"Science, Faculty of"@en . "Physics and Astronomy, Department of"@en . "DSpace"@en . "UBCV"@en . "Popil, Roman"@en . "2010-03-03T21:50:26Z"@en . "1979"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "The Rayleigh-Taylor instability of a water-air interface was investigated using electrical and photographic methods. An apparatus is described which accelerates a rectangular tank of water downwards and produces reproducible instabilities from a pure sinusoidal standing surface water wave of known phase and amplitude. The electrical measurements revealed that in addition to the bulk motion, films of water are produced on the walls of the water tank. The existence of these and other features of the instability are substantiated by photographs\r\nof the various instabilities that were produced. The electrical measurements led to a new scaling law for the phenomenon of climbing fluid films at accelerations greater than gravity. Several linear devices were also developed for measuring the amplitudes of surface water waves."@en . "https://circle.library.ubc.ca/rest/handle/2429/21437?expand=metadata"@en . "E X P E R I M E N T A L I N V E S T I G A T I O N OF THE R A Y L E I G H - T A Y L O R I N S T A B I L I T Y by Roman P o p i l B . S c , U n i v e r s i t y o f R e g i n a , ]977 A T H E S I S SUBMITTED I N P A R T I A L . F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF S C I E N C E i n THE FACULTY OF GRADUATE S T U D I E S D e p a r t m e n t o f P h y s i c s 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 U N I V E R S I T Y OF B R I T I S H COLUMBIA A p r i l , ] 9 7 9 @ Roman P o p i l , 1 9 7 9 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Co 1umbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my written permission. Department of Pti^SlC 9 The Univers i ty of B r i t i s h Co Iumbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS 5-ABSTRACT The R a y l e i g h - T a y l o r i n s t a b i l i t y o f a w a t e r -a i r i n t e r f a c e was i n v e s t i g a t e d u s i n g e l e c t r i c a l and p h o t o g r a p h i c m e t h o d s . An a p p a r a t u s i s d e s c r i b e d w h i c h a c c e l e r a t e s a r e c t a n g u l a r t a n k o f w a t e r d o w n w a r d s and p r o d u c e s r e p r o d u c i b l e i n s t a b i l i t i e s f r o m a p u r e s i n u s o i d a l s t a n d i n g s u r f a c e w a t e r wave o f known p h a s e a n d a m p l i t u d e . The e l e c t r i c a l m e a s u r e m e n t s r e v e a l e d t h a t i n a d d i t i o n t o t h e b u l k m o t i o n , f i l m s o f w a t e r a r e p r o d u c e d on t h e w a l l s o f t h e w a t e r t a n k . The e x i s t e n c e o f t h e s e and o t h e r f e a t u r e s o f t h e i n s t a b i l i t y a r e s u b s t a n t i a t e d by p h o t o -g r a p h s o f t h e v a r i o u s i n s t a b i l i t i e s t h a t w e r e p r o d u c e d . The e l e c t r i c a l m e a s u r e m e n t s l e d t o a new s c a l i n g l a w f o r t h e phenomenon o f c l i m b i n g f l u i d f i l m s a t a c c e l e r a t i o n s g r e a t e r t h a n g r a v i t y . S e v e r a l l i n e a r d e v i c e s were a l s o d e v e l o p e d f o r m e a s u r i n g t h e a m p l i t u d e s o f s u r f a c e w a t e r w a v e s . i i T A B L E OF CONTENTS 1) INTRODUCTION 1 2) CHAPTER I I E X P E R I M E N T A L APPARATUS A . T a n k a c c e l e r a t i o n s y s t e m 11 B . T a n k and A n a l y z i n g P l a t e s 12 . C . Wave e x c i t a t i o n 16 D . The e l e c t r i c a l s y s t e m . 20 E. The I . C . c a p a c i t a n c e m e a s u r i n g c i r c u i t . . 23 3) CHAPTER I I I E X P E R I M E N T A L R E S U L T S A . C a p a c i t a n c e m e a s u r e m e n t s . . . 26 B . R e s i s t a n c e m e a s u r e m e n t s 33 C . \" C a v i t a t i o n \" e f f e c t s 37 D . P h o t o g r a p h y o f t h e f l u i d m o t i o n 48 E . E l e c t r i c a l m e a s u r e m e n t s , n a r r o w t a n k . . . 63 F . T e s t f o r t h e a i r f i l m 68 G . C l i m b i n g f i l m s 73 4) CHAPTER I V CONCLUSION 81 5) B I B L I O G R A P H Y 84 6) A P P E N D I C E S A . D e t a i l s o f t h e w a v e f o r m g e n e r a t o r . . . . 85 B . O p t i c a l F o u r i e r a n a l y s i s 91 - i i i -L I S T OF T A B L E S I Wave E x c i t a t i o n E l e c t r o d e s 18 I I R e s i s t a n c e F o u r i e r A n a l y s i s 34 I I I C a v i t a t i o n R a t e and T a n k W i d t h 41 I V E f f e c t o f V i s c o s i t y on C a v i t a t i o n R a t e . . 44 V C l i m b i n g F i l m s R e s u l t s 75 V I D i g i t a l P r o g r a m f o r Wave G e n e r a t o r . . . . 90 - i v -L I S T OF F I G U R E S 1) P h o t o g r a p h o f a R a y l e i g h - T a y l o r I n s t a b i l i t y s h o w i n g t h e s p i k e and t h e b u b b l e . 7 2) S i d e v i e w o f t h e t a n k a c c e l e r a t o r 13 3) F r o n t v i e w o f t h e t a n k a c c e l e r a t o r 1A A) H o r i z o n t a l s e c t i o n o f t h e w a t e r t a n k . . . . 1 5 5) The f o u r t h h a r m o n i c F o u r i e r p l a t e 17 6) Wave g e n e r a t i n g e l e c t r o d e a s s e m b l y . . . . 19 7) B l o c k d i a g r a m o f t h e e l e c t r i c a l s y s t e m . . . 2 2 8) I . C . c a p a c i t a n e m e a s u r i n g c i r c u i t 2A 9) R e s p o n s e o f t h e I . C . t o d e p t h c h a n g e s . . . 25 10) O u t p u t o f t h e i m p e d a n c e b r i d g e 27 11) S p e c t r u m o f t h e s e c o n d h a r m o n i c wave . . . 28 12) R e s u l t s u s i n g t h e w a t e r as t h e d i e l e c t r i c 29 13) B r i d g e o u t p u t v s . t i m e o s c i l l o g r a p h s . . . 31 1A) C a p a c i t a n c e m e a s u r e m e n t s , d e c r e a s e i n mean l e v e l 32 15) R e s i s t a n c e F o u r i e r a n a l y s i s 35 16) L i n e a r d e c r e a s e o f t a n k c a p a c i t a n c e . . . 39 17 ) C a l i b r a t i o n o f t h e i m p e d a n c e b r i d g e . . . AO 18) \" C a v i t a t i o n \" r a t e , t a n k w i d t h 3 cm . . . . A2 19) C a v i t a t i o n r a t e f o r a 30% s u g a r s o l u t i o n . A5 20) I m p e d a n c e b r i d g e t e s t t o t r a n s i e n t c h a n g e s i n c a p a c i t a n c e A7 21) I n i t i a l u n p e r t u r b e d s u r f a c e p h o t o g r a p h . . A9 22) F i r s t h a r m o n i c wave i n s t a b i l i t y 51 23) F o u r t h h a r m o n i c i n s t a b i l i t y s h o t 53 24) F o u r t h h a r m o n i c i n s t a b i l i t y o f p h a s e 0 . . 55 25) P h o t o g r a p h o f t h e s t a n d i n g wave 56 26) S i d e on f l a s h p h o t o g r a p h s 57 27 ) \" W e t t i n g \" f l a s h p h o t o g r a p h s 58 28) E i g t h h a r m o n i c f l a s h p h o t o g r a p h 60 29 ) F o u r t h h a r m o n i c wave i n a n a r r o w t a n k . . 62 30) B r i d g e o u t p u t f o r f o u r t h h a r m o n i c . . . . 64 31) R e s i s t a n c e c h e c k o f c h a n g e i n mean l e v e l . 66 32) V a l i d i t y o f g r o w t h d i r e c t i o n t e s t s . . . . 67 33) T e s t f o r p r e s e n c e o f an a i r f i l m 70 34 ) P h o t o t r a n s i s t o r r e s u l t s f o r a i r f i l m . . . 71 35 ) B r i d g e r u n s f o r a p . c . b o a r d c a p a c i t o r p l a t e t a n k 74 36) C l i m b i n g f i l m s b e h a v i o u r , g r a p h . . . . . 76 37) L o g a r i t h m i c p l o t f o r c l i m b i n g f i l m s . . . 77 38) C l i m b i n g f i l m s r e s u l t s s c a l i n g l a w g r a p h . 78 39) S c h e m a t i c o f . t h e d i g i t a l w a v e f o r m g e n e r a t o r 86 40 ) W a v e f o r m g e n e r a t o r memory 87 41) U p / d o w n c o u n t e r a d d r e s s l o g i c 88 42) S e t - u p f o r o p t i c a l F o u r i e r a n a l y s i s . . . 92 - v i -ACKNOWLEDGEMENT T h i s p r o j e c t was c a r r i e d o u t u n d e r t h e a u s p i c e s o f D r . F . L . C u r z o n w h o s e k e e n i n t e r e s t i n my s t u g g l i n g e n d e a v o u r s and e n t h u s i a s m f o r s c i e n c e I f o u n d t o be b o t h e n c o u r a g i n g and i n s p i r i n g . A . C h e u c k ' s t e c h n i c a l a s s i s t a n c e w i t h t h e e l e c t r o n i c s and i n n u m e r a b l e o t h e r t e c h n i c a l i t i e s p r o v e d t o be i n v a l u a b l e . M . H e i n r i c h and G . A u c h i n l e u c k a s s i s t e d w i t h much o f t h e m a c h i n e shop w o r k . - v i i -1-CHAPTER I INTRODUCTION R a y l e i g h - T a y l o r i n s t a b i l i t i e s o c c u r w h e n e v e r t h e r e i s an a c c e l e r a t i o n o f t h e p l a n a r i n t e r f a c e o f two s u p e r i m p o s e d f l u i d s s u c h t h a t t h e a c c e l e r a t i o n i s d i r e c t e d f r o m t h e l e s s d e n s e t o t h e d e n s e r f l u i d . I f t h e b i n a r y f l u i d s y s t e m i s l e f t c o m p l e t e l y u n d i s t u r b e d a c o n d i t i o n o f an u n s t a b l e e q u i l i b r i u m w i l l e x i s t . H o w e v e r , i f t h e i n t e r f a c e i s i n i t i a l l y p e r t u r b e d p r i o r t o a c c e l e r a t i o n , t h e p e r t u r b a t i o n w i l l g row e x p o n e n t i a l l y w i t h t i m e , g i v i n g r i s e t o t h e s o c a l l e d R a y l e i g h - T a y l o r i n s t a b i l i t y . T h i s i n s t a b i l i t y i s a common o c c u r r e n c e i n n a t u r e as i n t h e f o r m a t i o n o f i c i c l e s , and t h e a c c e l e r a t i o n o f s t e l l a r m a t e r i a l on s t e l l a r s u r f a c e s . R a y l e i g h - T a y l o r i n s t a b i l i t i e s a r e o f c u r r e n t i n t e r e s t i n p l a s m a p h y s i c s w h e r e t h e u n s t a b l e s u r f a c e i s t h e p l a s m a vacuum i n t e r f a c e . T h e y o c c u r w h e n e v e r t h e a t t e m p t i s made t o a c c e l e r a t e t h e p l a s m a by a m a g n e t i c f i e l d as i n a z - p i n c h o r a t o k o m a k . P e r h a p s t h e mos t common e x a m p l e c a n be t a k e n t o be t h e c a s e o f a g l a s s o f w a t e r t u r n e d u p s i d e d o w n . H e r e , t h e a c c e l e r a t i o n i s t h a t due t o g r a v i t y and i s d i r e c t e d f r o m t h e w a t e r t o t h e a i r and t h u s t h e a i r - w a t e r i n t e r f a c e i s u n s t a b l e . The w a t e r , i n s t e a d o f m a i n t a i n i n g a n e a r l y p l a n e l o w e r i n t e r f a c e as i t f a l l s , w i l l t e n d t o be d e f o r m e d i n t o l o n g s p i k e s . I t i s t h e f o r m a t i o n and r a t e o f g r o w t h -2-o f t h e s e s p i k e s t h a t h a s b e e n t h e t o p i c o f s e v e r a l e x p e r i m e n t a l and n u m e r o u s t h e o r e t i c a l i n v e s t i g a t i o n s . S h o r t l y a f t e r t h e t h e o r e t i c a l p a p e r by G . I . T a y l o r ( 1 9 5 0 ) , t h e i n s t a b i l i t y was i n v e s t i g a t e d e x p e r i m e n t a l l y by D . J . L e w i s ( 1 9 5 0 ) , who s u b j e c t e d a s h o r t c o l u m n o f w a t e r t o a p r e s s u r e d i f f e r e n c e i n t h e a i r a b o v e and b e l o w t h e w a t e r . Emmons, C h a n g and W a t s o n ( 1 9 5 9 ) c a r r i e d o u t a s i m i l a r e x p e r i m e n t i n w h i c h a r e c t a n g u l a r t a n k o f w a t e r was a c c e l e r a t e d downward by t h e f o r c e o f s t r e t c h e d r u b b e r t u b i n g . A t a c c e l e r a t i o n s g r e a t e r t h a n t h a t o f g r a v i t y , t h e a t m o s p h e r i c p r e s s u r e o f t h e a i r t a k e s t h e r o l e o f t h e l i g h t e r f l u i d b e i n g a c c e l e r a t e d i n t h e d i r e c t i o n o f t h e h e a v i e r o n e . C o l e and T a n k i n ( 1 9 7 2 ) and R a t a f i a ( 1 9 7 3 ) h a v e p e r f o r m e d s i m i l a r e x p e r i m e n t s w h e r e a r e c t a n g u l a r t a n k o f w a t e r was a c c e l e r a t e d downward a t a c c e l e r a t i o n s s e v e r a l t i m e s g r e a t e r t h a n g r a v i t y . The f i r s t o r d e r t h e o r y as o r i g i n a l l y f o r m u l a t e d by G . I . T a y l o r f o l l o w s e a s i l y f r o m t h e c o n v e n t i o n a l h y d r o d y n a m i c s . The f r e e s u r f a c e o f t h e w a t e r i s g i v e n b y : y = n ( x , z , t ) w h e r e y i s t a k e n a l o n g t h e v e r t i c a l a x i s , and x , z a r e r e c t a n g u l a r c o o r d i n a t e s i n t h e h o r i z o n t a l p l a n e . D i f f e r e n -t i a t i n g t h i s e x p r e s s i o n w i t h r e s p e c t t o t i m e t , y i e l d s t h e f r e e s u r f a c e b o u n d a r y c o n d i t i o n : -3-- n t - u n x + v = 0 w h e r e t h e s u b s c r i p t s d e n o t e p a r t i a l d e r i v a t i v e s and u , v a r e t h e h o r i z o n t a l and v e r t i c a l c o m p o n e n t s o f t h e v e l o c i t y . B e r n o u l l i ' s l a w upon n e g l e c t i n g s e c o n d o r d e r t e r m s i n t h e v e l o c i t y g i v e s an e x p r e s s i o n f o r t h e p r e s s u r e : p = p 0 - (g + g x ) p y + P * t w h e r e (g + g^) i s t h e v i r t u a l a c c e l e r a t i o n s e e n by t h e i n t e r f a c e , p i s t h e d e n s i t y o f t h e f l u i d and * i s t h e v e l o c i t y p o t e n t i a l w h i c h s a t i s f i e s L a p l a c e ' s e q u a t i o n : 2 V * = 0 and whose g r a d i e n t g i v e s t h e c o m p o n e n t s o f t h e v e l o c i t y : y_= ( u , v , w ) = V$ The t h e o r y c o n s i d e r s two s u p e r i m p o s e d f l u i d s e a c h w i t h a p r e s s u r e , d e n s i t y and v e l o c i t y p o t e n t i a l . F o r t h e u p p e r f l u i d ( a i r ) , t h e v e l o c i t y p o t e n -t i a l and p r e s s u r e a r e : * . = A e x p { - k y } f ( t ) c o s { k x } P x = P 0 - (g + g i ) P i v + p i ( * i ) t S i m i l a r l y , f o r t h e l o w e r f l u i d t h e c o r r e s p o n d i n g q u a n t i t i e s a r e : * 2 = - A e x p { - k y } f ( t ) c o s { k x > P 2 = P 0 - (g \u00E2\u0080\u00A2 g ! ) p 2 y + P 2 ( * 2 ) t - 4 -w h e r e k = 2Tr /X i s t h e w a v e n u m b e r , and X i s t h e w a v e l e n g t h o f t h e p e r i o d i c i n s t a b i l i t y . The f o r m a d o p t e d f o r t h e v e l o c i t y p o t e n t i a l s i s t h a t f o r s t a n d i n g w a v e s t h a t s a t i s f y t h e b o u n d a r y c o n d i t i o n s t h a t t h e v e l o c i t i e s be f i n i t e a t y = \u00C2\u00B1 \u00C2\u00BB , t h a t t h e y be e q u a l a t t h e i n t e r f a c e y = 0 and t h a t y_ \u00E2\u0080\u00A2 n = 0 on t h e t a n k w a l l s w h e r e n_ i s t h e n o r m a l v e c t o r t o a w a l l . The n o n l i n e a r t e r m n x u i n t h e f r e e s u r f a c e c o n d i t i o n i s n e g l e c t e d y i e l d i n g n t - v - ( * . ) y = - k A f ( t ) c o s {kx} w h i c h u p o n i n t e g r a t i o n b e c o m e s : n = - k A ( t / t f ( t ) d t ) c o s { k x } o -A t t h e i n t e r f a c e , t h e p r e s s u r e s P j and p 2 a r e e q u a l . T h i s c o n d i t i o n , when u s e d w i t h t h e e x p r e s s i o n f o r n , l e a d s t o : -Cg + gj) ( P 2 - P x ) k f ( t ) - ( p 2 + P l ) f \u00C2\u00BB ( t ) = 0 The t h e o r y i s c o n s i d e r a b l y s i m p l e r a t t h i s p o i n t , i f t h e r e q u i r e m e n t i s made t h a t a t t = 0 and y = 0 t h e f l u i d v e l o c i t y i s z e r o . T h i s i s s a t i s f i e d i f t h e t i m e d e p e n d e n c e i s f ( t ) = s i n h { n t } w h e r e n i s t h e g r o w t h r a t e o f t h e i n s t a b i l i t y . The c o n d i t i o n f o r e q u a l p r e s s u r e s now g i v e s an e x p l i c i t e x p r e s s i o n f o r t h e g r o w t h r a t e : 2 n = - Cg + g x ) ( P 2 - P j ) l c ( P 2 + P j ) H e n c e , t h e i n t e r f a c e as a f u n c t i o n o f t i m e i s g i v e n by t h e e x p r e s s i o n : n = k A n \" * c o s h { n t } c o s {kx} T h e r e f o r e , i f t h e q u a n t i t y (g \u00E2\u0080\u00A2 g j ) < 0 t h e n t h e g r o w t h r a t e n i s p o s i t i v e and t h e d i s t u r b a n c e g r o w s e x p o n e n t i a l l y . B e l l m a n n and P e n i n i g t o n ( 1 9 5 4 ) c o n s i d e r e d t h e e f f e c t s o f s u r f a c e t e n s i o n and v i s c o s i t y . F o r an a i r -w a t e r i n t e r f a c e , t h e e f f e c t s o f v i s c o s i t y a r e n e g l i g i b l e . S u r f a c e t e n s i o n T , i s i n t r o d u c e d t h r o u g h t h e e q u a l p r e s s u r e s c o n d i t i o n : P 2 \" ? ! + T n x x = 0 The r e s u l t i n g e x p r e s s i o n f o r t h e g r o w t h r a t e b e c o m e s : 2 n = - (g + g1)(.P2 - P 2 ) k _ ^ 3 ( p j + P 2 ) (Pj^ + P 2 ) 2 I f n > 0 t h e s u r f a c e i s u n s t a b l e . I t i s e v i d e n t f r o m t h e a b o v e e x p r e s s i o n t h a t t h e s u r f a c e t e n s i o n s t a b i l i z e s modes w i t h a w a v e l e n g t h : X < 2ir, k-Cg \u00E2\u0080\u00A2 gj) ( P 2 - P j ) - 6 -2 s i n c e f o r s u c h w a v e l e n g t h s n < 0 . T h u s f o r i n i t i a l d i s t u r b a n c e s whose w a v e l e n g t h s a r e s m a l l e r t h a n t h e a b o v e , t h e r e i s no i n s t a b i l i t y . T h i s e x p l a i n s why s m a l l d r o p l e t s c l i n g t o t h e u n d e r s i d e o f a h o r i z o n t a l s u r f a c e s u c h as a c e i l i n g . F o r w a t e r , u n d e r t h e a c t i o n o f g r a v i t y , t h e c r i t i c a l w a v e l e n g t h i s 1 .73 cm so t h a t d r o p l e t s o f a l a r g e r r a d i u s w i l l t e n d t o d r i p w h i l e s m a l l e r o n e s w i l l h a n g . The f i r s t o r d e r t h e o r y as f o r m u l a t e d b y G . I . T a y l o r d o e s n o t a c c o u n t f o r t h e l a t e r s t a g e s o f t h e i n s t a b i l i t y w h e r e t h e c r e s t s d e v e l o p i n t o l o n g s p i k e s and t h e t r o u g h s i n t o r o u n d e d b u b b l e s ( F i g u r e 1 ) . L e w i s ' s i n v e s t i g a t i o n s l e d t o t h e c o n c l u s i o n t h a t t h e f i r s t o r d e r t h e o r y i s v a l i d o n l y u n t i l t h e a m p l i t u d e o f t h e s p i k e s i s a b o u t 0 . 4 X . By a s s u m i n g t h a t t h e s u r f a c e d i s t u r b a n c e and v e l o c i t y p o t e n t i a l s c a n be e x p a n d e d i n a p o w e r s e r i e s , Emmons e t a l . d e t e r m i n e d how t h e l a t e r s t a g e s o f t h e i n s t a b i l i t y e v o l v e . The h a r m o n i c c o n t e n t o f t h e d i s t o r t e d g r o w i n g d i s t u r b a n c e i s g i v e n by t h e F o u r i e r s e r i e s : OC* n = I . A ( t ) c o s { m x ) m= 1 m The o r i g i n a l p u r p o s e o f t h i s e x p e r i m e n t was t o f o l l o w t h e d e v e l o p m e n t o f t h e F o u r i e r s p e c t r u m o f t h e i n i t i a l p e r t u r b a t i o n w i t h t i m e . To t h i s p u r p o s e , a c a p a c i t a t i v e F o u r i e r a n a l y z i n g t e c h n i q u e as d e v e l o p e d b y C u r z o n and L a n g i l l e ( 1 9 7 2 ) was u s e d . T h i s m e t h o d Figure 1.Photograph of a Ray 1eigh-Tay1or i n s t a b i l i t y a r i s i n g from the fourth harmonic standing wave. The \"sp i k e \" and \"bubble\" arc i n d i c a t e d . -8-a n a l y z e s e a c h h a r m o n i c mode s e p a r a t e l y i . e . , i t e s s e n t i a l l y m e a s u r e s A ^ f t ) f o r s e l e c t e d m. The a s s u m p t i o n i s made t h a t t h e v a r i a t i o n i n t h e a r e a o f t h e p l a t e s and d e p t h o f t h e w a t e r i s i n t h e x d i r e c t i o n o n l y so t h a t t h e c a p a c i t a n c e i s c a l c u l a t e d as an i n t e g r a l o v e r t h e x v a r i a b l e . F o r t h e s y s t e m u s e d i n t h e e x p e r i m e n t s , t h e w a t e r , whose c o n d u c t i v i t y i s i n c r e a s e d b y a d d i n g some a c i d f o r m s one p l a t e o f a p a r a l l e l p l a t e c a p a c i t o r . The o t h e r p l a t e , s e p a r a t e d f r o m t h e f i r s t by a t h i n p o l y e t h y l e n e s h e e t f o r m s one w a l l o f t h e w a t e r t a n k . I t c o n s i s t s o f a number o f e q u a l l y s p a c e d v e r t i c a l s t r i p s whose w i d t h W(x) i s a f u n c t i o n o f x and i s g i v e n b y : K ' (x ) = W q(1 - c o s { q i r x / L } ) (1) w h e r e W q = c o n s t a n t , q i s an i n t e g e r e q u a l t o t h e mode number t h a t i s t o be a n a l y z e d and L i s t h e l e n g t h o f t h e t a n k . H e n c e , f o r a s m a l l i n c r e m e n t d x , t h e c a p a c i t a n c e i s g i v e n b y : dC = K E o ( N W ( x ) y ( x ) ) d x (2) d w h e r e K i s t h e d i e l e c t r i c c o n s t a n t , e Q t h e p e r m i t t i v i t y o f f r e e s p a c e , d i s t h e t h i c k n e s s o f t h e d i e l e c t r i c s h e e t and N i s t h e number o f s t r i p s p e r u n i t l e n g t h . The e l e v a t i o n o f t h e w a t e r s u r f a c e c a n g e n e r a l l y be e x p r e s s e d b y a \" F o u r i e r s e r i e s e x p a n s i o n : -9-y = y + Z -5 c o s{rTrx / L } f ( t ) J Jo r=0 r r v ' The y Q t e r m i s t h e mean l e v e l o f t h e f l u i d w h i c h i s a s s u m e d t o be c o n s t a n t and i n d e p e n d e n t o f t h e t i m e i f t h e m o t i o n i s p u r e l y two d i m e n s i o n a l . 5^ i s t h e a m p l i t u d e o f t h e r t h h a r m o n i c o f t h e s u r f a c e d i s t u r b a n c e a n d f ( t ) i s t h e t i m e d e p e n d e n c e . T h e r e f o r e , t h e i n c r e m e n t i n c a p a c i t a n c e i s : dC = ( y Q + I = 0 c : r c o s { r T r x / L } f r ( t ) ) X (1 - c o s{qnx / L } ) d x The i n t e g r a t i o n o v e r x l e a d s t o C = A ( y o L - \u00C2\u00A3 L E = 1 c ; r c o s { q i T x / L } x c o s { r i r x / L } f ( t ) d x ) r = M y D L - 5 q L f q ( t ) ) w h e r e t h e o r t h o g o n a l i t y o f t h e c o s i n e s i s u s e d i n t h e l a s t s t e p and A i s a c o n s t a n t . T h u s t h e t i m e d e p e n d e n c e o f t h e c a p a c i t a n c e i s p r o p o r t i o n a l t o t h e t i m e d e p e n d e n c e o f t h e q t h h a r m o n i c o f t h e s u r f a c e d i s t u r b a n c e . The s e n s i t i v i t y o f t h e t e c h n i q u e i s d e t e r m i n e d by n o t i n g t h a t t h e c h a n g e i n c a p a c i t a n c e AC due t o an o v e r a l l i n c r e a s e o f w a t e r d e p t h o f amount h i s t w i c e t h e c h a n g e i n c a p a c i t a n c e due t o a q t h h a r m o n i c wave o f t h e same a m p l i t u d e : AC w a t e r d e p t h = AC h a r m o n i c wave - 1 0 -tce N ~ b\" o ( 1 \" C O S ^ x / U j h d x < E Q N L / W (1 - c o s { q i r x / L } h c o s { q i r x / L } d x , 0 0 d = 2 A r e s i s t a n c e F o u r i e r a n a l y z i n g t e c h n i q u e w o r k s a l o n g t h e same p r i n c i p l e s . I n t h i s m e t h o d , two F o u r i e r a n a l y z i n g p l a t e s a r e p l a c e d a l o n g two s i d e s o f t h e w a t e r t a n k w i t h t h e c o n d u c t i n g w a t e r b e t w e e n t h e m . T h e n t h e r e s i s t a n c e i s g i v e n b y : R = p 1 A w h e r e p i s t h e r e s i s t i v i t y , 1 t h e d i s t a n c e b e t w e e n t h e two p l a t e s and A i s t h e a r e a g i v e n b y t h e p r o d u c t o f t h e s t r i p w i d t h and d e p t h o f t h e w a t e r as i n e q u a t i o n (2). I n t h e c a p a c i t a n c e m e a s u r i n g t e c h n i q u e , t h e c a p a c i t a n c e i s m e a s u r e d b y t h e o u t p u t s i g n a l o f an i m p e d a n c e b r i d g e w h i l e i n t h e r e s i s t a n c e m e t h o d t h e v o l t a g e a c r o s s t h e t a n k i s o b s e r v e d . D e t a i l s o f t h e e q u i p m e n t r e q u i r e d t o p r o d u c e and m e a s u r e R - T i n s t a b i l i t i e s a r e now p r e s e n t e d i n t h e f o l l o w i n g c h a p t e r . -11-CHAPTER I I E X P E R I M E N T A L A P P A R A T U S A . Tank a c c e l e r a t i o n s y s t e m < The P l e x i g l a s t a n k c o n t a i n i n g t h e w a t e r - a i r i n t e r f a c e i s a c c e l e r a t e d d o w n w a r d by t h e f o r c e s u p p l i e d by a d r i v e n p i s t o n w h i c h i s s u p p l i e d w i t h c o m p r e s s e d a i r f r o m a l a r g e r e s e r v o i r and a i r t a n k . The r e s e r v o i r e n s u r e s t h a t t h e c h a n g e i n p r e s s u r e s u p p l i e d t o t h e p i s t o n i s i n t h e o r d e r o f o n l y a few p e r c e n t d u r i n g t h e l e n g t h o f i t s r u n . The p i s t o n ' s c y l i n d e r i s as l o n g as t h e t a n k ' s l e n g t h o f t r a v e l (450 mm) so t h a t t h e r e i s a n e a r l y c o n s t a n t f o r c e s u p p l i e d t o t h e t a n k t h r o u g h o u t i t s m o t i o n . By u s i n g a l a s e r beam and a p h o t o t r a n s i s t o r t o d e t e r m i n e t i m e s o f t r a v e l u n d e r v a r i o u s a p p l i e d a i r p r e s s u r e s . , t h e a c c e l e r a t i o n was f o u n d t o i n c r e a s e l i n e a r l y w i t h t h e a i r p r e s s u r e . F o r t h e c y l i n d e r o f i n n e r b o r e o f 2 5 . 4 mm t h e i n c r e m e n t was f o u n d t o be O . l g / p . s . i . , g b e i n g t h e a c c e l e r a t i o n i n f r e e f a l l . The maximum p r e s s u r e n o r m a l l y a p l l i e d was 25 p . s . i . so t h a t t h e maximum v i r t u a l a c c e l e r a t i o n ( i . e . , t h e a c c e l e r a t i o n o f t h e t a n k m i n u s t h e a c c e l e r a t i o n o f f r e e f a l l ) was 2 . 5 g . S i n c e any i n i t i a l s u r f a c e p e r t u r b a t i o n w i l l g row e x p o n e n t i a l l y u n d e r s u c h an a p l l i e d a c c e l e r a t i o n , c a r e mus t be t a k e n t o r e l e a s e t h e t a n k a t t = 0 i n s u c h a f a s h i o n t h a t t h e w a t e r i s n o t a p p r e c i a b l y d i s t u r b e d . To t h i s e n d , a t a n k r e l e a s e m e c h a n i s m was d e v i s e d b y - 1 2 -c u t t i n g away p a r t o f t h e w a l l o f a b r a s s t u b e . The r e m a i n i n g p o r t i o n s u b t e n d s an a n g l e o f 90 d e g r e e s a t t h e a x i s and s u p p o r t s a h o r i z o n t a l b r a s s r o d w h i c h f a s t e n s o n t o t h e b a s e o f t h e w a t e r t a n k . The r e l e a s i n g b r a s s t u b e i s i n t u r n m o u n t e d o n t o t h e a l u m i n u m f r a m e o f t h e t a n k a c c e l e r a t o r . To r e l e a s e t h e t a n k , t h e r e l e a s e t u b e i s r o t a t e d b y a s o l e n o i d r e l a y , so t h a t t h e b r a s s r o d i s no l o n g e r s u p p o r t e d b y t h e t u b e . T h e t a n k s l i d e s on two v e r t i c a l s t e e l r a i l s a t e i t h e r s i d e b e f o r e c o m i n g i n t o c o n t a c t w i t h t h e s h o c k a b s o r b e r s a t t h e end o f t h e s h a f t , ( F i g u r e s 2 and 3 ) . The s h o c k a b s o r b e r s a r e s p r i n g l o a d e d p i s t o n s w h i c h a r e c o n n e c t e d t o g e t h e r b y an e x h a u s t a i r v a l v e w h i c h i s a d j u s t e d t o a c h i e v e o p t i m u m s t o p p i n g c o n d i t i o n s . B . Tank and A n a l y z i n g P l a t e s The w a t e r t a n k ' s s i d e s and b a s e a r e c o n s t r u c t e d f r o m h i n c h and 3 / 4 i n c h P l e x i g l a s s h e e t r e s p e c t i v e l y . The l e n g t h , w i d t h and h e i g h t a r e 188 x 107 x 200 mm. When u s e d i n c a p a c i t a n c e m e a s u r e m e n t s t h e w a t e r i n c o n t a c t w i t h t h e l a r g e s t w a l l s o f t h e r e c t a n g u l a r t a n k ( F i g u r e 4 ) f o r m s one p l a t e o f a c a p a c i t o r a n d t h e s e c o n d p l a t e i s a d j a c e n t t o one o f t h e 188 x 107 mm s i d e s . A r u b b e r 0 - r i n g i s f i t t e d a l o n g t h e p e r i m e t e r o f t h i s t o make t h e t a n k l e a k p r o o f . The c a p a c i t o r p l a t e i s i n s u l a t e d f r o m t h e w a t e r by a t h i n p o l y e t h y l e n e s h e e t o f t h i c k n e s s -13-SOLENOID VALVE | TIMING CIRCUIT . T O T C0MRES5ED AIR P I S T O N TANK TANK R E L E A S E MECHANISM SOLENOID RELAY TO | TIMING \u00E2\u0082\u00AC> CIRCUIT Figure 2. Side view of the tank accelerator, - 1 4 --15-THREADED HOLES LUCITE CAPACITOR PLATE POLYETHYLENE SHEET Figure 4a) H o r i z o n t a l s e c t i o n o f t h e w a t e r t a n k . D e t a i l o f one s i d e c o n t a i n i n g t h e c a p a c i t o r and d i e l e c t r i c s h e e t . - 1 6 -0 . 0 7 7 mm p l a c e d b e t w e e n t h e 0 - r i n g and t h e c a p a c i t o r p l a t e . I n e a r l i e r e x p e r i m e n t s , v a r n i s h and e p o x y r e s i n w e r e u s e d f o r t h e i n s u l a t o r b u t t h e s e m a t e r i a l s p r o v e d t o be i n e f f e c t i v e b e c a u s e o f s m a l l p i n h o l e s w h i c h l e a k e d c u r r e n t . To a n a l y z e t h e w a t e r w a v e s , t h e c a p a c i t o r p l a t e c o n s i s t s o f v e r t i c a l s t r i p s o f c o p p e r w h o s e w i d t h v a r i e s s i n u s o i d a l l y as g i v e n b y e q u a t i o n (1) i n t h e I n t r o d u c t i o n . I n t h e e x p e r i m e n t s , W q was c h o s e n t o be 0 . 5 cm and t h e mode number q was v a r i e d f r o m 1 t o 5 ( F i g u r e 5 ) . The F o u r i e r a n a l y z i n g p l a t e s a r e made f r o m p r i n t e d c i r c u i t b o a r d u s i n g t h e s t a n d a r d p h o t o g r a p h i c e t c h i n g t e c h n i q u e . C . Wave E x c i t a t i o n S t a n d i n g p u r e modes a r e g e n e r a t e d on t h e w a t e r s u r f a c e by a p p l y i n g an a l t e r n a t i n g h i g h v o l t a g e o f t h e f o r m : h V = V Q ( {1 - c o s (cot) } / 2 ) w h e r e V i s c o n s t a n t e q u a l t o 4 . 7 K V , and OJ i s t h e r e s o n a n t o ^ f r e q u e n c y o f t h e d e s i r e d s u r f a c e wave i n t h e w a t e r . The r e s o n a n t f r e q u e n c y i s d e t e r m i n e d t o an e x c e l l e n t a p p r o x i -m a t i o n by t h e d i s p e r s i o n r e l a t i o n f o r s t a n d i n g w a t e r w a v e s w h i c h may be t a k e n t o be : 2 u> = g k t a n h ( k h ) w h e r e h i s t h e d e p t h o f t h e f l u i d i n t h e t a n k . Figure 5. The fourth harmonic (q=4) Fourier analyzing p l a t e . - 1 8 -To e x c i t e t h e w a v e s u s i n g t h e p e r i o d i c h i g h v o l t a g e , h o r i z o n t a l e l e c t r o d e s a r e p o s i t i o n e s o v e r t h e w a t e r s u r f a c e . The d i m e n s i o n s and p o s i t i o n s o f t h e e l e c t r o d e s v a r i e d w i t h t h e mode e x c i t e d and a r e s u m m a r i z e d i n t h e t a b l e b e l o w : T A B L E I : WAVE E X C I T A T I O N ELECTRODES MODE NUMBER NUMBER OF ELECTRODES LENGTH P O S I T I O N * 1 1 L / 2 L / 2 2 1 L / 2 L / 4 4 2 + L / 4 L / 8 8 4 + L / 8 L / 1 6 * D i s t a n c e f r o m r i g h t ' end o f t a n k t o t h e edge o f t h e f i r s t e l e c t r o d e . t E l e c t r o d e s p a c i n g = e l e c t r o d e l e n g t h . The w i d t h o f t h e e l e c t r o d e s a r e a b o u t e q u a l t o t h e w i d t h o f t h e p a r t i c u l a r t a n k u s e d . I n a l l c a s e s , t h e e l e c t r o d e s w e r e p o l i s h e d and t h e e d g e s r o u n d e d t o m i n i m i z e s p a r k i n g t o t h e g r o u n d e d w a t e r . To a d j u s t t h e d i s t a n c e b e t w e e n t h e e l e c t r o d e and t h e w a t e r s u r f a c e , t h e f o r m e r was s u p p o r t e d by a v e r t i c a l t h r e a d e d r o d ( F i g u r e 2 ) . The p l a n e o f t h e e l e c t r o d e c o u l d be a d j u s t e d by l e v e l i n g s c r e w s ( F i g u r e 6 ) . -19-THREADED ROD SUPPORT e TO HV SUPPLY 2 0 M n 3 e ' o / / o 0 ' o o ' o >^ o 1 I I H H k / L ^ LEVELING \u00E2\u0080\u00A2SCREWS \u00E2\u0080\u00A2BUBBLE \u00E2\u0080\u00A2ELECTRODE L / 8 ; L/4 3L/4 WATER SURFACE L / 8 K F i g u r e 6 . The e l e c t r o d e u s e d t o h a r m o n i c s t a n d i n g s u r f a c e wave. g e n e r a t e t h e f o u r t h - 2 0 -The a l t e r n a t i n g h i g h v o l t a g e w a v e f o r m i s p r o d u c e d b y a p r o g r a m m a b l e d i g i t a l w a v e f o r m g e n e r a t o r t h e d e t a i l s o f w h i c h a r e g i v e n i n t h e A p p e n d i x . The w a v e f o r m g e n e r a t o r s t o r e s one h a l f o f t h e d e s i r e d w a v e f o r m w h i c h i s e n t e r e d m a n u a l l y on two s t a n d a r d 4 x 16 RAM memory I . C . c h i p s . An u p / d o w n c o u n t e r t o g e t h e r w i t h a l o g i c c i r c u i t t h a t p e r i o d i c a l l y r e v e r s e s t h e d i r e c t i o n o f t h e c o u n t s w e e p s t h e 16 w o r d s o f t h e m e m o r y . The o u t p u t o f t h e memory i s t h e n c o n v e r t e d t o an a n a l o g s i g n a l w h i c h i s a m p l i f i e d b y an o p e r a t i o n a l a m p l i f i e r ( K e p c o ) t o p r o d u c e a p e r i o d i c h i g h v o l t a g e a t t h e e l e c t r o d e s a b o v e t h e w a t e r . A 20Mfl r e s i s t o r i s u s e d b e t w e e n t h e e l e c t r o d e s and t h e a m p l i f i e r f o r p r o t e c t i o n o f t h e c i r c u i t r y i n t h e e v e n t o f s p a r k i n g . D . T h e E l e c t r i c a l S y s t e m The d i s p l a c e m e n t n o f t h e w a t e r - a i r i n t e r f a c e e x c i t e d by t h e e l e c t r o s t a t i c f i e l d i s o f t h e f o r m : n = A c o s ( u t + o>)cos (kx ) w h e r e $ i s t h e p h a s e o f t h e w a v e . I f a t t h e t i m e o f r e l e a s e o f t h e t a n k ( t = 0) t h e p h a s e i s 0 o r IT so t h a t n i s a t a max imum, t h e n t h e s u b s e q u e n t u n s t a b l e wave g r o w s a t t h e r a t e g i v e n by t h e f i r s t o r d e r t h e o r y - 2 1 -d e s c r i b e d i n t h e I n t r o d u c t i o n . The r e l e a s e o f t h e t a n k a t t h i s f i x e d p h a s e ' has t h e p a r t i c u l a r a d v a n t a g e t h a t t h e r e s u l t i n g i n s t a b i l i t i e s a r e r e p r o d u c i b l e . To r e l e a s e t h e t a n k a t t h e r i g h t i n s t a n t o f t i m e , a c o m p a r a t o r t i m i n g c i r c u i t i s u s e d . The e l e c t r c a l s y s t e m i s shown i n F i g u r e 7 . The t i m i n g c i r c u i t t r i g g e r s on t h e o u t p u t v o l t a g e o f t h e d i g i t a l w a v e f o r m g e n e r a t o r when i t s o u t p u t v o l t a g e m a t c h e s t h a t p r e s e t on t h e c o m p a r a t o r . A t i m e d e l a y i s t h e n a d j u s t e d so t h a t t h e t a n k r e l e a s e p u l s e o p e r a t e s a t t h e i n s t a n t t h e s t a n d i n g wave i s a t maximum a m p l i t u d e . T h i s done by h a v i n g two F o u r i e r a n a l y z i n g p l a t e s i m m e r s e d i n t h e w a t e r a t o p p o s i t e s i d e s o f t h e t a n k . The p l a t e s a r e c o n n e c t e d t o a s e r i e s r e s i s t a n c e o f 2Kft and a 50 V d . c . p o w e r s u p p l y . The v o l t a g e a c r o s s t h e 200 ohms o f a c i d i f i e d w a t e r i s o b s e r v e d on a s t o r a g e o s c i l l o s c o p e . S i n c e t h e r e s i s t a n c e v a r i e s i n v e r s e l y w i t h t h e c r o s s s e c t i o n a l a r e a o f w a t e r b e t w e e n t h e p l a t e s , an i n c r e a s e o f a r e a due t o wave m o t i o n r e s u l t s i n a d e c r e a s e i n v o l t a g e . T h u s , t h e d e l a y on t h e t i m i n g c i r c u i t was a d j u s t e d so t h a t w h e n -e v e r t h e t a n k r e l e a s e p u l s e f i r e d , t h e m i n i m u m o f t h e s i n u s o i d a l s i g n a l was d i s p l a y e d a t t = 0 . The c o m p a r a t o r t i m i n g c i r c u i t p r o d u c e s two s i m u l t a n e o u s p u l s e s , one o f w h i c h i s u s e d t o r e l e a s e t h e t a n k and t h e o t h e r t o o p e n t h e s o l e n o i d v a l v e and CLOCK OSCILLATOR WAVEFORM GENERATOR COMPARATOR TIMING CIRCUIT KEPCO OP AMP TANK RELEASE PULSE 500 msec. TO SCOPE EXTERNAL TRIGGER SOLENOID VALVE PULSE 50-500 msec A C RELAY 5MQ 1 H.V. ELECTRODE RELAY RELEASE MECHANISM SOLENOID AIR VALVE F i g u r e 7. B l o c k d i a g r a m o f t h e e l e c t r i c a l s y s t e m . - 2 3 -and s u p p l y t h e p i s t o n w i t h c o m p r e s s e d a i r . The l a t t e r p u l s e h a s i t s w i d t h a d j u s t a b l e i n t h e r a n g e 50 - 500 msec so t h a t t h e s o l e n o i d v a l v e w i l l be o p e n f o r t h e e n t i r e d u r a t i o n o f t h e m o t i o n o f t h e t a n k . The d i g i t a l p u l s e s a r e u s e d t o s w i t c h t h e m a i n s v o l t a g e t o t h e s o l e n o i d s b y means o f s o l i d s t a t e r e l a y u n i t s . E . The I . C . C a p a c i t a n c e M e a s u r i n g C i r c u i t E e s i d e s u s i n g an i m p e d a n c e b r i d g e t o m e a s u r e c h a n g e s i n c a p a c i t a n c e w i t h t i m e , a s i m p l e c a p a c i t a n c e m e a s u r i n g c i r c u i t was a l s o d e v e l o p e d and u t i l i z e d ( P o p i l and C u r z o n 1979) . The c i r c u i t ( F i g u r e 8) e s s e n t i a l l y c o n s i s t s o f one i n t e g r a t e d c i r c u i t c h i p t h a t d e t e r m i n e s t h e u n k n o w n c a p a c i t a n c e by means o f a f r e q u e n c y t o v o l t a g e c o n v e r s i o n . The b e s t s e n s i t i v i t y o f t h e c i r c u i t was f o u n d t o be a t an i n p u t f r e q u e n c y o f 3kHz ' . F o r g o o d l i n e a r i t y , C^ was s e t a t 1 yF and a t 500 Kfi (10 t u r n H e l i p o t ) . R j was a d j u s t e d s o t h a t t h e o u t p u t was 4 v o l t s when t h e t a n k was f i l l e d t o d e p t h o f 9 cm f r o m t h e b o t t o m . L i n e a r i t y i s a c h i e v e d o n l y i f t h e c o n d u c t i v i t y o f . t h e w a t e r i s i n c r e a s e d by a d d i n g 1 ml o f 18 m o l a r s u l f u r i c a c i d t o 2 l i t e r s o f w a t e r ( F i g u r e 9) . U n d e r t h e s e c o n d i t i o n s , t h e s e n s i t i v i t y o f t h e c i r c u i t i s 38 mV/mm o f w a t e r . The c a l i b r a t i o n was done s i m p l y by n o t i n g t h e o u t p u t v o l t a g e o f t h e c i r c u i t as a f u n c t i o n o f t h e w a t e r d e p t h . 15V 1 UNKNOWN CAPACITY (TO TANK) 14 13 12 11 10 9 8 LM2907N 1 2 3 4 5 6 7 100 K 100K rti Ri 500K 10K 1K5 1K5 OUTPUT \u00E2\u0080\u00A2e CA3140 860 Figure 8. The I.C. capacitance measuring c i r c u i t . -25-HEIGHT c.m. Figure 9. Response of the I .e . capacitance measuring c i r c u i t to changes in water depth. - 2 6 -CHAPTER I I I E X P E R I M E N T A L RESULTS A . C a p a c i t a n c e M e a s u r e m e n t s D u p l i c a t i o n o f t h e o r i g i n a l e x p e r i m e n t a l s e t - u p d e v i s e d by L a n g i l l e (1970) y i e l d e d r e s u l t s t h a t a p p e a r e d a t f i r s t i n s p e c t i o n t o be i n a c c o r d a n c e w i t h t h e g e n e r a l t h e o r y . The a r r a n g e m e n t u s e d d i s t i l l e d w t e r as t h e d i e l e c t r i c b e t w e e n two V a r a t h a n e v a r n i s h c o v e r e d F o u r i e r a n a l y z i n g p l a t e s . As shown i n F i g u r e 1 0 , t h e o u t p u t s i g n a l s f r o m t h e i m p e d a n c e b r i d g e a p p e a r t o be e x p o n e n t i a l w i t h t i m e and a r e l a r g e r f o r r u n s w i t h an i n i t i a l l y p e r t u r b e d s u r f a c e t h a n f o r an u n p e r t u r b e d s u r f a c e . B e c a u s e o f s p u r i o u s e l e c t r i c a l n o i s e p r o b l e m s , I r e p l a c e d t h e t h e e l e c t r o s t a t i c wave g e n e r a t o r by a p u l s e d a i r j e t w h i c h e x c i t e d t h e s e c o n d h a r m o n i c (q = 2) wave as t h e i n i t i a l p e r t u r b a t i o n . H o w e v e r , t h i s m e t h o d o f wave e x c i t a t i o n r e s u l t s i n w a v e s w h i c h a r e r a t h e r r i c h i n h a r m o n i c c o n t e n t as F i g u r e 11 s h o w s . P l o t t i n g t h e r e s u l t s f o r e a c h mode v e r s u s c o s h (UK t) w h e r e UK i s t h e a n g u l a r f r e q u e n c y o f t h e p a r t i c u l a r mode a n a l y z e d , y i e l d e d c u r v e s t h a t do n o t d e v i a t e s i g n i f i c a n t l y f r o m s t a i g h t l i n e s a t e a r l y t i m e s ( F i g u r e 12) . To e l i m i n a t e t h e n o i s e p r o b l e m s c a u s e d by t h e e l e c t r i c f i e l d o f t h e wave g e n e r a t o r , I r e b u i l t L a n g i l l e ' s a p p a r a t u s u s i n g t h e w a t e r as a c a p a c i t o r p l a t e and a -27-F i g u r e 10. O u t p u t o f t h e i m p e d a n c e b r i d g e a a f u n c t i o n o f t i n e . a) G r o w t h o f a p e r t u r b e d s u r f a c e . b) G r o w t h o f an u n p e r t u r b e d s u r f a c e G a i n i s 0 . l v / d i v , t i m e b a s e i s 20 m s e c / d i v . HISTOGRAM SHOWING THE AMPLITUDES OF THE FIVE HARMONICS DURING EXCITATION F i g u r e 1 1 . S p e c t r u m o f t h e s e c o n d h a r m o n i c wave g e n e r a t e d by a p u l s e d a i r j ' e t . The h o r i z o n t a l a x i s i s number o f t h e q t h h a r m o n i c o f t h e F o u r i e r a n a l y z i n g p l a t e u s e d t o d e t e c t t h e w a v e s . -29-Figure 12. Results using the water as a d i e l e c t r i c . The output signals obtained by using different harmonic plates are plotted versus coshCuKt) where x ~11\u00E2\u0080\u00A2*\u00C2\u00BB5# - 3 0 -0.077mm t h i c k p o l y e t h y l e n e s h e e t as t h e d i e l e c t r i c as d e s c r i b e d i n C h a p t e r I I p a r t B. I n i t i a l l y , a l l e x p e r i m e n t s w e r e p e r f o r m e d u s i n g t h e f i r s t h a r m o n i c s t a n d i n g w a v e s as t h e i n i t i a l p e r t u r b a t i o n s i n c e t h e a i m o f t h e e x p e r i m e n t was t o f o l l o w t h e d e v e l o p m e n t o f h i g h e r o r d e r modes due t o t h e d i s t o r t i o n o f t h e g r o w t h o f a l o w e r o r d e r mode . The i m p e d a n c e b r i d g e u s e d i n t h e s e e x p e r i m e n t s i s t h e G e n e r a l R a d i o 1 5 0 0 - A . F o r o p t i m u m r e s u l t s , t h e s e n s i t i v i t y c o n t r o l o f t h e b r i d g e was s e t a t h e 9 o ' c l o c k p o s i t i o n , t h e CRL s e l e c t o r s w i t c h was s e t a t \" C s \" and t h e DQ d i a l i s t y p i c a l l y s e t a t < 0 . 1 (on t h e \" l o w D \" s c a l e ) f o r b a l a n c e . When r u n s w e r e made u s i n g t h e i m p e d a n c e b r i d g e , i t was f o u n d t h a t t h e g r o w t h s w e r e no l o n g e r e x p o n e n t i a l a s t h e y w e r e i n L a n g i l l e ' s s e t - u p w h e r e t h e w a t e r was u s e d as t h e d i e l e c t r i c . A l s o , i t was s e e n t h a t t h e e f f e c t o f \" p h a s i n g \" ( i . e . , r e l e a s i n g t h e t a n k when $ = 0 , IT o r i r / 2 i n n = A c o s (wt \u00E2\u0080\u00A2 )cos(kx)) , h a s n e g l i g i b l e e f f e c t on t h e s e g r o w t h s as i n F i g u r e 1 3 . T h e s e r e s u l t s w e r e c o n f i r m e d when t h e I . C . c a p a c i t a n c e m e a s u r i n g c i r c u i t was u s e d i n p l a c e o f t h e i m p e d a n c e b r i d g e . T h e s t r i k i n g f e a t u r e o f t h e s e r e s u l t s i s t h e f a c t t h a t t h a t t h e c a p a c i t a n c e d e c r e a s e s i n a n o n - e x p o n e n t i a l f a s h i o n i n a l l c a s e s i r r e s p e c t i v e o f t h e p h a s i n g o f t h e i n i t i a l p e r t u r b a t i o n , ( F i g u r e s 14a> and 14b)) . -31-a) b) c) F i g u r e 1 3 . I n p e d a n c e b r i d g e o u t p u t v e r s u s t i m e s h o w i n g t h a t t h e g r o w t h r a t e o f t h e i n s t a b i l i t y i s i n d e p e n d e n t o f t h e p h a s e , * . a ) P h a s e = 0 b ) * - T T / 2 C) = TT C a i n i s O . l v / d i v T i m e b a s e i s 20 m s e c / d i v . -32-time dec reas ing capac i tance F i g u r e 1 4 . C a p a c i t a n c e m e a s u r e m e n t s s h o w i n g t h e d e c r e a s e i n mean l e v e l c a u s e d b y t h e g r o w t h of i n s t a b i l i t i e s . a ) I . C . c i r c u i t o u t p u t , g a i n i s 0 . 2 v / d i v , t i m e b a s e i s 20 m s e c / d i v . b ) I . C . c i r c u i t o u t p u t , g a i n i s 0 . 5 v / d i v t i m e b a s e i s 0 . 2 s e c / d i v . c ) C a p a c i t a n c e b r i d g e o u t p u t , g a i n i s O . l v / d i v a n d t h e t i m e b a s e i s 2 0 m s e c / d i v . -33-The d i r e c t i o n o f t h e g r o w t h was l a t e r v e r i f i e d u s i n g t h e i m p e d a n c e b r i d g e . T h i s was done by s e t t i n g t h e r e f e r e n c e c a p a c i t a n c e on t h e i m p e d a n c e b r i d g e b e l o w t h a t o f t h e t a n k w i t h t h e u n p e r t u r b e d w a t e r i n i t . T h i s o f c o u r s e , r e s u l t s i n an o u t p u t s i g n a l t h a t i s a p p r e c i a b l y -g r e a t e r t h a n t h e n u l l b a l a n c e s i g n a l . When t h e t a n k was a c c e l e r a t e d d o w n w a r d , t h e o u t p u t i s s e e n t o d e c r e a s e t o t h e n u l l s i g n a l l e v e l o n c e t h e c a p a c i t a n c e o f t h e t a n k ha s d i m i n i s h e d t o t h a t s e t on t h e i m p e d a n c e b r i d g e . A f u r t h e r r e d u c t i o n i n c a p a c i t a n c e t h e n c a u s e s t h e o u t p u t t o g row w i t h t i m e , ( F i g u r e 14c ) ) . The r e s u l t s f r o m t h e i m p e d a n c e b r i d g e and I . C . c i r c u i t s u g g e s t t h a t t h e r e i s a c h a n g e i n t h e mean l e v e l o f t h e c o n t a c t o f t h e w a t e r w i t h t h e w a l l s o f t h e w a t e r t a n k . B e c a u s e t h e c a p a c i t a n c e d e c r e a s e s w i t h t i m e , i t w o u l d t h e n seem t h a t w a t e r i s s e p a r a t i n g f r o m t h e w a l l s and h e n c e t h e mean l e v e l d e c r e a s e s . B. R e s i s t a n c e M e a s u r e m e n t s O b s e r v i n g t h e t v o l t a g e s i g n a l b e t w e e n two F o u r i e r a n a l y z i n g p l a t e s i m m e r s e d i n i o n i z e d w a t e r y i e l d e d r a t h e r s u r p r i s i n g r e s u l t s when t h e t a n k was a c c e l e r a t e d d o w n w a r d . S i n c e t h e a r e a o f c o n t a c t w i t h t h e w a l l s i s e x p e c t e d t o i n c r e a s e due t o t h e g r o w t h o f t h e i n s t a b i l i t y , t h e v o l t a g e b e t w e e n t h e p l a t e s s h o u l d c o r r e s p o n d i n g l y d e c r e a s e s i n c e i t i s i n v e r s e l y p r o p o r t i o n a l t o t h e a r e a . H o w e v e r , - 3 4 -as s e e n i n F i g u r e 1 5 , t h e v o l t a g e i n c r e a s e s i n an e x p o n e n t i a l f a s h i o n . U s i n g t h e 1 s t h a r m o n i c wave as t h e i n i t i a l p e r t u r b a t i o n , s e m i - l o g a r i t h m i c p l o t s o f t h e a m p l i t u d e v e r s u s t i m e y i e l d e d l i n e a r g r a p h s w h o s e s l o p e s t h e n g a v e t h e g r o w t h r a t e . I t was f o u n d t h a t u s i n g t h e v a r i o u s h a r m o n i c p l a t e s p r o d u c e d a p p r o x i m a t e l y t h e same g r o w t h r a t e . The r e s u l t s a r e p r e s e n t e d b e l o w : TABLE III : RESISTANCE FOURIER ANALYSIS HARMONIC NUMBER (q) 1 2 3 4 5 p l a i n c o p p e r s h e e t GROWTH RATE n ( s e c ) 2 3 . 5 \u00C2\u00B1 13% 2 3 . 4 \u00C2\u00B1 14% 2 3 . 4 \u00C2\u00B1 15% 2 5 . 4 \u00C2\u00B1 17% 1 9 . 8 \u00C2\u00B1 18% 2 0 . 7 \u00C2\u00B1 11% The g r o w t h o f t h e v o l t a g e s i g n a l s i n t h e o p p o s i t e d i r e c t i o n o f t h a t e x p e c t e d c a n be e x p l a i n e d b y a s s u m i n g o n c e a g a i n t h a t t h e a r e a o f c o n t a c t o f t h e w a t e r w i t h t h e w a l l s d e c r e a s e s l i n e a r l y w i t h t i m e a s t h e c a p a c i t a n c e o b s e r v a -t i o n s i n d i c a t e . S i n c e t h e i n v e r s e o f a l i n e a r l y d e c r e a s i n g f u n c t i o n i s a r i s i n g h y p e r b o l a , t h e v o l t a g e s i g n a l s w h i c h a r e p r o p o r t i o n a l t o t h e i n v e r s e o f t h e c o n t a c t a r e a t h e n a p p e a r as p o s i t i v e l y g o i n g e x p o n e n t i a l s . - 3 5 -F i g u r e 1 5 . The r e s i s t a n c e F o u r i e r a n a l y s i s . V o l t a g e a c r o s s t h e t a n k as a f u n c t i o n o f t i m e . G a i n i s 0 . 0 5 v / d i v and t h e t i m e b a s e i s 5 0 m s e c / d i v . -36-The v o l t a g e s i g n a l s o b s e r v e d h e r e c a n be u s e d t o i n t e r p r e t t h e e x p o n e n t i a l g r o w t h s t h a t a r e o b s e r v e d when t h e w a t e r i s u s e d as t h e d i e l e c t r i c b e t w e e n two F o u r i e r a n a l y z i n g p l a t e s . I n t h i s c o n f i g u -r a t i o n , ( C u r z o n and L a n g i l l e 1 9 7 2 ) , t h e r e i s c o n s i d e r -a b l e l e a k a g e o f c u r r e n t a c r o s s t h e t a n k s i n c e t h e w a t e r and t h e v a r n i s h on t h e c a p a c i t o r p l a t e s a r e i n e f f e c t i v e as i n s u l a t o r s . T h u s , i t i s t h e r e s i s t a n c e b e t w e e n t h e p l a t e s t h a t i s a c t u a l l y b e i n g m e a s u r e d b y t h e i m p e d a n c e b r i d g e and h e n c e t h e e x p o n e n t i a l g r o w t h s a r e o b t a i n e d . L a n g i l l e (1970) n o t e s t h a t t h e r e i s d i f f i c u l t y i n m a i n t a i n i n g t h e n u l l b a l a n c e s i g n a l c o n s t a n t a n d l o w f r o m one e x p e r i m e n t a l r u n t o t h e n e x t . H o w e v e r , when t h e t a n k i s u s e d as a p r o p e r c a p a c i t o r ( see F i g u r e 4) , t h e r e i s no p r o b l e m t o b a l a n c e t h e b r i d g e t o o b t a i n a n u l l s i g n a l o f l e s s t h a n 1 mV r . m . s . I n s u m m a r y , t h e r e s i s t a n c e and c a p a c i t a n c e m e a s u r e m e n t s y i e l d u n e x p e c t e d r e s u l t s . The f a c t t h a t t h e r e s i s t a n c e i n c r e a s e s and t h e c a p a c i t a n c e d e c r e a s e s as t h e t a n k i s a c c e l e r a t e d i n d i c a t e s t h a t t h e r e i s a c h a n g e i n t h e mean l e v e l o f t h e w a t e r t h a t i s i n c o n t a c t w i t h t h e w a l l s . T h u s , t h e m e a s u r e m e n t s do n o t g i v e an i n d i c a t i o n o f t h e i n s t a b i l i t y b u t d e s c r i b e t h e s i t u a t i o n a t t h e w a l l s o f t h e c o n t a i n e r . \u00C2\u00AB -37-C . \" C a v i t a t i o n \" E f f e c t s I r r e s p e c t i v e o f t h e mode i n i t i a l l y on t h e s u r f a c e , t h e s i g n a l s f r o m t h e i m p e d a n c e b r i d g e a l w a y s i n d i c a t e a l i n e a r g r o w t h due t o a d e c r e a s e i n c a p a c i t a n c e . S i n c e t h e c a p a c i t a n c e m e a s u r e d i s a v e r a g e d o v e r t h e a r e a o f t h e p l a t e C = / ( e /d ) dA t h e o n l y way t h a t t h e c a p a c i t a n c e c a n d e c r e a s e i s i f d , t h e s e p a r a t i o n b e t w e e n t h e two c a p a c i t o r p l a t e s i n c r e a s e s . T h i s means t h a t i f t h e w a t e r p u l l s away f r o m t h e w a l l s so t h a t a i r g e t s b e t w e e n t h e w a t e r and t h e p o l y e t h y l e n e s h e e t , d w i l l i n c r e a s e and t h e c a p a c i t a n c e w i l l be s e e n t o d e c r e a s e . I f t h e t h i c k n e s s o f t h e a i r f i l m i s l a r g e c o m p a r e d t o t h e t h i c k n e s s o f t h e d i e l e c t r i c s h e e t , t h e n C = / ( e /d ) dA\u00C2\u00BB A ' w h e r e A ' i s t h e a r e a o f t h e w a t e r i n c o n t a c t w i t h t h e d i e l e c t r i c s h e e t . I f t h e o t h e r p l a t e o f t h e c a p a c i t o r i s a m e t a l s h e e t , t h e n t h e a b o v e e x p r e s s i o n i n d i c a t e s t h a t C w i l l be p r o p o r t i o n a l t o t h e a v e r a g e depth~D~ o f t h e w a t e r w h i c h t o u c h e s t h e d i e l e c t r i c o f t h e c a p a c i t o r . F o r g i v e n a p p l i e d v i r t u a l a c c e l e r a t i o n , i t was o b s e r v e d t h a t d C / d t ( i . e . dD\"/dt) i s c o n s t a n t . The p o s s i b i l i t y t h a t t h i s phenomenon i s t h e same as t h e f o r m a t i o n o f t h e b u b b l e s o f t h e i n s t a b i l i t y c a n be e l i m i n a t e d s i n c e -38-o t h e r s (Emmons e t a l 1959) , h a v e f o u n d t h a t t h e b u b b l e v e l o c i t y i s p r o p o r t i o n a l t o t h e s q u a r e r o o t o f t h e v i r t u a l a c c e l e r a t i o n . F o r c o n v e n i e n c e , t h i s a p p a r e n t r e p l a c e n e n t o f w a t e r by an a i r f i l m w i l l h e n c e f o r t h be c a l l e d \" c a v i t a t i o n \" . To d e t e r m i n e how t h e c a v i t a t i o n p h e n o m e n o n i s a f f e c t e d by t h e i n i t i a l s h a p e o f t h e s u r f a c e , e x p e r i -m e n t s w e r e p e r f o r m e d w h e r e an a d j u s t a b l e P l e x i g l a s b a f f l e was p l a c e d i n t h e t a n k . The b a f f l e was v e r t i c a l , w i t h i t s p l a n e p a r a l l e l t o t h e c a p a c i t o r p l a t e . The d i s t a n c e o f t h e w a t e r b e t w e e n t h e c a p a c i t o r p l a t e and t h e b a f f l e was v a r i e d f r o m 0 . 5 cm t o 1 0 . 5 cm. A t 0 . 5 cm t h e r e i s c o n s i d e r a b l e c u r v a t u r e o f t h e w a t e r s u r f a c e due t o t h e m e n i s c u s . I n v e s t i g a t i o n s w e r e p e r f o r m e d w i t h no e x c i t e d w a v e s on t h e i n i t i a l w a t e r s u r f a c e . The c a v i t a t i o n r a t e s w e r e d e t e r m i n e d s i m p l y by m e a s u r i n g t h e s l o p e s o b t a i n e d f r o m t h e b r i d g e s i g n a l s ( F i g u r e 16) . The s e n s i t i v i t y o f t h e c a p a c i t a n c e b r i d g e was e s t a b l i s h e d by s l o w l y f i l l i n g t h e t a n k and o b s e r v i n g t h e o u t p u t v o l t a g e as a f u n c t i o n o f w a t e r d e p t h . The s e n s i t i v i t y was 6 . 6 7 c m / V p \u00E2\u0080\u0094 p and was o b s e r v e d t o be l i n e a r t o a n e t c h a n g e o f 9cm o f w a t e r p r o v i d e d t h a t t h e s e n s i t i v i t y c o n t r o l on t h e b r i d g e i s m a i n t a i n e d a t one q u a r t e r o f a t u r n o r l e s s ( F i g u r e 17) . ^ A t y p i c a l g r a p h o f t h e m e a s u r e d c a v i t a t i o n - 3 9 -F i g u r e 16. O s c i l l o g r a p h s h o w i n g t h e l i n e a r d e c r e a s e o f t a n k c a p a c i t a n c e w i t h t i m e . T h e t a n k w i d t h w a s 2 cm i n t h i s i n s t a n c e and t h e v i r t u a l a c c e l e r a t i o n 1 . 0 g ' s . G a i n i s 0 . 2 v / d i v , t i m e b a s e 2 0 m s e c / d i v , T h e c a l i b r a t i o n i s 6.67cm w a t e r / V P\"P - 4 0 -F i g u r e 17. C a l i b r a t i o n o f t h e GR b r i d g e . W a t e r h e i g h t i s m e a s u r e d e q u i l i b r i u m l e v e l . -1500A i m p e d a n c e upwards f r o m t h e - 4 1 -v e l o c i t y (dtT/dt) as a f u n c t i o n o f v i r t u a l a c c e l e r a t i o n i s shown i n F i g u r e 1 8 . The e r r o r i n c r e a s e s f o r t h e l a r g e r a c c e l e r a t i o n s b e c a u s e t h e g r o w t h i s o b s e r v e d f o r c o n s i d e r a b l y s h o r t e r t i m e s and t h e r e i s a l s o a n i n c r e a s e i n s p u r i o u s n o i s e s i g n a l s i n t h e e n v e l o p e o f t h e b r i d g e o u t p u t . The r e s u l t s f o r t h e i n v e s t i g a t i o n o f t h e e f f e c t o f t h e t a n k w i d t h a r e p r e s e n t e d b e l o w : T A B L E I I I : C A V I T A T I O N RATE AND TANK WIDTH TANK WIDTH (cm) RATE ( c m / s e c - g ) 0 . 5 37 1 30 2 38 3 32 4 39 6 27 8 34 1 0 . 5 32 As s e e n f r o m t a b l e I I I , t h e r e i s no o b s e r v a b l e d e p e n d e n c e o f t h e c a v i t a t i o n v e l o c i t y on t h e t a n k w i d t h ; t h e n e t r e s u l t i s t h a t t h e v e l o c i t y o f t h e d e c r e a s i n g mean l e v e l i s r e l a t e d t o t h e v i r t u a l a c c e l e r a t i o n b y : dfT/dt = v = ( 3 4 c m / s e c - s \u00C2\u00B1 4 c m / s e c - g ) x g w h e r e g i s t h e v i r t u a l a c c e l e r a t i o n o r e x c e s s a c c e l e r a t i o n -42-F i g u r c 18. \" C a v i t a t i o n \" r a t e f o r a t a n k w i d t h o f 3 c m . - 4 3 -i n t e r m s o f f r e e f a l l a c c e l e r a t i o n i . e . , g = a g Q , 2 g Q = 980 c m / s e c , a > 0 . D i m e n s i o n a l i n s p e c t i o n o f t h e p r o p o r t i o n a l i t y b e t w e e n t h e v e l o c i t y and t h e a p p l i e d a c c e l e r a t i o n l e a d s t o s c a l i n g l a w s , f r o m w h i c h t h e d e p e n d e n c e o f t h e c a v i t a t i o n r a t e on v a r i o u s p h y s i c a l q u a n t i t i e s may be d e d u c e d . I n p a r t i c u l a r , when 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 was c o n s i d e r e d as b e i n g a f u n c t i o n o f t h e d e n s i t y and v i s c o s i t y , t h e 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 t h e p l a u s i b l e r e s u l t t h a t t h e c a v i t a t i o n r a t e s h o u l d be i n v e r s e l y p r o p o r t i o n a l t o t h e v i s c o s i t y : dD~ p g L 2 dT = v = \u00E2\u0080\u0094 \u00C2\u00B0 -w h e r e L Q i s a s c a l i n g l e n g t h and n t h e v i s c o s i t y . To c h e c k t h e v i s c o s i t y e f f e c t , a q u e o u s s u g a r s o l u t i o n s we re p r e p a r e d o f d i f f e r e n t c o n c e n t r a t i o n s r a n g i n g f r o m 20 t o 60 p e r c e n t s u g a r by w e i g h t . The v i s c o s i t i e s o f t h e s e s o l u t i o n s a t 25 C i s g i v e n by t h e H a n d b o o k o f C h e m i s t r y and P h y s i c s , 4 3 r d e d i t i o n and a r e l i s t e d i n t a b l e IV t o g e t h e r 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 : - 4 4 TABLE IV : EFFECT OF V I SCOS ITY ON THE CAV ITAT ION RATE PERCENT SUGAR (by w e i g h t ) 20 30 40 50 60 V I SCOS ITY ( c P ) 1 .695 2 . 735 5. 164 1 2 . 4 0 4 4 . 0 3 CAV ITAT ION RATE ( c m / s e c - g ) 32 31 4 6 35 77 From t h e t a b l e i t i s i m m e d i a t e l y a p p a r e n t t h a t t h e c a v i t a t i o n r a t e d o e s n o t v a r y i n v e r s e l y w i t h t h e v i s c o s i t y as s u p p o s e d . The r a t e s a b o v e y i e l d a mean v a l u e o f 44 c m / s e c - g w i t h a s t a n d a r d d e v i a t i o n o f 17 c m / s e c - g so t h a t t h e r a t e s o f t h e v i s c o u s s o l u t i o n s a r e c o m p a r a b l e t o t h o s e f o r p u r e w a t e r . The l a r g e s c a t t e r i n t h e d a t a f o r t h e s u g a r s o l u t i o n s i s due t o t h e e r r a t i c i r r e g u l a r i t i e s o f t h e b r i d g e s i g n a l s t h a t a p p e a r a t a c c e l e r a t i o n s g r e a t e r t h a n 1.5 g . The r a t e s we re d e t e r m i n e d by d r a w i n g t h e b e s t f i t l i n e t h r o u g h t h e p o i n t s as i n F i g u r e 19 . T h i s n e c e s s i t a t e d t h e e x c l u s i o n o f t h e o r i g i n a t 0 a c c e l e r a t i o n , N o n e t h e l e s s , e v e n i f t h e o r i g i n i s i n c l u d e d i n t h e d t e r m i n a t i o n o f t h e c a v i t a t i o n r a t e s , t h e c o n c l u s i o n s t i l l r e m a i n s t h a t t h e r e i s no i n v e r s e v a r i a t i o n w i t h v i s c o s i t y . - 4 5 -VIRTUAL ACCERATION (g's) 0.5 1.5 2.5 F i g u r e 19. C a v i t a t i o n r a t e v e r s u s t h e v i r t u a l a c c e l e r a t i o n f o r a 30% s u g a r s o l u t i o n . - 4 6 -Th e r e s p o n s e o f t h e b r i d g e t o t r a n s i e n t c h a n g e s i n c a p a c i t a n c e was c h e c k e d t o e n s u r e t h a t t h e l i n e a r g r o w t h s o b t a i n e d i n t h e r u n s a r e t r u l y r e p r e s e n t a t i v e o f t h e p h y s i c a l s i t u a t i o n and a r e n o t due t o some i n a b i l i t y o f t h e b r i d g e t o f o l l o w f a s t c h a n g e s i n c a p a c i t a n c e . One c a p a c i t o r p l a t e was f i x e d t o t h e t a n k s u p p o r t f r a m e and was h e l d a g a i n s t t h e d i e l e c t r i c s h e e t and t h e o t h e r c a p a c i t o r p l a t e i n t h e w a t e r t a n k . I n t h i s s e t - u p , t h e maximum c a p a c i t a n c e was 320 pF when t h e two c a p a c i t o r p l a t e s f u l l y o v e r l a p p e d one a n o t h e r . By d r o p p i n g t h e t a n k , t h e c a p a c i t a n c e was r e d u c e d t o 250 pF when t h e two p l a t e s no l o n g e r had a common a r e a o f o v e r l a p . C a l i b r a t i o n o f t h e s i g n a l o u t p u t f o r t h i s t e s t gave a c o n v e r s i o n f a c t o r o f 11 mV/cm b e t w e e n t h e o u t p u t a m p l i t u d e and t h e r e l a t i v e v e r t i c a l d i s p l a c e m e n t o f t h e c a p a c i t o r p l a t e s . The t a n k was t h e n a l l o w e d t o f a l l , so t h a t t h e c a p a c i t a n c e o f t h e two p l a t e s c o u l d be o b s e r v e d as a f u n c t i o n o f t i m e . P l o t t i n g t h e a m p l i t u d e a g a i n s t t h e t i m e s q u a r e d y i e l d a s t a i g h t l i n e t o an e x c e l l e n t a p p r o x i m a t i o n ( F i g u r e 2 0 ) . T h e s l o p e y i e l d s 2 t h e v a l u e o f 547 cm/ sec as t h e e f f e c t i v e a c c e l e r a t i o n . T h u s , one may s a f e l y c o n c l u d e t h a t t h e b r i d g e d o e s i n d e e d a c c u r a t e l y f o l l o w a b r u p t c h a n g e s i n c a p a c i t a n c e . I n v i e w o f t h e u n e x p e c t e d r e s u l t s f r o m t h e e l e c t r i c a l m e a s u r e m e n t s , I d e c i d e d t o p h o t o g r a p h t h e -47-F i g u r e 2 0 . T e s t o f t h e i m p e d a n c e b r i d g e to t r a n s i e n t change s i n c a p a c i t a n c e . - 4 8 -f l u i d m o t i o n i n v a r i o u s ways so as t o d i s c o v e r wha t s o r t o f i n s t a b i l i t i e s d i d o c c u r and i f an a i r f i l m e x i s t s b e t w e e n t h e c a p a c i t o r p l a t e s and t h e w a t e r i n t h e t a n k . D. P h o t o g r a p h y o f t h e F l u i d M o t i o n To d e t e r m i n e v i s u a l l y t h e p r e s e n c e o f i n s t a b i l i -t i e s , v a r i o u s modes w e r e e x c i t e d e l e c t r o s t a t i c a l l y on t h e w a t e r s u r f a c e and t h e r e s u l t i n g i n s t a b i l i t y was p h o t o g r a p h e d . A m e c h a n i c a l s w i t c h l o c a t e d a l o n g s i d e one o f t h e r a i l s n e a r t h e end o f t h e t a n k ' s t r a v e r s e , t r i g g e r e d a s t a n d a r d p h o t o g r a p h i c f l a s h g u n a i m e d a t t h e t a n k . The e x p e r i m e n t s we re p e r f o r m e d i n t o t a l d a r k n e s s and t h e c a m e r a s h u t t e r was h e l d o p e n t h r o u g h o u t t h e d u r a t i o n o f t h e r u n (500 msec o r l e s s ) . The i n s t a b i l i t i e s we re a l l p h o t o g r a p h e d i n t h e i r l a t e r s t a g e s o f g r o w t h a t t h e i n s t a n t w h e r e t h e y h a v e r e a c h e d maximum a m p l i t u d e j u s t b e f o r e t h e m o t i o n o f t h e t a n k i s s t o p p e d by t h e s h o c k t a b s o r b e r s . The p h o t o g r a p h s we re a l l t a k e n by a N i k o r m a t EL c a m e r a w i t h a 50 mm f / 2 l e n s u s i n g I l f o r d PANF 50 A .S . A . f i lm . When t h e f i r s t h a r m o n i c wave was e x c i t e d and t h e t a n k a c c e l e r a t e d , i t was f o u n d t h a t t h e r e was n e g l i g i b l e g r o w t h i n t h e v e r t i c a l d i r e c t i o n and i n s t e a d , mos t o f t h e d i s t u r b a n c e a p p e a r e d a c r o s s t h e t a n k ( F i g u r e 22a) ) . The p h o t o g r a p h s do show h o w e v e r , t h a t t h e r e i s a c l i m b i n g F i g u r e 2 1 . The i n i t i a l l u n p e r t u r b e d s u r f a c e a t an a c c e l e r a t i o n o f 1.5 g ' s . - 5 0 -f i l m o f w a t e r t h a t b u l g e s i n t h e f o r m o f d r o p s a t t h e e d g e s . T h i s f i l m e x i s t s on a l l f o u r s i d e s o f t h e o f t h e t a n k b u t i s m o s t l y s e e n on t h e s i d e w a l l s o f t h e t a n k i n t h e p h o t o g r a p h . T h e r e i s a l s o t h e a p p e a r a n c e o f \" b u b b l e s \" a t t h e c o r n e r s o f t h e t a n k as was f i r s t n o t e d by Emmons e t a l . F i g u r e s 21 and 22a) show t h a t t h e d i s t u r b a n c e o f t h e u n p e r t u r b e d s u r f a c e and t h a t o f t h e f i r s t h a r m o n i c e x c i t e d wave r e s p e c t i v e l y , a r e e s s e n t i a l l y t h e same. T h e r e f o r e , t h e c a p a c i t a n c e and r e s i s t a n c e d a t a o b t a i n e d f o r t h e i n s t a b i l i t y t h o u g h t t o a r i s e f r o m t h e f i r s t h a r m o n i c wave a l s o a p p l y t o t h e i n i t i a l l y u n p e r t u r b e d s u r f a c e as w e l l . P h o t o g r a p h s o f t h e i n s t a b i l i t y e v o l v i n g f r o m t h e s e c o n d h a r m o n i c e x c i t e d wave w h i c h was u s e d by L a n g i l l e (1970) show s i m i l a r r e s u l t s as t h o s e o b t a i n e d f o r t h e f i r s t h a r m o n i c wave ( F i g u r e 22b) ) . T h e r e i s some g r o w t h i n t h e v e r t i c a l d i r e c t i o n , b u t o n c e a g a i n t h e t r a n s v e r s e d i s t u r b a n c e a p p e a r s t o d o m i n a t e . S u c c e s s f u l i n s t a b i l i t i e s we re a c h i e v e d when t h e 4 t h h a r m o n i c wave was e x c i t e d u s i n g t h e e l e c t r o d e shown p r e v i o u s l y i n F i g u r e 5 . U s i n g t h e t i m i n g c i r c u i t d e s c r i b e d i n C h a p t e r I I p a r t D, r e p r o d u c i b l e 4 t h h a r m o n i c i n s t a b i l i t i e s w e r e o b t a i n e d on e a c h r u n ( F i g u r e 23) . The e f f e c t o f p h a s i n g was a l s o s u c c e s s f u l i n t h i s c a s e . The t a n k was a c c e l e r a t e d a t t h e i n s t a n t when t h e e x c i t e d s t a n d i n g wave had a p h a s e o f IT o r 0 and t h e r e s u l t s F i g u r e 22. a) F l a s h p h o t o g r a p h s o f the water s u r f a c e The f i r s t h a r m o n i c e x c i t e d wave at 1.5 g's v i r t u a l a c c e l e r a t i o n . F i g u r e 22. b) T n e second harmonic i n s t a b i l i t y at 1.5 g ' s . - 5 3 -Figure 23. The fourth harmonic i n s t a b i l i t y . The h o r i z o n t a l l i n e in the photographs indicates the i n i t i a l water level.The two consecutive shots indicate the degree of r c p r o d u c i b i l t y of the i n s t a b i l i t y . The i n i t i a l phase for these photographs was *=TT. - 5 4 -a r e as e x p e c t e d ( F i g u r e s 23 and 24) . The p h o t o g r a p h s i n d i c a t e t h a t t h e maximum a m p l i t u d e a t t a i n e d by t h e s p i k e s as m e a s u r e d f r o m t h e i n i t i a l w a t e r l e v e l i s a b o u t 12 cm. I n c o n t r a s t , t h e i n i t i a l p e r t u r b a t i o n h a s an a m p l i t u d e o f t y p i c a l l y 1.5 mm. I n F i g u r e 2 3 , t h e r e i s a l s o s e e n some d i s t u r b a n c e a c r o s s t h e s u r f a c e as w e l l as a c r o s s t h e t o p o f t h e s p i k e s p e r p e n d i c u l a r t o t h e p l a n e o f t h e p h o t o g r a p h . A p h o t o g r a p h o f t h e u n a c c e 1 e r a t e d s u r f a c e w i t h t h e s t a n d i n g wave on i t ( F i g u r e 25) , i n d i c a t e s t h a t a s o u r c e o f t h i s t r a n s v e r s e d i s t u r b a n c e i s due t o some r e s o n a n t e x c i t a t i o n o f a \" c r o s s - t a n k \" mode a l o n g w i t h t h e d e s i r e d 4 t h h a r m o n i c a l o n g t h e l e n g t h o f t h e t a n k . S i d e - o n p h o t o g r a p h s ( F i g u r e 26 a) and b) ) , i n d i c a t e t h a t t h e r e i s a n e t t e n d e n c y o f t h e w a t e r t o b u l g e o u t w a r d i n t h e m i d d l e o f t h e t a n k w h e r e a s t h e w a t e r t h a t i s i n c o n t a c t w i t h t h e w a l l s i s r e s t r i c t e d i n i t s m o t i o n b e c a u s e o f i t s a d h e s i o n t o t h e w a l l s . F u r t h e r e v i d e n c e t h a t t h e c o n t a c t a r e a o f t h e w a t e r w i t h t h e w a l l s l a g s b e h i n d t h e g r o w t h o f t h e i n s t a b i l i t y i s p r o v i d e d by F i g u r e 2 7 . H e r e , t h e l e f t h a l f o f t h e f r o n t w a l l o f t h e t a n k ha s b e e n f i t t e d w i t h a t h i n s h e e t o f s a n d b l a s t e d P l e x i g l a s . The s a n d -b l a s t e d s u r f a c e i s i n c o n t a c t w i t h t h e w a t e r and becomes d a r k upon b e i n g w e t t e d by t h e c o l o u r e d w a t e r . I n a l l c a s e s , t h e s a n d b l a s t e d s i d e shows t h a t t h e w e t t i n g due Figure 24. The same fourth harmonic i n s t a b i l i t y as in figure 23, except that the tank was released when the phase of the standing wave was = 0 . - 5 6 -F i a u r e 2 5 . F l a s h p h o t o g r a p h o f t h e e x c i t e d f o u r t l h a r m o n i c s t a n d i n g wave a t p h a s e $ = TT i n f r e e f a l l . - 5 7 -Figure 26. Side on f l a s h photographs of the fourth harmonic i n s t a b i l i t y showing the upward bulging of the f l u i d in the transverse d i r e c t i o n across the t ank. -58-Figure 2 7 . The fourth harmonic i n s t a b i l i t y with the l e f t h a l f of the tank showing the wetting of a translucent screen. a) at 1.5 g's a c c e l e r a t i o n b) at 1.0 g's. - 5 9 -t o t h e g r o w t h o f t h e s p i k e i s n a r r o w e r and s m a l l e r t h a n t h e s p i k e w h i c h a p p e a r s on t h e r i g h t h a n d s i d e o f t h e p h o t o g r a p h s . H e n c e , i n t h e t a n k o f w i d t h 1 0 . 5 cm i t ha s b e e n s e e n t h a t a l t h o u g h t h e i n s t a b i l i t i e s r e a c h h i g h a m p l i t u d e s t h e w a t e r t h a t i s i n c o n t a c t w i t h t h e w a l l s doe s n o t f o l l o w t h e d e v e l o p m e n t o f t h e i n s t a b i l i t y . The t e n d e n c y o f t h e w a t e r s u r f a c e t o b u l g e u p w a r d mus t t h e n n e c e s s a r i l y p u l l i n w a t e r away f r o m t h e w a l l s as t h e i n s t a b i l i t y g r o w s . S i n c e t h e e l e c t r i c a l m e a s u r e -men t s e m p l o y e d o n l y m e a s u r e t h e c o n t a c t a r e a o f t h e w a t e r w i t h a w a l l , t h e y do n o t t r u l y r e p r e s e n t t h e g r o w t h o f t h e i n s t a b i l i t y , b u t i n s t e a d i n d i c a t e t h e a p p a r e n t l y l i n e a r d e c r e a s e o f t h e c o n t a c t a r e a w i t h t i m e . L a s t l y , t h e 8 t h h a r m o n i c mode was e x c i t e d and t h e r e s u l t a n t i n s t a b i l i t y o b s e r v e d . The m o t i v a t i o n f o r e x c i t i n g t h i s h i g h mode i s t h a t i t s w a v e l e n g t h a p p r o a c h e s t h a t u s e d by t h e o t h e r i n v e s t i g a t o r s m e n t i o n e d p r e v i o u s l y . I t was f o u n d t h a t t h e i n s t a b i l i t y i n t h i s c a s e doe s r e a c h an a p p r e c i a b l e a m p l i t u d e ( F i g u r e 28) , h o w e v e r t h e maximum a m p l i t u d e a c h i e v e d i s a b o u t h a l f t h a t a c h i e v e d by t h e 4 t h h a r m o n i c w a v e . I n v i e w o f t h e c o m p l e x s t r u c t u r e and n a t u r e o f t h e f l u i d s u r f a c e f o r t h i s p a r t i c u l a r mode, i t was c o n s i d e r e d u n p r o f i t a b l e t o do f u r t h e r s t u d i e s o f i t . F i g u r e 28. The e i g h t h h a r m o n i c i n s t a b i l i t y w i t h t h e i n i t i a l p h a s e * = i r . - 6 1 -I n o r d e r t o s u p p r e s s t h e mode a c r o s s t h e n a r r o w e s t d i m e n s i o n o f t h e w a t e r s u r f a c e a n o t h e r t a n k was c o n s t r u c t e d s u c h t h a t i t s w i d t h was l e s s t h a n t h e c u t - o f f w a v e l e n g t h f o r t h i s mode: X c = 2u "Thesis/Dissertation"@en . "10.14288/1.0085276"@en . "eng"@en . "Physics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Experimental investigation of the Rayleigh-Taylor instability"@en . "Text"@en . "http://hdl.handle.net/2429/21437"@en .