n s i o n a l v a r i a b l e s s u c h as w a t e r f l o w v e l o c i t y V, w a t e r c d e p t h Y, o r b e d f o r m m i g r a t i o n v e l o c i t y B. A l t h o u g h , most e x p e r i m e n t s a r e o f s u b - c r i t i c a l r a n g e , t h e r e s u l t s a r e s t i l l v e r y e n c o u r a g i n g . DISCUSSION OF EXPERIMENTAL RESULTS B o t h s e d i m e n t p a r t i c l e movement and b e d l o a d t r a n s p o r t depend on t h e i n t e n s i t y o f bed s h e a r s t r e s s i n t h e v i c i n i t y o f t h e s e d i m e n t c h a n n e l b e d . The t r a c t i v e bed s h e a r s t r e s s a c t i n g on s e d i m e n t p a r t i c l e s i s g e n e r a l l y p r o p o r t i o n a l t o t h e 2 v a l u e o f V , so t h a t i f w a t e r f l o w v e l o c i t y i n c r e a s e s and e x c e e d s t h e c r i t i c a l v e l o c i t y , s e d i m e n t p a r t i c l e movement w i l l commence. I f t h e f l o w i s c a u s e d t o a c c e l e r a t e , e i t h e r by u n s t e a d y f l o w o r by c o n v e c t i v e a c c e l e r a t i o n o v e r a b e d f o r m , t h e b o u n d a r y l a y e r s w i l l become t h i n and e n e r g e t i c , w h i c h w i l l p r o d u c e more i n t e n s e s h e a r s t r e s s t o enhance t h e s e d i m e n t p a r t i c l e movement. The c r i t i c a l o r minimum w a t e r f l o w v e l o c i t y n e e d e d f o r t h e o n s e t o f s e d i m e n t p a r t i c l e movement w i l l c o r r e s p o n d t o a c r i t i c a l t r a c t i v e b e d s h e a r s t r e s s , and w i l l depend on t h e s e d i m e n t p r o p e r t i e s . I n t h e r a n g e o f r i p p l e s , t h e r e i s no i n t e r a c t i o n b etween s e d i m e n t c h a n n e l b e d and w a t e r s u r f a c e . A t t h i s s t a g e , w a t e r f l o w v e l o c i t y V p l a y s t h e d o m i n a n t r o l e 32 i n d e t e r m i n i n g t h e r i p p l e d i m e n s i o n s . R i p p l e d i m e n s i o n s a r e F r o u d e number i n d e p e n d e n t . I t i s p o s s i b l e t o keep a g i v e n w a t e r f l o w v e l o c i t y c o n s t a n t b u t v a r y t h e v a l u e o f t h e F r o u d e number by s i m p l y c h a n g i n g t h e w a t e r d e p t h . The v a l u e o f F r o u d e number becomes s i g n i f i c a n t i n t h e r a n g e o f F r o u d e - d u n e s . The u p s t r e a m f a c i n g s l o p e o f F r o u d e - d u n e s i n d u c e s a c o n v e c t i v e a c c e l e r a t i o n , and bed s h e a r s t r e s s . T h e s e h i g h s h e a r s t r e s s e s a r e r e s p o n s i b l e f o r t h e h i g h s e d i m e n t t r a n s p o r t r a t e on t h e u p s t r e a m f a c e o f t h e F r o u d e - d u n e s . Thus, F r o u d e number a c t s as an i n t e n -s i f i e r o f bed s h e a r s t r e s s w h i c h m o d i f i e s t h e s e d i m e n t b e d f o r m b e h a v i o r i f t h e F r o u d e number i s h i g h enough, N e s p e c i a l l y i f f e x c e e d s 0.5. Thus, t h e i n t e r a c t i o n b etween s e d i m e n t c h a n n e l b e d f o r m s and t h e w a t e r s u r f a c e e x i s t s . F r o u d e - d u n e d i m e n s i o n s a r e F r o u d e number d e p e n d e n t . When t h e F r o u d e number i n c r e a s e s and a p p r o x i m a t e l y e x c e e d s 0.7, t h e c o n v e c t i v e a c c e l e r a t i o n w h i c h i s p r o p o r -t i o n a l t o t h e bed s h e a r s t r e s s i s g e t t i n g t o be t o o l a r g e , f o r i n s t a n c e , c o n v e c t i v e a c c e l e r a t i o n w i l l a p p r o a c h an N i n f i n i t e v a l u e when f a p p r o a c h e s t o one, t h e s e d i m e n t c h a n n e l b e d w i l l become f l a t a g a i n . The r e s u l t s o f p r e s e n t e x p e r i m e n t s were s t u d i e d and t h e r e l a t i o n s h i p s among b e d f o r m d i m e n s i o n s and d i m e n s i o n l e s s p a r a m e t e r s o r v a r i o u s d i m e n s i o n a l v a r i a b l e s a r e shown i n t h e f i g u r e s . 33 THE RELATIONSHIP BETWEEN f AND CB F o r p o r t a g e s i l i c a s a n d and F r a s e r R i v e r s a n d b ed, i t i s f o u n d t h a t when t h e w a t e r f l o w v e l o c i t y i s l e s s t h a n 30 cm/ s e c . , and t h e v a l u e o f F r o u d e number i s l e s s t h a n 0.2, s e d i m e n t p a r t i c l e s do n o t move a t a l l . I t must be remembered t h a t i f t h e w a t e r d e p t h was l a r g e enough, a F r o u d e number o f 0.2 c o u l d c o r r e s p o n d t o a h i g h e r v e l o c i t y and s e d i m e n t c o u l d move f r e e l y , i n f l u e n c e d by t h e h i g h e r s h e a r s t r e s s . When t h e f l o w i s i n c r e a s e d t o a v a l u e o f v e l o c i t y e x c e e d i n g 30 cm/sec., and w i t h t h e same d e p t h , so t h a t t h e v a l u e o f F r o u d e number i s a p p r o x i m a t e l y between 0.2 and 0.3, r i p p l e s s t a r t t o g e n e r a t e and e s t a b l i s h a s t a b l e f o r m . The b e d f o r m m i g r a t i o n v e l o c i t y B i s a b o u t z e r o a t t h i s s t a g e . The F r o u d e number v a l u e o f 0.2 t o ' 0 . 3 and mean w a t e r f l o w v e l o c i t y v a l u e o f 30 cm/sec., i m p l y t h e c r i t i c a l v e l o c i t y o r c r i t i c a l s h e a r s t r e s s a t c e r t a i n w a t e r d e p t h t o i n i t i a t e t h e s e d i m e n t p a r t i c l e movement. As F r o u d e number i n c r e a s e s and e x c e e d s 0.4 o r 0.5, e x p e r i m e n t a l r e s u l t s show t h a t F r o u d e number i s g e t t i n g t o be v e r y s i g n i f i c a n t , t h a t i s , b o t h V and Y a r e i m p o r t a n t i n d e t e r m i n i n g t h e b e d f o r m d i m e n s i o n s . I t i s f o u n d t h a t i n t h e r a n g e o f a p p r o x i m a t e l y 0 . 4 £ N f £ 0.65, l o n g e r w a v e l e n g t h dunes e x i s t and m i g r a t e downstream. F i g . 11 shows t h a t t h e r e e x i s t s a l o g a r i t h m i c r e l a t i o n s h i p b etween R C B , w h i c h i s CB^50/v and l o g ^ Q N f i n t h i s r a n g e o f dunes, t h i s l o g a r i t h m i c r e l a t i o n s h i p c a n be 34 m a t h e m a t i c a l l y r e p r e s e n t e d by, R O T D = 2.25 l o g , _ N,. + 0.99 (1.19) t h e c o n s t a n t s i n t h i s e q u a t i o n , s u c h as 2.25 o r 0.99 s h o u l d be f u n c t i o n s o f s e d i m e n t p r o p e r t i e s and a r e t o be d e t e r m i n e d b y e x p e r i m e n t a l r e s u l t s . The b e d f o r m s d i s a p p e a r and c h a n n e l bed becomes f l a t a g a i n when F r o u d e number i n c r e a s e s t o e x c e e d 0.7 b u t l e s s t h a n one. F o r B a k e l i t e s e d i m e n t bed, s i n c e t h e v a l u e o f s e d i m e n t d e n s i t y i s s m a l l e r t h a n b o t h P o r t a g e S i l i c a s a n d o r F r a s e r R i v e r s a n d d e n s i t y , B a k e l i t e s a n d p a r t i c l e s s t a r t m o v ing a t s m a l l e r v a l u e o f v e l o c i t y and c o r r e s p o n d i n g c r i t i c a l t r a c t i v e s t r e s s . F o r t h e c o n d i t i o n s u s e d i n t h e t e s t , t h e v e l o c i t y was a b o u t 26 cm/sec. and F r o u d e number was 0.1. T h i s i n d i c a t e s t h e r e q u i r e d s h e a r s t r e s s o r c r i t i c a l v e l o c i t y f o r i n i t i a t i n g t h e s a n d movement a r e much s m a l l e r f o r t h i s l i g h t e r and c o a r s e r B a k e l i t e s e d i m e n t p a r t i c l e s . R i p p l e s s t a r t e d g e n e r a t i n g a t a v e l o c i t y a b o u t 26 cm/sec and F r o u d e number a b o u t 0.16 w i t h w a t e r f l o w d e p t h e q u a l t o 28 cm. When t h e w a t e r f l o w v e l o c i t y and F r o u d e number were i n c r e a s e d t o e x c e e d a b o u t 87 cm/sec and 0.73 r e s p e c t i v e l y , b e d f o r m s d i s a p p e a r e d and t h e s e d i m e n t c h a n n e l bed became f l a t a g a i n . The B a k e l i t e s e d i m e n t b e d f o r m s m i g r a t e downstream i m m e d i a t e l y a f t e r b e i n g g e n e r a t e d , and t h e m i g r a t i o n v e l o c i t y o f B a k e l i t e s e d i m e n t b e f o r m i s much g r e a t e r t h a n t h e m i g r a t i o n v e l o c i t i e s o f o t h e r b e d f o r m s f o r s e d i m e n t s o f g r e a t e r d e n s i t y u n d e r s i m i l a r w a t e r f l o w c o n d i t i o n s . F o r B a k e l i t e s e d i m e n t b e d f o r m s , a l o g a r i t h m i c N . R r e l a t i o n s h i p b etween f a n d ' CB a l s o e x i s t s , w h i c h c a n be shown i n F i g . 12 o r c a n be w r i t t e n a s : l o g 1 R C B = 2.53 l o g 1 ( ) N f + 1.4-1 (1.20) t h e c o n s t a n t s i n t h i s e q u a t i o n s u c h as 2.53 o r 1.41 a r e a l s o f u n c t i o n s o f s e d i m e n t p r o p e r t i e s , and c a n o n l y be d e t e r m i n e d b y e x p e r i m e n t a l r e s u l t s . However, l o g a r i t h m i c r e l a t i o n s h i p s (1.19) and (1.20) c a n o n l y be t r u e i n t h e r a n g e o f f o r m a t i o n and m i g r a t i o n o f l o n g e r w a v e l e n g t h F r o u d e - d u n e s . THE RELATIONSHIPS BETWEEN Y/L„ AND N^ OR R__ A n d e r s o n ( 1 9 5 3 ) , o b t a i n e d an e q u a t i o n , w h i c h s u g g e s t e d t h e r e l a t i o n s h i p between F r o u d e number and t h e r e l a t i v e w a v e l e n g t h Y / L R a s f o l l o w s : k ' ^ ' ( » * * ^ - 5 T O ^ > « • » > F o r t h e deep w a t e r s i t u a t i o n o r when Y » L , A n d e r s o n ' s e q u a t i o n c a n be s i m p l i f i e d by, V = ( g L R / 2% ) ° ' 5 (1.22) However, A n d e r s o n ' s e q u a t i o n i s n o t a p p l i c a b l e t o t h e s h a l l o w w a t e r s i t u a t i o n o r when w a t e r d e p t h Y « L . ri S u p e r i m p o s i n g A n d e r s o n ' s e q u a t i o n on t h e p r e s e n t e x p e r -i m e n t a l r e s u l t s , F i g . 13 i n d i c a t e s t h a t A n d e r s o n ' s e q u a t i o n i s i n good agreement w i t h p r e s e n t e x p e r i m e n t a l r e s u l t s f o r P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d b e d f o r m s . As f o r 36 y / L B a k e l i t e s e d i m e n t b e d f o r m s , t h e e s t i m a t e d v a l u e s o f R f r o m A n d e r s o n ' s e q u a t i o n a r e a l w a y s l a r g e r t h a n t h e e x p e r i m e n t a l r e s u l t s . I n o t h e r words, A n d e r s o n ' s e q u a t i o n a l w a y s u n d e r e s t i m a t e s t h e v a l u e s o f b e d f o r m w a v e l e n g t h s when s e d i m e n t m a t e r i a l s a r e c o a r s e a n d / o r l i g h t , t h i s i s due t o A n d e r s o n ' s l a c k o f c o n s i d e r a t i o n o f s e d i m e n t p r o p e r t i e s . S i n c e t h e r e l a t i o n s h i p s between F r o u d e number N f and m o d i f i e d by R e y n o l d ' s number CB a r e o b t a i n e d and c a n be r e p r e s e n t e d by e q u a t i o n s (1.19) and ( 1 . 2 0 ) , s u b s t i t u t i n g t h e s e two e q u a t i o n s i n t o A n d e r s o n ' s e q u a t i o n , two m o d i f i e d e q u a t i o n s R t h a t s u g g e s t t h e r e l a t i o n s h i p s between Y / L R a n d CB f o r e a c h t y p e s o f s e d i m e n t i n t h e r a n g e o f f o r m a t i o n and m i g r a t i o n o f dunes c a n be o b t a i n e d . A f t e r r e w r i t i n g i n more e x p l i c i t f o r m s , t h e m o d i f i e d e q u a t i o n s c a n be r e p r e s e n t e d a s , R C B = -1.125 l o g 1 ( ) K + 0.99 (1.23) f o r P o r t a g e S i l i c a s a n d b e d f o r m s and F r a s e r R i v e r s a n d b e d f o r m s , and l o g 1 0 R C B = -1.265 l o g l Q K + 1.41 (1.24) f o r B a k e l i t e s a n d b e d f o r m s . Where 21W, 271 v 2 K - v^ '<: L ~ ' - , M , W / L R » F i g . 14 i n d i c a t e s t h a t t h e r e l a t i o n s h i p s b etween Y / L R and CB f o r a l l t h r e e t y p e s o f s e d i m e n t b e d f o r m s i n p r e s e n t s t u d y . E q u a t i o n (1.23) s u c c e s s f u l l y e s t i m a t e s t h e b e d f o r m w a v e l e n g t h L R o f b o t h P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d b e d f o r m s . But e q u a t i o n (1.24) i s t h e same as A n d e r s o n ' s 37 e q u a t i o n , w h i c h a l w a y s u n d e r e s t i m a t e s t h e v a l u e s o f b e d f o r m w a v e l e n g t h s , so t h a t , a l t h o u g h m o d i f i e d e q u a t i o n s (1.23) and (1.24) t a k e t h e e f f e c t s o f s e d i m e n t p r o p e r t i e s i n t o c o n s i d -e r a t i o n , e q u a t i o n (1.24) i s s t i l l n o t i n v e r y good agreement w i t h t h e e x p e r i m e n t a l r e s u l t s a t l i g h t B a k e l i t e s e d i m e n t b e d f o r m s due t o t h e b a s i c s h o r t c o m i n g o f A n d e r s o n ' s e q u a t i o n . THE RELATIONSHIPS BETWEEN BEDFORM WAVELENGTHS AND V OR C B The f l o w o f w a t e r and s e d i m e n t in an open c h a n n e l depends on t h e i n t e r a c t i o n o f t h e w a t e r f l o w w i t h t h e d e f o r m a b l e s e d i m e n t b o u n d a r y . M.C. Q u i c k s u g g e s t e d t h a t f r o m f r e e s u r f a c e wave t h e o r y , t h e s p e e d o f wave, C, c a n be e x p r e s s e d by C = ( - a U L t a n h ^ ) ) 0 ' 5 (1.25) IT L W where LW i s w a t e r wave l e n g t h and Y i s w a t e r d e p t h . Water i s assumed t o be a t r e s t . F o r t h e c a s e o f deep w a t e r wave o r when LW << 2 Y, (1.25) c a n be s i m p l i f i e d as C = ( g Lw/ 2i7.) 0 - 5 (1.26) F o r t h e c a s e o f s h a l l o w w a t e r o r when W^ >> Y. (1.25) c a n be s i m p l i f i e d as C = ( g Y ) 0 , 5 — — ( 1 . 2 7 ) R e f e r r i n g t o F i g . 15, t h e p r o g r e s s i v e s u r f a c e wave i s shown i n F i g . 15A. I n F i g . 15B, t h e wave h a s b e e n a r r e s t e d b y s u p e r -i m p o s i n g a w a t e r v e l o c i t y o r V = C i n t h e o p p o s i t e d i r e c t i o n t o wave m o t i o n . I n F i g . 15C, t h e s i t u a t i o n h a s b e e n i n v e r t e d so t h a t t h e f r e e s u r f a c e i s t h e o r i g i n a l b e d . The q u e s t i o n 38 r e m a i n i n g t o be answered i s t h a t , u n d e r what c o n d i t i o n i s t h i s i n v e r s i o n o f t h e wave p i c t u r e p o s s i b l e ? F o r deep w a t e r c a s e o r i f t h e wave i s s h o r t , t h e w a t e r p a r t i c l e m o t i o n b e l o w t h e wave w i l l be i n c i r c u l a r o r b i t s , w h i c h d i e away e x p o n e n t i a l l y w i t h d e p t h . A t a d e p t h o f Y> LW/2, t h e w a t e r p a r t i c l e m o t i o n i s a l m o s t i m p e r c e p t i b l e and, t h e r e f o r e , t h e p r e s s u r e a t t h i s d e p t h i s c o n s t a n t . The c h a n n e l b e d i s , t h e r e f o r e , a s t r e a m -l i n e o f c o n s t a n t p r e s s u r e and c a n be i n v e r t e d t o f o r m a f r e e s u r f a c e . From s u c h an argument i t c a n be a p p r e c i a t e d t h a t s h o r t b e d f o r m s o r r i p p l e s c a n e x i s t i n deep w a t e r . Y> and l e a v e t h e f r e e s u r f a c e u n d i s t u r b e d . The a v e r a g e w a t e r v e l o c i t y c a n be o b t a i n e d by e q u a t i o n (1.22) V = ( g LR/ 2JI ) 0 - 5 T h i s e q u a t i o n i s v a l i d i f LW i s l e s s t h a n 2Y, t h u s , i t i s r e a s o n a b l e t o s t a t e t h a t s h o r t w a v e l e n g t h r i p p l e l e n g t h s a r e 2 p r o p o r t i o n a l t o V , and mean p a r t i c l e s i z e i s p e r m i s s i v e . F i g . 16 i n d i c a t e s t h e v a r i a t i o n o f b e d f o r m l e n g t h s L K w i t h r e s p e c t t o t h e v a r i a t i o n o f m i g r a t i o n v e l o c i t y C . S i n c e B t h e v a r i a t i o n o f w a t e r f l o w d e p t h Y i n p r e s e n t e x p e r i m e n t s i s s m a l l , i t i n d i c a t e s t h a t a t t h e same v a l u e o f m i g r a t i o n v e l o c i t y and w a t e r d e p t h , l i g h t B a k e l i t e s e d i m e n t r i p p l e l e n g t h s a r e much g r e a t e r t h a n P o r t a g e S i l i c a s a n d r i p p l e l e n g t h s o r F r a s e r R i v e r s a n d r i p p l e l e n g t h s . 39 F i g . 17 i n d i c a t e s t h e v a r i a t i o n o f b e d f o r m l e n g t h s L w i t h r e s p e c t t o t h e v a r i a t i o n o f mean w a t e r f l o w v e l o c i t y V. Due t o t h e s m a l l v a l u e o f B a k e l i t e s e d i m e n t d e n s i t y , B a k e l i t e s e d i m e n t b e d f o r m l e n g t h s a r e l o n g e r t h a n P o r t a g e S i l i c a s a n d b e d f o r m l e n g t h s o r F r a s e r R i v e r s a n d b e d f o r m l e n g t h s a t t h e same v a l u e s o f Y and V. F i g . 17 shows t h a t b e d f o r m w a v e l e n g t h b e h a v i o r c a n be s u b d i v i d e d i n t o t h r e e r a n g e s , i . e . r a n g e o f r i p p l e s , r a n g e o f r i p p l e - d u n e s o r t r a n s i t i o n a l r a n g e and r a n g e o f dunes o r F r o u d e - d u n e s . F i g . 18 shows d i f f e r e n t b e d f o r m w a v e l e n g t h b e h a v i o r s i n t h e r a n g e s o f r i p p l e s and d u n e s . I n F i g . 19, c u r v e A i n d i c a t e s t h a t f o r s h o r t w a v e l e n g t h r i p p l e s w h i c h a r e b e l i e v e d t o be i n d e p e n d e n t o f F r o u d e number, however, i t c o u l d s t i l l mean t h a t r i p p l e w a v e l e n g t h s depend on e i t h e r o r b o t h w a t e r f l o w v e l o c i t y V and w a t e r d e p t h Y. I n o t h e r words, " F r o u d e number i n d e p e n d e n t " , means t h a t t h e r e i s n o t i n t e r a c t i o n between s e d i m e n t r i p p l e s and f r e e w a t e r s u r f a c e waves. On t h e o t h e r hand, c u r v e B shows t h a t l o n g e r w a v e l e n g t h dunes a r e " F r o u d e number d e p e n d e n t " , i n t e r a c t i o n s e x i s t between s e d i m e n t b e d f o r m s and f r e e w a t e r s u r f a c e waves. V and Y a r e b o t h i m p o r t a n t i n d e t e r m i n i n g t h e s e d i m e n t b e d f o r m d i m e n s i o n s . R a u d k i v i s t a t e d t h a t r i p p l e l e n g t h i s p r o p o r t i o n a l t o t h e p a r t i c l e s i z e and i s i n d e p e n d e n t o f t h e d e p t h o f w a t e r f l o w ; and dune l e n g t h i s p r o p o r t i o n a l t o t h e d e p t h o f w a t e r f l o w and o n l y s l i g h t l y d e p e n d e n t on p a r t i c l e s i z e . However, 4-0 u s i n g t h e argument o f i n v e r s i o n a f r e e s u r f a c e g r a v i t y wave, we c a n r e l a t e bed w a v e l e n g t h t o deep w a t e r wave t h e o r y . F i g . 20 and F i g . 21, i n d i c a t e t h e v a r i a t i o n o f b e d f o r m l e n g t h s w i t h r e s p e c t t o t h e v a r i a t i o n o f V o r p / p g I t i s f o u n d t h a t i n t h e r a n g e o f s h o r t w a v e l e n g t h r i p p l e s , p r e s e n t e x p e r i m e n t a l r e s u l t s w h i c h a r e i n good agreement w i t h t h e argument o f i n v e r s i o n o f t h e wave p i c t u r e . Thus, " R i p p l e 2 l e n g t h i s p r o p o r t i o n a l t o V , and mean p a r t i c l e s i z e i s permissive'.', c a n be c o n f i r m e d . The r e a s o n t h a t r i p p l e l e n g t h d epends on mean p a r t i c l e s i z e a s R a u d k i v i c o n c l u d e d , i s t h a t c r i t i c a l v e l o c i t y f o r r i p p l e f o r m a t i o n i s l e s s f o r f i n e r and l i g h t e r s e d i m e n t p a r t i c l e s . Simons e t a l . , a l s o r e p o r t e d i n cl 1963, t h a t no r i p p l e s e x i s t when 50 i s g r e a t e r t h a n o.6 mm. However, i t i s f o u n d t h a t i f s e d i m e n t d e n s i t y i s much l i g h t e r , s u c h as B a k e l i t e s e d i m e n t , t h a n r e g u l a r s a n d , o r i f i n deep w a t e r s i t u a t i o n , r i p p l e s c a n be g e n e r a t e d f o r c o a r s e r s e d i m e n t c h a n n e l b e d p r o v i d e d t h a t w a t e r f l o w v e l o c i t y i s l a r g e enough, t h u s , i t i s v e r y d i f f i c u l t t o a c c e p t R a u d k i v i ' s s t a t e m e n t due t o i t s o v e r s i m p l i f i c a t i o n . The i m p o r t a n c e o f F r o u d e number i n t h e r a n g e o f dunes i m p l i e s t h a t t h e b e d f o r m f e a t u r e s a r e r e l a t e d t o b o t h V and Y. But t h e r e s t i l l r e m a i n s a q u e s t i o n t o be answered c o n c e r n i n g t h e e f f e c t o f w a t e r d e p t h Y on b e d f o r m l e n g t h . T h i s q u e s t i o n c a n be a n s w e r e d o n l y b y c a r r y i n g o u t a number o f e x p e r i m e n t s u s i n g a l a r g e r a n g e o f w a t e r f l o w d e p t h s and s e d i m e n t p r o p e r t i e s . . 41 U n f o r t u n a t e l y , due t o the p h y s i c a l l i m i t a t i o n , the water depths used f o r most experiments, i n c l u d i n g the present study, were too shallow t o be able to answer t h i s q u e s t i o n . THE RELATIONSHIPS BETWEEN Y/H AND N_ OR R K r OB There are o n l y a few p u b l i c a t i o n s a v a i l a b l e /which d i s c u s s bedform h e i g h t behavior under v a r i o u s s i t u a t i o n s . T h i s c o u l d be because the s c a l e o f bedform h e i g h t i s s m a l l e r than the s c a l e of bedform l e n g t h , e s p e c i a l l y i n the range of r i p p l e s . In a d d i t i o n , bedform height i s not v e r y s t a b l e d u r i n g the gener-a t i o n of bedform f e a t u r e s . F i g . 22 i n d i c a t e s t h a t the r e l a t i o n -s h i p between Y/LJ^ and N^ .. I t i s found t h a t t h i s r e l a t i o n s h i p i s v e r y s i m i l a r t o the curve r e p r e s e n t i n g the r e l a t i o n s h i p between Y/L_, and N f. S i m i l a r e x p l a n a t i o n s and a n a l y s i s can be H x made f o r bedform waveheight b e h a v i o r s . But the s c a l e of the bedform h e i g h t behavior i s much s m a l l e r than the water flow depth Y, so a l l experimental r e s u l t s f a l l v e r y c l o s e l y onto the same curve, thus, i t i s v e r y d i f f i c u l t t o i n v e s t i g a t e the e f f e c t s of sediment p r o p e r t i e s on the bedform h e i g h t behaviors from t h i s f i g u r e . F i g . 23 and F i g . 24, r e p r e s e n t the v a r i a t i o n of bedform he i g h t H and H /Y, with r e s p e c t t o the v a r i a t i o n of Froude R R number. I t i s found t h a t the v a l u e of l i g h t B a k e l i t e sediment bedform h e i g h t s are always l a r g e r than those of Portage S i l i c a sand and F r a s e r R i v e r sand at the same va l u e of Froude number. 42 B a k e l i t e s e d i m e n t b e d f o r m h e i g h t i n c r e a s e s w i t h i n c r e a s i n g N^, bu t a f t e r r e a c h i n g t h e maximum v a l u e o f b e d f o r m h e i g h t , t h e n t h e b e d f o r m h e i g h t r e m a i n s a b o u t c o n s t a n t . B a k e l i t e b e d f o r m s d i s a p p e a r when w a t e r f l o w F r o u d e number a p p r o x i m a t e l y e x c e e d s 0.6. P o r t a g e S i l i c a s a n d b e d f o r m h e i g h t and F r a s e r R i v e r s a n d b e d f o r m h e i g h t i n c r e a s e w i t h i n c r e a s i n g F r o u d e number, and t h e y d i s a p p e a r i f F r o u d e number e x c e e d s 0.7, w h i c h i s h i g h e r t h a n t h a t o f B a k e l i t e s e d i m e n t b e d f o r m s (N^ = 0 . 6 ) . T h i s i s b e c a u s e B a k e l i t e s e d i m e n t i s r o u g h e r and t h i s r o u g h n e s s a s s i s t s i n m i x i n g e n e r g y i n t o t h e b o u n d a r y l a y e r . From e q u a t i o n (1.19) and ( 1 . 2 0 ) , t h e w a t e r f l o w v e l o c i t y C R V o r F r o u d e number n e e d e d t o p r o d u c e a c e r t a i n v a l u e o f B o r CB i s s m a l l e r f o r B a k e l i t e s e d i m e n t b e d i f t h e v a r i a t i o n o f w a t e r d e p t h Y i s s m a l l . As f o r t h e v a r i a t i o n o f R , F i g . 25 and F i g . 0 B i n d i c a t e t h a t s i n c e t h e d e n s i t y o f B a k e l i t e s e d i m e n t i s much l i g h t e r t h a n t h o s e o f P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d f o r t h e same b e d f o r m h e i g h t , B a k e l i t e s e d i m e n t b e d f o r m s m i g r a t e much f a s t e r t h a n P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d b e d f o r m s . F i g . 1.27 shows t h e r e l a t i o n s h i p s between dunes m i g r a t i o n v e l o c i t y C,, and mean w a t e r f l o w v e l o c i t y V. B a k e l i t e s e d i m e n t dunes m i g r a t e much f a s t e r t h a n P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d dunes ^at'-'thesasame v a l u e o f V. 4-3 THE RELATIONSHIPS BETWEEN BEDFORM STEEPNESS AND N_ OR R__ x CB F i g . 28 and F i g . 2 9 , ' i n d i c a t e t h a t t h e v a l u e o f b e d f o r m s t e e p n e s s S„ o r H /L_ i n c r e a s e s w i t h i n c r e a s i n g o f N^ o r R ^ „ , R R R a f CB however, a f t e r r e a c h i n g t h e maximum s t e e p n e s s , b e d f o r m s t e e p -n e s s s t a r t s d e c r e a s i n g . The v a l u e s o f maximum s t e e p n e s s and c o r r e s p o n d i n g N^. o r R v a r i e s w i t h t h e v a r i a t i o n o f s e d i m e n t x C D p r o p e r t i e s . We have l e a r n e d t h a t a t t h e same v a l u e s o f V and Y, b o t h v a l u e s o f B a k e l i t e s e d i m e n t b e d f o r m l e n g t h and h e i g h t a r e g r e a t e r t h a n t h o s e o f P o r t a g e S i l i c a s a n d o r F r a s e r R i v e r s a n d b e d f o r m . S i n c e t h e r a n g e o f v a r i a t i o n o f b e d f o r m h e i g h t i s s m a l l e r t h a n t h o s e o f b e d f o r m l e n g t h , t h u s , p r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t t h e v a l u e o f B a k e l i t e s e d i m e n t b e d f o r m s t e e p n e s s i s a l w a y s l e s s t h a n t h o s e of- t h e o t h e r two t y p e s o f s e d i m e n t b e d f o r m s u n d e r s i m i l a r w a t e r f l o w c o n d i t i o n s . THE RELATIONSHIP BETWEEN SEDIMENT AND WATER FLOW TRANSPORT RATE T h e r e a r e v a r i o u s s e d i m e n t b e d l o a d t r a n s p o r t f o r m u l a e a v a i l a b l e f o r t h e s e d i m e n t b e d l o a d d i s c h a r g e e s t i m a t i o n , s u c h a s S h i e l d s ' , Du Boys, E i n s t e i n - B r o w n , and K a l i n s k e b e d l o a d t r a n s p o r t f o r m u l a e . I f t h e shape and t h e bed s l o p e o f w a t e r c h a n n e l r e m a i n unchanged, i t c a n be f o u n d that/the s e d i m e n t b e d l o a d t r a n s p o r t r a t e q w i l l v a r y w i t h some power o f w a t e r d i s c h a r g e Q o r w a t e r f l o w v e l o c i t y V. T h a t i s , q s oC Q m o r V m (1 .28) 44 C o m p a r i s o n s o f b e d l o a d t r a n s p o r t f o r m u l a e were made by R a u d k i v i ( 1 9 6 3 ) . W i t h a i d o f t h e S h i e l d s ' e n t r a i n m e n t f u n c t i o n T c/(y -y)d a p p r o x i m a t e l y e q u a l t o 0.056, and s p e c i f i c g r a v i t y o f s a n d p a r t i c l e e q u a l s 2.6, t h e w a t e r d i s c h a r g e r a t e 3/2 1/2 Q i n t e r m s o f C h a z y 1 s f o r m u l a i s p r o p o r t i o n a l t o Y S , 5/3 1/2 o r e x p r e s s e d i n t e r m s o f Manning's f o r m u l a , Q o< Y S , where S means c h a n n e l bed s l o p e . Thus, f o r a c o n s t a n t v a l u e o f S, v a l u e o f m c a n be e v a l u a t e d . The f o l l o w i n g t a b l e shows t h e v a l u e s o f m f r o m R a u d k i v i ' s s t u d y : m Q i s r e p r e s e n t e d by Q i s r e p r e s e n t e d by C h e z y 1 s F o r m u l a Manning's F o r m u l a S h i e l d ' s f o r m u l a 5/3 8/5 Du Boy's f o r m u l a 4/3 6/5 E i n s t e i n -Brown ' s f o r m u l a 2 9/5 K a l i n s k e ' s f o r m u l a 5/3 8/5 R a u d k i v i n o t e d t h a t " i f one p l o t s t h e s e r e l a t i o n s h i p s b e tween q s and Q on l o g a r i t h m i c s c a l e s , t h e n v a r i a t i o n s a r e a p p a r e n t b o t h i n t h e s l o p e and i n t h e l o c a t i o n on t h i s p l a n e . D a t a o b t a i n e d f r o m o b s e r v a t i o n s on n a t u r a l w a t e r c o u r s e s , however, a r e e q u a l l y w i d e l y s p r e a d f r o m r i v e r t o r i v e r . W i t h most f o r m u l a e t h e s l o p e becomes t o o f l a t w i t h h i g h d i s c h a r g e and 45 s e d i m e n t l o a d " . T h i s u n d e r l i e s t h e f a c t t h a t most o f t h e s e f o r m u l a e a r e o n l y a p p l i c a b l e t o t h e b e d l o a d t r a n s p o r t . A t h i g h w a t e r d i s c h a r g e r a t e o r a t l a r g e mean w a t e r f l o w v e l o c i t y , much o f t h e s e d i m e n t w i l l be t r a n s p o r t e d i n s u s p e n s i o n . However, R a u d k i v i a l s o n o t e d t h a t " e x p e r i e n c e shows t h a t i n a p p l y i n g f o r m u l a e o f t h i s k i n d t o p a r t i c u l a r c a s e s , e r r o r o f 100 p e r c e n t a r e t o be e x p e c t e d " . I n o t h e r words, t h e f o r m u l a e p r o v i d e e s t i m a t e s o n l y . T h e r e f o r e , i t i s most d e s i r a b l e t h a t more t h a n one f o r m u l a be u s e d . As f o r p r e s e n t e x p e r i m e n t s , t h e s e d i m e n t t r a n s p o r t r a t e was a p p r o x i m a t e d by q s = \ P § 9 C B H R and t h e w a t e r d i s c h a r g e r a t e was a p p r o x i m a t e d by Q = V Y. P r e s e n t e x p e r i m e n t s show t h a t i n t h e r a n g e o f r i p p l e s , t h e s e d i m e n t t r a n s p o r t r a t e i s v e r y s m a l l . F u r t h e r i n c r e a s e o f t h e w a t e r f l o w v e l o c i t y V as w e l l as t h e F r o u d e number N f , t h e r a n g e o f F r o u d e dunes w i l l be r e a c h e d . The b e d l o a d t r a n s p o r t r a t e °^ s i n c r e a s e s w i t h w a t e r d i s c h a r g e r a t e Q t o t h e power o f a b o u t 7 t o 8. T h i s d oes n o t a g r e e w i t h R a u d k i v i ' s s t u d y w h i c h t h e o r e t i c a l l y shows t h i s v a l u e t o be b etween one and two. The p o s s i b l e p r i m a r y c a u s e f o r t h i s d i s a g r e e m e n t i s t h a t t h e b a s i c a s s u m p t i o n s o f R a u d k i v i ' s s t u d y may n o t be t r u e f o r p r e s e n t e x p e r i m e n t s o r f o r o t h e r o b s e r v a t i o n s on n a t u r a l w a t e r c o u r s e s . The b a s i c a s s u m p t i o n s o f R a u d k i v i 1 s s t u d y a r e t h a t t h e e f f e c t i v e c h a n n e l s l o p e i s c o n s t a n t and C h e z y and M anning f o r m u l a e a r e a p p l i c a b l e . However, s i n c e t h e r e e x i s t v a r i o u s t y p e s o f s e d i m e n t b e d f o r m s u n d e r d i f f e r e n t w a t e r f l o w c o n d i t i o n s , 46 i t i s d i f f i c u l t t o know t h e v a l u e o f t h e e f f e c t i v e s l o p e o f t h e c h a n n e l , w h i c h may n o t be c o n s t a n t . F u r t h e r m o r e , t h e s e d i m e n t c h a n n e l b e d may n o t be p a r a l l e l t o t h e w a t e r f l o w s u r f a c e and t h e r e e x i s t s w a t e r f l o w a c c e l e r a t i o n . T h e r e f o r e , t h e a s s u m p t i o n s o f u n i f o r m f l o w w i l l n o t be v a l i d , and t h e C h e z y and M a n n i n g f o r m u l a e w i l l n o t be a p p l i c a b l e . S i n c e t h e b o u n d a r y l a y e r t h e o r y i n d i c a t e s t h a t i s n o t u n r e a s o n a b l e t o assume t h a t i s p r o p o r t i o n a l t o t h e 2 v a l u e o f V , t h e n i t f o l l o w s t h a t t h e E i n s t e i n - B r o w n b e d l o a d t r a n s p o r t e q u a t i o n g i v e s q $o< V 6 (1.29) F i g . 30 i n d i c a t e s t h e r e l a t i o n s h i p o f s e d i m e n t t r a n s -p o r t r a t e q s a g a i n s t w a t e r f l o w v e l o c i t y V. P r e s e n t e x p e r i -m e n t a l r e s u l t s show t h a t r e s u l t s f r o m d i f f e r e n t w a t e r f l o w d e p t h s s h o u l d c o l l a p s e t o one l i n e w i t h t h e s l o p e o f a b o u t s e v e n , w h i c h i s i n good agreement w i t h t h e e q u a t i o n ( 1 . 2 9 ) . 4-7 CHAPTER 1.5 CONCLUSIONS (1) Two c r i t e r i a g o v e r n t h e mechanism o f b e d f o r m b e h a v i o u r , t h e w a t e r f l o w v e l o c i t y V, and t h e F r o u d e number N^ .. The mean w a t e r f l o w d e p t h Y i s a v a r i a b l e l i n k i n g t h e s e two f a c t o r s . The w a t e r f l o w v e l o c i t y must be h i g h enough t o p r o d u c e s e d i m e n t m o t i o n and y e t n o t so h i g h t h a t t h e s e d i m e n t i s c a r r i e d i n s u s p e n s i o n . R i p p l e s do n o t a p p e a r u n t i l t h e c h a n n e l w a t e r f l o w v e l o c i t y i s a b o u t 50% g r e a t e r t h a n t h e c r i t i c a l mean w a t e r f l o w v e l o c i t y f o r o n s e t o f s e d i m e n t m o t i o n . A t h i g h e r v e l o c i t y r a n g e s , t h e r i p p l e s w i l l become v e r y a c t i v e and c a n p r o d u c e l a r g e s e c o n d a r y f e a t u r e s w h i c h w i l l be c a l l e d r i p p l e - d u n e s . F u r t h e r i n c r e a s e o f w a t e r f l o w v e l o c i t y c a u s e s a l l r i p p l e s t o d i s a p p e a r and o n l y p u r e dunes r e m a i n and t h e s e f e a t u r e s w i l l be c a l l e d F r o u d e - d u n e s . T h e r e e x i s t s an u p p e r and l o w e r l i m i t ' i r g F r o u d e number f o r t h e r a n g e o f F r o u d e - d u n e s , b u t t h e r e i s o n l y an u p p e r l i m i t o f F r o u d e number f o r r i p p l e s . F o r r a n g e o f r i p p l e s , t h e l o w e r l i m i t i s s e t by t h e minimum v e l o c i t y t o i n i t i a t e t h e s e d i m e n t p a r t i c l e s on c h a n n e l b e d t o move. The u p p e r l i m i t i n g F r o u d e number f o r t h e r a n g e o f r i p p l e s c o r r e s p o n d s t o t h e minimum a l l o w a b l e N^. f o r t h e r a n g e o f F r o u d e dunes, w h i c h i s a b o u t 0.5. Bedforms w i l l become F r o u d e - d u n e s when t h e F r o u d e number i n c r e a s e s and e x c e e d s t h e l o w e r l i m i t i n g F r o u d e number o f t h e r a n g e o f F r o u d e - d u n e s . The u p p e r l i m i t i n g F r o u d e number i s t h e maximum a l l o w a b l e N^. f o r r a n g e o f F r o u d e - d u n e s a v a l u e o f 0.7, l a r g e r F r o u d e numbers w i l l f l a t t e n t h e san d bedwaves. 48 I n t h e r a n g e o f r i p p l e s , w a t e r f l o w v e l o c i t y V i s t h e d o m i n a n t f a c t o r i n d e t e r m i n i n g t h e b e d f o r m b e h a v i o r s , and t h e r e i s no i n t e r a c t i o n between r i p p l e s and f r e e s u r f a c e waves, t h u s t h e r i p p l e l e n g t h s a r e F r o u d e number i n d e p e n d e n t . By u s i n g t h e argument o f i n v e r s i o n o f wave p i c t u r e , w a t e r f l o w v e l o c i t y L 0 5 V= ( g R / 2 u ) i s t r u e , p r o v i d e d L,, i s l e s s t h a n 2 Y. w P r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t , i n t h e r a n g e o f r i p p l e s 2 s q u a r e o f w a t e r f l o w v e l o c i t y , V , i s a l m o s t l i n e a r l y r e l a t e d t o t h e r i p p l e l e n g t h s L . Thus, e x p e r i m e n t a l r e s u l t s a r e i n v e r y good agreement w i t h M.C. Q u i c k ' s argument o f wave p i c t u r e i n v e r s i o n i n t h e r a n g e o f r i p p l e s . R i p p l e l e n g t h i s p r o p -2 o r t i o n a l t o t h e v a l u e o f V , and, t h e mean p a r t i c l e s i z e i s p e r m i s s i v e . When w a t e r f l o w v e l o c i t y i n c r e a s e s so t h a t t h e F r o u d e number e x c e e d s t h e l o w e r l i m i t i n g F r o u d e number o f t h e r a n g e o f F r o u d e - d u n e s , r i p p l e s d i s a p p e a r and t h e b e d f o r m f e a t u r e s become s t a b l e l o n g w a v e l e n g t h p u r e dunes. The F r o u d e number now becomes an i m p o r t a n t f a c t o r i n d e t e r m i n i n g t h e b e d f o r m b e h a v i o r s . The F r o u d e number a l s o d e f i n e s t h e w a t e r s u r f a c e r e s p o n s e t o t h e change i n s e d i m e n t bed l e v e l so t h a t t h e r e e x i s t s an i n t e r a c t i o n between s e d i m e n t F r o u d e - d u n e s and w a t e r s u r f a c e waves. (2) The v e l o c i t y f o r o n s e t o f r i p p l e s and t h e v a l u e s o f u p p e r and l o w e r l i m i t i n g F r o u d e numbers f o r t h e - r a n g e o f r i p p l e s and t h e r a n g e o f F r o u d e - d u n e s a r e m a i n l y d e p e n d e n t on s e d i m e n t p a r t i c l e s i z e and d e n s i t y . I n t h e p r e s e n t e x p e r i m e n t s , s i n c e B a k e l i t e s e d i m e n t i s much l i g h t e r and c o a r s e r t h a n t h e o t h e r two t y p e s o f s e d i m e n t , B a k e l i t e s e d i m e n t b e d f o r m s change f r o m t h e r a n g e o f r i p p l e s t o t h e r a n g e o f F r o u d e -dunes a t s m a l l e r v a l u e o f F r o u d e number o r mean w a t e r f l o w v e l o c i t y . T h i s i n d i c a t e s t h e r e q u i r e d s h e a r s t r e s s o r c r i t i c a l v e l o c i t y f o r i n i t i a t i n g t h e s a n d movement a r e much s m a l l e r f o r l i g h t e r a n d / o r c o a r s e r s e d i m e n t p a r t i c l e s . A l s o , c o a r s e r s e d i m e n t b e d f o r m s change t o t h e r a n g e o f F r o u d e - d u n e s a t much s m a l l e r v a l u e o f V o r N^ . t h a n t h o s e b e d f o r m s o f same t y p e b u t f i n e r s e d i m e n t s a n d . Simons e t a l ( 1 9 6 3 ) , r e p o r t e d t h a t no r i p p l e s e x i s t when 50 i s g r e a t e r t h a n 0.6 mm. B u t i f we t a k e t h e e f f e c t o f w a t e r d e p t h Y i n t o c o n s i d e r a t i o n , we c a n have same v e l o c i t y i n t h e v i c i n i t y o f t h e c h a n n e l b e d and v a r y t h e v a l u e o f F r o u d e number by c h a n g i n g t h e w a t e r d e p t h . S i n c e t h e f l o w v e l o c i t y n e a r t h e s e d i m e n t c h a n n e l b e d i s t h e domi n a n t f a c t o r i n g e n e r a t i n g r i p p l e s , t h u s , i f t h e w a t e r d e p t h Y, c a n be i n c r e a s e d s u f f i c i e n t l y , we s h o u l d be a b l e t o p r o d u c e r i p p l e s f o r c o a r s e s e d i m e n t s (^50 >0.6 mm) by i n c r e a s i n g b o t h V and Y and y e t m a i n t a i n i n g N^ . i n t h e r a n g e o f r i p p l e s . P r e s e n t experiments'-show t h a t f o r B a k e l i t e s e d i m e n t p a r t i c l e s , i n w h i c h ^50 e q u a l 0.89 mm, t h e s e d i m e n t c h a n n e l bed d i d become r i p p l e d o v e r a c e r t a i n s m a l l w a t e r f l o w v e l o c i t y r a n g e and w i t h N^. l e s s t h a n 0.4. (3) L o g a r i t h m i c r e l a t i o n s h i p s c a n be f o u n d between C D °*50/v and l o g ^ (N^.) , f o r v a r i o u s t y p e s o f s e d i m e n t c h a n n e l b e d f o r m s . C o m b i n i n g t h e s e l o g a r i t h m i c r e l a t i o n s h i p s w i t h A n d e r s o n ' s e q u a t i o n , w h i c h d e s c r i b e s t h e r e l a t i o n s h i p between Y/L and N.p, t h e r e l a t i o n s h i p s between Y/L and C D d [ : . / v f o r v a r i o u s t y p e s o f s e d i m e n t c h a n n e l b e d f o r m s t h e n c a n be o b t a i n e d T h e s e r e l a t i o n s h i p s c a n e s t i m a t e t h e b e h a v i o r s o f s e d i m e n t b e d f o r m s s u c c e s s f u l l y . But b e c a u s e A n d e r s o n ' s e q u a t i o n does n o t t a k e t h e s e d i m e n t p r o p e r t i e s i n t o c o n s i d e r a t i o n , i f t h e s e d i m e n t s a n d i s t o o l i g h t , t h e s e r e l a t i o n s h i p s a l w a y s u n d e r -e s t i m a t e t h e v a l u e s o f b e d f o r m l e n g t h s . (4) A t t h e same v a l u e o f o r V, t h e B a k e l i t e s e d i m e n t b e d f o r m l e n g t h and h e i g h t a r e g r e a t e r t h a n P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d b e d f o r m l e n g t h and h e i g h t r e s p e c t i v e l y . B a k e l i t e s e d i m e n t b e d f o r m h e i g h t i n c r e a s e s w i t h i n c r e a s i n g N^ ., however, a f t e r r e a c h i n g maximum v a l u e o f b e d f o r m h e i g h t i n t h e r a n g e o f F r o u d e - d u n e s , t h e n i t r e m a i n s n e a r l y c o n s t a n t and f i n a l l y , b e d f o r m s d i s a p p e a r when a p p r o x i m a t e l y e x c e e d s 0.6. P o r t a g e S i l i c a s a n d and F r a s e r R i v e r s a n d b e d f o r m h e i g h t s a l s o i n c r e a s e w i t h i n c r e a s i n g and b e d f o r m s d i s a p p e a r i f N^ . a p p r o x i m a t e l y e x c e e d s 0.7. U s i n g F r o u d e number as a b e d f o r m c r i t e r i o n , F i g . 11, F i g . 13, F i g . 23 and F i g . 24, e t c . , i n d i c a t e t h e e x p e r i m e n t a l r e s u l t s i n t h e r a n g e o f r i p p l e s and F r o u d e -dunes, and, t h u s , p r o v i d e t h e means t o p r e d i c t t h e p o s s i b l e t y p e ' o r t y p e s o f b e d f o r m and t h e a p p r o x i m a t e b e d f o r m d i m e n s i o n s i f s e d i m e n t p a r t i c l e p r o p e r t i e s and w a t e r f l o w c h a r a c t e r i s t i c s a r e known. F i g . 17 and F i g . 20 a r e u s e d t o c o n f i r m t h e p r e d i c -t i o n f r o m F i g . 11, F i g . 13, F i g . .23 and F i g . 24, by u s i n g mean w a t e r f l o w v e l o c i t y as a b e d f o r m c r i t e r i o n f o r g i v e n s e d i m e n t p r o p e r t i e s . More w a t e r f l u m e e x p e r i m e n t s o v e r a l a r g e r r a n g e o f w a t e r d e p t h s h o u l d be done i n t h e " f u t u r e t o f i n d o u t t h e e f f e c t o f t h e w a t e r d e p t h on s e d i m e n t t r a n s p o r t and b e d f o r m f o r m a t i o n . 51 (5) A l l b e d f o r m s t e e p n e s s o f v a r i o u s t y p e s o f s e d i m e n t b e d f o r m s w i l l i n c r e a s e w i t h i n c r e a s i n g o r R u n t i l r e a c h i n g t h e maximum v a l u e o f b e d f o r m s t e e p n e s s i n t h e r a n g e o f F r o u d e - d u n e s , t h e n t h e y s t a r t d e c r e a s i n g . The v a l u e o f B a k e l i t e s e d i m e n t b e d f o r m s t e e p n e s s i s a l w a y s l e s s t h a n t h o s e o f P o r t a g e S i l i c a s a n d o r F r a s e r r i v e r s a n d b e d f o r m s u n d e r t h e same w a t e r f l o w c o n d i t i o n s . (6) S i n c e t h e w a t e r d e p t h s u s e d i n t h e e x p e r i m e n t s were l i m i t e d , i n t h e r a n g e o f r i p p l e s , s e d i m e n t t r a n s p o r t r a t e q s was v e r y s m a l l . F o r t h e r a n g e o f F r o u d e - d u n e s , t h e s e d i m e n t t r a n s p o r t r a t e v a r i e d w i t h w a t e r d i s c h a r g e r a t e Q t o t h e power o f m, t h a t i s °*s °* Q m. From R a u d k i v i ' s c o m p a r i s o n s t u d y , t h e v a l u e o f m i s t h e o r e t i c a l l y e x p e c t e d t o be i n t h e r a n g e o f one and two. However, R a u d k i v i a l s o n o t e d t h a t " e x p e r i e n c e shows t h a t i n a p p l y i n g f o r m u l a e o f t h i s k i n d t o p a r t i c u l a r c a s e s , e r r o r o f 100 p e r c e n t a r e t o be e x p e c t e d " . P r e s e n t e x p e r i m e n t a l r e s u l t s a r e n o t i n agreement s i n c e t h e b a s i c a s s u m p t i o n s o f R a u d k i v i ' s s t u d y , f o r example, e f f e c t i v e s l o p e f o r t h e s e d i m e n t c h a n n e l bed i s c o n s t a n t and w a t e r f l o w i s u n i f o r m and s t e a d y , do n o t h o l d t r u e o v e r whole w a t e r f l u m e a t a l l t h e t i m e . But t h e p r e s e n t r e s u l t s do a g r e e w i t h a v e l o c i t y a d a p t a t i o n o f t h e E i n s t e i n - B r o w n e q u a t i o n as d e s c r i b e d b e l o w . S i n c e b o u n d a r y l a y e r t h e o r y i n d i c a t e s t h a t T Q i s assumed 2 t o be p r o p o r t i o n a l t o t h e v a l u e o f V , t h u s f r o m t h e E i n s t e i n -Brown b e d l o a d e q u a t i o n , a r e l a t i o n s h i p i s o b t a i n e d as q g oi, 6, ( e g . [ 1 . 2 9 ] ) . P r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t r e s u l t s f r o m 52 d i f f e r e n t w a t e r f l o w d e p t h s s h o u l d c o l l a p s e t o one l i n e w i t h t h e s l o p e o f a b o u t s e v e n , o r c i s 0 < ^ v ' 7 > w h i c h i s i n good agreement w i t h t h e e q u a t i o n ( 1 . 2 9 ) . 53 CHAPTER 1.6 BIBLIOGRAPHY B a g n o l d , R.A., The ffLow? o f c o h e s i o n l e s s g r a i n s i n f l u i d s . P h i l o s o p h i c a l T r a n s a c t i o n s ( V o l . 24-9). London: R o y a l S o c i e t y , 1956-1957. B a s c h i n , 0. D i e E n t s t e h u n g W e l l e n a h n l i c h e r O b e r f l a c h e n f o r m e n Z. G e s e l l s c h f t , E r d k u n d e , BO. XXXIV 1899. E i n s t e i n , H.A. & C h i e n N. M e c h a n i c s o f s t r e a m s w i t h m o v a b l e be d s o f f i n e s a n d s , a d i s c u s s i o n . T r a n s Am. S o c i e t y  C i v i l E n g i n e e r s ( V o l . 1 2 3 ) , 1958. E x n e r , F.M., Z u r T h e s r i e d e r F l u s s m a a n d e r , S i t z b e r . Akad  W i s s . Wien, p t I l a , ( V o l . 128) H e f t 10. G r a f , W.H. H y d r a u l i c s o f s e d i m e n t t r a n s p o r t . McGraw H i l l Book Company, 1971. I n g l i s , C.C. The e f f e c t o f v a r i a t i o n s i n c h a r g e and g r a d e on t h e s l o p e s and s h a p e s o f c h a n n e l s . I n t e r n A s s o c . H y d r . Res. g r e n o b l e 3. K o n d r a t ev, N. R i v e r F low and R i v e r C h a n n e l F o r m a t i o n . t r a n s l a t i o n fromRussian ( 1 9 5 9 ) , N a t l . S c i e n c e F o u n d a t i o n  & U.S. D e p t . I n t e r i o r s . 1962. L i u , H.K. The m e c h a n i c s o f s e d i m e n t r i p p l e f o r m a t i o n . J o u r n a l  o f t h e h y d r a u l i c d i v i s i o n , 1957, ASCE 83, No. HY2, P r o c . p a p e r 1197. N o r d i n , C.F. A s p e c t s o f f l o w r e s i s t a n c e and s e d i m e n t t r a n s p o r t R i o Grande n e a r B e r n a l i l l o , New M e x i c o , U.S. G e o l . S u r v e y  Water S u p p l y P a p e r . 14-98-H. Q u i c k , M.C. S e d i m e n t t r a n s p o r t by r i p p l e s and d u n e s . U n p u b l i s h e d p r e l i m i n a r y d r a f t , U n i v e r s i t y o f B r i t i s h C o l o m b i a . R a u d k i v i , A . J . S t u d y o f s e d i m e n t r i p p l e f o r m a t i o n . J o u r n a l o f t h e h y d r a u l i c d i v i s i o n , 1963, ASCE 89, No. HY6, P r o c . p a p e r 3692. R a u d k i v i , A . J . L o o s e B o u n d a r y H y d r a u l i c s . Pregamon p r e s s . R i c h a r d s o n , E.V. & Simons, D.B. R e s i s t a r c e t o f l o w i n s a n d c h a n n e l s . I n t e r n A s s o c . H y d r . Res. 12. F o r t C o l l i n s , C o l o r a d o . 1967. 54 Simons, D.B. & R i c h a r d s o n , E.V. Forms o f b e d r o u g h n e s s i n a l l u v i a l c h a n n e l s . P r o c . Am. S o c i e t y C i v i l E n g i n e e r s ( V o l . 8 7 ) , 1961. T a z i o H a y a s h i , F. F o r m a t i o n o f dunes and a n t i d u n e s i n open c h a n n e l . J o u r n a l o f H y d r a u l i c D i v i s i o n , 1970, 9_6 ASCE No. HY2, P r o c . p a p e r 7056. V a n o n i , V.A. F a c t o r s d e t e r m i n i n g b e d f o r m s o f a l l u v i a l s t r e a m s . J o u r n a l o f H y d r a u l i c D i v i s i o n , 1974, 100, ASCE, No. HY3, P r o c . p a p e r 10396. V a n o n i , V.A. & Brooks, N.H. The m e c h a n i c s o f s e d i m e n t r i p p l e  f o r m a t i o n ; a d i s c u s s i o n . I n L i u , H.K. 84 ASCE, No. HY1, P r o c . p a p e r 1558. V a n o n i , V.A. ( E d i t o r ) S e d i m e n t a t i o n E n g i n e e r i n g 54, ASCE M&R . W i l l i a m s , P.B. & Kemp, P.H. I n i t i a t i o n o f r i p p l e s on t h e f l a t s e d i m e n t b e d s . J o u r n a l o f H y d r a u l i c D i v i s i o n , 97 ASCE No. HY4, P r o c . p a p e r 8042, 1971. 55 APPENDIX I EXPERIMENTAL RESULTS AND FIGURES (1) EXPERIMENTAL RESULTS OF PORTAGE SILICA SEDIMENT BEDFORMS Y (cm) V (cm/sec) H_, (cm) L n (cm) N,. C„ (cm/sec) h i K I D 21.5 29.98 0.35 8.0 0.21 0.0 21.0 36.08 0.6 9.5 0.25 0.0014 20.5 36.39 0.7 9.5 0.26 0.0021 20.5 46.38 0.6 10.39 0.33 0.287 20.5 52.55 1.1 12.14 0.37 0.380 21.0 54.61 1.1 14.0 0.38 0.05 20.5 59.76 1.3 15.5 0.42 0.096 21.0 72.12 1.6 22.5 0.50 0.125 ** 18.2 76.32 2.7 104.1 0.57 0.2 ** 20.5 85.5 2.3 107.0 0.60 0.325 37.0 53.58 1.55 9.5 0.28 0.0112 37.0 46.38 0.7 8.25 0.24 0.0068 37.0 31.96 0.3 8.5 0.167 0.0 17.0 90.02 2.65 100.0 0.7 0.13 21.0 72.29 2.5 50.0 0.5 0.15 21.0 76.0 2.5 90.0 0.53 0.15 (2) EXPERIMENTAL RESULTS OF FRASER RIVER SAND BEDFORMS 20:.0 43.23 1.75 17.5 0.31 0.0056 21.0 50.06 1.85 18.1 0.35 0.0075 21.0 53.03 1.85 21.0 0.37 0.015 21.0 55.19 1.9 30.0 0.384 0.0208 56 APPENDIX I CONT Y (cm) V (cm/sec) H (cm) L (cm) N^ . C (cm/sec) 20.5 56.30 2.1 48.3 0.397 0.066 21.0 58.30 2.2 57.0 0.406 0.075 21.0 63.74 2.3 65.0 0.44 0.098 21.0 70.58 2.5 65.0 0.49 0.10 ** 21.5 73.99 2.5 65.0 0.51 0.16 ** 21.0 78.60 2.5 67.24 0.548 0.16 21.0 84.25 2.8 79.5 0.587 0.17 20.3 84.82 3.3 78.0 0.6 0.106 21.0 85.69 2.85 78.0 0.6 0.2 18.9 84.34 3.1 105.3 0.62 0.192 21.0 90.6 2.85 110.25 0.63 0.21 21.0 97.93 3.5 105.0 0.68 0.25 (3) EXPERIMENTAL RESULTS OF BAKELITE SEDIMENT BEDFORMS 28.0 26.13 0.001 100.0 0.157 0.001 20.0 27.64 0.001 100.0 0.197 0.001 21.0 30.13 0.9 15.5 0.21 0.03 20.5 33.5 1.0 67.5 0.236 0.04 21.0 41.51 2.0 87.5 0.289 0.04 28.0 48.04 2.5 87.5 0.29 0.125 26.0 65.45 5.5 127.5 0.41 0.29 15.0 52.54 4.5 125.0 0.433 0.25 20.0 67.15 4.5 125.0 0.48 0.5 57 APPENDIX I CONT... Y (cm) V (cm/sec) 20.0 87.67 21.0 87.93 15.0 87.67 20.0 103.05 ** R i p p l e - d u n e s , o r r i p p l e e x i s t on t h e u p s t r e a m f a c e o f d u n e s . (4) FIGURES: FOLLOWING FIGURES ARE PLOTTED FROM THE EXPERIMENTAL RESULTS. H R (cm) 4.5 3 . 5 2 . 5 0.001 L R (cm) 130 .0 192 . 5 190.0 100 .0 N^. Cg (cm/sec) 0.62 0.74 0.612 1.515 0.72 1.67 0.736 3.0 58 F I G . 1.1 59 2 CORRECTION TERM x = Y R,S , l b / f t o h FOR ANTIDUNE BED CONFIGURATION (AFTER RICHARDSON ET.AL ( 1 9 6 7 ) . ) F I G . 1 . 3 61 FIG.1.5A F I G . 1. SB-PROFILES OF MEAN VELOCITY, LONGITUDINAL AND TRANSVERSE COMPONENTS OF VELOCITY FLUCTUATIONS. (AFTER RAUDKIVI, A . J . ( 1 9 6 3 ) . ) V -V -1 1 1 - 1 240- 200. 160. 120-DISTANCE CM 80. 40-F I G . l . 6 20. 15. 10. 5. 2.5 0. Y CM PORTAGE S I L I C A SANDBED Y = 18.2 CM ABOVE CREST POINTS V = 76.3 CM/SEC FROUDE NUMBER N = 0.57 BEDFORM AT TIME = 0.0 SEC BEDFORM AT TIME = 216.0 SEC t o y . v . : . . , . . - ; - . - . v v - s a n d • f-; / ~ / Y J \ ; < * t r a p w a t e r t a n k pump # 1 = 0 : f l u m e e n t r a n c e honeycomb s e d i m e n t c h a n n e l bed = 0 pump # 2 Topv i e w e n l a r g e d s e d i m e n t b e d f o r m s 0 pump # pump # 2 S i d e v i e w F I G . l . 7 65 67 PROGRESSIVE SURFACE WAVE C FREE SURFACE — V — -BED ( A ) '//////////////s/////777rr/1////////"'//"I"'""""""' ARRESTED SURFACE WAVE ZERO VELOCITY, - ^ ^ ^ ^ ^ F R E E SURFACE C=V (B) BED w t w i i i i i i i n i i i i i i i i m i w i n n i i n i i m i h i INVERTED AND ARRESTED BED WAVE FREE SURFACE V V=C (C) m r n r n L ^ ^rrrrriTTTmni TTTTTTTJl F I G . l . 1 5 F I G . l .16 F I G . l .17 160 3-7 3.2 PORTAGE S I L I C A SAND DUNES 80. CM' 2.4 Y 1.6 RIPPLES 0.8 20. 80. V CM — L 0.0 \ 140. 0. F I G . l . 1 8 72 75 10- 20. 30- 40- 50. 60.70.8090.100. V CM/SEC F I G . l . 3 0 76 PART I I : RIPPLES GENERATED BY OSCILLATORY FLOW  CHAPTER 2.1 INTRODUCTION Water waves a r e t h e m a j o r f a c t o r i n d e t e r m i n i n g t h e g e o m e t r y and c o m p o s i t i o n o f b e a c h e s . Waves p r o v i d e an i m p o r t a n t e n e r g y s o u r c e f o r t r a n s p o r t i n g b o t t o m s e d i m e n t m a t e r i a l s o n s h o r e , o f f s h o r e , and a l o n g s h o r e . I t i s i m p o r t a n t t o d i s t i n g u i s h between v a r i o u s t y p e s o f w a t e r waves t h a t may be g e n e r a t e d and p r o p a g a t e d . Water waves may be c l a s s i f i e d by v a r i o u s ways, s u c h as by wave p e r i o d T, o r by t h e r e c i p -r o c a l o f T, t h e wave f r e q u e n c y f , o r by wave l e n g t h . I f t h e wave p e r i o d T i s l e s s t h a n 0.1 s e c . o r t h e wave l e n g t h i s l e s s t h a n 1 i n c h , w a t e r waves p r o p a g a t e m a i n l y u n d e r t h e i n f l u e n c e o f s u r f a c e t e n s i o n f o r c e , so t h a t t h e y h a v e b e e n named c a p i l l a r y waves. I t i s f o u n d t h a t o n l y a s m a l l amount o f t h e t o t a l w a t e r wave e n e r g y i s a s s o c i a t e d w i t h c a p i l l a r y waves, and t h e y a r e n o t c a p a b l e o f moving b o t t o m s e d i m e n t p a r t i c l e s . I f t h e wave p e r i o d i s g r e a t e r t h a n 1 s e c o n d , o r wave l e n g t h i s g r e a t e r t h a n a b o u t 2 i n c h e s , t h e a c t i o n o f g r a v i t a t i o n a l f o r c e i s t h e p r i n c i p a l f o r c e w h i c h i n f l u e n c e s t h e s u r f a c e f o r m and t h e p r o p a g a t i o n o f t h e w a t e r waves, so t h a t t h e y h a v e b e e n named g r a v i t y waves. The l a r g e s t amount o f t o t a l wave e n e r g y i s a s s o c i a t e d w i t h g r a v i t y waves. The w a t e r p a r t i c l e s a r e p r e d i c t e d t o move a l o n g c l o s e d o r b i t s b y l i n e a r wave t h e o r y . Water p a r t i c l e s g e n e r a l l y move a l o n g e l l i p t i c a l p a t h s i n s h a l l o w w a t e r , where w a t e r s u r f a c e wave l e n g t h /^ w i s g e n e r a l l y g r e a t e r t h a n 25 t i m e s o f w a t e r 77 d e p t h . The o r b i t s o f w a t e r p a r t i c l e m o t i o n a r e f u n c t i o n o f w a t e r wave p e r i o d , wave l e n g t h , wave h e i g h t and w a t e r d e p t h . Thus, t h e more s h a l l o w t h e w a t e r , t h e f l a t t e r t h e e l l i p e s . The h o r i z o n t a l w a t e r p a r t i c l e d i s p l a c e m e n t n e a r t h e b o t t o m c a n be l a r g e and t h e v e r t i c a l w a t e r p a r t i c l e d i s p l a c e m e n t n e a r t h e b o t t o m a p p r o a c h e s t o z e r o , t h a t i s t h e w a t e r p a r t i c l e m o t i o n n e a r t h e b o t t o m becomes a l i n e a r o s c i l l a t o r y m o t i o n . I t i s t h e s e a l m o s t l i n e a r p a r t i c l e o r b i t s a t t h e s e a o r open c h a n n e l b e d w h i c h p r o d u c e an o s c i l l a t o r y s h e a r s t r e s s and i n d u c e t u r b u l e n c e w h i c h a r e c a p a b l e o f moving t h e b o t t o m s e d i m e n t p a r t i c l e s and p r o d u c i n g s m a l l s e d i m e n t r i p p l e s . I n deep w a t e r , t h a t i s f o r a w a t e r d e p t h w h i c h i s g e n e r a l l y g r e a t e r t h a n h a l f o f t h e w a t e r wave l e n g t h , w a t e r p a r t i c l e s g e n e r a l l y move a l o n g c l o s e d c i r c u l a r p a t h s . The a m p l i t u d e o f w a t e r p a r t i c l e m o t i o n o r b i t d e c r e a s e s e x p o n e n -t i a l l y w i t h d e p t h . Wather p a r t i c l e m o t i o n o r b i t s d e c a y t o n e a r z e r o i n a b o u t h a l f a wave l e n g t h . T h i s i m p l i e s t h a t t h e r e i s no w a t e r p a r t i c l e movement a t t h e b o t t o m , and t h u s , deep w a t e r g r a v i t y waves do h o t move t h e bed s e d i m e n t p a r t i c l e s . B e d forms r e s u l t i n g f r o m w a t e r s u r f a c e wave a c t i o n a r e a t l e a s t as s i g n i f i c a n t as t h o s e f o r m e d by u n i d i r e c t i o n a l w a t e r f l o w s , w h i c h have been d i s c u s s e d i n t h e f i r s t p a r t o f t h i s t h e s i s , b e c a u s e t h e y e x e r t a s t r o n g i n f l u e n c e on s e d i m e n t t r a n s p o r t and b e d f o r m f o r m a t i o n . F o r example, i t i s f o u n d i n o r d e r t o d e s i g n v a r i o u s c o a s t a l s t r u c t u r e s , o r t o p r e d i c t t h e Change o f b o t t o m c o n f i g u r a t i o n s , i t i s n e c e s s a r y t o u n d e r t a n d t h e mechanism o f s e d i m e n t b e d f o r m f o r m a t i o n due t o t h e a c t i o n o f w a t e r s u r f a c e waves. 78 I n t h e v i c i n i t y o f an open c h a n n e l bed, t h e w a t e r p a r t i c l e m o t i o n u n d e r t h e w a t e r s u r f a c e ;\ waves c a n be t h e o r -e t i c a l l y assumed t o be a s y m m e t r i c a l o s c i l l a t i o n and c o u l d r e s u l t i n an a l m o s t e q u a l l y s y m m e t r i c m o t i o n o f t h e s e d i m e n t p a r t i c l e s . I f t h e w a t e r has a s m a l l a d d i t i o n a l u n i d i r e c t i o n a l v e l o c i t y i n a d d i t i o n t o t h e w a t e r wave m o t i o n , s u c h as l i t t o r a l c u r r e n t s , t h e n l a r g e n e t t r a n s p o r t o f s e d i m e n t c a n r e s u l t , s m a l l s a n d r i p p l e s w i l l m i g r a t e downstream i n t h e same d i r e c t i o n as t h e w a t e r s u r f a c e wave m o t i o n . I n t h i s s e c o n d p a r t o f t h e t h e s i s , a number o f e x p e r i m e n t s h a ve b e e n c a r r i e d o u t t o f i n d o u t t h e g e n e r a l i z e d r e l a t i o n s h i p s among r i p p l e d i m e n s i o n s , w a t e r s u r f a c e wave c h a r a c t e r i s t i c s , and s e d i m e n t p a r t i c l e p r o p e r t i e s by u s i n g v a r i o u s d i m e n s i o n a l v a r i a b l e s and d i m e n s i o n l e s s p a r a m e t e r s f r o m d i m e n s i o n a l a n a l y s i s . P r e v i o u s s t u d i e s done by Homma and H o r i k a w a (1964-), H o r i k a w a and Watanabe ( 1 9 6 7 ) , and M o g r i d g e and Kamphuis ( 1 9 7 2 ) , a r e a l s o u s e d f o r c o m p a r i s o n s t u d y p u r p o s e s . 79 CHAPTER 2.2 LITERATURE REVIEW The s t u d i e s on s a n d r i p p l e s f o r m e d by t h e a c t i o n o f w a t e r s u r f a c e waves have b e e n made by manyy i n v e s t i g a t o r s . The m a j o r i t y o f t h e s e s t u d i e s h a v e been b a s e d on e x p e r i m e n t s c a r r i e d o u t i n l a b o r a t o r y w a t e r wave f l u m e s o r u s i n g o s c i l l a t o r y p l a t e s o r w a t e r t u n n e l s . One o f t h e e a r l i e r s t u d i e s was made by B a g n o l d ( 1 9 4 5 ) , and s t i l l a k e y r e f e r e n c e f o r r i p p l e b e h a v i o r . L a t e r s t u d i e s i n c l u d e works by Y a l i n and R u s s e l l ( 1 9 6 2 ) , Kennedy and F a l c o n ( 1 9 6 5 ) , Homma and H o r i k a w a ( 1 9 6 2 ) , H o r i k a w a and Watanabe ( 1 9 6 7 ) , and Mogridgex-and Kamphuis ( 1 9 7 2 ) . B a g n o l d assumed t h a t t h e b o t t o m - w a t e r o s c i l l a t i o n i s n e a r l y s i m p l e h a r m o n i c , so t h a t t h e r i p p l e mechanism c o u l d be s t u d i e d by o s c i l l a t i n g a s e c t i o n o f bed t h r o u g h s t i l l w a t e r . I n a d d i t i o n t o o b s e r v i n g t h e g e n e r a t i o n o f r i p p l e s i n s a n d , B a g n o l d a l s o o b s e r v e d t h e w a t e r m o t i o n u s i n g f i x e d a r t i f i c i a l r i p p l e s and c o n c l u d e d t h a t " t h e w a t e r m o t i o n d i f f e r s a c c o r d i n g as t h e s t r o k e l e n g t h 2 ? i s g r e a t e r o r l e s s t h e r i p p l e l e n g t h L ( c r e s t t o c r e s t ) . I f i t i s g r e a t e r ( h i g h m i d - s t r o k e v e l o c i t i e s and low e n d - s t r o k e a c c e l e r a t i o n s ) , t h e t r o u g h i s o c c u p i e d d u r i n g most o f t h e s t r o k e by s m a l l e d d i e s w h i c h have d e v e l o p e d f r o m t h e l e e c r e s t " , b u t "when 2 £ i s l e s s t h a n L , t h e t r o u g h s a r e empty o f e d d i e s t i l l t h e l a s t q u a r t e r ( o f t h e s t r o k e ) i s r e a c h e d " . A g a i n i t s h o u l d be s t r e s s e d t h a t B a g n o l d ' s a r t i f i c i a l r i p p l e s may n o t c o r r e s p o n d t o t h e n a t u r a l m aking o f s a n d and w a t e r m o t i o n , so t h a t 2 E, < L may 8 0 n e v e r e x i s t i n n a t u r e , e x c e p t as a r e s i d u a l f e a t u r e . B a g n o l d c o n t i n u e s , " d u r i n g r e t a r d a t i o n a t t h e end o f t h e l a s t q u a r t e r o f t h e s t r o k e , a s i n g l e eddy i n t h e m i d d l e o f t h e t r o u g h r a p i d l y grows i n t o a v e r y l a r g e v o r t e x , w h i c h f i l l s t h e whole t r o u g h . A t t h e end o f t h e s t r o k e , i t s i n e r t i a c a u s e s t h e v o r t e x t o l a g b e h i n d , so t h a t i t l e a v e s t h e m i d d l e o f i t s t r o u g h and mounts up t h e s l o p e p a s t t h e c r e s t and o u t i n t o f r e e w a t e r " . However, as m e n t i o n e d e a r l i e r , t h e a r t i f i c i a l f i x e d r i p p l e s were u s e d d u r i n g c a r r y i n g o u t B a g n o l d ' s s t u d y , o t h e r w i s e t h e s i t u a t i o n o f 2 K < L w i l l n o t a p p e a r i n p r e s e n t o r any o t h e r w a t e r wave f l u m e s e d i m e n t b e d f o r m s . B a g n o l d a l s o r e p o r t e d t h a t , " t h e r e a r e a t l e a s t two d i s t i n c t t y p e s o f r i p p l e , e a c h h a v i n g i t s own f o r m a t o r y mechanism. Though t h e two r i p p l e t y p e s may i n some c a s e s be s i m i l a r i n a p p e a r a n c e , t h e y d i f f e r g r e a t l y i n c h a r a c t e r . F o r t h e r o l l i n g - g r a i n r i p p l e , B a g n o l d s t a t e d t h a t , " a t c r i t i c a l s p e e d o f t h e w a t e r m o t i o n a t w h i c h g r a i n s on a smoothed s u r f a c e f i r s t b e g i n t o move, g r a i n s s t a r t t o be r o l l e d t o and f r o o v e r t h e s u r f a c e , b u t a r e n o t l i f t e d o f f i t " , and, " t h o u g h i n i t i a l l y d i s t r i b u t e d a t random, t h e r o l l i n g g r a i n s become o r g a n i z e d as t i m e goes on, and t e n d t o come t o r e s t i n p a r a l l e l t r a n s v e r s e z o n e s . More g r a i n s r e a c h t h e s e z o n e s t h a n l e a v e them", t h e r e f o r e , " t h e z o n e s s o o n become l i t t l e wavy r i d g e s a few g r a i n s h i g h , w h i c h c r e s t s sway f r o m l e e s i d e t o l e e s i d e d u r i n g s u c c e s s i v e s t r o k e r e v e r s a l s " . F o r t h e v o r t e x r i p p l e , B a g n o l d s t a t e d t h a t , " i f t h e s p e e d e x c e e d s t w i c e t h e c r i t i c a l s p e e d , t h e r i p p l e v o r t e x s u d d e n l y 81 a p p e a r s , and t h e whole r e g i m e b r e a k s down", "At t h e end o f t h e s t r o k e when t h e f u l l y grown v o r t e x b e g i n s t o o v e r r u n t h e r i p p l e , i t comes i n t o c o n t a c t w i t h t h e san d a t t h e r i p p l e ' s f o o t . G r a i n s a r e s c o o p e d o u t f r o m h e r e and s h o t upwards p a r a l l e l w i t h t h e s u r f a c e o f t h e s l o p e " . Homraa and H o r i k a w a ( 1 9 6 2 ) , u s e d n a t u r a l s a n d p a r t i c l e s and t h e i r 25 m e t e r s l o n g w a t e r wave f l u m e t o c a r r y o u t a s e r i e s o f e x p e r i m e n t s and p r o p o s e d t h e g e n e r a l i z e d r e l a t i o n s h i p s among d i m e n s i o n s o f w a t e r wave g e n e r a t e d b e d f o r m s , w a t e r s u r f a c e wave d i m e n s i o n s , and s e d i m e n t p r o p e r t i e s u s i n g f i v e d i m e n s i o n l e s s p a r a m e t e r s , S , H A , L /F , U F / \> and Wo^50/v • H ti H m H o r i k a w a and Watanabe ( 1 9 6 7 ) , c a r r i e d o u t a n o t h e r s e r i e s o f e x p e r i m e n t s i n t h e i r w a t e r wave f l u m e by u s i n g low d e n s i t y S u l f o n i c p o l y s t y r o l p e l l e t s i n s t e a d o f n a t u r a l s a n d p a r t i c l e s t o c o n f i r m t h e a p p l i c a b i l i t y o f t h o s e r e l a t i o n s h i p s o b t a i n e d by Homma and H o r i k a w a ( 1 9 6 2 ) . By u s i n g d i m e n s i o n a l a n a l y s i s , M o g r i d g e and Kamphuis ( 1 9 7 2 ) , g r o u p e d a s e t o f s u i t a b l e i n d e p e n d e n t d i m e n s i o n l e s s p a r a m e t e r s , p ^ , (2£ + | s | ) / d 5 0 , ((p s-p)gd 5 0 3 /(pv 2)) . a n d pdgQ/( (ps-p)gT2) > where, £ i s d e f i n e d as a m p l i t u d e o r s e m i -o r b i t a l d i a m e t e r o f f l u i d m o t i o n , i m m e d i a t e l y o u t s i d e t h e b o u n d a r y l a y e r f r o m i t s mean p o s i t i o n , % i s d e f i n e d as t h e n e t mass t r a n s p o r t d i s t a n c e moved by t h e f l u i d i n a wave p e r i o d i m m e d i a t e l y o u t s i d e t h e b o u n d a r y l a y e r . S i s o f m i n o r i m p o r t a n c e i n san d wave f o r m a t i o n , and i s v e r y s m a l l compared t o K, ( r e f e r t o A p p e n d i x I V ) , t h u s ( 2 ? + I'<$ I )/d c a n be r o u g h l y a p p r o x i m a t e d by 2 5 / d 5 Q . The v a l u e o f Pg ^ p was v a r i e d by u s i n g d i f f e r e n t s e d i m e n t p a r t i c l e s , t h e v a l u e o f 2 ? / d r ^ was 50 v a r i e d by i n c r e a s i n g t h e w a t e r s u r f a c e wave h e i g h t o r a l t e r i n g 3 2 t h e w a t e r d e p t h , t h e v a l u e o f ( ( P s ? - P ) g d 5 Q /(pv )) was v a r i e d 2 by c h a n g i n g t h e w a t e r t e m p e r a t u r e , and P.d^^/( (P - P) gT ) was v a r i e d by a l t e r i n g t h e o s c i l l a t i o n p e r i o d . The o r b i t a l d i a m e t e r o f f l u i d p a r t i c l e m o t i o n i n t h e v i c i n i t y o f t h e b e d f o r m s 2 ? were measured by r e c o r d i n g t h e p a t h s o f a l m o s t n e u t r a l l y b u o y a n t p a r t i c l e s i n t h e f l u i d on v i d e o t a p e . I n o r d e r t o d e t e r m i n e t h e r e l a t i o n s h i p s o f b e d f o r m d i m e n s i o n s w i t h r e p s e c t t o t h e v a r i a t i o n s o f t h e s e d i m e n s i o n -l e s s p a r a t m e t e r s , M o g r i d g e and Kamphuis have done a number o f e x p e r i m e n t s i n t h e i r 50 m e t e r s l o n g , 1 m e t e r wide and 1 . 2 m e t e r deep w a t e r wave f l u m e . T h e s e e x p e r i m e n t s were s u p p l e -mented by e x t e n d i n g t h e r a n g e o f w a t e r wave a m p l i t u d e and p e r i o d i n a 12 m e t e r l o n g , 1 meter deep and 0.5 m e t e r wide o s c i l l a t o r y w a t e r wave t u n n e l . A number o f wave f l u m e e x p e r i m e n t s were r e p e a t e d i n t h e o s c i l l a t o r y w a t e r wave t u n n e l t o c o n f i r m t h a t , s t a t i s t i c a l l y , a l l e x p e r i m e n t a l r e s u l t s b e l o n g t o t h e same p o p u l a t i o n . From t h e i r e x p e r i m e n t a l r e s u l t s f r o m l a b o r a t o r y w a t e r wave f l u m e and o s c i l l a t o r y w a t e r wave t u n n e l , M o g r i d g e and Kamphuis c o n c l u d e d t h a t : 8 3 ( 1 ) The d i f f e r e n c e i n b e d f o r m d i m e n s i o n s c a u s e d by 3 c h a n g i n g (pg ~ p)9^gQ /(pv 2) i s so s m a l l as t o be u n i m p o r t a n t . ( 2 ) A t v a l u e s o f P ^^g / ( ( p s _ p )gT ) g r e a t e r t h a n a p p r o x i m a t e l y 2 0 0 . 1 0 , b e d f o r m s do n o t a p p e a r on t h e s e d i m e n t c h a n n e l bed, b u t t h i s c o n c l u s i o n a p p e a r s d o u b t f u l as w i l l be d i s c u s s e d l a t e r i n C h a p t e r 2 . 5 . ( 3 ) P g / p w a s shown t o be an i m p o r t a n t p a r a m e t e r a t t h e v a l u e s o f pd^Q/( (ps~p)gT ) g r e a t e r t h a n a p p r o x i m a t e l y 2 0 . 1 0 '. R e s u l t s f o r p s/p = 2 . 6 8 and p s/p = 1 . 6 0 showed a d i f f e r e n c e o f a p p r o x i m a t e l y 1 5 % i n b e d f o r m l e n g t h and h e i g h t . (4) F o r t h e v a r i a t i o n o f b e d f o r m l e n g t h L w i t h K ( 2 £ + ]&'I)/d--. , as ( 2 f + :|S|.)/d,_^ i n c r e a s e s , b e d f o r m l e n g t h b ( J 1 D O i s i n i t i a l l y d e p e n d e n t o n l y on t h e f l u i d m o t i o n o r b i t a l l e n g t h , b u t t h i s c o n c l u s i o n may be t o o s t r o n g t o a c c e p t a s w i l l be d i s c u s s e d l a t e r i n C h a p t e r 2 . 5 . A t l a r g e v a l u e s o f ( 2 £ + .|.sr-)/d t h e b e d f o r m l e n g t h a p p e a r s t o become d i r e c t l y p r o p o r t i o n a l t o t h e g r a i n s i z e d i a m e t e r . The c o n s t a n t o f p r o p o r t i o n a l i t y i s a 2 f u n c t i o n o f p d ^ / t f p ^ -p )gT ) • Bedform h e i g h t and s t e e p n e s s a t f i r s t i n c r e a s e s w i t h i n c r e a s i n g ( 2 £ + / d ^ , r e a c h e s a maximum v a l u e and t h e n d e c r e a s e s u n t i l t h e b e d f o r m d i s a p p e a r s . ( 5 ) D e s i g n c u r v e s have been drawn by means o f w h i c h i t i s p o s s i b l e t o p r e d i c t t h e l e n g t h and h e i g h t o f b e d f o r m t h a t w i l l d e v e l o p on any s p e c i f i e d s e d i m e n t b e d f o r g i v e n c o n d i t i o n s o f f l u i d o s c i l l a t i o n a m p l i t u d e and p e r i o d . B o t h e x p e r i m e n t a l r e s u l t s and c o n c l u s i o n s f r o m H o r i k a w a and Watanabe, and M o g r i d g e and Kamphuis a r e compared w i t h p r e s e n t e x p e r i m e n t a l r e s u l t s i n l a t e r c h a p t e r s . 84 CHAPTER 2.5 EXPERIMENTAL TECHNIQUE A w a t e r wave f l u m e o f a b o u t 20 m e t e r s l o n g , 62 cm wide and 75 cm deep i n t h e H y d r a u l i c E n g i n e e r i n g L a b o r a t o r y a t t h e U n i v e r s i t y o f B r i t i s h C o l o m b i a was u s e d f o r t h e p r e s e n t s t u d y . A w a t e r wave g e n e r a t o r o f t h e f l a p t y p e c a n g e n e r a t e r e g u l a r waves w i t h wave p e r i o d f r o m a b o u t 0.7 t o 1.4 s e c o n d o r more. A wave e n e r g y a b s o r b e r was p l a c e d on t h e o p p o s i t e end o f t h e wave f l u m e i n o r d e r t o a b s o r b t h e wave e n e r g y a t t h e end o f t h e f l u m e , t h u s , t h e wave r e f l e c t i o n c o u l d be a v o i d e d . ( Ref. to F i g . 2.1 ). Sed i m e n t m a t e r i a l was u n i f o r m l y d i s t r i b u t e d i n t h e m i d d l e s e c t i o n o f t h e w a t e r wave f l u m e . The s e d i m e n t bed was a b o u t 4 m e t e r s l o n g . T h r e e t y p e s o f s e d i m e n t p a r t i c l e s , P o r t a g e S i l i c a s a n d , F r a s e r R i v e r s a n d and B a k e l i t e s and, were u s e d f o r t h e p u r p o s e o f s t u d y i n g b e d f o r m b e h a v i o r s on d i f f e r e n t t y p e s o f s e d i m e n t m a t e r i a l s . D e n s i t i e s and g r a i n s i z e d i s t r i -b u t i o n s w h i c h d e t e r m i n e d by t h e method o f s i e v i n g o f e a c h t y p e o f s e d i m e n t p a r t i c l e s were d i s c u s s e d i n t h e f i r s t p a r t o f t h i s t h e s i s . The w a t e r t e m p e r a t u r e was i n t h e r a n g e o f 8 t o 12°C, and was assumed t o be c o n s t a n t . To d e s c r i b e a s u r f a c e wave, i t i s n e c e s s a r y t o measure wave p e r i o d T, wave l e n g t h L^, wave h e i g h t H^, and w a t e r d e p t h Y. A d d i t i o n a l d a t a , s u c h as wave c e l e r i t y C, and w a t e r p a r t i c l e m o t i o n o r b i t i n t h e v i c i n i t y o f t h e c h a n n e l b e d a r e a l s o d e s i r a b l e t o compare w i t h t h e t h e o r e t i c a l a s s u m p t i o n s . The w a t e r s u r f a c e wave h e i g h t , wave l e n g t h and wave p e r i o d were 85 m easured by w a t e r wave p r o b e s . The p r o b e s were u s e d i n c o n j u n c t i o n w i t h a d j u s t a b l e r e c o r d i n g s p e e d H o n e y w e l l r e c o r d e r w h i c h w i l l i n d i c a t e t h e v a l u e o f o u t p u t v o l t a g e . The d i r e c t o p e r a t i o n a l mode was u s e d f o r a l l o f t h e e x p e r i m e n t s o f t h e p r e s e n t s t u d y . The v a l u e o f o u t p u t v o l t a g e o b t a i n e d f r o m t h e H o n e y w e l l r e c o r d e r i s i n d i r e c t p r o p o r t i o n t o t h e w a t e r s u r f a c e wave h e i g h t . The c o n s t a n t o f p r o p o r t i o n a l i t y c a n be d e t e r m i n e d d u r i n g t h e c a l i b r a t i o n o f w a t e r wave p r o b e . The d e t a i l c a l i b r a t i o n p r o c e d u r e i s d i s c u s s e d i n A p p e n d i x H I . Thus, t h e w a t e r s u r f a c e wave h e i g h t c a n be d e t e r m i n e d . Wave p e r i o d c a n be measured f r o m t h e r e c o r d i n g s p e e d o f t h e H o n e y w e l l r e c o r d e r and t h e number o f peak o u t p u t v o l t a g e s d u r i n g a c e r t a i n p e r i o d o f t i m e . A s t o p w a t c h i s a l s o u s e d t o c h e c k t h e v a l u e o f wave p e r i o d o b t a i n e d f r o m t h e H o n e y w e l l r e c o r d e r . Water wave l e n g t h c a n be o b t a i n e d by u s i n g two w a t e r wave p r o b e s s i m u l t a n e o u s l y , p l a c i n g t h e s e two w a t e r wave p r o b e s a p a r t and a d j u s t i n g t h e d i s t a n c e i n between t i l l two r e c o r d e r s i n d i c a t e n e a r l y e x a c t s i m u l t a n e o u s peak o u t p u t v o l t a g e o r t h e c o n t o u r s o f wave w a t e r m o t i o n f r o m two r e c o r d e r s a r e i n p h a s e . The w a t e r s u r f a c e wave l e n g t h e q u a l s t h e d i s t a n c e between t h e s e two w a t e r wave p r o b e s . A s t o p watch i s u s e d t o measure t h e wave c e l e r i t y C. A l l t h e v a l u e s o f wave c e l e r i t i e s o b t a i n e d by s t o p w a t c h measurement a r e r e c h e c k e d w i t h t h e r a t i o o f w a t e r s u r f a c e wave l e n g t h and wave p e r i o d . Water f l o w d e p t h Y i s measured t h r o u g h t h e g l a s s s i d e w a l l o f w a t e r wave f l u m e b e f o r e t h e w a t e r s u r f a c e waves were g e n e r a t e d . 86 The measurements o f w a t e r p a r t i c l e m o t i o n o r b i t s i n t h e v i c i n i t y o f t h e s e d i m e n t b e d f o r m s a r e , c o m p a r a t i v e l y d i f f i c u l t . S i n c e b a s i c a s s u m p t i o n s w h i c h commonly made i n -d e v e l o p i n g l i n e a r wave t h e o r y a r e a c c e p t a b l e f o r p r e s e n t e x p e r i m e n t s , t h e v a l u e s o f £ , a m p l i t u d e o f w a t e r p a r t i c l e m o t i o n o r b i t i n t h e v i c i n i t y o f t h e s e d i m e n t b e d f o r m s , and Um, t h e maximum h o r i z o n t a l w a t e r p a r t i c l e m o t i o n v e l o c i t y i n t h e v i c i n i t y o f t h e s e d i m e n t b e d f o r m s were e v a l u a t e d b y e q u a t i o n s d e v e l o p e d f r o m l i n e a r wave t h e o r y , and w i l l be d i s c u s s e d i n t h e n e x t c h a p t e r . S e d i m e n t b e d f o r m d i m e n s i o n s , s u c h as b e d f o r m l e n g t h and b e d f o r m h e i g h t , were measured by b o t h measurement t h r o u g h t h e g l a s s s i d e w a l l o f w a t e r wave: f l u m e and t h e d i r e c t measurement. S e v e r a l measurements a t v a r i o u s p o s i t i o n on s e d i m e n t c h a n n e l bed were t a k e n i n e a c h r u n o f e x p e r i m e n t , i f t h r e e o r f o u r r e s u l t s were c o n s i s t a n t , t h a t i s t h e d i f f e r e n c e s b etween t h e s e measurements a r e a p p r o x i -m a t e l y w i t h i n 10%, t h e n t h e a v e r a g e v a l u e o f t h e s e measure-ments was c o n s i d e r e d t o be s a t i s f a c t o r y and was t a k e n as e x p e r i m e n t a l r e s u l t . 87 CHAPTER 2.4 ANALYSIS OF EXPERIMENTAL RESULTS: DIMENSIONAL ANALYSIS The v a r i a b l e n e e d e d t o c h a r a c t e r i z e t h e w a t e r s u r f a c e H L wave a r e w a t e r s u r f a c e wave h e i g h t w and l e n g t h w, w a t e r s u r f a c e wave p e r i o d T and w a t e r d e p t h Y. S i n c e t h e wave c e l e r i t y , C = Lw/T, t h e w a t e r s u r f a c e wave v e l o c i t y C c a n be e x c l u d e d i f b o t h Lw andT a r e i n c l u d e d i n t h e d i m e n s i o n a l a n a l y s i s . A l s o s i n c e l i n e a r wave t h e o r y i s a c c u r a t e f o r H L p r e s e n t e x p e r i m e n t s , t h e f o u r v a r i a b l e s w, w, T and Y c a n be s u b s t i t u t e d by t h e maximum h o r i z o n t a l w a t e r p a r t i c l e v e l o c i t y Um and t h e maximum h o r i z o n t a l w a t e r p a r t i c l e d i s -p l a c e m e n t f r o m mean p o s i t i o n K i n t h e v i c i n i t y o f t h e s e d i m e n t c h a n n e l b e d . V a l u e s o f Um and E, c a n be c a l c u l a t e d by e q u a t i o n s d e v e l o p e d f r o m l i n e a r wave t h e o r y , w h i c h g i v e s U m = M S i Cosh(2uH R/L w) -(2 1) m 2 L w C o s h ( 2 T T Y / L w . t 2 - 1 ) and _ = Cosh (2T T H R / L W ) (2.2) K ~ 2 S i n h ( 2 T T ; Y / L W ) T h e s e two e q u a t i o n s a r e a c c e p t a b l e f o r t r a n s i t i o n a l w a t e r s u r f a c e wave, 0.04 < ,^/'-w < 0.50, and a r e assumed t o a p p l y t o a l l p r e s e n t e x p e r i m e n t s . The v a l u e s o f b e d f o r m H U h e i g h t s R a r e u s e d i n d e t e r m i n i n g t h e v a l u e s o f m and ? , t h a t means t h e v a l u e s o f ^m and £ a r e e s t i m a t e d maximum 88 h o r i z o n t a l water, p a r t i c l e v e l o c i t y and d i s p l a c e m e n t f r o m mean p o s i t i o n o f i t s m o t i o n o r b i t i n t h e v i c i n i t y o f t h e c r e s t p o i n t s o f t h e s e d i m e n t b e d f o r m s . M o g r i d g e and Kamphuis a r g u e d t h a t i f t h e f l u i d i s s i n u s o i d a l , t h e r e l a t i o n s h i p between Um and 5 c a n be e x p r e s s e d a s : Um = Constant. -S- (2.3) t h e r e f o r e , o n l y K was i n v o l v e d i n ' " t h e i r a n a l y s i s o f d e v e l o p -i n g d i m e n s i o n l e s s p a r a m e t e r s . F o r p r e s e n t s t u d y , Um was u s e d i n o u r d i m e n s i o n a l a n a l y s i s t o d e v e l o p i n d e p e n d e n t d i m e n s i o n -l e s s p a r a m e t e r s . I n a d d i t i o n , d i m e n s i o n l e s s p a r a m e t e r s d e v e l o p e d f r o m t h e work done by M o g r i d g e and Kamphuis a r e u s e d as w e l l . As f o r t h e a c c u r a c y o f u s i n g e q u a t i o n (2.1) and (2.2) i n d e t e r m i n i n g t h e v a l u e s o f Um and £ , a l t h o u g h t h e b a s i c a s s u m p t i o n s w h i c h commonly made i n d e v e l o p i n g l i n e a r wave t h e o r y a r e a c c e p t a b l e t o a l l p r e s e n t e x p e r i m e n t s , b u t l i n e a r wave t h e o r y a p p l i e s t o a wave w h i c h i s s y m m e t r i c a l a b o u t s t i l l -w a t e r l e v e l and h a s w a t e r p a r t i c l e s t h a t move i n c l o s e d o r b i t s . However, t h e f l u i d p a r t i c l e m o t i o n u n d e r t h e wave a c t i o n i s r a r e l y s y m m e t r i c a l . I n a d d i t i o n , w a t e r p a r t i c l e m o t i o n o r b i t s a r e g e n e r a l l y open, t h u s , t h e r e a r e v e r y s m a l l d i s c r e p e n c i e s between c a l c u l a t e d v a l u e s and t r u e a m p l i t u d e o f w a t e r p a r t i c l e m o t i o n o r b i t s i n t h e v i c i n i t y o f t h e s e d i m e n t b e d f o r m s . I n o r d e r t o g e t r i d o f t h i s i n a c c u r a c y , M o g r i d g e and Kamphuis 89 d e f i n e d a new v a r i a b l e 2£ + \S\ and named as e f f e c t i v e f l u i d o r b i t l e n g t h , w h i c h were measured by r e c o r d i n g t h e p a t h s o f a l m o s t n e u t r a l l y b u o y a n t p a r t i c l e s i n t h e f l u i d on v i d e o t a p e . But t h e p r o b l e m s s u c h as l o c a l i r r e g u l a r i t i e s and e f f e c t s o f w a t e r wave f l u m e s i d e w a l l a r e i n v o l v e d . A l s o , t h e s e p r o b l e m s w i l l c a u s e some s m a l l i n a c c u r a c i e s . However, a l l t h e s e v a l u e s o f i n a c c u r a c i e s i n method o f c a l c u l a t i o n f r o m e q u a t i o n s (2.1) and (2.2) o r i n t h e method o f v i d e o t a p e r e c o r d i n g measurement a r e c o n s i d e r e d t o be r e a s o n a b l y s m a l l and t h u s a c c e p t a b l e . The v a r i a b l e s n e e d e d t o c h a r a c t e r i z e t h e s e d i m e n t p a r t i c l e p r o p e r t i e s a r e s e d i m e n t d e n s i t y P s , and mean g r a i n d i a m e t e r d^Q . The v a r i a b l e s n e e d e d t o c h a r a c t e r i z e t h e w a t e r a r e i t s d e n s i t y p , 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 a t i o n a l a c c e l e r a t i o n g . S i n c e t h e w a t e r t e m p e r a t u r e i s assumed t o be c o n s t a n t , t h u s , t h e q u a n t i t a t i v e p r o p e r t y o f t h e phenomena o f s e d i m e n t b e d f o r m s f o r m a t i o n u n d e r t h e w a t e r s u r f a c e waves may be d e f i n e d by t h e s e v a r i a b l e s a s : Q = f l ( Um' T ' p s ' d 50 ' p ' v ' 9 ' P ~ (2.4) D i m e n s i o n a l a n a l y s i s l e a d s n - f r P d 5 0 U n " Umd^ n / pUm x 2 / pUm . \ Q " f 2 ( ( p s - p ) g ^ ' * v » ( ( p s - P ) g Y ) , ( ( P S - P ) g d 5 0 ) ] (2.5) The f i r s t d i m e n s i o n l e s s p a r a m e t e r i s t h e same as one o f d i m e n s i o n l e s s p a r a m e t e r s f r o m t h e work done by M o g r i d g e and Kamphuis ( 1 9 7 2 ) . The s e c o n d and t h i r d d i m e n s i o n l e s s 90 p a r a m e t e r s a r e m o d i f i e d R e y n o l d ' s numbers i n t e r m s o f w a t e r d e p t h Y and mean g r a i n d i a m e t e r d ^ ^ . The f o u r t h and f i f t h d i m e n s i o n l e s s p a r a m e t e r s a r e m o d i f i e d F r o u d e numbers i n t e r m s o f w a t e r d e p t h and mean g r a i n d i a m e t e r . I n o r d e r t o compare p r e s e n t e x p e r i m e n t a l r e s u l t s w i t h p r e v i o u s works done by H o r i k a w a and Watanabe ( 1 9 6 7 ) , o r b y M o g r i d g e and Kamphuis ( 1 9 7 2 ) , d i m e n s i o n l e s s p a r a m e t e r s d e v e l o p e d by them s u c h as ( P s - p ) g d 5 0 3 2g _R_s _Hj£_ L R , Umg are pv?* ' d 5 Q ' P ' £ > ? ' a n a v a l s o u s e d . DISCUSSIONS OF EXPERIMENTAL RESULTS RELATIONSHIPS BETWEEN BEDFORM LENGTHS AND MAXIMUM HORIZONTAL WATER PARTICLE MOTION VELOCITIES OR DISPLACEMENTS FROM ITS MEAN POSITION IN THE V I C I N I T Y OF THE SEDIMENT CHANNEL BED M.C. Q u i c k (1977-1978), s u g g e s t s t h a t t h e a l m o s t l i n e a r h o r i z o n t a l w a t e r p a r t i c l e m o t i o n o r b i t s a t t h e s e d i m e n t c h a n n e l bed p r o d u c e an o s c i l l a t o r y s h e a r s t r e s s w h i c h i s c a p a b l e o f p r o d u c i n g s m a l l s a n d r i p p l e s . As e a c h w a t e r s u r f a c e wave p a s s e s , an o s c i l l a t o r y f l o w o c c u r s a t t h e c h a n n e l bed w h i c h has an a m p l i t u d e K . The v a l u e o f K i s s m a l l compared t o t h e w a t e r s u r f a c e wave l e n g t h L . So t h a t t h e w hole r e g i o n o f s e d i m e n t c h a n n e l bed c a n be c o n s i d e r e d t o be s u b j e c t e d t o a s i m u l t a n e o u s o s c i l l a t o r y b e d s h e a r s t r e s s o f a m p l i t u d e ? . When s u b j e c t e d t o s u c h a bed s h e a r s t r e s s o f s u f f i c i e n t i n t e n s i t y t o p r o d u c e s e d i m e n t m o t i o n , s a n d p a r t i c l e s w i l l f i r s t o f a l l s i m p l y o s c i l l a t e t o and f r o on t h e f l a t b e d, and s o o n an i n s t a b i l i t y w i l l d e v e l o p and s m a l l 91 b e d f o r m s w i l l s t a r t t o d e v e l o p . F i g . 2 shows t h a t r i p p l e l e n g t h s keep a b o u t c o n s t a n t a t f i r s t u n t i l t h e v a l u e o f K i n c r e a s e s and a p p r o a c h e s t o some t r a n s i t i o n a l v a l u e s o f K f o r e a c h t y p e o f s e d i m e n t r i p p l e s . T h e s e c o n s t a n t l e n g t h r i p p l e s a r e named r o l l i n g g r a i n r i p p l e s by B a g n o l d ( 1 9 4 6 ) . F u r t h e r i n c r e a s i n g t h e v a l u e o f £ t i l l i t r e a c h e s and e x c e e d s t h e t r a n s i t i o n a l v a l u e s o f K , t h e b e d f o r m s become v o r t e x r i p p l e s and t h e i r l e n g t h s w i l l i n c r e a s e t i l l r e a c h i n g t h e maximum v a l u e o f r i p p l e l e n g t h f o r e a c h t y p e o f s e d i m e n t r i p p l e s . The t r a n s i t i o n a l v a l u e s o f £ v a r y w i t h d i f f e r e n t s e d i m e n t p r o p e r t i e s . I n t h e r a n g e o f v o r t e x r i p p l e s , F i g . 2 i n d i c a t e s t h a t t h e r e e x i s t s a l o g a r i t h m i c r e l a t i o n s h i p b etween b e d f o r m l e n g t h s and t h e v a l u e s o f £ , t h e maximum h o r i z o n t a l w a t e r p a r t i c l e d i s p l a c e m e n t f r o m mean p o s -i t i o n i n t h e v i c i n i t y o f t h e s e d i m e n t c h a n n e l b e d . M a t h e m a t i -c a l l y , i t c a n be r e p r e s e n t e d a s : l o g 1 0 L R = 0.83 l o g 1 Q 5 + 0.18 (2.6) T h i s i s r o u g h l y a p p l i c a b l e t o a l l t h r e e t y p e s o f s e d i m e n t b e d f o r m s . B o t h F i g . 2 and e q u a t i o n (2.6) show t h a t t h e v a l u e s o f 2 £ a r e a l w a y s g r e a t e r t h a n t h e b e d f o r m l e n g t h s L . M o g r i d g e and Kamphuis ( 1 9 7 2 ) , c o n c l u d e d t h a t a t low v a l u e s o f £, ( a l l p r e s e n t e x p e r i m e n t s b e l o n g t o t h i s c a t e g o r y ) , t h e b e f o r m l e n g t h i s i n d e p e n d e n t o f t h e s e d i m e n t g r a i n s i z e and i s a f u n c t i o n o f £ o n l y . The r e a s o n f o r t h e i r c o n c l u s i o n i s t h a t M o g r i d g e and K a m p h u i s 1 e x p e r i m e n t a l r e s u l t s g i v e l i n e a r r e l a t i o n s h i p o f s l o p e 1:1 between t h e v a l u e s o f l o g , _ 1 ( L /id ) X L ) K D U and l o g l n ( 2 £ / d n ) a t t h e r a n g e o f low v a l u e s o f £ . Mathemat-92 i c a l l y , t h a t i s 1 O g l 0 ( V d 5 0 ) = 1 0 9 10 (2 tVdcn) + C (2.7) o r l o g 10 L R = l o g 10 K + ( l o g 10 2+ e •) (2.7A) where v a l u e s o f C' a r e t o be d e t e r m i n e d f r o m t h e e x p e r i m e n t a l r e s u l t s . However, f o r v a r i o u s t y p e s o f s e d i m e n t b e d f o r m s , M o g r i d g e and Kamphuis' r e s u l t s show t h a t t h e v a l u e s o f C' a r e n o t a l w a y s t h e same, t h e y v a r y w i t h r e s p e c t t o t h e v a r i -a t i o n o f s e d i m e n t p r o p e r t i e s . F u r t h e r m o r e , t h e s l o p e o f l i n e a r r e l a t i o n s h i p between l o g , (L / d c r i ) and l o g , (2 g/dc-.) a r e n o t a l w a y s e q u a l t o 1:1, s m a l l v a r i a t i o n s a r e i n v o l v e d due t o v a r i o u s d i f f e r e n t s e d i m e n t p r o p e r t i e s . P r e s e n t e x p e r i m e n t a l r e s u l t s g i v e t h e v a l u e o f s l o p e o f l i n e a r r e l a t i o n s h i p between log.. L and l o g £ o r l o g (L /d ) l u R J. U X O R J U and l o g 1 0 ( 2 £ / d 5 Q ) e q u a l 1: 0.83. Thus, t h e e f f e c t s o f v a r i a t i o n o f s e d i m e n t p a r t i c l e p r o p e r t i e s s u c h as p g and d,_Q a r e c e r t a i n l y n o t n e g l i g i b l e , e s p e c i a l l y when s e d i m e n t p a r t i c l e s i z e s a r e u n u s u a l l y c o a r s e o r f i n e , o r when s e d i m e n t d e n s i t i e s a r e u n u s u a l l y l a r g e o r s m a l l . P r e s e n t e x p e r i m e n t a l r e s u l t s w i l l be compared w i t h t h e work done by M o g r i d g e and Kamphuis i n t h e n e x t c h a p t e r . B a g n o l d ( 1 9 4 6 ) , u s i n g an o s c i l l a t o r y b e d i n s t i l l w a t e r , and M o g r i d g e and Kamphuis ( 1 9 7 2 ) , u s i n g an o s c i l l a t o r y w a t e r t u n n e l , b o t h showed t h a t r i p p l e l e n g t h s become d e p e n d e n t o n l y on s e d i m e n t g r a i n s i z e d i a m e t e r f o r h i g h v a l u e s o f £ . However, i n p r e s e n t w a t e r wave f l u m e e x p e r i m e n t s , t h e v a l u e s o f K i s l i m i t e d . 93 S i n c e t h e r a n g e o f v a r i a t i o n o f w a t e r wave p e r i o d T i s smalll i n p r e s e n t s t u d y , t h u s , a l i n e a r r e l a t i o n s h i p b etween U and K c a n be o b t a i n e d f r o m e q u a t i o n ( 2 . 3 ) , m M U = c o n s t a n t . • K/T. From t h i s l i n e a r r e l a t i o n s h i p and m e q u a t i o n ( 2 . 6 ) , i t i s r e a s o n a b l e t o s t a t e t h a t , i n t h e r a n g e o f v o r t e x r i p p l e s , a l i n e a r r e l a t i o n s h i p b etween l o g ^ ^ L ^ and l o g ^ 0 U " m w i t h a b o u t t h e same s l o p e as e q u a t i o n (2.6) c a n be o b t a i n e d . F i g . 3 shows t h a t r o l l i n g g r a i n r i p p l e l e n g t h s keep a b o u t c o n s t a n t . However, i n t h e r a n g e o f v o r t e x r i p p l e s , t h e r i p p l e l e n g t h s w i l l i n c r e a s e w i t h i n c r e a s i n g t h e v a l u e o f U"m t i l l r e a c h i n g t h e maximum v a l u e o f r i p p l e l e n g t h f o r e a c h t y p e o f s e d i m e n t r i p p l e s . P r e s e n t e x p e r i m e n t a l r e s u l t s i n t h e r a n g e o f v o r t e x - r i p p l e s c a n be a p p r o x i m a t e d as l o g , ^L. = 0 . 8 6 l o g - . . U - 0.5 (2.8) a 1 0 R a 1 0 m w h i c h s l o p e i s i n good agreement w i t h e q u a t i o n ( 2 . 6 ) . RELATIONSHIPS BETWEEN BEDFORM LENGTHS AND MODIFIED FROUDE NUMBERS OR MODIFIED REYNOLD'S NUMBERS F i g . A i n d i c a t e s t h e r e l a t i o n s h i p between b e d f o r m l e n g t h s and M o d i f i e d F r o u d e Number i n t e r m s o f mean g r a i n d i a m e t e r d 5 Q . P r e s e n t e x p e r i m e n t a l r e s u l t s c a n be a p p r o x i m a t e d by l o g a r i t h m i c r e l a t i o n s h i p l og 1 0 L R = 0.79 log 1 Q ( p U m 2 / ( ( p s - P ) g d 5 0 ) ) ° ' 5 + 0.2 (2.9) 94 w h i c h i s r o u g h l y a p p l i c a b l e t o a l l t h r e e t y p e s o f s e d i m e n t b e d f o r m s , and c a n be r e w r i t t e n i n a more e x p l i c i t f o r m as l o g l O L R = " ° ' 3 9 5 l o g l 0 ( P S " P ) + ° - 7 9 l o S i o Um - 0.395 l o g 1 0 d 5 0 - 0.98 (2.9A) E q u a t i o n (2.9A) c o n f i r m s t h e i m p o r t a n c e o f t h e maximum h o r i z o n t a l w a t e r p a r t i c l e v e l o c i t y i n t h e v i c i n i t y o f s e d i m e n t c h a n n e l bed U"m i n d e t e r m i n i n g t h e b e d f o r m l e n g t h s . However, t h e n u m e r i c a l d a t a f r o m o t h e r two t e r m s i n e q u a t i o n (2.9A) show t h a t t h e e f f e c t s o f s e d i m e n t p r o p e r t i e s s u c h as s e d i m e n t p a r t i c l e d e n s i t y and mean g r a i n d i a m e t e r a r e n o t n e g l i g i b l e . As m e n t i o n e d e a r l i e r , s i n c e t h e v a r i a t i o n o f t h e w a t e r s u r f a c e wave p e r i o d T i s v e r y s m a l l d u r i n g c a r r y i n g o u t a l l p r e s e n t e x p e r i m e n t s , and t h e w a t e r s u r f a c e waves a r e assumed t o be s i n u s o i d a l , t h u s , t h e h o r i z o n t a l w a t e r p a r t i c l e m o t i o n a m p l i t u d e £ and v e l o c i t y U m a r e r e l a t e d b y e q u a t i o n ( 2 . 3 ) , e i t h e r U .. o r £ c a n s u c c e s s f u l l y r e p r e s e n t t h e b e h a v i o r s o f w a t e r s u r f a c e wave m o t i o n . T h e r e f o r e , U a r e as i m p o r t a n t as m ^ t h e v a l u e o f £ , w h i c h p r o d u c e t h e n e c e s s a r y o s c i l l a t o r y s h e a r s t e s s i n t h e v i c i n i t y o f t h e c h a n n e l bed t o f o r m t h e s e d i m e n t b e d f o r m s . I n o r d e r t o i n v e s t i g a t e i n d i v i d u a l b e d f o r m l e n g t h b e h a v i o r s o f e a c h t y p e o f s e d i m e n t c h a n n e l bed, t h e r e l a t i o n -s h i p s b etween b e d f o r m l e n g t h s and d i m e n s i o n l e s s p a r a m e t e r 2 0 5 ( ( pU m/ (PS - P ) gY„- ) * ) a r e p l o t t e d i n F i g . 5. S i n c e t h e 95 v a r i a t i o n o f w a t e r d e p t h Y i s s m a l l , i t i s f o u n d t h a t r o l l i n g g r a i n r i p p l e l e n g t h s w i l l keep a b o u t c o n s t a n t a t f i r s t , u n t i l t h e v a l u e o f U i n c r e a s e s and e x c e e d s t h e v a l u e o f t r a n s i t i o n a l m U f o r e a c h t y p e o f s e d i m e n t r i p p l e s , t h e n t h e b e d f o r m s w i l l m become v o r t e x r i p p l e s and r i p p l e l e n g t h s w i l l i n c r e a s e w i t h i n c r e a s i n g U . The v a l u e s o f t r a n s i t i o n a l U s h o u l d m a i n l y a m m J depend upon t h e s e d i m e n t p a r t i c l e p r o p e r t i e s . T h i s c a n be c o n f i r m e d by F i g . 6 and F i g . 7, w h i c h show t h e r e l a t i o n s h i p s b etween r i p p l e l e n g t h s and m o d i f i e d R e y n o l d ' s numbers U d^^/V and U Y/jJ . F o r e a c h t y p e o f s e d i m e n t c h a n n e l bed, m 50 m t h e r o l l i n g g r a i n r i p p l e l e n g t h s a t f i r s t keep a b o u t t h e same v a l u e w i t h i n c r e a s i n g v a l u e s o f m o d i f i e d R e y n o l d ' s numbers U d- / i ^ and U Y/y . A f t e r r e a c h i n g t h e t r a n s i t i o n a l m 50 m v a l u e s o f m o d i f i e d R e y n o l d ' s number U d^^/v o r U Y/v , t h e J m 50 ^ m ' b e d f o r m s become v o r t e x r i p p l e s and t h e r i p p l e l e n g t h s b e g i n t o i n c r e a s e w i t h i n c r e a s i n g m o d i f i e d R e y n o l d ' s numbers. I t c a n be f o u n d f r o m F i g . 6 and F i g . 7 t h a t a l l t h r e e t y p e s o f s e d i m e n t r i p p l e s have t h e i r own v a l u e s o f t r a n s i t i o n a l v a l u e s o f m o d i f i e d R e y n o l d ' s numbers. Thus, i t i m p l i e s t h a t t h e t r a n s i t i o n a l v a l u e s o f U o r m o d i f i e d R e y n o l d ' s numbers U d^^/U m J m 50 and U^Y/J^ m a i n l y depend on t h e s e d i m e n t p a r t i c l e p r o p e r t i e s . RELATIONSHIPS BETWEEN BEDFORM HEIGHTS AND MODIFIED FROUDE NUMBERS, MODIFIED REYNOLD'S NUMBERS, MAXIMUM HORIZONTAL WATER PARTICLE VELOCITY U AND AMPLITUDE "f . m 3 M.C. Q u i c k a l s o s u g g e s t e d t h a t r i p p l e s g e n e r a t e d by t h e a c t i o n o f t h e w a t e r wave w i l l a c h i e v e a c e r t a i n l i m i t i n g h e i g h t , and, on t h e f l a t bed, w i l l be s l i g h t l y a s y m e t r i c a l and w i l l m i g r a t e s l o w l y i n t h e w a t e r s u r f a c e wave d i r e c t i o n . 96 A l t h o u g h i t i s p o s s i b l e t o i n c l u d e t h e v e r t i c a l v e l o c i t y component, M.C. Q u i c k assumes t h a t t h e h o r i z o n t a l v e l o c i t i e s p r e d o m i n a t e , and t h e v a l u e s o f h o r i z o n t a l v e l o c i t i e s c a n be r e p r e s e n t e d by: U = U m (Cos (wti) + (KA) Cos (K x) Cos ( w t ) ) (2.10) where K i s t h e r i p p l e wave number o r 2 T T / L _ , A i s t h e r i p p l e K wave a m p l i t u d e o r H / , and W i s t h e w a t e r s u r f a c e wave f r e q u e n c y o r 2 i r / T • The f i r s t t e r m i n e q u a t i o n , U m C o s ( w t ) , i s p r o d u c e d d i r e c t l y b y t h e s u r f a c e waves and t h e s e c o n d t e r m , U KA Cos (KX ) Cos ( w t ) , i s c a u s e d b y t h e f i r s t t e r m , U Cos (w t) m m and t h e s e d i m e n t r i p p l e i t s e l f . The b o u n d a r y l a y e r t h i c k n e s s ± j_ c a n be a p p r o x i m a t e d as ( v/W) 2 o r (2TTV / T ) 2 . S i n c e i t i s v e r y t h i n so t h a t i t h a s b e e n i g n o r e d . The o s c i l l a t o r y v e l o c i t y f i e l d p r o d u c e d by t h e w a t e r s u r f a c e waves i s i n t e n s i f i e d by t h e s a n d r i p p l e s , and i t i s p r e c i s e l y what i s n e e d e d t o f o r m t h e s a n d i n t o r i p p l e s . S i n c e t h e bed s h e a r s t r e s s i s l i k e l y 2 911 t o be p r o p o r t i o n a l t o U g r a d i e n t U-^— , ,. . . ,, ^ ^ a 3x, r i p p l e d i m e n s i o n s w i l l 2 p r o d u c e a U g r a d i e n t w h i c h i n c r e a s e s a p p r o x i m a t e l y u n i f o r m l y w i t h r i p p l e h e i g h t . Such a g r a d i e n t w o u l d p r o d u c e a u n i f o r m i n c r e a s e i n bed s h e a r s t r e s s . The i m p o r t a n c e o f h o r i z o n t a l v e l o c i t i e s i n t h e v i c i n i t y o f t h e s e d i m e n t c h a n n e l b e d t o t h e r i p p l e h e i g h t s c a n be f o u n d f r o m t h e p r e s e n t e x p e r i m e n t a l r e s u l t s . I t i s o b s e r v e d t h a t t h e b e d f o r m h e i g h t s i n c r e a s e w i t h t h e i n c r e a s i n g v a l u e o f o r E, . The d i f f e r e n c e b etween t h e r i p p l e h e i g h t b e h a v i o r s i n t h e r a n g e 97 o f r o l l i n g g r a i n r i p p l e s and i n t h e r a n g e o f v o r t e x r i p p l e s , i s n o t as n o t i c e a b l e as t h e d i f f e r e n c e between t h e r i p p l e l e n g t h b e h a v i o r s . F i g . 8 i n d i c a t e s t h a t p r e s e n t e x p e r i m e n t a l r e s u l t s c a n be r o u g h l y a p p r o x i m a t e d by a l i n e a r r e l a t i o n s h i p b e tween l o g ^ Q H R and t h e d i m e n s i o n l e s s p a r a m e t e r l o g ^ Q 2 0 5 (pU / ( ( p - p )gd._^) ' ) and t h i s c a n be r e p r e s e n t e d mathe-m s 50 m a t i c a l l y a s : l o g 1 0 H R = 1.92 l o g 1 0 ( p U m 2 / ( ( Ps - p ) g d 5 0 ) ° - 5 ) - 1.3 (2.11) i t i s r o u g h l y a p p l i c a b l e t o a l l t h r e e t y p e s o f s e d i m e n t c h a n n e l b e d f o r m s i n t h e p r e s e n t s t u d y . E x a m i n i n g c l o s e l y t h e e x p e r i m e n t a l r e s u l t s and F i g . 8, i t c a n be f o u n d t h a t f o r e a c h t y p e o f s e d i m e n t c h a n n e l b e d f o r m , i t has i t s own b e d f o r m h e i g h t b e h a v i o r . S i n c e t h e v a r i a t i o n o f t h e w a t e r wave p e r i o d T i s s m a l l and t h e r e e x i s t s a l i n e a r r e l a t i o n s h i p between U and r , m ^ t h u s , t h e maximum h o r i z o n t a l w a t e r p a r t i c l e d i s p l a c e m e n t f r o m i t s mean p o s i t i o n i n t h e v i c i n i t y o f t h e s e d i m e n t c h a n n e l b e d £ i s a l s o i m p o r t a n t i n d e t e r m i n i n g t h e b e d f o r m wave h e i g h t . R e p l a c i n g d,_. by t h e w a t e r d e p t h Y i n d i m e s n i o n l e s s p a r a m e t e r , 50 F i g . 9 i n d i c a t e s t h a t t h e r e e x i s t s , f o r e a c h t y p e o f s e d i m e n t c h a n n e l b e d f o r m , a l i n e a r r e l a t i o n s h i p b e t w e e n log., _ H and 3 1 0 R 2 - 0 5 l o g ^ 0 ( p U m / ( ( p g - p ) gY') " ). The v a r i a t i o n o f t h e s l o p e s o f t h e s e l i n e a r r e l a t i o n s h i p s i n d i c a t e t h e e f f e c t s o f t h e s e d i m e n t p r o p e r t i e s s u c h as d,_^ o r p . 50 s 98 The r e l a t i o n s h i p s between b e d f o r m h e i g h t and m o d i f i e d R e y n o l d ' s number U d,__/,, and U Y/ a r e i n d i c a t e d i n F i g . 10 J m 50 V m ' v a and F i g . 11. I n a d d i t i o n , t h e r e l a t i o n s h i p s between b e d f o r m h e i g h t and maximum h o r i z o n t a l w a t e r p a r t i c l e m o t i o n v e l o c i t y U"m and d i s p l a c e m e n t f r o m t h e mean p o s i t i o n i n t h e v i c i n i t y o f s e d i m e n t c h a n n e l b e d f o r m s a r e shown i n F i g . 12 and F i g . 13. I t i s f o u n d t h a t b e d f o r m h e i g h t i n c r e a s e s w i t h i n c r e a s i n g U"m o r K . Thus, l i n e a r r e l a t i o n s h i p s between l o g , H and l o g J. U K ID U d r _ / , log_,-U Y/ , l o g , _ U , o r l o g , n E, , c a n be a p p r o x -m 50 v ' a 1 0 m ' v ' a 1 0 m' M 1 0 ^ ' ^ K i m a t e l y e s t i m a t e d f o r e a c h t y p e o f s e d i m e n t b e d f o r m s . The i m p o r t a n c e o f t h e e f f e c t s o f s e d i m e n t p r o p e r t i e s c a n be r e a l i z e d f r o m t h e v a r i a t i o n o f t h e s l o p e s o f t h e s e l i n e a r r e l a t i o n s h i p s and t h e r e q u i r e d minimum v a l u e s o f U o r t o i n i t i a t e t h e M m b e d f o r m f o r m a t i o n w h i c h have b e e n shown i n F i g s . 10, 11, 12 and 13. F o r example, t h e s l o p e s o f t h e l i n e a r r e l a t i o n s h i p s b etween l o g 1 0 H R and l o g 1 Q U ^ / v , l o g ^ U j / v , l o g ^ l l . o r log-^ 0£ f o r P o r t a g e S i l i c a s e d i m e n t b e d f o r m s i s f l a t t e r t h a n t h o s e f o r F r a s e r R i v e r sand and B a k e l i t e s e d i m e n t b e d f o r m s . RELATIONSHIPS BETWEEN BEDFORM STEEPNESS S AND MAXIMUM HORIZONTAL WATER PARTICLE DISPLACEMENT FROM ITS MEAN POSITION IN THE V I C I N I T Y OF THE SEDIMENT CHANNEL BED F i g . 14 i n d i c a t e s t h a t i n p r e s e n t e x p e r i m e n t s , t h e r a t i o o f b e d f o r m h e i g h t and l e n g t h , o r t h e b e d f o r m s t e e p n e s s S„ f i r s t o f a l l i n c r e a s e s w i t h i n c r e a s i n g v a l u e o f 5 u n t i l t h e R maximum l i m i t i n g s t e e p n e s s i s r e a c h e d . 99 M.C. Q u i c k ( 1 9 7 7-1978), a r g u e s t h a t t h e o s c i l l a t o r y -v e l o c i t y f i e l d p r o d u c e d by t h e w a t e r s u r f a c e waves i s i n t e n s i f i e d by t h e s a n d r i p p l e s , and t h e i n t e n s i f i c a t i o n o f v e l o c i t y U and t h e r e s u l t i n g i n t e n s i f i c a t i o n o f s h e a r s t r e s s d e pend on t h e r a t i o o f H_ : L. . F o r r i p p l e s u n d e r w a t e r ti ri s u r f a c e waves, w i t h no a d d i t i o n a l s t e a d y f l o w , t h i s r a t i o o f r i p p l e s t e e p n e s s S_, a p p e a r s t o r e a c h a maximum v a l u e o f a b o u t ti 1: 6.2, u n l e s s s t e a d y f l o w c u r r e n t s a l s o e x i s t , i n w h i c h c a s e t h e b e d f o r m s t e e p n e s s w i l l be l o w e r . P r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t r i p p l e s t e e p n e s s e s a r e i n t h e r a n g e o f 1: 6.2 t o 1: 10.0. EXPERIMENTAL RESULTS AT SLOPING-1 CHANNEL BED As t h e w a t e r s u r f a c e waves move s h o r e w a r d s o v e r a s h e l v i n g , l i n e a r wave t h e o r y shows t h a t w a t e r s u r f a c e wave s t e e p n e s s , maximum h o r i z o n t a l w a t e r p a r t i c l e m o t i o n v e l o c i t y U and ^ J m d i s p l a c e m e n t £ f r o m i t s mean p o s i t i o n i n t h e v i c i n i t y o f s e d i m e n t c h a n n e l b e d w i l l i n c r e a s e w i t h d e c r e a s i n g w a t e r d e p t h . S i n c e t h e r e e x i s t l o g a r i t h m i c r e l a t i o n s h i p s between b e d f o r m l e n g t h o r h e i g h t and t h e v a l u e o f U^ o r ? , i t i s e x p e c t e d t h a t r i p p l e l e n g t h and h e i g h t w i l l i n c r e a s e w i t h d e c r e a s i n g w a t e r d e p t h . P r e s e n t e x p e r i m e n t a l r e s u l t s do a g r e e w i t h t h i s p o i n t . When t h e w a t e r s u r f a c e waves move i n t o s h o a l i n g w a t e r t h e y become i n c r e a s i n g l y more a s y m m e t r i c a l . R a u d k i v i , A . J . ( 1 9 6 7 ) , s t a t e d t h a t " q u a l i t a t i v e l y s e e n , t h e t r o u g h i n f r o n t o f 100 a c r e s t i s i n s h a l l o w e r w a t e r t h a n t h e t r o u g h b e h i n d i t and moves more s l o w l y so t h a t t h e c r e s t b e h i n d i t c a t c h e s up on t h e t r o u g h i n f r o n t . T h i s l e a d s t o a s t e e p e r l e a d i n g f a c e o f t h e c r e s t . The asymmetry i n c r e a s e s w i t h t h e s l o p e o f t h e b e a c h " , and, " e v e n u a l l y , t h e wave h e i g h t becomes so g r e a t i n r e l a t i o n t o t h e w a t e r d e p t h t h a t t h e wave becomes u n s t a b l e and b r e a k s . T h i s o c c u r s when t h e r a t i o o f w a t e r d e p t h t o wave h e i g h t i s a p p r o x i m a t e l y A:3. A t t h i s p o i n t t h e v e l o c i t y o f w a t e r p a r t i c l e s i n t h e wave c r e s t e x c e e d s t h e wave f o r m c e l e r i t y and t h e wave b r e a k s " . When t h e w a t e r s u r f a c e wave b r e a k s , p a r t o f w a t e r s u r f a c e wave e n e r g y w i l l be l o s t d u r i n g i t s b r e a k i n g . U n s t a b l e r e v e r s e d w a t e r f l o w w i l l t h e n d e s t r o y t h e s t a b l e mechanism o f o s c i l l a t o r y v e l o c i t y f i e l d n e a r t h e b o t t o m . Thus, as shown i n F i g . 15, t h e r i p p l e l e n g t h and h e i g h t w i l l d e c r e a s e when w a t e r s u r f a c e waves b r e a k . SUMMARY A l m o s t l i n e a r h o r i z o n t a l w a t e r p a r t i c l e m o t i o n o r b i t s a t h e s e d i m e n t c h a n n e l b e d p r o d u c e d an o s c i l l a t o r y s h e a r s t r e s s w h i c h i s c a p a b l e o f p r o d u c i n g s m a l l s a n d b e d f o r m s . R e l a t i o n -s h i p s c a n be f o u n d f r o m p r e s e n t e x p e r i m e n t a l r e s u l t s b etween b e d f o r m d i m e n s i o n s s u c h as H , L_. o r H /L and v a r i o u s d imen-K K K K s i o n a l v a r i a b l e s u c h as U o r £ o r d i m e n s i o n l e s s p a r a m e t e r s m ^ s u c h as m o d i f i e d F r o u d e numbers and m o d i f i e d R e y n o l d ' s numbers E q u a t i o n s and f i g u r e s were u s e d t o r e p r e s e n t t h e s e r e l a t i o n s h i 101 The c h a r a c t e r i s t i c s o f w a t e r s u r f a c e waves s u c h as £ and U"m a r e t h e most i m p o r t a n t f a c t o r s i n d e t e r m i n i n g t h e b e d f o r m d i m e n s i o n s . However, t h e e f f e c t s o f s e d i m e n t p a r t i c l e p r o p e r t i e s c a n n o t be n e g l e c t e d . F o r example, t h e minimum r e q u i r e d v a l u e s o f £ o r U t o i n i t i a t e b e d f o r m M m f o r m a t i o n and t h e r a t e s o f v a r i a t i o n o f b e d f o r m d i m e n s i o n s w i t h r e s p e c t t o d i m e n s i o n a l v a r i a b l e s o r d i m e n s i o n l e s s p a r a m e t e r s w i l l be a f f e c t e d by s e d i m e n t p a r t i c l e p r o p e r t i e s . B o t h l i n e a r wave t h e o r y and p r e s e n t e x p e r i m e n t a l r e s u l t s f o r a s l o p i n g s e d i m e n t c h a n n e l bed show t h a t t h e w a t e r s u r f a c e wave d i m e n s i o n s s u c h as H /L , U , and £ o r s e d i m e n t r i p p l e w w m d i m e s n i o n L D and H_, w i l l i n c r e a s e w i t h d e c r e a s i n g w a t e r d e p t h . The w a t e r s u r f a c e wave b r e a k s when t h e r a t i o o f w a t e r d e p t h t o wave h e i g h t a p p r o a c h e s 4:3. P a r t o f w a t e r s u r f a c e wave e n e r g y w i l l be d i s s i p a t e d , t h e s t a b l e mechanism o f o s c i l l a t o r y v e l o c i t y f i e l d n e a r t h e b o t t o m w i l l be d e s t r o y e d and r i p p l e d i m e n s i o n s w i l l d e c r e a s e d u r i n g t h e b r e a k i n g o f w a t e r s u r f a c e waves.. 102 CHAPTER 2.5 COMPARISON STUDY R.A. B a g n o l d ( 1 9 4 6 ) , s t a t e d t h a t " c l a s s i c a l wave t h e o r y , t h o u g h i t n e g l e c t s d r a g , was assumed t o g i v e w i t h s u f f i c i e n t a c c u r a c y t h e w a t e r m o t i o n n e a r t h e s e d i m e n t c h a n n e l b o t t o m . F o r waves i n deep w a t e r and a l s o t h o s e s h a l l o w w a t e r waves i n w h i c h t h e wave h e i g h t i s s m a l l compared w i t h t h e w a t e r d e p t h , t h e h o r i z o n t a l o s c i l l a t i o n s o f t h e w a t e r n e a r t h e b o t t o m a r e known t o a p p r o x i m a t e c l o s e l y t o s i m p l e h a r m o n i c m o t i o n " . Thus, t h e mechanism o f w a t e r s u r f a c e wave a c t i o n on t h e s e d i m e n t c h a n n e l b e d c a n be a p p r o x i m a t e d b y t h e mechanism o f o s c i l l a t i n g a s e c t i o n o f s e d i m e n t b e d t h r o u g h s t i l l w a t e r . From t h e i r e x p e r i m e n t a l r e s u l t s , M o g r i d g e and Kamphuis ( 1 9 7 2 ) , d e v e l o p e d a s e t o f d e s i g n c u r v e s w h i c h make i t p o s s i b l e t o p r e d i c t t h e l e n g t h and h e i g h t o f w a t e r s u r f a c e wave g e n e r a t e d b e d f o r m o f any s p e c i f i e d s e d i m e n t bed i f c o n d i t i o n s o f f l u i d o s c i l l a t i o n a m p l i t u d e and p e r i o d a r e g i v e n . Some o f M o g r i d g e and K amphuis 1 e x p e r i m e n t s were c a r r i e d o u t i n an o s c i l l a t o r y w a t e r t u n n e l i n o r d e r t o e x t e n d t h e r a n g e o f w a t e r wave m o t i o n a m p l i t u d e and p e r i o d o f o s c i l l a t i o n . B e c a u s e t h e s e o s c i l l a t o r y w a t e r t u n n e l e x p e r i m e n t s o f l a r g e v a l u e o f w a t e r wave m o t i o n p e r i o d T (up t o 1 4 . s e c . ) , c a n n o t be r e p e a t e d i n t h e w a t e r wave f l u m e , M o g r i d g e and Kamphuis r e p e a t e d a number o f w a t e r wave f l u m e e x p e r i m e n t s i n t h e o s c i l l a t o r y w a t e r t u n n e l t o make s u r e t h a t a l l e x p e r i m e n t s f r o m b o t h w a t e r wave f l u m e and o s c i l -l a t o r y w a t e r t u n n e l b e l o n g t o t h e same p o p u l a t i o n . However, 103 s i m p l e h a r m o n i c m o t i o n c a n o n l y be an a p p r o x i m a t i o n o f t h e w a t e r p a r t i c l e h o r i z o n t a l o s c i l l a t i o n n e a r t h e s e d i m e n t b o t t o m . The w a t e r p a r t i c l e m o t i o n o r b i t s a r e n o t e x a c t l y c l o s e d o r b i t s , t h e r e a r e some l i n e a r n e t t r a n s p o r t s e x i s t i n t h e w a t e r wave m o t i o n d i r e c t i o n . I n a d d i t i o n , l a r g e o s c i l l a t i n g p e r i o d w a t e r waves, s u c h as wave p e r i o d T > 10 s e c . , c a n n o t be r e p r o d u c e d i n any l a b o r a t o r y open c h a n n e l s . A l l p r e s e n t e x p e r i m e n t s a r e f r o m w a t e r wave f l u m e , so t h e c o m p a r i s o n s t u d y c a n o n l y be made w i t h M o g r i d g e and Kamphuis' s m a l l £ e x p e r i m e n t a l r e s u l t s and t h e i r d e s i g n c u r v e s w h i c h a r e a p p l i -c a b l e t o t h e w a t e r wave f l u m e e x p e r i m e n t s o n l y . One o f t h e c o n c l u s i o n s w h i c h i s drawn f r o m M o g r i d g e o and Kamphuis' s t u d y i s t h a t " a t v a l u e s o f P ^ Q A ( P S ~ P ) gT ) ^ g r e a t e r t h a n a p p r o x i m a t e l y 200 x 10 , b e d f o r m s do n o t a p p e a r on a s e d i m e n t b ed", b u t p r e s e n t s t u d y does n o t a g r e e w i t h t h i s s i n c e e x p e r i m e n t a l r e s u l t s on B a k e l i t e s a n d b e d f o r v a l u e s o f pd 5 0/( (p s~p)gT ) e q u a l t o e i t h e r 244 . 1 0 ~ 6 o r 339 . 1 0 ~ 6 , t h e B a k e l i t e s e d i m e n t r i p p l e s e x i s t . F i g u r e s 16, 17, 18 and 19, u s i n g M o g r i d g e and Kamphuis' d e s i g n c u r v e s as r e f e r e n c e s , i n d i c a t e t h e s e e x p e r i m e n t a l r e s u l t s on B a k e l i t e s e d i m e n t c h a n n e l b e d . F o r t h e s h a l l o w w a t e r waves, l i n e a r wave t h e o r y g i v e s c = ( g Y ) ° ' 5 = Lw/T (2.12) and (2 .13) 104 t h e r e f o r e , one c a n show t h a t _ _ g _ /HWx Jl _ g / Hw s j (9 i s ) ^ - ^ k ( L w " ) T -"47(7g^-)T (2-14) M o g r i d g e and K a m p h u i s 1 c o n c l u s i o n a c t u a l l y i m p l i e s t h a t f o r a g i v e n ; s e d i r a e n t p a r t i c l e p r o p e r t i e s s u c h as d ^ ^ and o , w a t e r s u r f a c e wave p e r i o d T must be l a r g e enough s and so t h e v a l u e o f 5 w i l l be l a r g e enough t o i n i t i a t e t h e r i p p l e f o r m a t i o n . I n f a c t , i f t h e v a l u e o f H w / L w o r Hw/(gy.)0-5 i s l a r g e enough, t h e minimum r e q u i r e d v a l u e o f £ t o i n i t i a t e b e d f o r m f o r m a t i o n c a n be r e a c h e d w i t h a s m a l l w a t e r s u r f a c e wave p e r i o d T. However, t h e r a t i o o f w a t e r d e p t h Y t o wave H h e i g h t w must n o t e x c e e d t h e a p p r o x i m a t e l y 4 : 3 t o a v o i d t h e w a t e r s u r f a c e wave b r e a k i n g . F o r t h e deep w a t e r waves, l i n e a r wave t h e o r y shows t h a t 5 i s m a i n l y a f u n c t i o n o f w. Water s u r f a c e wave p e r i o d T d o e s n o t have t h e s i m p l e s i g n i f i c a n c e . T h e r e i s no s e d i m e n t b e d f o r m f o r m a t i o n u n d e r t h e a c t i o n o f deep w a t e r s u r f a c e waves. A n o t h e r c o n c l u s i o n made by M o g r i d g e and Kamphuis i s t h a t " f o r v a r i a t i o n o f b e d f o r m l e n g t h w i t h (2 5 + IS'I ) / d 5 Q , as (2 5 + |'<5 |' )/d Q i n c r e a s e s , b e d f o r m l e n g t h i s i n i t i a l l y d e p e n d e n t o n l y on t h e f l u i d o r b i t a l l e n g t h ? . A t l a r g e v a l u e o f (2? + )6| )/d,_^, t h e b e d f o r m becomes d i r e c t l y p r o p o r -t i o n a l : ' . t p t h e g r a i n s i z e d i a m e t e r . The c o n s t a n t o f p r o p o r -2 t i o n a l i t y i s a f u n c t i o n o f P d 5 Q / ( ( P g -P ) 9 T ) " . As d i s -c u s s e d b e f o r e , M o g r i d g e and Kamphuis made t h i s c o n c l u s i o n 105 by t h e i r e x p e r i m e n t a l r e s u l t s a t low v a l u e s o f E, , w h i c h g i v e s l i n e a r r e l a t i o n s h i p o f s l o p e 1 : 1 between l o g . ^ ( L R / d 5 Q ) and l o g 1 Q (2£ / d 5 Q ) . T h i s means t h a t , L = C o n s t a n t - Sj (2.15) H P r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t t h e s l o p e s o f t h e l i n e a r r e l a t i o n s h i p s between l o g - ^ 0 ( L R / d 5 Q ) and l o g ^ 0 (2 E, / d 5 Q ) i s n o t f a r f r o m 1 : 1 , b u t f o r s e d i m e n t d e n s i t y o f u n u s u a l l y l a r g e o r s m a l l and f o r g r a i n s i z e s o f u n u s u a l l y f i n e o r c o a r s e , t h e v a l u e o f t h e s l o p e may n o t be c l o s e t o 1 : 1 . F u r t h e r m o r e , e v e n i f t h e v a l u e o f t h e s l o p e i s 1 : 1, s i n c e t h e v a l u e o f t h e c o n s t a n t i n e q u a t i o n ( 2 . 15 ) v a r i e s w i t h t h e v a r i a t i o n o f t h e s e d i m e n t p a r t i c l e p r o p e r t i e s . Thus, when t h e v a l u e o f E, i s s m a l l , t h e e f f e c t s o f s e d i m e n t p r o p e r -t i e s on s e d i m e n t b e d f o r m l e n g t h s a r e n o t n e g l i g i b l e . M o g r i d g e and Kamphuis' c o n c l u d e d t h a t "when E, i s s m a l l , r i p p l e l e n g t h d epends on w a t e r wave m o t i o n o r b i t o n l y " i s t o o s t r o n g t o a c c e p t . However, when c o m p a r i n g p r e s e n t e x p e r i m e n t a l r e s u l t s w i t h M o g r i d g e and Kamphuis' d e s i g n c u r v e s , F i g . 20 i n d i c a t e s t h a t e v e n t h e p r e s e n t e x p e r i m e n t a l r e s u l t s s c a t t e r a r o u n d t h e d e s i g n c u r v e s , g e n e r a l l y s p e a k i n g , t h e y a r e i n good a g r e e -ment w i t h t h e d e s i g n c u r v e s . B a g n o l d (1946) u s i n g an o s c i l -l a t o r y b e d i n s t i l l w a t e r and M o g r i d g e and Kamphuis (1972) u s i n g an o s c i l l a t o r y w a t e r t u n n e l , b o t h showed t h a t t h e b e d -f o r m l e n g t h becomes d e p e n d e n t o n l y on g r a i n s i z e d i a m e t e r f o r h i g h v a l u e o f £ . However, t h e v a l u e o f 5 i s l i m i t e d 106 i n p r e s e n t s t u d y . M o g r i d g e and Kamphuis a l s o c o n c l u d e d t h a t " b e d f o r m h e i g h t and s t e e p n e s s a t f i r s t i n c r e a s e w i t h i n c r e a s i n g (2£ + |6j )/dr.r., r e a c h maximum v a l u e s and t h e n d e c r e a s e u n t i l t h e b e d f o r m s d i s a p p e a r " . S i n c e t h e s i t u a t i o n s o f l a r g e v a l u e s o f % o r T do n o t e x i s t i n t h e p r e s e n t w a t e r wave f l u m e e x p e r i m e n t s , p r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t r i p p l e h e i g h t and s t e e p n e s s i n c r e a s e w i t h i n c r e a s i n g £ . Thus, i n t h e r a n g e o f s m a l l v a l u e s o f K o r T, p r e s e n t e x p e r i m e n t a l r e s u l t s r o u g h l y a g r e e w i t h M o g r i d g e and Kamphuis' d e s i g n c u r v e s and c a n be shown i n F i g . 21. Homma and H o r i k a w a (1964) and H o r i k a w a and Watanabe (1967) c a r r i e d o u t a number o f e x p e r i m e n t s i n w a t e r wave f l u m e o f 25 m e t e r l o n g , 60 cm wide and 80 cm deep. G e n e r a l i z e d r e l a t i o n s h i p s among w a t e r w a v e - g e n e r a t e d r i p p l e d i m e n s i o n s , w a t e r s u r f a c e waves, and t h e s e d i m e n t p r o p e r t i e s were p r o p o s e d . They p l o t t e d t h e i r e x p e r i m e n t a l r e s u l t s and f o u n d o u t t h e r e l a t i o n s h i p s b etween K /L„ and U £ / v , H_,/L_, and H_/_ by R m R R R t ; l e a s t s q u a r e method, w h i c h c a n be r e p r e s e n t e d by ? / L R = 0.0795 ( U m 5 / v ) 0 - 3 3 2 (2.16) f o r S u l f o n i c p o l y s t y r o l p e l l e t s (p = 1.21 and d 5 Q = 0.054 cm) s e d i m e n t c h a n n e l b e d . ? / L D = 0.0985 (U K / v ) 0 - 2 8 9 (2.17) K m and H R / L R = 0.231 ( H / £ ) ° - 1 6 — — (2.18) 107 f o r n a t u r a l s a n d ( p = 2.6 and d 5 Q = 0.021 cm) c h a n n e l b e d . The weak p o i n t s o f H o r i k a w a and W a t a n a b e 1 s s t u d y a r e two f o l d , t h e r a n g e o f v a r i a t i o n o f w a t e r d e p t h i s f r o m 22.9 cm t o 24.1 cm. T h i s s m a l l d i f f e r e n c e on w a t e r d e p t h may n o t be s i g n i f i c a n t a t a l l . F u r t h e r m o r e , o n l y two t y p e s o f s e d i m e n t p a r t i c l e s were u s e d and s e d i m e n t p a r t i c l e p r o p e r t i e s a r e n o t i n v o l v e d i n t h e i r g e n e r a l i z e d r e l a t i o n s h i p s . N e v e r t h e l e s s , t h e i r e x p e r i m e n t a l r e s u l t s and g e n e r a l i z e d e q u a t i o n s a r e n o t f a r f r o m t h e M o g r i d g e and Kamphuis' c o n c l u s i o n s o r p r e s e n t e x p e r i m e n t a l r e s u l t s , and c a n be shown i n f i g u r e s 22 and 23. H o r i k a w a and Watanabe (1967) n o t e d t h e " c a r e f u l i n v e s -t i g a t i o n o f a l l o f t h e s e t h r e e r e l a t i o n s h i p s " i n d i c a t e s t h e f o l l o w i n g m a t t e r s : (1) Due t o t h e l i g h t s e d i m e n t d e n s i t y , b e h a v i o u r o f t h e S u l f o n i c P o l y s t y r o l p e l l e t b e d f o r m s c a n be r e p r e s e n t e d by two d i f f e r e n t l i n e s i n F i g . 22 o r F i g . 23 d e p e n d i n g upon w h e t h e r t h e c o n d i t i o n i s L^> 2£ o r L R< 2 £ . T h e s e two c u r v e s a r e d i f f e r e n t f r o m t h e l i n e f o r n a t u r a l s a n d r i p p l e s . (2) The i n t e r s e c t i o n p o i n t b etween two l i n e s f o r S u l f o n i c P o l y s t y r e n e p e l l e t b e d f o r m s i n F i g . 22 may g i v e t h e maximum v a l u e o f t h e r i p p l e s t e e p n e s s . H o r i k a w a and Watanabe m e n t i o n e d t h a t t h e c o n d i t i o n o f L R < 2 £ i s a l w a y s t r u e f o r n a t u r a l s a n d r i p p l e s , b u t n o t f o r S u l f o n i c P o l y s t y r e n e p e l l e t r i p p l e s . T h i s c a n be c o n f i r m e d by some o f M o g r i d g e and Kamphuis' e x p e r i m e n t a l r e s u l t s o f P o l y s t y r e n e s e d i m e n t ( P = 1 . 0 5 and d n = 0.154mm ) r i p p l e s . 108 However, p r e s e n t e x p e r i m e n t a l r e s u l t s and M o g r i d g e and Kamphuis' e x p e r i m e n t a l r e s u l t s on B a k e l i t e s e d i m e n t (p = 1.33 and d_^ = 0.52 mm) b e d o r o t h e r s a n d b e d s show t h a t a l l t h e y bO b e l o n g t o t h e c o n d i t i o n o f L < 2 K . SUMMARY S i n c e l a r g e o s c i l l a t i n g p e r i o d w a t e r s u r f a c e waves a r e s e l d o m a b l e t o a p p e a r a t any t y p e o f open c h a n n e l , p r e s e n t e x p e r i m e n t s a r e l i m i t e d i n t h e r a n g e o f s m a l l v a l u e o f w a t e r wave m o t i o n a m p l i t u d e and p e r i o d . I n p r e s e n t e x p e r i m e n t s , B a k e l i t e s e d i m e n t r i p p l e s e x i s t a t t h e v a l u e o f P d 5 Q / ( ( P g - p ) g T 2 ) g r e a t e r t h a n 200 x 1 0 - 6 . T h i s does n o t a g r e e w i t h M o g r i d g e and K a m p h u i s 1 c o n c l u s i o n ' w h i c h s t a t e d t h a t b e d f o r m s do n o t a p p e a r on a s e d i m e n t bed a t t h e v a l u e o f Pd / ( ( P ~ p ) 9"^2) g r e a t e r t h a n 200 x 1 0 ~ 6 . From M o g r i d g e and K a m p h u i s 1 e x p e r i m e n t a l r e s u l t s , r e l a t i o n s h i p b e t w e e n L_, and K c a n be r e p r e s e n t e d as (L = c o n s t a n t • £) i f H K t h e v a l u e o f K i s s m a l l . Thus, M o g r i d g e and Kamphuis c o n c l u d e d t h a t "as v a l u e s o f (2 5+ l'<S.-l..)/dC(_ i n c r e a s e s , b e d f o r m l e n g t h i s ou i n i t i a l l y d e p e n d e n t o n l y on t h e f l u i d o r b i t a l l e n g t h . " However, s i n c e t h e c o n s t a n t o f p r o p o r t i o n a l i t y o f t h e r e l a t i o n s h i p b etween L and 5 and t h e minimum r e q u i r e d v a l u e o f K t o i n i t i a t e b e d -R f o r m f o r m a t i o n depend upon t h e s e d i m e n t p a r t i c l e p r o p e r t i e s , so t h e e f f e c t s o f s e d i m e n t p a r t i c l e p r o p e r t i e s a r e n o t n e g l i g i b l e . I n g e n e r a l , when t h e v a l u e o f £ i s s m a l l , p r e s e n t e x p e r i m e n t a l r e s u l t s a g r e e w i t h M o g r i d g e and Kamphuis' d e s i g n c u r v e s w h i c h make i t p o s s i b l e t o p r e d i c t t h e r i p p l e d i m e n s i o n 109 u n d e r v a r i o u s c o n d i t i o n s o f w a t e r s u r f a c e waves. H o r i k a w a u a n d Watanabe ( 1 9 6 7 ) , f o u n d some g e n e r a l i z e d r e l a t i o n s h i p among b e d f o r m d i m e n s i o n s and w a t e r s u r f a c e wave d i m e n s i o n s f r o m t h e i r w a t e r wave f l u m e e x p e r i m e n t s . I t was a l s o f o u n d t h a t f o r l i g h t d e n s i t y s e d i m e n t p a r t i c l e , b e d f o r m b e h a v i o r may be d i v i d e d i n t o two p a r t s d e p e n d i n g upon where t h e c o n d i t i o n i s L < 2 K o r L >2 K. The e x p e r i m e n t a l r e s u l t s K K f r o m H o r i k a w a and Watanabe g e n e r a l l y a g r e e w i t h M o g r i d g e and Kamphuis' s t u d y and p r e s e n t e x p e r i m e n t a l r e s u l t s . 110 CHAPTER 2.6 CONCLUSIONS (1) A l m o s t l i n e a r h o r i z o n t a l w a t e r p a r t i c l e o r b i t s a t t h e s e d i m e n t c h a n n e l b e d w i l l p r o d u c e an o s c i l l a t o r y b e d s h e a r s t r e s s w h i c h i s c a p a b l e o f p r o d u c i n g s m a l l s a n d r i p p l e s . The whole r e g i o n o f s e d i m e n t c h a n n e l b e d c a n be c o n s i d e r e d t o be s u b j e c t e d t o a s i m u l t a n e o u s o s c i l l a t o r y b e d s h e a r s t r e s s o f a m p l i t u d e % . When t h e s e d i m e n t c h a n n e l bed i s s u b j e c t e d t o a b ed s h e a r s t r e s s o f s u f f i c i e n t i n t e n s i t y t o p r o d u c e s e d i m e n t m o t i o n , s a n d p a r t i c l e s w i l l o s c i l l a t e on t h e c h a n n e l bed and s o o n an i n s t a b i l i t y w i l l d e v e l o p and s m a l l b e d f o r m s w i l l s t a r t t o d e v e l o p . R i p p l e s may a c h i e v e a c e r t a i n l i m i t i n g h e i g h t , and w i l l m i g r a t e s l o w l y i n t h e w a t e r wave d i r e c t i o n . The o s c i l l a t o r y v e l o c i t y f i e l d p r o d u c e d by t h e w a t e r s u r f a c e waves i s i n t e n s i f i e d b y t h e s e d i m e n t r i p p l e s , and t h i s i n t e n -s i f i c a t i o n i s what i s n e e d e d t o f o r m s e d i m e n t p a r t i c l e s i n t o a r i p p l e . (2) Assume t h e h o r i z o n t a l l i n e a r w a t e r p a r t i c l e v e l o c i t i e s p r e d o m i n a t e , w h i c h c a n be w r i t t e n a s , (Ref. t o Quick (1977-1978). U = U ( C o s ( w t ) + (KA) Cos (Kx) Cos ( w t ) ) m where K i s t h e r i p p l e wave number o r 2TT/L , A i s t h e r i p p l e R wave a m p l i t u d e o r H_/2 and W i s t h e w a t e r s u r f a c e wave f r e q u e n c y o r 2^/^,. The f i r s t t e r m o f r i g h t hand s i d e i s p r o d u c e d b y t h e w a t e r s u r f a c e waves, and t h e s e c o n d t e r m i s c a u s e d b y t h e w a t e r s u r f a c e waves and t h e s e d i m e n t r i p p l e s i t s e l f . S i n c e b e d s h e a r s t r e s s i s l i k e l y t o be p r o p o r t i o n a l t o U g r a d i e n t o r U-TTT, r i p p l e d i m e n s i o n s w i l l p r o d u c e a U O A g r a d i e n t w h i c h i n c r e a s e s a p p r o x i m a t e l y u n i f o r m l y w i t h r i p p l e h e i g h t and p r o d u c e s a u n i f o r m i n c r e a s e i n bed s h e a r s t r e s s . (3) I t i s o b s e r v e d t h a t t h e r i p p l e l e n g t h s a r e a b o u t c o n s t a n t a t f i r s t , u n t i l t h e v a l u e o f E o r U i n c r e a s e s and m a p p r o a c h e s t o some t r a n s i t i o n a l v a l u e o f E o r U f o r e a c h ^ s m t y p e o f s e d i m e n t r i p p l e s . T h e s e c o n s t a n t l e n g t h r i p p l e s a r e named r o l l i n g g r a i n r i p p l e s by B a g n o l d ( 1 9 4 6 ) . F u r t h e r i n c r e a s i n g t h e v a l u e o f E o r U , s e d i m e n t b e d f o r m s w i l l become a ^ m v o r t e x r i p p l e s , and t h e r i p p l e l e n g t h w i l l i n c r e a s e w i t h i n c r e a s i n g E, o r t i l l r e a c h i n g t h e maximum v a l u e o f r i p p l e l e n g t h f o r e a c h t y p e o f s e d i m e n t r i p p l e s . A c c o r d i n g t o M o g r i d g e and K a m p h u i s 1 s t u d y , a f t e r r e a c h i n g t h e maximum v a l u e o f r i p p l e l e n g t h , f u r t h e r i n c r e a s i n g o f E, o r U^, b e d f o r m w i l l d i s a p p e a r , p r e s u m a b l y b e c a u s e t h e s h e a r s t r e s s e s and c o r r e s -p o n d i n g s e d i m e n t m o t i o n become so s t r o n g ' t h a t b e d f o r m s a r e w i p e d o u t . However, t h i s s i t u a t i o n d i d n o t o c c u r i n p r e s e n t s t u d y s i n c e t h e v a l u e o f £ i s l i m i t e d f o r w a t e r wave f l u m e e x p e r i m e n t s . From t h e e x p e r i m e n t a l r e s u l t s , r e l a t i o n s h i p s b etween L and p o r U c a n be a p p r o x i m a t e l y e s t i m a t e d f o r R s m J e a c h t y p e o f s e d i m e n t r i p p l e s . T h e s e r e l a t i o n s h i p s i n d i c a t e t h e i m p o r t a n c e o f t h e v a l u e o f g o r U M i n d e t e r m i n i n g t h e r i p p l e l e n g t h . However, t h e r a t e o f v a r i a t i o n o f r i p p l e l e n g t h w i t h r e s p e c t t o £ o r U , t h e i n i t i a l c o n s t a n t s t a b l e r o l l i n g g r a i n r i p p l e l e n g t h , t h e t r a n s i t i o n a l v a l u e o f j- o r U , and t h e minimum r e q u i r e d v a l u e o f £ t o i n i t i a t e r i p p l e f o r m a t i o n f o r e a c h t y p e o f s e d i m e n t r i p p l e s a r e d e p e n d i n g upon t h e s e d i m e n t p a r t i c l e p r o p e r t i e s . M o g r i d g e and Kamphuis' 112 c o n c l u s i o n s t a t e d t h a t "when £ i s s m a l l , r i p p l e l e n g t h depends on £ o n l y " , m i g h t be t o o s t r o n g , and t h e e f f e c t s o f t h e s e d i m e n t p a r t i c l e p r o p e r t i e s a r e n o t n e g l i g i b l e . R e l a t i o n -s h i p s b etween r i p p l e h e i g h t s and v a r i o u s d i m e n s i o n l e s s p a r a m e t e r s s u c h as m o d i f i e d F r o u d e numbers and m o d i f i e d R e y n o l d ' s numbers o r d i m e n s i o n a l v a r i a b l e s s u c h as £ and c a n be a p p r o x i m a t e l y e s t i m a t e d . E x p e r i m e n t a l r e s u l t s show t h a t r i p p l e h e i g h t i n c r e a s e s w i t h i n c r e a s i n g o r E, , u n t i l t h e maximum l i m i t i n g r i p p l e h e i g h t i s r e a c h e d . T h i s c o n f i r m s t h e i m p o r t a n c e o f and £ i n d e t e r m i n i n g t h e r i p p l e h e i g h t . I t i s s i m i l a r t o t h e b e h a v i o r o f r i p p l e l e n g t h , t h e e f f e c t s o f s e d i m e n t p a r t i c l e p r o p e r t i e s a r e n o t n e g l i g i b l e . (4-) The i n t e n s i f i c a t i o n o f h o r i z o n t a l v e l o c i t y U and r e s u l t i n g i n t e n s i f i c a t i o n o f bed s h e a r s t r e s s depend on t h e r i p p l e s t e e p n e s s . P r e s e n t e x p e r i m e n t a l r e s u l t s show t h a t t h e r i p p l e s t e e p n e s s S R i n i t i a l l y i n c r e a s e s w i t h i n c r e a s i n g o f £ u n t i l t h e maximum l i m i t i n g r i p p l e s t e e p n e s s i s r e a c h e d . P r e s e n t e x p e r i m e n t a l r e s u l t s a l s o show t h a t S D i s a p p r o x -i m a t e l y i n t h e r a n g e o f 0.10 t o 0.16, t h a t i s , t h e v a l u e o f t h e r a t i o o f r i p p l e h e i g h t t o r i p p l e l e n g t h H : L i s i n t h e R R r a n g e o f 1 : 6.2 t o 1 : 10. (5) F o r some s i t u a t i o n s , s u c h as f o r l i g h t B a k e l i t e s e d i m e n t bed, r i p p l e s e x i s t e v e n when P d ^ Q / ( ( P s - P) gT ) i s g r e a t e r t h a n 200 . 10 6 . T h i s d o e s n o t a g r e e w i t h M o g r i d g e and K a m p h u i s 1 c o n c l u s i o n , w h i c h s t a t e d t h a t b e d f o r m s do n o t a p p e a r on s e d i m e n t bed a t v a l u e s o f p d 5 0 / ( ( P s - P) gT ) g r e a t e r t h a n a p p r o x i m a t e l y 200 . 10 6 . M o g r i d g e and K a m p h u i s 1 113 c o n c l u s i o n a c t u a l l y i m p l i e d t h a t f o r a g i v e n s e d i m e n t p a r t i c l e p r o p e r t i e s s u c h as p g and d 5 Q , w a t e r s u r f a c e wave p e r i o d T must be l a r g e enough t o i n i t i a t e t h e b e d f o r m f o r m a t i o n . However, one c a n show t h a t i n s h a l l o w w a t e r , E = -rr^- ( - 7 = ^ ) T. TT V g y I f t h e v a l u e o f J^L- i s l a r g e enough, minimum r e q u i r e d v a l u e o f 5 t o i n i t i a t e b e d f o r m f o r m a t i o n c a n be r e a c h e d w i t h a s m a l l w a t e r s u r f a c e wave p e r i o d T. However, t h e r a t i o o f w a t e r d e p t h Y t o w a t e r s u r f a c e wave h e i g h t must n o t e x c e e d a b o u t 4 : 3 t o a v o i d t h e b r e a k i n g o f w a t e r s u r f a c e waves. (6) When £ i s s m a l l , p r e s e n t e x p e r i m e n t a l r e s u l t s g e n e r a l l y a g r e e w i t h M o g r i d g e and Kamphuis' d e s i g n c u r v e s , w h i c h make i t p o s s i b l e t o p r e d i c t r i p p l e l e n g t h and h e i g h t f o r g i v e n c o n d i t i o n s o f f l u i d o s c i l l a t i o n and s e d i m e n t p r o p e r t i e s . S i n c e t h e v a l u e s o f K o f p r e s e n t w a t e r wave f l u m e e x p e r i m e n t a l r e s u l t s a r e a l l l e s s t h a n 6 cm, i t i s n o t a b l e t o compare p r e s e n t e x p e r i m e n t a l r e s u l t s w i t h M o g r i d g e and Kamphuis' d e s i g n c u r v e s i n t h e r a n g e o f l a r g e v a l u e s o f E, However, s i n c e e x p e r i m e n t a l r e s u l t s f o r r i p p l e d i m e n s i o n s a r e s c a t t e r e d , so t h a t t h e M o g r i d g e and Kamphuis' d e s i g n c u r v e s c a n p r o v i d e a p p r o x i m a t e v a l u e s o f r i p p l e d i m e n s i o n s o n l y . (7) F o r r e g u l a r s e d i m e n t c h a n n e l b e d , s u c h as P o r t a g e S i l i c a s a n d bed o r F r a s e r R i v e r s a n d bed, r i p p l e l e n g t h s a r e a l w a y s s m a l l e r t h a n t h e v a l u e s o f 2 u n d e r t h e same f l u i d o s c i l l a t i o n c o n d i t i o n . However, b o t h M o g r i d g e and Kamphuis, o r H o r i k a w a and Watanabe, o b t a i n e d some o f t h e i r e x p e r i m e n t a l r e s u l t s a t p o l y s t y r e n e c h a n n e l b e d, w h i c h h a ve t h e s i t u a t i o n o f L >2 E,. S i n c e t h e v a l u e o f s e d i m e n t R d e n s i t y p o f p o l y s t y r e n e p e l l e t s i s o n l y s l i g h t l y g r e a t e r 114 than the water density, suspension load becomes very important, and the experimental r e s u l t s under t h i s s i t u a t i o n may be not s i g n i f i c a n t , because the mechanism o f sediment motion and bedform formation might be d i f f e r e n t . However, only when bedload remains as the major part of the sediment transportation i n open channel, the experimental r e s u l t s on sediment channel bed w i l l be s i g n i f i c a n t . Otherwise, the r i p p l e s w i l l disappear immediately a f t e r they have been generated by the a c t i o n of water surface waves o f large value of and suspension load w i l l become the important part of sediment transportation, thus, the experimental r e s u l t s are not s i g n i f i c a n t under t h i s s i t u a t i o n . I t i s found that Bakelite sediment p a r t i c l e s can be used t o obtain s i g n f i c a n t experimental r e s u l t s , but as f o r l i g h t e r sediment material p a r t i c l e s , such as Sulf o n i c P o l y s t y r o l sediment (f = 1.21) used by Horikawa and Watanabe or Polystyrene sediment (P = 1.05) used by Mogridge and Kamphuis, i t i s necessary t o f i n d out i f the sediment bedload remains as the major part of sediment tran s p o r t a t i o n p r i o r t o c a r r y out any experiments. In summary, we f i n d that the present experiments agree with the works done by Bagnold or Mogridge and Kamphuis, e t c . Theory developed by M.C. Quick, explains the mechanism which produces r i p p l e s and i s consistant with the experimental r e s u l t s . Future work should study the influence of a superimposed current, i n a d d i t i o n t o the wave induced o s c i l l a t o r y flow. This net transport s i t u a t i o n i s very important f o r co a s t a l s i t u a t i o n s and forms an i n t e r -mediate s i t u a t i o n between the steady flow bedforms discussed i n Part I of the t h e s i s and the o s c i l l a t o r y flow bedforms of the second part. CHAPTER 2.7 BIBLIOGRAPHY B a g n o l d , R.A., M o t i o n o f waves i n s h a l l o w w a t e r , i n t e r a c t i o n b etween waves and s a n d b o t t o m s . P r o c . R o y a l S o c i e t y o f  London, S e r i e s A. 1946, 187, 1-15 G r a f , W.H. H y d r a u l i c s o f s e d i m e n t t r a n s p o r t . McGraw H i l l Book Company, 1971. H o r i k a w a K. & Watanabe A. A s t u d y on s a n d movement due t o wave a c t i o n . J o u r n a l o f C o a s t a l E n g i n e e r i n g i n J a p a n , 1967 10, 39-57. Kennedy, J . F . & F a l c o n M. Wave g e n e r a t e d s e d i m e n t r i p p l e s . M.I.T. H y d r o d y n a m i c s Lab. R e p o r t . 1965, 86. I s c a a s o n , M i c h a e l de -St. Q. Mass t r a n s p o r t i n s h a l l o w w a t e r waves. J o u r n a l o f t h e waterway p o r t C o a s t a l and Ocean  D i v i s i o n , 1978, 05. M o g r i d g e , G.R. & Kamphuis J.W. E x p e r i m e n t s on b e d f o r m g e n e r a t i o n by wave a c t i o n . P r o c e e d i n g s .of .the 1 3 t h C o a s t a l E n g i n e e r i n g C o n f e r e n c e , 1972, 2, 1123-1142. Q u i c k , M.C. U n p u b l i s h e d p r e l i m i n a r y d r a f t , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1977-1978. R a u d k i v i , A . J . L o o s e b o u n d a r y h y d r a u l i c s . Pergamon P r e s s , 1967. Sheen, S . J . T u r b u l e n c e o v e r a s a n d r i p p l e . T h e s i s s u b m i t t e d as p a r t r e q u i r e m e n t f o r t h e d e g r e e o f M a s t e r o f E n g i n e e r i n g , U n i v e r s i t y o f A u c k l a n d , 1964. U.S. Army C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r . S h o r e P r o t e c t i o n  M a n u a l, 1973. Y a l i n , M.S. & R u s s e l R.C.H. S i m i l a r i t y i n s e d i m e n t t r a n s p o r t due t o waves. P r o c e e d i n g s o f t h e 8 t h C o a s t a l E n g i n e e r i n g  C o n f e r e n c e , 1962, 151-167. 116 APPENDIX I I EXPERIMENTAL RESULTS AND FIGURES (1) PORTAGE S I L I C A SEDIMENT RIPPLES Y (cm) T ( s e c ) 30 .0 30 .0 30 .0 36 .0 36 .0 25 .0 30 .0 30 .0 30 .0 36 .0 36 .0 36 .0 25 .0 30 .0 30 .0 30.0 36 .0 36 .0 36 .0 25.0 0.9 0.9 0.9 0.9 0.9 0.9 1 .06 1 .06 1 .06 1 .06 1 .06 1 .06 1 .06 1 .21 1 .21 1 .21 1 .21 1 .21 1 .21 1 .21 C (cm/sec) 14-0 .4-5 140.45 140.45 140.45 153 .50 153.50 130.51 148.10 148.07 161.32 161.32 161.32 134.22 154.66 154.66 154.66 168.64 168.64 168.64 142.61 H (cm) w 10 .4 9 .92 8 .64 12 . 69 11 .34 9 . 32 12 .83 10.94 9 . 38 13 .23 12 .83 8 .91 .8 .64 10.53 7 .21 .7 .16 13 .77 11 .21 7 .56 5.13 H R (cm) O .49 0.49 O . 38 0.32 0.34 0.3 0.6 0.54 0 .49 0.53 0.38 0.34 0.45 0.83 0.68 0.01 0.68 0.64 0 .01 0.38 L R (cm) 3 .38 3 .68 2.9 3 . 3 3 .46 3 .64 4 .27 3 . 75 3 . 65 4 .74 4.0 3 .61 4 .06 4.50 3 . 70 3.0 5 .52 4.68 4.0 3 .58 117 (2) FRASER RIVER SEDIMENT RIPPLES Y (cm) T ( s e c ) C (cm/sec) H (cm) w H R (cm) L R (cm) 30.0 30.0 30 .0 30.0 307.0 30.0 1 .06 1 .06 1 .06 0.9 0.9 0 . 9 148 .07 148.07 148.07 140.45 140.45 140.45 6.54 9 .25 10.94 8 .37 10.76 10.82 0 .01 0 . 56 0 . 75 O .075 0 .23 O . 34 3.0 3.25 3 .88 3.0 3 .0 3 .3 (3) BAKELITE SEDIMENT RIPPLES 30 .0 30 .0 30.0 30.0 30 .0 30 .0 30 .0 30.0 0.9 0.9 0.9 0.9 1 .06 1 .06 1 .06 1 .06 140.45 140.45 140.45 140.45 148.07 148.07 148.07 148.07 12 .07 11 .18 7 . 87 5.1 10.67 8 . 64 7 .37 4 . 57 0 .48 0.38 0.15 0.01 O . 83 0 .48 O . 16 0.01 3 .56 3 . 3 2 . 79 2 .0 4.13 4 .09 2.8 2.0 (4) FIGURES: FOLLOWING FIGURES ARE PLOTTED FROM THE EXPERIMENTAL RESULTS c o a r s e s p a n a d j u s t f i n e s p a n a d j u s t f o r h i g h w a t e r l e v e l n u l l a d j u s t c o a r s e z e r o a d j u s t f i n e z e r o a d j u s t f o r low w a t e r l e v e l d i r e c t (DIR) o p e r a t i o n a l mode s h o r t i n g b a r s r e v e r s e o p e r a t i o n a l mode s h o r t i n g b a r wat e r wave p r o b e a d j u s t m e n t motor w a t e r wave p r o b e w a t e r t a n k f l a p t y p e wave g e n e r a t o wave e n e r g y a b s o r b e r s i n k 1 1 ) 1 1 1 1 0 0 oo H o n e y w e l l r e c o r d e r power s u p p l y F i g . 2 . 1 15.Q1 U d /v m 50 FIG.2 .6 FIG.2 . 8 FIG.2 .9 FIG.2.10 FIG.2.11 o o 20.0 10.0 5.0 V ° H 2'° o * o R o % X o A Q % 1.0 ° °oo 0 x * 0 ° C M 0.5 0.1 0.05 0-01 3.0 1 Q > 0 15-0 20.0 25.0 30.0 1.0 U CM/SEC m FIG.2.12 o Ac ° o O O A * 0 0 ° ° 0 o X A 4 X 1 I 1 I — 1 — I — I — 2.0 2.5 3.0 3.5 4.0 4.5 5.0 g CM FIG.2 .13 20.0 h 15.0' Pd 50 ( P s _ P ) g T 2 4 . .10 =100.0 200 .0 Pd 50 ( P s-P ) g T ' 150.0 100.0 90-0 80.0 70-0 60.0 d 5 0 50.0 40-0 10 =244 .9_ 30.0 J L I I I P/P =1.33 s 20.0 1 1-5.0 20.0 30.0 40.0 50.0 100.0 150.0 200.0 20.0 2 5 / d 5 Q FIG.2 .16 100 .0 /-200 .0 P D P J Q ( P 2 - p ) g T Z .10 =339.79 p / p =1.33 s I | I I 1 I 1 I I | 30.0 40.0 50.0 100.0 150.0 200.0 2 5 / d 5 Q to cn FIG.2.17 20.0 10-0 H P 5.0 '50 2.0 1-0 P d 50 10 =50.0 200-o/ / / / 1 7 ( P S - P ) g r x?c / / \ 100.0/ / \ / x \ \ \ \ \ \ \ \ \ \ Pd 50 20.0 10-0 H R 5.0 d 50 \ ( Ps-P )gT 2.0 2-. 10 =339 . 7$ p /p = 1 . 33 s J I I 1 1 1 L 1.0 50 .0/-' 1 0 0 . 0 / / \ / . A / \ ' \ 200.0' x, / \ / ' \ / / \ \ \ \ Pd \ (P - P ) g T z \ 5 0 10 6=224.927 \ P /p = 1 . 33 s J 1 1 I I I 1 L 20.0 30.0 40.0 50.0 100-0 150-0 200-0 20.0 30.0 40-0 50.0 100.0 150.0 200.0 2 5 /d 50 2 5 /d 50 FIG.2.18 FIG.2.19 1000. 500. L R 100. 50 50 (p _ p ) g T 2 - 1 0 -p- 2 .0 15 .26 11.0 8 .44 18.05 25.04. o X 2 .65 2 .65 2 . 65 2 . 66 2 . 66 / 75.0 211.0 30.0 ° * 50 .0 H R 50. 20.0 Pd 4 * 50 100.0 = ( p s - p ) g T 2 .10 10.0' 1°. \ \ * \ \ \ \ \ \ 1(^.0 / . / X / o o \ \ \ ^20 .0 \ \ X \ ko.o = Pd 50 , ^ 6 ^10 ( Ps-p )gT JX)0 .0 80.0 100.0 200.0 . 500.0 2 5 / d 5 Q 1000.0 2000.0 10Q0 FIG.2 .20 200-0 300.0 500-0 2 5 / d 5 0 1000.0 1500.0 FIG.2.21 to co H 1.0 0.5 0-1 j- 0-05 0-01 0-005 0.001 0.0005 0.0001 0.001 0.005 0.01 0.05 0.1 H R / 2 £ 0.5 1.0 1000.0 2X .L R 100.0 10.0 s e d i m e n t S u l f o n i c p o l y s t y r o l --L R < 2 ? 50 1.21 - 0.54 mm 5.0 10.0 1 0 . 0 FIG.2.22 S u l f o n i c p o l y s t y r o l -- 1.21 - 0.54 mm - L R > 2 ? o P o r t a g e S i l i c a s a n d --x F r a s e r R i v e r sand A B a k e l i t e s a n d 2.65 - 0.20 mm 2.66 - 0.33 mm 1.33 - 0.89 mm 1000.0 10000-0 100000.0 2U E/ v m FIG.2.23 to 1 3 0 NT <N CM O H o o o* ° 6 h ( A 6 ) 2 ( A / - ' H ) l 9 131 APPENDIX I I I CALIBRATION PROCEDURE OF WATER WAVE MEASURING EQUIPMENT The w a t e r s u r f a c e wave c h a r a c t e r i s t i c s were measured by t h e model 157 L e v e l - T e l w a t e r wave prober. As shown i n F i g . 2.1, t h e model 157 L e v e l - T e l w a t e r wave p r o b e i s d e s i g n e d w i t h c h a n g e a b l e o p e r a t i o n a l mode p r o v i s i o n s t o a l l o w e i t h e r d i r e c t (DIR) o r r e v e r s e (REV) o p e r a t i o n a l mode. I n t h e d i r e c t mode, t h e o u t p u t v o l t a g e v a r i e s i n d i r e c t p r o p o r t i o n t o t h e i n p u t s i g n a l . I n t h e r e v e r s e mode, t h e o u t p u t v o l t a g e i n c r e a s e s w i t h d e c r e a s i n g ' i n p u t s i g n a l . C a l i b r a t i o n s h o u l d be done e v e r y t i m e b e f o r e u s i n g model 157 L e v e l - T e l w a t e r wave p r o b e s i n m e a s u r i n g w a t e r s u r f a c e wave c h a r a c t e r i s t i c s . C a l i b r a t i o n p r o c e d u r e s f o r model 157 L e v e l - T e l w a t e r wave p r o b e i n d i r e c t mode o p e r a t i o n w h i c h i s u s e d d u r i n g c a r r y i n g o u t a l l e x p e r i m e n t s o f p r e s e n t s t u d y a r e as f o l l o w i n g : (1) The model 157 L e v e l - T e l w a t e r wave p r o b e were u s e d i n c o n j u n c t i o n w i t h H o n e y w e l l r e c o r d e r , w h i c h r e c o r d i n g s p e e d c a n be a d j u s t e d . C o n n e c t a d j u s t a b l e power s u p p l y t o t h e wave p r o b e and a d j u s t t h e power s u p p l y t o a c o n s t a n t o u p u t v o l t a g e , w h i c h i s 26.5 v o l t s f o r p r e s e n t s t u d y . (2) R e f e r t o F i g . 1, a 700 ohm r e s i s t o r i s c o n n e c t e d between t h e wave; p r o b e and t h e r e c o r d e r . A t c o n s t a n t low w a t e r d e p t h l e v e l , s a y Y = 100 cm, z e r o a d j u s t r e c o r d e r by s h o r t i n g o u t p u t t e r m i n a l s and t h e n a d j u s t i n g zero-, r a n g e 132 by a d j u s t i n g c o a r s e and f i n e z e r o . C l o c k w i s e r o t a t i o n w i l l i n c r e a s e o u t p u t v o l t a g e and c o u n t e r c l o c k w i s e r o t a t i o n w i l l d e c r e a s e t h e o u t p u t v o l t a g e . A d j u s t i n g t h e o u t p u t v o l t a g e t i l l i t r e a c h e s 2.8 v o l t s . (3) A f t e r i n c r e a s i n g t h e w a t e r d e p t h t o a n o t h e r c o n s t a n t h i g h w a t e r l e v e l , s u c h as Y = 50 com, t h e n s e t t h e s p a n a d j u s t m e n t by r o t a t i n g t h e c o a r s e and f i n e s p a n s . A g a i n c l o c k w i s e r o t a t i o n o f s p a n a d j u s t m e n t w i l l i n c r e a s e o u t p u t v o l t a g e , and c o u n t e r c l o c k w i s e r o t a t i o n o f s p a n a d j u s t m e n t w i l l d e c r e a s e o u t p u t v o l t a g e . A d j u s t i n g t h e o u t p u t v o l t a g e t i l l i t r e a c h e s 14 v o l t s . I f a n o t h e r r e s i s t o r i s u s e d t o c o n n e c t between t h e w a t e r wave p r o b e and t h e r e c o r d e r , t h e n t h e v a l u e o f o u t p u t v o l t a g e s h o u l d be a l t e r e d a c c o r d i n g l y by Ohm's law. (4) Reduce t h e w a t e r l e v e l f r o m t h i s h i g h w a t e r d e p t h t o low w a t e r d e p t h , r e c o r d t h e o u t p u t v o l t a g e s a t s e v e r a l w a t e r d e p t h s i n between. F i n a l l y , r e c h e c k i n g t h e ? . l i n e a r r e l a t i o n s h i p b etween t h e w a t e r d e p t h and t h e o u t p u t v o l t a g e . Water wave h e i g h t s c a n be o b t a i n e d f r o m t h i s l i n e a r r e l a t i o n s h i p and t h e v a r i a t i o n o f o u t p u t v o l t a g e when t h e w a t e r s u r f a c e waves p a s s t h e w a t e r wave p r o b e . 133 APPENDIX IV RELATIONSHIP BETWEEN THE MAXIMUM HORIZONTAL t h e b o t t o m b o u n d a r y l a y e r h as been s t u d i e d by I s a a c s o n ( 1 9 7 8 ) , 2 0 5 and a s e t o f c u r v e s s h o w i n g <S,/'( T(Hw/y) (gY) ' ) v e r s u s 2 0 5 (Y/gT )) * f o r v a r i o u s v a l u e s o f Hw/Y was p r e s e n t e d . ( R e f . t o F i g . 2 . 2 4 ) . The r a t i o o f t h e n e t mass t r a n s p o r t d i s t a n c e 6 and t h e maximum h o r i z o n t a l w a t e r p a r t i c l e m o t i o n d i s p l a c e m e n t 2 5 c a n be o b t a i n e d w i t h a i d o f t h i s s e t o f c u r v e s . A t t h e o u t e r edge o f t h e b o t t o m b o u n d a r y l a y e r , t h e v a l u e o f 2£ c a n be a p p r o x i m a t e d by l i n e a r wave t h e o r y as WATER PARTICLE MOTION DISPLACEMENT, 2 5, AND THE MASS TRANSPORT DISTANCE,5 . The b e h a v i o r o f mass t r a n s p o r t a t t h e o u t e r edge o f 2F, = Hw/sinh ( 2 T T Y / L W ) (2.16) t h u s , t h e r a t i o n o f S a n d 2£ c a n be w r i t t e n as 6 ^ - s i n h (2TY/Lw) ( T ( H w / ) ^ ( g Y ) " . b > < — > <9Y> TT S i n h ( Y w 2n Y Lw lT(Hw/ Y (gY)"-* ' W ' K (Y/gT )0.b (2.17) 1 3 4 F o r t h e c a s e o f s h a l l o w w a t e r , o r when Hw/Y< 0 . 0 4 , s i n h ( 2 H Y / L w ) c a n be a p p r o x i m a t e d as 2 TTY/ , t h e wave L i W 0 5 s p e e d C=(gY) ' and wave p e r i o d T=Lw/ c, t h u s , g T 2 / Y = L w 2 / Y 2 and s i n h ( 2 T r Y/Lw) == P^- = 2TT ( Y / g T 2 ) 0 ' 5 ( 2 . 1 8 ) Lw S u b s t i t u t e e q u a t i o n ( 2 . 1 8 ) i n t o e q u a t i o n ( 2 . 1 7 ) and y i e l d s IT = ( 2 ^ ( T ( H w / Y / ( g Y ) 0 . b ) <Hw/y) ( 2 . 1 9 ) The v a l u e s o f <S/25 f o r v a r i o u s v a l u e s o f Hw/Y c a n be o b t a i n e d f r o m e q u a t i o n ( 2 . 1 9 ) and F i g . 2 . 2 4 . I t i s f o u n d t h a t t h e maximum p o s s i b l e v a l u e o f 6 / 2 ? i s i n t h e o r d e r o f 0 . 2 . I t i s a l s o f o u n d t h a t t h e v a l u e o f 6 / 2 £ w i l l i n c r e a s e w i t h i n c r e a s i n g w a t e r d e p t h . However, s i n c e v a l u e s o f b o t h ^ and 2 £ d e c r e a s e w i t h i n c r e a s i n g w a t e r d e p t h , a l a r g e r a t i o o f &/2E, does n o t have any p h y s i c a l meaning, and t h e bo t t o m s e d i m e n t t r a n s p o r t a t i o n does n o t o c c u r a t a l l . 

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