THE USE OF LIGHT REFRACTION FOR THE INVESTIGATION OF STATIONARY ULTRASONIC WAVES t>y A d o lp h P a u l L o e b e r AN ABSTRACT S u b m itte d to th e S chool o f G ra d u a te S tu d ie s o f M ic h ig a n S t a t e C o l l e g e o f A g r i c u l t u r e sued A p p l i e d S c i e n c e i n p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts f o r th e degree o f DOCTOR OF PHILOSOPHY D e p a r t m e n t o f P h y s i c s a n d A stro n o m y 1954 A p p ro v e d £ - JK ' Adolph P. Loeber ABSTRACT Tho r e f r a c t i o n o f l i g h t b y a n u l t r a s o n i c wave c a n b e u s e d a s a t o o l f o r i n v e s t i g a t i n g th e sound f i e l d , p a r t i c u l a r l y i n t h e r e g i o n b e lo w one m e g a c y c le /se c o n d * An e f f e c t d i s c o v e r e d by L u c a s a n d B i q u a r d ( 1 ) , n a m e l y , t h e b r o a d e n i n g o f a n a r r o w l i g h t beam a s i t p a s s e s t h r o u g h a s o u n d f i e l d w as e x p l a i n e d b y th em a s b e i n g due t o r e f r a c t i o n * H e u t e r a n d P o h lm a n ( 2 ) u s e d a m o d if ie d e x p e r im e n ta l a rra n g e m e n t to m easu re sound a b s o r p tio n * A s i m p l e e x p l a n a t i o n h a s b e e n made f o r t h e a b o v e p h en o m en o n b a s e d on an e x p e r i m e n t a l m e th o d due t o W ie n e r ( 3 ) * A sim p le m a th e m a tic a l t h e o ­ r y i s p r e s e n t e d w h ich d e s c r i b e s th e v a r i a t i o n i n i n t e n s i t y o f t h e u n d e ­ f l e c t e d beam a s t h e l i g h t beam i s p a s s e d t h r o u g h v a r i o u s p o s i t i o n s o f a s t a t i o n a r y s i n u s o i d a l s o u n d wave* A s im ila r b u t q u a l i t a t i v e ap p ro ach has b e e n m ade t o t h e c a s e o f a s i n u s o i d a l wave i n t e r f e r i n g w i t h a n o p p o s i t e l y d i r e c t e d s a w to o th w ave. E x p e r i m e n t a l l y , u s e h a s b e e n made o f v a r i a t i o n s o f b o t h t h e m e th o d o f L u c a s a n d B i q u a r d a s w e l l a s t h a t o f H u e t e r a n d P o h lm a n . These v a r i ­ a t i o n s w e re u s e d t o o b t a i n p h o t o g r a p h i c r e c o rd s o f th e s p a t i a l p r e s s u r e d is tr ib u tio n in a s ta tio n a ry u ltra s o n ic w ave. A m e th o d i s d e sc rib e d fo r m a k in g r a p i d m e a s u r e m e n t s o f s o u n d v e l o c i t y i n l i q u i d s w h ic h y i e l d s r e ­ s u l t s a c c u r a t e to w i t h i n 1 . 5 p e r c e n t . fo rm d e t e r m i n a t i o n i n An a p p r o a c h t o t h e p r o b l e m o f wave l i q u i d s h a s b e e n m a d e , a n d a m e th o d f o r m e a s u r i n g t h e p r e s s u r e a m p l i t u d e i n a s t a t i o n a r y u l t r a s o n i c wave i s d e sc rib e d . Adolph P . Loeber L i t e r a t u r e O ite d R. L u c a s a n d P . B i q u a r d , CJompt. r e n d . ( P a r i s ) (1952). 194, 21^2, T . H u e t e r a n d R. P o h lm a n , Z. Angew. P h y s i k , _1, 4 0 5 , ( 1 9 4 9 ) . 0 . W i e n e r , Ann. P h y s i k ( 5 F ) {*2.* 1 0 5 ,(1 8 9 5 )* THE USE OF LIGHT REFRACTION FOR THE INVESTIGATION OF STATIONARY ULTRASONIC WAVES by A d o lp h P a u l L o e b e r A THESIS S u b m itte d to th e S chool o f G ra d u a te S tu d ie s o f M ic h ig a n S t a t e C o ll e g e o f A g r i c u l t u r e and A p p lie d S c ie n c e in p a r tia l f u l f i l l m e n t o f th e re q u ire m e n ts f o r th e degree o f DOCTOR OF PHILOSOPHY D e p a r t m e n t o f P h y s i c s an d A stro n o m y 1954 ProQuest Number: 10008459 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10008459 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346 ACKNOWLEDGEMENTS The a u t h o r t a k e s t h i s o p p o r t u n i t y t o e x p r e s s h i s s i n c e r e th a n k s to Dr* E . A. H ied em an n who f i r s t s u g g e s t e d t h e p o s s i b i l i t y o f a p p l y i n g t h e m e th o d o f W ie n e r t o t h e i n v e s t i g a t i o n o f u l t r a s o n i c w a v e s , a n d who by h i s c o n t i n u i n g i n t e r e s t a n d g u i d a n c e h a s made p o s s i b l e t h e r e s u l t s w h ic h have been ach iev ed * A ck n o w led g em en t o f a d e b t o f g r a t i t u d e i s a l s o due Dr* J o s e f K o lb who w as a s s o c i a t e d w i t h t h e a u t h o r d u r i n g t h e e a r l y p a r t o f t h i s i n v e s t i g a t i o n a n d w hose t i r e l e s s e n t h u s i a s m was a s o u r c e o f i n ­ s p ira tio n . T h an k s i s a l s o due Dr* A* L e i t n e r who g a v e v a l u a b l e a s s i s ­ ta n c e w ith r e s p e c t to th e m a th e m a tic s* O t h e r m em bers o f t h e P h y s i c s D ep artm en t h av e a i d e d th ro u g h h e l p f u l d i s c u s s i o n s . F in a lly , th a n k s i s d u e t o M. A* B r e a z e a l e who made t h e p h o t o g r a p h i c r e p r o d u c t i o n s . A d o lp h P a u l L o e b e r c a n d id a te f o r th e degree o f D o cto r o f P h ilo s o p h y F i n a l e x a m i n a t i o n , A u g u s t 6 , 1 9 5 4 , 1 0 : 0 0 A.M. C o n f e r e n c e ro o m , P h y s i c s M a th e m a tic s B u i l d i n g . D isse rta tio n : The Use o f L i g h t R e f r a c t i o n f o r t h e I n v e s t i g a t i o n o f S t a t i o n a r y U l t r a s o n i c W aves. O u tlin e o f S tu d ie s M a jo r s u b j e c t : M in o r s u b j e c t : P h y sic s M a th e m a tic s B io g r a p h ic a l Ite m s B orn, F eb ru ary 1, 19^0, D e t r o i t , M ic h ig a n . U n d e r g r a d u a t e S t u d i e s , Wayne U n i v e r s i t y , 1957-1941. G ra d u a te S t u d i e s , U n iv e r s it y o f C h ic a g o , I n s t i t u t e o f M e te o ro lo g y , 1 9 4 2 - 1 9 4 5 ; Wayne U n i v e r s i t y , 1 9 4 7 - 1 9 4 8 ; M i c h i g a n S t a t e C o l l e g e , 1950-1954. E x p erien ce: T e c h n ic ia n , C h ry s le r C o rp o ra tio n M e ta l lu r g ic a l L abora­ t o r y , 1941-1942; M e t e o r o l o g i s t , Second A ir D iv is io n , E ig h th A ir F orce, 1945-1945; E n g in e e r, C h r y s le r T ruck E n g in e e r in g , 1 9 4 5 - 1 9 4 7 ; I n s t r u c t o r i n P h y s i c s , Wayne U n i v e r s i t y , 1 9 4 7 - 1 9 5 0 ; I n s t r u c t o r i n P h y s i c s , ( p a r t tim e ) M arygrove C o l l e g e , 1 9 4 8 1 9 5 0 ; G r a d u a t e A s s i s t a n t a n d S p e c i a l G r a d u a t e R e s e a r c h As­ s i s t a n t , M ic h ig a n S t a t e C o l l e g e , 1951-1954. Member o f A m e r ic a n M e t e o r o l o g i c a l S o c i e t y , The A c o u s t i c a l S o c i e t y o f A m e r i c a , S o c i e t y o f t h e S i g n a X i , S igm a P i S i g n a , P i Mu E p silo n . TABLE OF CONTENTS Page INTRODUCTION I O p t i c a l M e th o d s i n U l t r a s o n i c s 1 L i g h t R e f r a c t i o n M e th o d s (M e th o d I ) 2 P u rp o se and Scope o f t h i s I n v e s t i g a t i o n 2 THEORY 5 T h e o r y o f Ray D e f l e c t i o n b y Medium H a v in g R e f r a c t i v e I n d e x G ra d ie n t 5 A p p l i c a t i o n o f T h e o r y t o U l t r a s o n i c Waves 7 M o d i f i c a t i o n o f t h e M e th o d U s i n g D e c r e a s e i n I n t e n s i t y o f U n d e f l e c t e d Beam (M e th o d I I ) 11 T h e o r y o f I n t e n s i t y D e c r e a s e o f U n d e f l e c t e d Beam A p p l i e d t o S i n u s o i d a l S t a t i o n a r y U l t r a s o n i c Wave 11 T h e o r y o f S a w t o o t h Wave F o r m a t i o n i n Waves o f F i n i t e A m p litu d e 25 T h e o r y o f I n t e n s i t y D e c r e a s e o f U n d e f l e c t e d Beam A p p l i e d t o a S i n u s o i d a l Wave I n t e r f e r i n g w i t h a S a w t o o t h Wave T r a v e l i n g i n O p p o site D ir e c tio n 28 EXPERIMENTAL APPARATUS 55 O sc illa to r 55 T ran sd u cers 55 O p tic a l A rran g em en ts 58 F re q u e n c y and C u r r e n t M e te r s 4l METHODOLOGY AND RESULTS P h o to g r a p h ic R ecord o f S p a t i a l P r e s s u r e D i s t r i b u t i o n i n a S t a t i o n a r y U l t r a s o n i c Wave 42 42 Page M e a s u r e m e n t o f S o und V e l o c i t y 48 A p p r o a c h t o Wave Form D e t e r m i n a t i o n 52 A p p ro ach to Sound P r e s s u r e M easu rem e n ts 57 SUGGESTIONS FOR FURTHER WORK 65 SUMMARY 66 BIBLIOGRAPHY 67 LIST OF FIGURES F ig u re 1. 2 . 5. 4. 5. 6. 7* 8. 910, 11 . 12. Page B r o a d e n i n g o f a s l i t im a g e due t o l i g h t r e f r a c t i o n by a n u l t r a s o n i c w ave. R e f r a c t i o n o f a l i g h t beam i n a l i q u i d w i t h c o n t i n u o u s l y v a ry in g r e f r a c t i v e in d e x . 6 S c h e m a tic a r ra n g e m e n t u s e d i n c a l c u l a t i n g i n t e n s i t y d e c r e a s e o f u n d e f l e c t e d beam p a s s i n g t h r o u g h s i n u s o i d a l s t a t i o n a r y sound w ave. 12 T h e o re tic a l curves o f r e la tiv e s l i t S l^ v s . p o s i t i o n i n w ave. 26 l i g h t en erg y th ro u g h f i n a l R e s u l t a n t o f s i n u s o i d a l wave t r a v e l i n g t o r i g h t a n d s a w t o o t h wave t r a v e l i n g t o l e f t ( t i m e * 0 ) . 50 R e c o r d e r c u r v e s h o w in g t h e d i s t r i b u t i o n o f l i g h t i n t h e c e n t r a l o r d e r o f t h e d i f f r a c t i o n p a t t e r n o f s l i t S 1^ i n th e p la n e o f s l i t 52 C urve o f t h e o r e t i c a l " r e c o r d e r r e l a t i v e i n t e n s i t y " v s . p o s i t i o n i n wave due t o s i n u s o i d a l wave i n t e r f e r i n g w i t h o p p o s i t e l y d i r e c t e d s a w to o th w ave. 55 B a sic o p t i c a l a rra n g e m e n t u s e d f o r l i g h t r e f r a c t i o n m e th o d (M e th o d I ) . 57 O p tic a l arrangem ent fo r o b ta in in g re c o rd e r cu rv es o f i n t e n s i t y c h a n g e i n u n d e f l e c t e d beam (M e th o d I I ) . 59 C a lib ra tio n cu rv e. read in g . 40 M ic ro p h o to m e te r r e a d in g v s . reco rd er S i m p l i f i e d s c h e m a t i c d ia g r a m o f o p t i c a l a r r a n g e m e n t f o r th e p h o to g ra p h ic r e c o r d in g o f th e s p a t i a l p r e s s u r e d i s t r i b u t i o n i n a s t a t i o n a r y u l t r a s o n i c wave b y t h e l i g h t r e f r a c t i o n m e th o d (M e th o d I ) . 45 P h o to g ra p h ic re c o rd o f s p a t i a l p re s s u re d i s t r i b u t i o n in a s t a t i o n a r y u l t r a s o n i c wave o b t a i n e d by l i g h t r e f r a c t i o n m e th o d (M e th o d I ) . 44 F ig u re 15 . Page O p tic a l a rra n g e m e n t f o r p h o to g r a p h ic a lly re c o rd in g th e s p a tia l p ressu re d is trib u tio n in a s ta tio n a ry u ltra s o n ic wave b y t h e m e th o d o f i n t e n s i t y v a r i a t i o n o f t h e u n d e f l e c t e d beam ( M e th o d I I ) . 46 P h o to g ra p h ic re c o rd o f s p a t i a l p r e s s u r e d i s t r i b u t i o n in a s t a t i o n a r y u l t r a s o n i c wave o b t a i n e d by t h e m e th o d o f i n t e n s i t y v a r i a t i o n o f t h e u n d e f l e c t e d beam ( M e th o d I I ) . 47 T y p i c a l r e c o r d e r t r a c e o b t a i n e d i n m a k in g s o u n d v e l o c i t y m e a s u r e m e n t s by t h e m e th o d o f i n t e n s i t y v a r i a t i o n o f t h e u n d e f l e c t e d beam . 50 R ecorder tr a c e o f " re c o rd e r r e l a t i v e in te n s ity " o f l i g h t t h r o u g h f i n a l s l i t v e r s u s p o s i t i o n i n t h e wave f o r a s t a t i o n a r y wave i n g l y c e r i n e . T r a n s d u c e r c u r r e n t a ) 100 m a . , b ) 2 0 0 m a . , c ) 500 m a . 55 17. R e c o r d e r t r a c e f o r s t a n d i n g wave i n b e n z e n e . 55 18. R e c o r d e r t r a c e f o r s t a n d i n g wave i n a c e t o n e . 55 19 . R e c o r d e r t r a c e f o r s t a n d i n g wave i n c a r b o n t e t r a c h l o r i d e . 56 V a r i a t i o n o f 3^ 59 14. 15. 16 . 20 . 21 . w i t h p a r a m e t e r A. m in T ran sd u cer c u r r e n t v s . p ara m e te r A f o r ru n s in g ly c e r in e and c a r b o n t e t r a c h l o r i d e . 60 INTRODUCTION O p t i c a l M etho d s i n U l t r a s o n i c s E a r l y o p t i c a l m e th o d s . Among th e m e th o d s u s e d f o r t h e q u a l i t a t i v e and q u a n t i t a t i v e i n v e s t i g a t i o n o f s o n i c an d u l t r a s o n i c f i e l d s , th e v a r i ­ o u s o p t i c a l m e th o d s a r e o f c o n s i d e r a b l e i m p o r t a n c e , e s p e c i a l l y a t u l t r a ­ s o n ic f r e q u e n c i e s . P e r h a p s t h e e a r l i e s t o p t i c a l i n v e s t i g a t i o n o f sound f i e l d s was made by T o p l e r who i n 1867 u s e d a s c h l i e r e n m eth o d t o r e n d e r so u n d w aves from a s p a r k s o u r c e v i s i b l e ( 1 ) . p r o v e d on t h i s m e th o d ( 2 ) , w h i l e s t i l l L a t e r , i n 1899* Wood im­ l a t e r F o le y an d S o u d e r p u b l i s h e d t h e i r m e th o d o f p h o t o g r a p h i n g sh ad o w g ra p h s o f w aves from a s p a r k s o u r c e by u s i n g a n o t h e r s p a r k a s t h e s o u r c e o f l i g h t f o r c a s t i n g t h e shadow and p ro d u c in g th e p i c t u r e (5)* L a t e r o p t i c a l m e th o d s . More r e c e n t l y , s i n c e t h e d i s c o v e r y by Debye a n d S e a r s ( 4 ) a n d L u cas and B iq u a r d ( 5 ) i n 1952 t h a t so un d w aves o f s h o r t eno ug h wave l e n g t h can a c t a s an o p t i c a l d i f f r a c t i o n g r a t i n g f o r l i g h t w h ic h p a s s e s t h r o u g h t h e sou nd f i e l d , m o s t o f t h e o p t i c a l m e th o d s w h ic h h a v e b e en d e v e lo p e d make u s e o f t h i s d i f f r a c t i o n g r a t i n g e f f e c t . F o r e x a m p le , t h e a d a p t a t i o n s o f B ar ( 6 ) , a s w e ll a s t h o s e o f H iedem ann and H oesh ( 7 ) o f T o p l e r 8s m etho d make u s e o f t h i s g r a t i n g e f f e c t . As a r e s u l t o f t h e d e p e n d e n c e o f t h e s e new er m e th o d s on t h e e x i s t a n c e o f an a c o u s t i c g r a t i n g , t h e s e m etho ds g e n e r a l l y f i n d t h e i r m o s t u s e f u l a p p l i ­ c a t i o n i n t h e u l t r a s o n i c f r e q u e n c y ra n g e above one m e g a c y c l e / s e c o n d . W h ile i t i s t r u e t h a t th e d e v e lo p m e n t o f c i r c u i t s f o r p r o d u c i n g s h o r t d u r a t i o n s p a r k l i g h t f l a s h e s ( 8 ) h ave p e r m i t t e d a d a p t a t i o n s o f t h e m e th o d o f F o le y a n d S o u d e r ( 5 ) t o h i g h e r f r e q u e n c i e s , t h e a p p l i c a t i o n s i n t h i s c a s e a l s o h a v e b e e n made i n t h e u l t r a s o n i c f r e q u e n c y r a n g e a b o v e one m e g ac y c le/se c o n d ( 9 ) . L i g h t R e f r a c t i o n M e th o d ( M e th o d I ) B e s i d e s t h e d i f f r a c t i o n e f f e c t w h i c h u l t r a s o n i c w av es c a n h a v e on l i g h t t r a v e r s i n g th e sound f i e l d , th e re e x is ts an o th er e f f e c t , d i s c o v e r e d b y L u c a s a n d B i q u a r d , a n d m e n t i o n e d by them i n t h e i r firs t fu n d a­ m e n t a l p a p e r ( 5 ) w h ic h c a n b e u s e d a s a t o o l f o r t h e i n v e s t i g a t i o n o f s o u n d f i e l d s when t h e wave l e n g t h o f t h e s o u n d i s g r e a t e r t h a n t h e w i d t h o f a l i g h t beam w h i c h p a s s e s t h r o u g h t h e s o u n d f i e l d * The e f f e c t r e ­ f e r r e d t o i s t h e b r o a d e n i n g o f t h e im a g e o f a s l i t o n a s c r e e n i f th e l i g h t w h ic h f o r m s i t i s c o n f i n e d t o a n a r r o w beam w h i c h p a s s e s t h r o u g h th e sound f i e l d * (S ee F ig u re 1 ) . Lucas and B iq u a rd e x p l a i n e d t h i s e f ­ f e c t a s due t o t h e r e f r a c t i o n o f t h e l i g h t by t h e s o u n d w a v e s . In a l a t e r t h e o r e t i c a l p a p e r , Lucas p o in t e d o u t t h a t th e r e f r a c t i o n o f l i g h t m ig h t be u s e d f o r m e a s u r in g sound a b s o r p t i o n and r e f l e c t i o n c o e f f i c i e n t s (1 0 ). The f i r s t u s e o f t h e p o s s i b i l i t i e s p r e d i c t e d by L u c a s a l r e a d y i n 1954 a p p e a r s t o h a v e b e e n made i n m uch l a t e r w ork by H u e t e r a n d P o h lm a n who i n 1949 u s e d t h i s e f f e c t t o m e a s u r e a b s o r p t i o n o f u l t r a s o n i c w av es ( 1 1 ) a n d by P o r r e c a , who i n 1952 m e a s u r e d t h e d i s t r i b u t i o n o f l i g h t i n ­ t e n s i t y a c r o s s t h e s l i t im ag e u n d e r v a r i o u s c o n d i t i o n s ( 1 2 ) . P u rp o s e and Scope o f t h i s I n v e s t i g a t i o n The p u r p o s e o f t h i s i n v e s t i g a t i o n i s t o e x a m in e t h e lig h t re fra c ­ t i o n m e th o d w i t h t h e p u r p o s e o f d e v e l o p i n g i t s u s e f u l n e s s a s a t o o l f o r 2 a) b) F i g u r e 1. B r o a d e n in g of a s l i t im a g e due to lig h t r e f r a c t i o n b y a n u l t r a s o n i c w av e. {Luc a s - B i q u a r d effect) a) sound off, b) so und on. 3 b o th th e q u a l i t a t i v e and q u a n t i t a t i v e i n v e s t i g a t i o n o f u l t r a s o n i c w av es, p r i m a r i l y s t a t i o n a r y u l t r a s o n i c w aves. S i n c e t h e m e th o d i s p a r t i c u l a r l y a d a p t a b l e t o t h e l o n g e r u l t r a s o n i c wave l e n g t h s , i t i s h o p e d t h a t t h i s i n v e s t i g a t i o n m ig h t h e l p i n o p e n in g th e d o or to o p t i c a l m e th o d s i n th e f r e q u e n c y r a n g e b e lo w o n e m e g a c y c l e / s e c o n d . 4 THEORY T h e o r y o f t h e Ray D e f l e c t i o n b y Medium H av in g R e f r a c t i v e In d e x G r a d i e n t The m e th o d o f W i e n e r . The o r i g i n a l e x p l a n a t i o n o f t h e b r o a d e n i n g o f t h e s l i t im a g e a s g i v e n b y L u c a s a n d B i q u a r d ( 5 ) , i s r a th e r in v o lv e d . H o w e v e r, t h i s e f f e c t c a n be e x p l a i n e d i n a s i m p l e m a n n e r , b a s e d u p o n a n e x p e r i m e n t a l m e th o d due t o W ie n e r ( 1 5 ) * W ie n e r h a s shown t h a t i f a h o r i z o n t a l l y d i r e c t e d l i g h t beam i m p i n g e s n o r m a l l y u p o n a t r a n s p a r e n t s u b ­ s ta n c e , i n o u r c a s e a l i q u i d , h a v in g a c o n tin u o u s ly v a ry in g in d e x o f r e ­ f r a c t i o n w hose g r a d i e n t d n /d x i s d ire c te d v e r tic a lly , q u ir e a c u rv a tu re i n p a s s in g th ro u g h th e s u b s ta n c e . t h e beam w i l l a c ­ The r a d i u s o f c u r v a t u r e R o f t h i s beam w i l l be g i v e n by Yv (i) w here x i s m e a su re d a lo n g th e d i r e c t i o n o f th e g r a d i e n t , and n i s th e in d e x o f r e f r a c t i o n a t th e p o i n t o f in c id e n c e . In p a s s in g th ro u g h a l a y e r o f l i q u i d o f le n g th £ , w i l l b e d e f l e c t e d fro m i t s o rig in a l d ire c tio n t h e beam o f l i g h t so t h a t t h e p o i n t w h e re i t i m p i n g e s o n a s c r e e n w i l l b e d i s p l a c e d from a p o s i t i o n S t o a p o s i t i o n S^. (See F ig u re 2 ) . L e t th e d i s t a n c e (S S ^ ) be c a l l e d d . S in c e , in p r a c t i c a l c a s e s , t h e v a l u e o f d n /d x i s v e r y s m a l l , i t i s p e r m i s s i b l e t o as s u m e t h e same v a l u e s o f n a t p o i n t s A and E an d t o u s e t h e v a l u e s o f th e i n c i d e n t and r e f r a c t i o n a n g l e s a t A i n p l a c e o f t h e i r s i n e and F ig u re 2 . R e f r a c t i o n o f a l i g h t beam i n a l i q u i d w i t h c o n tin u o u s ly v a ry in g r e f r a c t i v e in d e x . 6 ta n g e n t fu n c tio n s . U nder t h i s a s s u m p tio n , i t i s p o s s i b l e th e d isp la c e m e n t d, w ith th e a id o f e q u a tio n 2 . p la c e m e n t i s The v a l u e o f t h i s d is­ g iv e n by , w here g i s to c a l c u l a t e in r e a l ity c d<2_ , , t h e o p t i c a l p a t h l e n g t h fro m t h e c e n t e r M o f t h e t a n k t o t h e p o i n t 3^ o n t h e s c r e e n . H o w e v e r, f o r p r a c t i c a l p u r p o s e s , t h e g e o m e t r i c a l d i s t a n c e M!S^, o r e v e n t h e d i s t a n c e MS i s a s u f f i c i e n t l y a c ­ c u r a te a p p ro x im a tio n . The r e l a t i o n o f e q u a t i o n 2 i s a l s o due t o W ie n e r who u s e d t h e a b o v e d e s c r ib e d e f f e c t f o r th e m easurem ent o f d i f f u s i o n r a t e s o f one l i q u i d in to a n o th e r. M ore r e c e n t l y W o lin h a s a t t a c k e d t h i s p r o b l e m f r o m a som e­ w hat d i f f e r e n t ap p ro ach ( 1 4 ) , The e q u a t i o n w h ic h h e o b t a i n s h o w e v e r , r e ­ duces to e q u a tio n 2 f o r th e p a r t i c u l a r ca se c o n s id e re d h e r e . A p p l i c a t i o n o f T h e o r y t o U l t r a s o n i c Waves G e n e ra l, A l t h o u g h W i e n e r , a s m e n t i o n e d a b o v e , was p r i m a r i l y i n t e r ­ e s t e d i n t h e m e a s u r e m e n t o f d i f f u s i o n r a t e s i n l i q u i d s , a n d W o l i n 9s i n ­ te r e s t in t h e p r o b le m w as b r o u g h t a b o u t by t h e n e c e s s i t y o f k n o w in g t h e degree o f a n te n n a p a t t e r n d i s t o r t i o n c a u s e d by v a r i a t i o n i n th e d e n s ity i n s k i n a n d c o r e m a t e r i a l s o f radom e s a n d w i c h e s u s e d i n h i g h p r e c i s i o n t r a c k i n g r a d a r s , t h e a n a l y s i s i s a g e n e r a l on e w h ic h c a n b e a p p l i e d a l s o to any m a t e r i a l h av in g a g r a d ie n t o f r e f r a c t i v e a n is m w h i c h p r o d u c e s t h i s g r a d i e n t . in d e x , w h a te v e r th e m ech­ T h i s o f c o u r s e m a k es i t i m m e d i a t e l y a p p l i c a b l e t o a n u l t r a s o n i c w a v e , w h e re v a r i a t i o n s i n t h e p r e s s u r e i n 7 v a r i o u s p o r t i o n s o f t h e wave g i v e r i s e to v a r ia tio n s in th e r e f r a c t i v e in d e x * A p p lic a t i o n to p r o g r e s s iv e w aves. In a p la n e p ro g re s s iv e sin u ­ s o i d a l s o u n d wave t h e s o u n d p r e s s u r e a t a p a r t i c u l a r p o s i t i o n x i n s p a c e v a r ie s a c c o rd in g to th e r e l a t i o n (5) The d e r i v a t i v e d p / d x t h e n v a r i e s a c c o r d i n g t o t h e r e l a t i o n so t h a t a t an y p a r t i c u l a r p o s i t i o n x i n s p a c e a s t h e s o u n d wave p a s s e s th e g r a d i e n t o f p r e s s u r e w i l l v a ry in a s in u s o id a l m anner, r e t u r n i n g to its o r i g i n a l v a l u e a f t e r a tim e i n t e r v a l e q u a l t o one p e r io d * The v a r i ­ a t i o n i n th e p r e s s u r e w i l l produce a c o rre s p o n d in g v a r i a t i o n i n th e r e ­ fra c tiv e in d e x , so t h a t a l o n g t h e wave t h e r e w i l l e x i s t a g r a d i e n t o f r e f r a c t i v e i n d e x , w h ich i t s e l f w i l l v a r y s i n u s o i d a l l y * I f a beam o f l i g h t w h i c h i s th e sound i s s m a l l c o m p a re d t o t h e wave l e n g t h o f a llo w e d to t r a v e r s e th e sound f i e l d i n a d i r e c t i o n norm al t o t h a t o f t h e s o u n d wave p r o p a g a t i o n d i r e c t i o n , i t w i l l be d e f l e c t e d , f i r s t to one s id e and th e n to th e o p p o s ite s id e o f i t s u n d e f le c te d p o ­ s i t i o n a s t h e s o u n d wave p a s s e s . S in c e t h i s to - a n d - f r o m o tio n o f th e l i g h t beam t a k e s p l a c e a t t h e f r e q u e n c y o f t h e u l t r a s o n i c w a v e s , t h e e y e w i l l n o t be a b l e t o f o l l o w t h e i n d i v i d u a l d e f l e c t i o n s , a n d t h e im ag e p r o ­ d u c e d b y t h e l i g h t beam u p o n t h e s c r e e n w i l l a p p e a r t o be b r o a d e n e d . It s h o u ld be n o te d t h a t t h i s b r o a d e n in g w i l l be in d e p e n d e n t o f th e p o s i t i o n 8 x in t h e w ave f i e l d p r o v i d e d we a s s u m e a p l a n e w a v e , a n d no a b s o r p t i o n . I f a b so rp tio n is c o n s id e r e d , how ever, th e b ro a d e n in g w i l l d e c re a se as t h e l i g h t beam i s m oved f a r t h e r away fr o m t h e s o u n d s o u r c e . s e n tia lly I t was e s ­ a m o d i f i c a t i o n o f t h i s m e th o d w h ic h was u s e d b y H u t e r a n d P o h lm a n t o m e a s u r e s o u n d a b s o r p t i o n i n a n i m a l t i s s u e s ( 1 1 ) . A p p lic a tio n to s ta t io n a r y w aves. For a p la n e s in u s o id a l s t a t i o n a r y s o u n d w a v e , t h e s o u n d p r e s s u r e may b e w r i t t e n i n t h e f o l l o w i n g f o r m : p « P c o * C ^ t - ^ + (5) o r th e e q u i v a l e n t form : P = 2 P c o S < > t ) c o 5( ^ (6) So t h e e o u n d p r e s s u r e a t a p r e s s u r e l o o p w h ere c o s ( w x / c ) = 1 v a r i e s d u r ­ i n g o n e p e r i o d b e t w e e n +2P a n d - 2 P . s u r e i s alw ay s z e r o . At th e p r e s s u r e node th e sound p r e s ­ The g r a d i e n t d p / d x , h o w e v e r , v a r i e s b e t w e e n i t p o ­ s i t i v e a n d n e g a t i v e m axim a a t t h e p r e s s u r e n o d e s , a n d r e m a i n s z e r o a t t h e p re s s u re lo o p s . T h is i s i*£L — — in d ic a te d in th e e x p re ssio n cosC ^ ) s i (7) At th e p re ssu re loopB , s in ( t» ix /c ) i s a l w a y s z e r o w h i l e a t t h e p r e s s u r e n o d e s , w here s i n 4 ^ x / c ) - 1 th e p re s s u re The v a l u e o f t h i 3 s p a c e maximum v a r i e s v a l u e s —2P tu/c to-T 2P:'-^c» 9 g r a d i e n t h a s a maximum i n s p a c e . d u r i n g o ne p e r i o d b e t w e e n t h e T h u s i n a s t a t i o n a r y s o u n d wave t h e r e w i l l e x i s t , a s a r e s u l t o f th e p re s s u re g r a d ie n t, a g ra d ie n t o f re f r a c tiv e in d e x , th e m a g n itu d e o f w h i c h w i l l v a r y b e t w e e n z e r o a n d a maximum w i t h i n a q u a r t e r - w a v e l e n g t h . As i n d i c a t e d i n e q u a t i o n 7> t h e p e r i o d i c i t y i n t i m e i n t h e s t a t i o n ­ a r y w ave w i l l p r o d u c e a p e r i o d i c i t y i n t h e m a g n i t u d e a n d d i r e c t i o n o f t h e p r e s s u r e g r a d i e n t , and t h e r e f o r e a l s o i n t h e r e f r a c t i v e in d e x g r a d i ­ en t* I f a beam o f l i g h t o f a w i d t h w h ic h i s s m a l l c o m p a re d t o t h e wave l e n g t h o f t h e sound p a s s e s t h e sound f i e l d i n a d i r e c t i o n n o rm al to t h a t o f th e s t a t i o n a r y w a v e , f o r e x a m p l e , a t a n o d e , t h e n i t w i l l be d e f l e c t e d due t o t h e p e r i o d i c i t y i n t i m e o f t h e r e f r a c t i v e i n d e x g r a d i e n t s y m m e t r i ­ c a l l y to e i t h e r s id e o f th e o r i g i n a l p a th . T h is w i l l once a g a in p ro d u ce a b r o a d e n i n g o f t h e im a g e on t h e s c r e e n . I n t h e c a s e o f a p r o g r e s s i v e w a v e , i t w as p o i n t e d o u t t h a t t h e m ag­ n i t u d e o f t h i s b ro a d e n in g s h o u ld be in d e p e n d e n t o f th e p o s i t i o n i n th e w a v e , a s s u m i n g a p l a n e wave a n d no a b s o r p t i o n . I n t h e caBe o f a s t a n d i n g w ave h o w e v e r , t h e d e g r e e o f b r o a d e n i n g w i l l d e p e n d u p o n t h e p o s i t i o n i n t h e wave a t w h ic h t h e l i g h t beam p a s s e s , b e i n g g r e a t e s t a t t h e p r e s s u r e n o d es end l e a s t ( t h e o r e t i c a l l y z e ro ) a t th e p r e s s u r e lo o p s . I f t h e c o n t a i n e r o f l i q u i d i s m oved p a r a l l e l t o t h e s o u n d p r o p a g a ­ tio n d i r e c t i o n , w h i l e t h e p o s i t i o n o f t h e l i g h t beam r e m a i n s f i x e d , a p e r i o d i c b r o a d e n i n g a n d n a r r o w i n g o f t h e im ag e on t h e s c r e e n c a n be o b ­ serv ed . By m e a s u r i n g t h e d i s p l a c e m e n t o f t h e c o n t a i n e r a n d t h e n u m b e r o f c o r r e s p o n d i n g m axim a ( o r m in im a ) o f t h e im age w i d t h on t h e s c r e e n , one c a n d e t e r m i n e t h e u l t r a s o n i c wave l e n g t h i n t h e l i q u i d . I f th e fre q u e n c y o f th e s o u rc e i s a l s o m e a su re d , one can e a s i l y o b t a i n th e v e l o c i t y . 10 M o d i f i c a t i o n o f t h e M e th o d U s i n g D e c r e a s e i n I n t e n s i t y o f U n d e f l e c t e d Beam (M e th o d I I ) The l i g h t r e f r a c t i o n m e th o d w h ic h m a k e s u s e o f t h e b r o a d e n i n g o f t h e s l i t im a g e may be m o d i f i e d som ew h at t o y i e l d a m e th o d w h i c h seem s t o be m ore u s e f u l f o r c e r t a i n p u r p o s e s . In p la c e o f m e re ly o b s e r v in g th e s l i t im a g e b r o a d e n i n g o n a s c r e e n , t h e s c r e e n i s s lit is su b stitu te d fo r it* rem o v e d a n d a n a r r o w The s l i t i s p l a c e d s o t h a t t h e c e n t e r o f t h e u n d e v i a t e d beam i s a l l o w e d t o p a s s t h r o u g h . When t h e s o u n d i s t u r n e d o n , t h e beam w i l l becom e b r o a d e n e d , a n d t h e r e f o r e l e s s l i g h t w i l l p a s s th ro u g h th e s l i t . I n g e n e r a l t h e b r o a d e r t h e b eam , t h e s m a l l e r t h e i n t e n s i t y o f t h e l i g h t w h ic h p a s s e s t h r o u g h . In a p r o g r e s s iv e w ave, f o r e x a m p l e , t h e m o re i n t e n s e t h e s o u n d , t h e b r o a d e r w o u ld b e t h e b e a m , a n d th e le s s th e i n t e n s i t y o f th e l i g h t p a s s e d by th e s l i t ( 1 1 ) . In a s ta ­ t i o n a r y s o u n d w a v e , on t h e o t h e r h a n d , t h e s l i t w o u ld t r a n s m i t t h e g r e a t e s t a m o u n t o f l i g h t when t h e beam p a s s e s t h r o u g h a p r e s s u r e a n t i ­ node (w here th e b ro a d e n in g i s le a st) p r e s s u r e node (w h ere b ro a d e n in g i s a n d w o u ld t r a n s m i t t h e l e a s t a t t h e g re a te st). A m easure o f th e r e l a t i v e i n t e n s i t y o f t h e l i g h t t r a n s m i t t e d b y t h e s l i t a s t h e beam i s a l l o w e d t o t r a v e r s e v a r i o u s p o r t i o n s o f a s t a n d i n g wave s h o u l d y i e l d i n f o r m a t i o n a b o u t t h e wave fo rm o f t h e s t a n d i n g w a v e . I t may t h e r e f o r e be w e l l t o i n v e s t i g a t e a n a l y t i c a l l y t h e m a n n e r i n w h ic h t h e i n t e n s i t y o f l i g h t t r a n s m i t t e d b y t h e s l i t c a n b e e x p e c t e d t o v a r y a s t h e l i g h t beam i s a l ­ lo w e d t o p a s s t h r o u g h v a r i o u s p o r t i o n s o f t h e s t a n d i n g w av e. T h e o r y o f I n t e n s i t y D e c r e a s e o f U n d e f l e c t e d Beam A p p l i e d t o a S i n u s o i d a l S t a t i o n a r y U l t r a s o n i c Wave C o n s i d e r a n a r r a n g e m e n t a s shown i n F i g u r e 11 P a ra lle l lig h t is Figure % Schematic arrangement used passing through if) cr> 12 in c a l c u l a t i n g i n t e n s i t y decrease of u n d e f le c te d s in u s o id a l s t a t i o n a r y sound wave. _£f> beam i n c i d e n t u p o n t h e s l i t 3 1 ^ , w h i c h l i m i t s t h e w i d t h of* t h e l i g h t beam p a s s i n g t h r o u g h t h e c e l l c o n t a i n i n g t h e m edium i n w h ic h t h e s t a t i o n a r y u l t r a s o n i c w ave i s to be s e t u p . c e l l c o n tin u e s on to th e p la n e o f s l i t S l^ . p a t t e r n w i l l be o b s e rv e d . p a t t e r n w h ich i s The l i g h t beam p a s s i n g t h r o u g h t h e T h a t p o r tio n o f th e l i g h t i n th e d i f f r a c t i o n i n c i d e n t d i r e c t l y on t h e s l i t 3 1 ^ w i l l p a s s t h r o u g h , w h ile t h e re m a in d e r w i l l be i n t e r r u p t e d . is to H e re a s l i t d i f f r a c t i o n The p u r p o s e o f t h i s a n a l y s i s d e t e r m i n e t h e r a t i o o f l i g h t p a s s e d b y s l i t 3 1 ^ when t h e r e i s no sound i n th e c e l l , t o t h a t p a s s e d when t h e s o u n d i s tu rn e d on. C o n sid e r a p o i n t o n t h e d i f f r a c t i o n p a t t e r n t h e d i s t a n c e o f w h ic h fro m t h e o p t i c a x is i s d e s ig n a te d by d. L e t th e a n g le su b te n d e d by t h i s p o i n t and th e o p t i c a x i s a t t h e s l i t S ]^ b e c a l l e d ©, w h i l e t h e c o r r e s p o n d i n g a n g l e s u b te n d e d a t t h e c e n t e r o f th e sound c e l l be d e s ig n a t e d d i s t a n c e a l o n g t h e o p t i c a x i s b e t w e e n t h e two s l i t s , from t h e c e n t e r o f t h e c e l l to s l i t $1, b ? eji, s l i t S l^ . L e t r be t h e and g th e d is ta n c e L et th e le n g th o f th e f i n a l i t s w i d t h be s a n d t h e w i d t h o f s l i t S ip be a . L ig h t in t e n s i ty d i s t r i b u t i o n in th e s l i t d i f f r a c t i o n p a t t e r n . The i n t e n s i t y o f l i g h t I a t any g iv e n p o s i t i o n i n th e d i f f r a c t i o n p a t t e r n can be w r i t t e n a s fo llo w s : IT a ^ w here I 0 i s te rn , and $ •> t h e maximum i n t e n s i t y ( a t t h e c e n t e r o f t h e d i f f r a c t i o n p a t ­ is g iv e n by th e e x p r e s s i o n where © ^ is th e i n c i d e n c e a n g le o f th e l i g h t a t s l i t S l 0, and V i e * X wave l e n g t h o f t h e l i g h t . th e For th e p a r t i c u l a r case u n d e r c o n s id e r a tio n th e a n g le o f in c id e n c e ©^ i s z e r o . T h e r e f o r e e q u a t i o n 9 may be w r i t t e n : i sm B w h ic h s u b s t i t u t e d i n t o e q u a t i o n 8 y i e l d s f o r th e i n t e n s i t y I i n th e d i f ­ fra c tio n p a tte rn y ie ld s f — _ r- I s \n Tio"[T o. S in c e © i s a v e ry sm all a n g le , S in 0 a S mG 1*" sm © / . J th e a p p ro x im a tio n 0 ^ ? “fcav \ €* 12) may be u s e d , a n d s i n c e e q u a tio n (1 1 ) f i n a l l y may be w r i t t e n : f T a T 1 - - ™ ^ “^ 1 "vTixcL (14) ^ VVV 1T ad w h ic h y i e l d s th e i n t e n s i t y I i n th e d i f f r a c t i o n p a t t e r n a t any p o i n t d i n t e r m s o f t h e maximum i n t e n s i t y a n d c o n s t a n t s o f t h e s y s t e m . D isp la c e m e n t o f th e d i f f r a c t i o n th e sound i n th e c e l l i s p a t t e r n due t o s o u n d w a v e . When t u r n e d o n , t h e beam o f l i g h t p a s s i n g t h r o u g h 14 “t h e c e l l w i l l b e r e f r a c t e d t o a n d f r o a t t h e f r e q u e n c y o f t h e s o u n d . t h e a s s u m p t i o n b e made t h a t t h e e n t i r e Let d i f f r a c t i o n p a t t e r n m o v es a c r o s s th e p la n e o f th e s l i t a s a u n i t . A lth o u g h i n r e a l i t y i t i s fro , t h e d i f f r a c t i o n p a t t e r n w h ic h m ov es t o a n d f o r s i m p l i c i t y , c o n s i d e r t h a t t h e s l i t 5 1 ^ i s m o v in g t o a n d f r o a - c ro s s th e d i f f r a c t i o n p a t t e r n . t e r n , w h ic h i s The r e l a t i v e m o t i o n th e i m p o r t a n t c o n s i d e r a t i o n , w i l l be b etw een s l i t and p a t ­ unchanged by t h i s a s ­ su m p tio n . A c c o r d i n g t o t h e e q u a t i o n o f W ie n e r ( e q u a t i o n 2 ) , th e d is p la c e m e n t d o f t h e d i f f r a c t i o n p a t t e r n i n t h e p l a n e o f 5 1 ^ due t o t h e r e f r a c t i v e i n d e x g r a d i e n t d n / d x i n t h e s o u n d wave i s w here £ in th is case i s lig h t passes. g i v e n by t h e w i d t h o f t h e s o u n d beam th ro u g h w h ich th e F o r t h e g r a d i e n t o f p r e s s u r e i n t h e s t a n d i n g w a v e , we h av e th e r e l a t i o n o f e q u a tio n 7 dp. _ _ d r/j cosCujt') sir,(r5^) ( 16) ^ w h e r e 2P i s t h e p r e s s u r e a m p l i t u d e i n t h e s t a n d i n g w a v e . I t i s p o s s i b l e to o b t a i n a r e l a t i o n betw een th e p r e s s u r e g r a d i e n t d p /d x an d th e r e f r a c t i v e i n d e x g r a d i e n t d n /d x by u s e o f t h e L o r e n t z - L orenz r e l a t i o n 15 i n w h ic h n i s t h e r e f r a c t i v e i n d e x o f t h e m e d iu m , p i t s i s a c o n sta n t* d e n sity , end K E q u a t i o n 17 may be r e - w r i t t e n i n t h e fo rm ( 18) w here V i s t h e s p e c i f i c v o lu m e o f t h e l i q u i d b e i n g c o n s i d e r e d . By d i f ­ f e r e n t i a t i n g e q u a t i o n 18 we o b t a i n 0 (v^ o r, (i?) + 2.')7- re a rra n g in g ( 20) \ w h ic h y i e l d s (p V iilrv — — l ') ( v \ " L + z N) ^ ( 21 ) Now, fro m t h e d e f i n i t i o n o f c o m p r e s s i b i l i t y , t h e e x p r e s s i o n - Y (22) may b e o b t a i n e d , w h e r e K, t h e c o m p r e s s i b i l i t y , w i l l b e c o n s i d e r e d t o be a c o n sta n t in th is a n a ly s is . S o , c o m b i n i n g e q u a t i o n s 21 a n d 22 y i e l d s d f9t ( 25) te>V\ o r we may w r i t e % = c ty (ov\ 16 • (24) S in c e dn w i l l be v ery em ail w ith r e s p e c t to n , the n in th e r ig h t s id e o f eq u a tio n 24 may be co n sid ere d c o n sta n t and eq u a tio n 2 4 may be w r itte n (2 5 ) where k i s a c o n s ta n t g iv en by E quation 25 shows a d ir e c t p r o p o r t io n a lit y between th e p ressu re g r a d ie n t and th e r e f r a c t iv e in d ex grad ien t* Combining th e eq u a tio n o f Wiener (e q u a tio n 15 ) , th e eq u a tio n f o r th e i n t e n s i t y in a s l i t d if f r a c t io n p a t­ te r n (e q u a tio n 14), the eq u a tion fo r th e p r e ssu re g r a d ie n t in a s t a t i o n ­ a r y wave (e q u a tio n 1 6 ) , and th e eq u ation r e la t in g the p ressu re g r a d ie n t w ith th e r e f r a c t iv e index g r a d ie n t (eq u a tio n 24) w i l l y ie ld C o s( svv\ )\£>r (o Vl C. For s im p lic it y in n o ta tio n l e t 2. P (2 8 ) XjtV* (oh (L sind l e t ( 29) 17 'th e n f i n a l l y e q u a t i o n 27 b eco m es (5 0 ) E q u a t i o n 50 i s th e o p tic e s s e n t i a l l y an e x p r e s s i o n f o r th e i n t e n s i t y o f l i g h t on a x i s a t a n y ti m e t f o r a f i x e d v a l u e o f B. o f B f i x e s t h e p o s i t i o n i n t h e s t a n d i n g w ave. as a fa c to r* M oreover, B c o n ta in s A A i s made up e n t i r e l y o f c o n s t a n t s o f t h e s y s t e m e x c e p t f o r t h e f a c t o r P , w h ich i s tio n s F ix in g th e v a lu e th e h a l f a m p litu d e o f th e p r e s s u r e f l u c t u a ­ i n th e s t a t i o n a r y w ave. L ig h t e n e rg y re c e iv e d th ro u g h f i n a l s l i t 3 1 ^ * L e t th e a ssu m p tio n b e made t h a t t h e i n t e n s i t y o f l i g h t a t t h e c e n t e r o f t h e f i n a l e q u a l to th e a v e ra g e i n t e n s i t y o v e r th e w id th o f th e s l i t . s l i t is T h is a s ­ s u m p t i o n w o u ld m ake f o r t h e g r e a t e s t e r r o r w here t h e d i f f r a c t i o n p a t t e r n h a s e i t h e r a maximum o r m inim um , b u t s h o u l d n o t be e x c e s s i v e i f s , o f th e f i n a l s l i t i s p a tte rn . S I,. 5 s m a ll com pared to th e s i z e o f th e d i f f r a c t i o n Now c o n s i d e r t h e l i g h t e n e r g y w h ic h i s In a sm all in c re m e n t o f tim e , by S l^ i s th e w id th d t, tr a n s m itt e d th ro u g h s l i t th e in c re m e n t o f e n e rg y ds p a s s e d g iv e n by (5 1 ) I b e in g th e l i g h t i n t e n s i t y a t th e s l i t , th e s l i t * The t o t a l a n d X an d s t h e d i m e n s i o n s o f l i g h t e n e r g y S , p a s s e d by t h e s l i t d u r i n g o n e - h a l f p e r i o d w i l l t h e n be g i v e n b y : (5 2 ) 18 T b e in g th e p e r io d o f o s c i l l a t i o n o f th e sound wave. Using the valu e o f I from e q u a tio n 5 0 , eq u a tion 52 may be w r it t e n : 1 at. J0 ^ B c O < » C t o “fc) J (55) Now, f o r c o n v e n i e n c e , t h e f o l l o w i n g c h an g e i n v a r i a b l e i s i n t r o d u c e d , Let - co s (urfc'). (5 4 ) Then — lO S vO ^ (35) and o te ^ c te s _-< f e (36) M aking u s e o f t h e s e r e l a t i o n s ( e q u a t i o n s 5 4 , 55* 56) e q u a t i o n 55 becomes '' S i ~ b * i 2- sX b,O a*_ (57) '4-1 U s in g t h e t r i g o n o m e t r i c i d e n t i t y Bx- A- _ -L c o o ^^ ^_____ ~— ■=“ c LB ^ 2L 2- e q u a t i o n 57 may be w r i t t e n ( 56 ) w h ich , making us© o f th e c o s in e s e r i e s may be w r it te n ir S= —I i - <*i r \ ,I +1 n. i 2. 4 A _C>„C C , 2 b 2zV f ~zT + —ZT" " 6*. 2 L 2 7 fT T 2 &£ dlz. (40) -VI or i V . JUT, i V z W . 2 V > t+ -_ ' (4 1 ) w h ic h s i m p l i f i e s t o -I c _ 2 .9 s,X & ( w ( _ ^ )V a '.J P ? . Z 4- 4 '.J w T - b 2£r 4- . * c \x » (42) Iv T v-T 1 S u b s t i t u t i n g now f o r a an d da t h e i r e q u i v a l e n t s a s g i v e n i n e q u a t i o n s 54 and 5 5 , e q u a t i o n 42 becomes ' ^ __ 3l %s 3 I o cu 21 . c. CO a t — V * 2J *• -V co&n~ 2 ^ M 4-*. J t c o i ^ C(-U^ ' ) u> (43) 6! w h ich c an be w r i t t e n i n th e f o l l o w i n g fo rm : (44) - L) o z! ^ i0 0 20 * ^ 1 L e ttin g Y\ (-0 / T-va O Cos, (urt') CO * | 41 p At 4- ‘Z . | z j 4- Svvq ^ Z n 7? For n * 2 le have T S' -2.4 B* t o s ^ u> A C u /t} t --------------^ (49) which upon in t e g r a t io n becomes *t* —"T, ^ C.OS Sl*\ (<Q i t fc ( 50 ) -*o The f i r s t term ( i n th e b ra ck ets in eq u ation 50) w i l l alw ays be zero fo r th e l i m i t s in v o lv e d w h ile th e second term , e x c e p t fo r the c o n sta n t m u lti­ p l i e r s , i s i d e n t i c a l w ith th e in t e g r a l fo r th e case fo r n * 1* T herefore eq u a tio n 50 becomes o For n • ) we have m \ = ^ * <*t ^ (52) w hich upon in t e g r a t io n becomes -r C — c o ^ C u r b ) d T o j A t . ( 55 ) o In eq u a tio n 55 th e f i r s t term o f th e r ig h t s id e w i l l alw ays be zero fo r the l i m i t s in v o lv e d w h ile the second term i s again id e n t ic a l w ith the 22 n » 2 term e x c e p t fo r th e c o n sta n t m u lt i p l i e r s . So th e r e fo r e eq u ation 55 y i e l d s x- G (5 4 ) 3' and s i m i l a r i l y fo r h ig h er v a lu e s o f n . So th a t eq uation 46 which g iv e s th e en ergy p a s s in g th e f i n a l s l i t Sl^ becomes ^ A \ a t a (' U \ '3 1 5' " J (5 5 ) fir. £ / 2 i n t h e s e r i e s , and w r i t i n g s a s 2 T ak in g o u t th e f a c t o r in W/ T re su lts e q u a t i o n 5 5 t a k i n g t h e fo rm c _ / f f > T V 2"& v , 2 ^ 1 3 ] £ > i\> r A -\ B e lts ' 81 U \V c» (56) 25 w h i c h m ay a l s o b e w r i t t e n i n t h e g e n e r a l fo r m oo s» . ^ 13 y _ ( 57) V I— O E q u a t i o n 57 i s th e n th e e x p r e s s io n f o r th e l i g h t e n e rg y t r a n s m itt e d th ro u g h s l i t S l ^ i n a tim e i n t e r v a l o f o n e - h a l f p e r io d f o r a g iv e n v a lu e o f B. The s o u n d p r e s s u r e a m p l i t u d e , t h e p o s i t i o n i n t h e s t a t i o n a r y w a v e , a n d c o n s t a n t s o f t h e sy stem a r e a l l c o n t a i n e d i n B ( s e e e q u a t i o n s 26 and 2 9 ). Now, i n t h e same l e n g t h o f t i m e , i . e . o n e - h a lf p e r io d , th e lig h t e n e r g y 5^ , w h ic h w o u ld p a s s t h r o u g h t h e s l i t S l ^ i f no s o u n d w e r e i n t h e c e l l c a n be e x p r e s s e d b y t h e r e l a t i o n So t h a t t h e r a t i o o f t h e e n e r g y p a s s e d w i t h t h e s o u n d on t o t h a t p a s s e d w i t h t h e s o u n d o f f S / S 0 , o r t h e r e l a t i v e e n e r g y p a s s e d - S ^ i s g i v e n by th e e x p re ssio n c: = 430 ^ = v\- O p "7- ^ f ( 2.H +\") Y\\ (vvt- >1 \ (59) I f t h e f i r s t few t e r m s o f e q u a t i o n 59 a r e w r i t t e n o u t a n d i f its e q u i v a l e n t from e q u a t i o n 29 i s a p p e a rs in th e s u b stitu te d , fo r B th e e x p re s s io n f o r f o l l o w i n g fo rm (6 0) 24 From t h i s e x p r e s s i o n i t i s p o s s i b l e to c a l c u l a t e th e v a lu e s o f SD f o r d i f f e r e n t p o s i t i o n s x a l o n g t h e w ave, a ssu m in g c e r t a i n v a l u e s f o r t h e p a r a m e t e r A. S e v e ra l t h e o r e t i c a l cu rv es c o rre sp o n d in g to v a rio u s v a lu e s o f A a r e shown i n F i g u r e 4 . The minimum p o i n t s on t h e c u r v e s o f F i g u r e 4 c o rre s p o n d to th e n o d es o f p r e s s u r e i n th e s t a t i o n a r y w ave, i . e . , p o s i t i o n s w here s i n ^ x / c ) th e s 1 , o r w h e re t h e a e r i e s o f e q u a t i o n 60 b e ­ com es (61) T h e r e f o r e , i n t h e o r y a t l e a s t i f i t w e re p o s s i b l e t o o b t a i n e x p e r i ­ m e n ta lly , th e v alu e o f f o r v a r io u s p o s i t i o n s a lo n g th e w ave, i t s h o u l d b e p o s s i b l e n o t o n l y t o o b t a i n some i d e a a b o u t t h e fo rm o f t h e sound w ave, b u t , s i n c e t h e A c o u ld be d e t e r m in e d , and s i n c e i t c o n ta in s t h e p r e s s u r e a m p l i t u d e a s a f a c t o r , t h i s m e th o d s h o u l d p r o v i d e a n a p ­ p ro ach to th e d e te rm in a tio n o f th e p r e s s u r e a m p litu d e i n a s t a t i o n a r y s o u n d w av e. T h e o r y o f S a w t o o t h Wave F o r m a t io n i n Waves o f F i n i t e A m p l i t u d e I n t h e d i s c u s s i o n a b o v e t h e a s s u m p t i o n h a s b e e n made t h a t t h e w a v e s are sin u so id a l. T h is i s a u s u a l a ssu m p tio n , b u t i t s h o u l d be w o r t h w h i l e to r e c a l l u n d e r w h at c o n d i t i o n s su ch an a s s u m p tio n i s v a l i d , assu m in g th e o rig in a l s o u rc e to be s i n u s o i d a l . s o i d a l w ave fo rm a s i t The p r o b le m o f w h a t h a p p e n s t o a s i n u ­ i a p r o p a g a t e d i s on e w h ic h h a s b e e n c o n s i d e r e d f o r m any y e a r s , S t o k e s h a v i n g m e n t i o n e d t h e p r o b le m a s e a r l y a s 1848 ( 1 5 ) . I n v e s tig a tio n s w e r e s u b s e q u e n t l y made b y Riem ann ( 1 6 ) a n d S a r n s h a w ( 1 7 ) a n d m e n t i o n e d by R a l e i g h ( 1 8 ) i n h i s b o o k . 25 1*0 u* £ 5 ul CC O P o s it io n F ig u re 4 . in W ave. T h e o r e tic a l c u rv e s o f r e l a t i v e l i g h t en erg y th ro u g h f i n a l s l i t S l ^ v s . p o s i t i o n i n w av e. 26 The p r o b l e m a r i s e s f r o m t h e f a c t t h a t i n t h e d e v e l o p m e n t o f t h e u s u a l e q u a t i o n s f o r a p l a n e s o u n d w ave a n a p p r o x i m a t i o n i s made w h ic h lim its t h e a p p l i c a t i o n o f t h e e q u a t i o n s to c a s e s o f i n f i n i t e l y a m p litu d e s# sm all T h i s a m o u n ts t o n e g l e c t i n g a n y c h a n g e i n c o m p r e s s i b i l i t y o f t h e m edium u n d e r c o n s i d e r a t i o n w i t h c h a n g e i n p r e s s u r e # The r e s u l t i s t h a t th e w aves behave a 3 th e th e o ry p r e d i c t s i f th e c o m p r e s s ib i li ty i s a c o n sta n t* or i f i t i s n o t* t h a t th e a m p litu d e i s fille d , i.e ., th e e q u a tio n s a r e s t i l l a p p lic a b le p ro v id e d sm all enough. I f th e se c o n d itio n s a re n o t f u l ­ i f we h a v e a wave o f f i n i t e a m p l i t u d e i n a medium h a v i n g a n o n - c o n s ta n t c o m p r e s s i b i l i t y i n th e p r e s s u r e ran g e u n d e r c o n s i d e r a t io n t h e v e l o c i t y o f p r o p a g a t i o n o f t h e w ave i s n o l o n g e r a c o n s t a n t , b u t i s d i f f e r e n t f o r d i f f e r e n t p o r t i o n s o f th e w ave, th e r e s u l t b e in g t h a t th e c o n d e n s a t i o n s g a i n c o n t i n u a l l y on t h e r a r e f a c t i o n s i n t h e wave w i t h t h e r e s u l t t h a t t h e r e i s a te n d e n c y f o r th e c o n d e n s a t io n s to o v e r t a k e th e ra re fa c tio n s, t e n d i n g t o f o rm a s a w t o o t h s h a p e d w a v e . p o in te d o u t (18) As R a l e i g h h a s t h i s p r o c e s s ca n n o t go o n i n d e f i n i t e l y . Pay ( 1 9 ) a n d B i q u a r d ( 2 0 ) h a v e i n v e s t i g a t e d t h i s p r o b le m m a t h e m a t i ­ c a lly . F ay o b t a i n e d a n e x a c t s o l u t i o n t o t h e e q u a t i o n o f m o t i o n i n t h e form o f a F o u r i e r s e r i e s . When t h e r e i s a n o n - 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 p r e s s u r e c h a n g e a n d s p e c i f i c vo lu m e c h a n g e , i . e . , c o m p re ssib ility , a n o n —c o n s t a n t t h e r e .is f o u n d to b e a g r a d u a l t r a n s f e r o f e n e r g y fro m t h e low f r e q u e n c y c o m p o n e n ts to t h o s e o f h i g h e r f r e q u e n c y , a g a i n t e n d i n g t o fo rm a s a w t o o t h w a v e . S i n c e t h e h i g h f r e q u e n c i e s t e n d to be a b s o r b e d m o r e r a p i d l y t h a n do t h e low f r e q u e n c i e s , o n e m i g h t e x p e c t t h e wave form t o becom e s t a b i l i z e d a f t e r a t i m e . H o w e v e r, t h e c o n d i t i o n s 27 fo r s ta b i li t y v a ry w ith i n t e n s i t y , so t h a t n o p e r m a n e n t s t a b l e fo rm r e s u l t s , a s o - c a l l e d wm o s t s t a b l e fo rm " w h i c h c h a n g e s i t s b u t o n ly s h a p e m o re g r a d u a l l y t h a n a n y o t h e r f o r m o f t h e same wave l e n g t h a n d i n t e n s i t y . The m o s t s t a ­ b l e f o r m f o r v e r y lo w i n t e n s i t i e s b eco m es a s i n e f o r m . The d i s t o r t i o n o f w av es i n t o a s a w t o o t h fo rm h a s b e e n o b s e r v e d e x ­ p e r i m e n t a l l y i n a i r by H ubbard, e t . a l. ( 2 1 ) , w h ile M ik h a ilo v (2 2 ) c la im s t o h a v e o b s e r v e d a n o n - l i n e a r e f f e c t o f s e v e r a l l i q u i d s on s o u n d w aves t r a v e r s i n g th e l i q u i d s . By s u b j e c t i n g a l i q u i d s i m u l t a n e o u s l y t o s o u n d s o f two d i f f e r e n t f r e q u e n c i e s a n d t h e n d e t e c t i n g n o t o n l y t h e o r i g i n a l fre q u e n c ie s, but a lso sum a n d d i f f e r e n c e f r e q u e n c i e s , h e c o n c l u d e d t h a t th e l i q u i d s had n o n - li n e a r c h a r a c t e r i s t i c s * I f th is is t r u e , one m ig h t a l s o , i n a c c o r d a n c e w i t h t h e t h e o r y o u t l i n e d a b o v e , e x p e c t p o s s i b l e saw ­ t o o t h wave f o r m a t i o n i n liq u id s. T h e o r y o f I n t e n s i t y D e c r e a s e o f U n d e f l e c t e d Beam A p p l i e d t o a S i n u s o i d a l Wave I n t e r f e r i n g w i t h a S a w t o o t h Wave T r a v e l i n g I n O p p o s i t e D i r e c t i o n In th e m a th e m a tic a l th e o ry d ev elo p ed e a r l i e r , b een assum ed th r o u g h o u t. s i n u s o i d a l w av es h a v e I f , h o w e v e r , a s a w t o o t h wave form d o e s d e v e l o p f r o m a n o r i g i n a l s i n e w a v e , t h e c u r v e o f l i g h t i n t e n s i t y on t h e o p t i c a x i s a s a f u n c t i o n o f t h e p o s i t i o n i n t h e wave t h r o u g h w h ic h t h e l i g h t h a s p a s s e d w o u ld b e e x p e c t e d t o v a r y fro m t h e t y p e o f c u r v e p l o t t e d i n F ig u re 4 . J u s t how t h e f i n a l c u r v e s h o u l d l o o k w o u ld d e p e n d u p o n t h e d e g r e e t o w h ic h a s a w t o o t h fo rm h a d d e v e l o p e d , a n d w h e t h e r t h e wave w i t h w h ic h tu re . it was i n t e r f e r i n g was e s s e n t i a l l y s in u s o id a l o r saw to o th in n a ­ H o w e v e r, i f a s a w t o o t h fo rm d o e s d e v e l o p , i t w o u ld be m o r e l i k e l y 28 t o b e f o u n d a t some d i s t a n c e f r o m t h e s i n u s o i d a l s o u r c e t h a n v e r y n e a r th e so u rce. By t h e same t o k e n o n e m i g h t e x p e c t t o f i n d s i n u s o i d a l w aves n e a r th e so u rc e . one a n o th e r , Thus, f o r o p p o s ite ly th e re i s d i r e c t e d waves i n t e r f e r i n g w ith some p o s s i b i l i t y t h a t n ear th e source a s in u s o id a l w ave t r a v e l i n g i n o n e d i r e c t i o n m i g h t b e i n t e r f e r i n g w i t h a s a w t o o t h wave tr a v e lin g in th e o p p o s ite d ir e c tio n . I t m i g h t be o f g a te w hat ty p e o f a f i n a l cu rv e t h i s s i t u a t i o n m ig h t tic u la r, to see i f i n t e r e s t to i n v e s t i ­ p ro d u c e , and i n p a r ­ such a s i t u a t i o n p ro d u c e s a n o n -sy m m e tric a l c u rv e . W ith t h i s i n m i n d , a g r a p h i c a l a n a l y s i s h a s b e e n made o f s u c h a situ a tio n . An a r b i t r a r y s i n u s o i d a l p r e s s u r e wave t r a v e l i n g t o th e r i g h t h a s b e e n a s s u m e d t o i n t e r f e r e w i t h a n a r b i t r a r y s a w t o o t h p r e s s u r e wave o f t h e same a m p l i t u d e t r a v e l i n g t o t h e l e f t . (S ee F ig u re 5 ) . In th e p a r t i c u l a r c a s e c h o s e n an e x t r e m e t y p e o f s a w t o o t h h a s b e e n c h o s e n f o r s im p lic ity , th o u g h t h i s ex tre m e ty p e co u ld n o t o c c u r The a n a l y s i s w as c a r r i e d o u t a s f o l l o w s : in r e a l i t y . The s i n e p r e s s u r e c u r v e a n d t h e s a w t o o t h p r e s s u r e c u r v e w ere draw n a s i n F i g u r e 5 , w h e r e t h e i r p o s i t i o n s c o r r e s p o n d t o t h e tim e t — 0 . The s i n e c u r v e was a s s u m e d to move t o t h e r i g h t w i t h a d e f i n i t e v e l o c i t y , w h i l e t h e s a w t o o t h wave m oved to th e l e f t w ith a v e l o c i t y o f e q u a l m a g n itu d e . The r e l a t i v e p o s i t i o n s o f t h e w a v e s w e re t h e n draw n f o r t i m e s o f t * T / 1 8 , t — 2 T / 1 8 , e t c . , w here T i s th e p e rio d o f v i b r a t i o n . The r e s u l t a n t p r e s s u r e w i t h i n a wave l e n g t h w as t h e n o b t a i n e d f o r e a c h ti m e (T - 0 , T / 1 8 , 2 T / 1 8 , e t c . ) , b y a l g e b r a i c a l l y a d d i n g t h e p r e s s u r e s due t o t h e s i n e a n d s a w t o o t h com­ p o n e n t w aves. T h is r e s u l t e d in e ig h te e n d i f f e r e n t c u rv e s , each r e p r e ­ s e n tin g th e p re s s u re d i s t r i b u t i o n a l o n g a wave l e n g t h f o r a g i v e n i n s t a n t 29 F ig u re 5 , R e s u l t a n t o f s i n u s o i d a l wave t r a v e l i n g t o r i g h t a n d s a w t o o t h wave t r a v e l i n g t o l e f t ( t i m e t = 0 ) . i n tim e c o r r e s p o n d in g to one o f th e tim e s t * 0 , T /1 8 , 2 T /1 8 , e tc * Now, t h e d e f l e c t i o n o f t h e l i g h t beam t h r o u g h a p o i n t i n t h e s o u n d f i e l d a t a g iv e n tim e i s a d i r e c t f u n c t i o n , n o t o f th e p r e s s u r e b u t o f th e p r e s s u r e g ra d ie n t* T h e r e f o r e , i t was n e c e s s a r y t o d e te rm in e th e s lo p e o f th e p r e s s u r e c u r v e s f o r v a r i o u s p o s i t i o n s i n t h e w av e. An a p p r o x i m a t i o n t o t h e a v e r a g e s l o p e o v e r a s m a l l p o r t i o n o f a wave l e n g t h was o b t a i n e d by m e a s u r i n g g r a p h i c a l l y t h e s l o p e o f t h e c h o r d c o n n e c t i n g two p o i n t s o n t h e p r e s s u r e c u r v e s e p a r a t e d by a d i s t a n c e o f X / 1 8 , le n g th . X T hus, f o r each p r e s s u r e c u r v e , a v alu e f o r i t s b e i n g t h e wave s l o p e w as o b ­ t a i n e d f o r e i g h t e e n e q u a l l y s p a c e d p o s i t i o n s w i t h i n o n e wave l e n g t h . The d e f l e c t i o n d o f t h e l i g h t beam t h r o u g h a g i v e n p o s i t i o n i n t h e wave a t any i n s t a n t i s d i r e c t l y p r o p o r ti o n a l to t h i s s lo p e o f th e p r e s s u r e cu rv e a t th e p o in t* The d e f l e c t i o n d c a u s e d by t h e maximum s l o p e o b max t a i n e d i n t h e a b o v e a n a l y s i s was a r b i t r a r i l y 50 s e t eq u a l to o n e - h a l f th e w i d t h o f t h e c e n t r a l maximum o f t h e d i f f r a c t i o n p a t t e r n o f s l i t S I th e p la n e o f s l i t S l^ , a x i s t o becom e z e r o in T h i s c a u s e d t h e l i g h t i n t e n s i t y on t h e o p t i c a l f o r t h i s case* B ecause o f th e p r o p o r t i o n a l i t y e x ­ i s t i n g b etw een d and th e s lo p e o f th e p r e s s u r e c u r v e , t h i s p ro c e d u re a u t o m a t i c a l l y t i e d any g iv e n v a lu e f o r th e slo p e to a d e f i n i t e v a lu e o f d* The v a l u e s f o r d c o r r e s p o n d i n g t o t h e m e a s u r e d v a l u e s o f t h e s l o p e s w ere th e n c o m p u ted . The a m o u n t o f l i g h t p a s s i n g t h e f i n a l s l i t S l ^ f o r a n y g i v e n d v a l u e c o u l d t h e n be d e t e r m i n e d i f t h e d i f f r a c t i o n p a t t e r n w e re known* t r i b u t i o n w as o b t a i n e d a s f o l l o w s : th e l i g h t d i s t r i b u t i o n in A reco rd er tra c e o f th is lig h t d is­ A p h o t o m u l t i p l i e r tu b e o f a m ic r o ­ p h o t o m e t e r w as p l a c e d b e h i n d s l i t S l ^ a n d c o n n e c t e d t o a r e c o r d e r * S lit 3 1 ^ t o g e t h e r w i t h t h e p h o t o m u l t i p l i e r t u b e w e re m o u n te d on a m i c r o m e t e r s c r e w w h ic h w as d r i v e n by a s y n c h r o n o u s m o t o r a c r o s s t h e s l i t d i f f r a c t i o n p a t t e r n a t a r a t e o f one m i lli m e t e r p e r m in u te . M e a n w h ile t h e r e c o r d e r p a p e r was m o v i n g a t a r a t e o f two i n c h e s p e r m i n u t e . T h u s , a t r a c e was o b t a i n e d o n t h e r e c o r d e r p a p e r w h ic h show ed l i g h t i n t e n s i t y v e r s u s p o ­ s itio n in th e d i f f r a c t i o n p a t t e r n . (S ee F ig u re 6 ) . S in c e th e r e c o r d e r Bc&le was a l i n e a r o n e , w h i l e t h a t o f t h e m i c r o p h o t o m e t e r was n o t q u i t e lin e a r, th e r e c o rd e r a f t e r be r e f e r r e d t o d id n o t r e a d a t r u e i n t e n s i t y , and i t as " reco rd er in te n s ity .” Now, s i n c e t h e d e f l e c t i o n s g iv e n p o r t i o n o f th e s h a ll h e re ­ d fo r a lig h t wave w e re known f o r ti m e I t was a s i m p l e m a t t e r t o o b t a i n beam p a s s i n g t h r o u g h a t = 0 , T /1 8 , 2 T /1 8 , e t c . th e " r e c o r d e r i n t e n s i t y " a x i s a t th e s e tim e s w ith th e a i d o f F ig u re 6 . o b ta in e d o f th is " re c o rd e r in te n s ity " 51 on t h e o p t i c a l From t h e e i g h t e e n v a l u e s f o r a g i v e n p o s i t i o n i n t h e wave a -B h tm iin g ith e I u O 1 L : t so* > z ui £ 70- UJ > I 6 6°- of ui s 8 u» £ 50- O \ F i g u r e 7* C u rv e o f t h e o r e t i c a l 11r e c o r d e r r e l a t i v e i n t e n s i t y 18 v s . p o s i t i o n i n wave due t o s i n u s o i d a l wave i n t e r f e r i n g w i t h o p ­ p o s i t e l y d i r e c t e d s a w t o o t h w av e. 55 m ean v a l u e o v e r t h e p e r i o d T w as c o m p u te d . r e c o r d e r in te n sity * * T h i s r e p r e s e n t e d t h e "mean f o r one p o s i t i o n i n th e w ave. S im ila rly v a lu e s of m ean r e c o r d e r i n t e n s i t y 18 w e r e o b t a i n e d f o r e a c h o f t h e e i g h t e e n e q u a l l y s p a c e d p o s i t i o n s a l o n g a wave l e n g t h . te n sity " v a l u e s b y t h e maximum " r e c o r d e r i n t e n s i t y 18 a t t h e c e n t r a l m a x i ­ mum o f t h e c u r v e o f F i g u r e co rd er in te n s ity " le n g th . D i v i d i n g t h e s e "m ean r e c o r d e r i n ­ 6, p r o d u c e d w h a t m i g h t be c a l l e d " r e l a t i v e r e ­ r e a d i n g s f o r e a c h o f t h e e i g h t e e n p o i n t s a l o n g t h e wave T h e s e r e a d i n g s w e r e p l o t t e d a g a i n s t p o s i t i o n i n t h e wave a n d r e ­ s u l t e d i n t h e c u r v e shown i n F i g u r e 7* r e t i c a l cu rv e, y e t i t te n sitie s A lt h o u g h t h i s c u r v e i s a t h e o ­ show s 18r e c o r d e r i n t e n s i t i e s " sin c e a l l i t s r a t h e r th a n a c t u a l i n ­ i n t e n s i t i e s w e re o b t a i n e d fro m F i g u r e i t s e l f a r e c o r d e r r e c o r d , and th e r e f o r e a l l i t s 6, w h ic h i s in te n s itie s are "reco rd er i n t e n s i t i e s . 18 A lth o u g h th e m anner o f a p p ro a c h in t h i s a n a l y s i s has been r a t h e r a r b i t r a r y , and a lth o u g h i t f i n a l curve w ill is r e a l i z e d t h a t th e p a r t i c u l a r shape o f th e d e p e n d u p o n some f a c t o r s w h ic h h a v e b e e n a r b i t r a r i l y a ssu m e d , a s f o r e x a m p le , th e a m p litu d e o f th e com ponent w aves, and th e p a rtic u la r d e f l e c t i o n c o rre s p o n d in g to a g iv e n p r e s s u r e g r a d i e n t , i t is n e v e r t h e l e s s h o p e d t h a t some s i g n i f i c a n c e c a n be a t t a c h e d to t h e asy m ­ m e tr y and th e g e n e r a l shape o f th e c u r v e . & EXPERIMENTAL APPARATUS O sc illa to r The o s c i l l a t o r u s e d f o r d r i v i n g t h e t r a n s d u c e r s u s e d i n t h i s i n ­ v e s t i g a t i o n w as a c o m m e r c i a l t y p e m a n u f a c t u r e d b y B r u s h E l e c t r o n i c s Company u n d e r t h e t r a d e name " H y p e r s o n i c G e n e r a t o r , m o d e l B U -2 0 4 ." T h is m odel can be e q u ip p e d w ith d i f f e r e n t tu n in g u n i t s f o r c o v e rin g d i f f e r e n t freq u en cy ra n g e s. I n t h i s i n v e s t i g a t i o n t h e t u n i n g u n i t u s e d w as t h a t d e s i g n a t e d b y t h e B r u s h Company a s " T u n i n g D r a w e r , m o d e l BU-4 o 4-B " , a n d c o v e re d th e f r e q u e n c y ra n g e 0 .5 m e g a c y c le /s e c o n d to one m e g a c y c le /se c o n d * T h is g e n e r a to r i s e s s e n t i a l l y a s e l f - e x c i t e d H a rtle y o s c i l l a t o r u s i n g a t y p e 8 1 0 t u b e a s t h e o s c i l l a t o r a n d u s i n g two t y p e 866 -A t u b e s as r e c t i f ie r s . Its n o m in al r a d io f r e q u e n c y o u t p u t i s r a t e d b y t h e m anu­ f a c t u r e r s a s 250 w a t t s . T ran sd u cers The t r a n s d u c e r s u s e d i n t h i s i n v e s t i g a t i o n w e r e p r e - p o l a r i z e d b a r i ­ um t i t a n a t e c e r a m i c e l e m e n t s m a n u f a c t u r e d by B r u s h E l e c t r o n i c s Company. F o r t h e p h o t o g r a p h i n g o f t h e s t a t i o n a r y w aves a s w e l l a s some p r e ­ l i m i n a r y w ork o n t h e o t h e r a s p e c t s o f t h i s i n v e s t i g a t i o n a b a r iu m t i t a ­ n a t e ( B r u s h c e r a m i c A) d i s c , u s e d a s a t h i c k n e s s v i b r a t o r , was e m p lo y e d . The n o m i n a l d i a m e t e r o f t h e d i s c was o n e i n c h a n d i t s n o m i n a l r e s o n a n c e f r e q u e n c y was 0 . 4 m e g a c y c l e / s e c o n d . For t h e r e m a i n i n g p o r t i o n o f t h e i n v e s t i g a t i o n a b ariu m t i t a n i t e p l a t e 55 (B ru sh ce ra m ic B ), a ls o a t h i c k n e s s v i b r a t o r , w as u s e d . I t s n o m i n a l r e s o n a n c e f r e q u e n c y was l i k e w i s e 0.4- m e g a c y c l e / s e c o n d a n d i t s n o m i n a l s i z e o r i g i n a l l y was 2 - 5/16 i n c h e s square. T h i s was c u t i n t o two p i e c e s so t h a t t h e p i e c e a c t u a l l y u s e d i n t h e w o rk w as 2 - 5 / 1 6 i n c h e s b y a p p r o x i m a t e l y 1 - 1 / 1 6 i n c h e s . B o th t r a n s ­ d u c e r s w hen u s e d w e r e m o u n t e d i n h o l d e r s w i t h a n a i r b a c k i n g f o r r a d i ­ a t i n g to one s i d e . O p tic a l A rran g em en ts S ev eral s lig h tly d i f f e r e n t o p t i c a l a r r a n g e m e n t s w e re u s e d . For th e p h o t o g r a p h i n g o f t h e s o u n d p r e s s u r e d i s t r i b u t i o n i n t h e s o u n d w ave by t h e l i g h t r e f r a c t i o n m e th o d (M e th o d I ) , t h a t shown i n F ig u re 8 . t h e b a s i c a r r a n g e m e n t u s e d was A source o f in te n s e l i g h t L illu m in a te d a s l i t S l^ th ro u g h a c o n d e n se r le n s 0 . The l e n s w as u s e d t o i l l u m i n a t e t h e s l i t 31^ w i t h a f a i r l y n a r r o w beam , w h i l e l e n s w as u s e d t o f o c u s a s h a r p im a g e o f t h e e d g e s o f s l i t S l^ o n t h e s c r e e n S c , w h ic h w as r e p l a c e d in some c a s e s by a p h o t o g r a p h i c f i l m . The l i g h t fro m 31^ was a l l o w e d to p a s s t h r o u g h t h e c e l l c o n t a i n i n g t h e s o u n d wave o n i t s way t o t h e s c r e e n . The n e e d f o r t h i s f a i r l y c o m p l i c a t e d o p t i c a l a r r a n g e m e n t a r o s e fro m th e ty p e o f l i g h t so u rc e u s e d . v o lt sin g le I t was a G e n e r a l E l e c t r i c t y p e 52C 6-8 f ila m e n t h e a d lig h t b u lb . W ith t h i s s o u r c e , i t was v e r y d i f ­ f i c u l t to o b t a i n ev en i l l u m i n a t i o n o v e r a l a r g e enough a r e a by i l l u m i n ­ a t i n g s l i t S l^ d i r e c t l y w ith th e c o n d en ser in th e o r d in a r y m anner, i . e . , b y f o r m i n g a n im a g e o f t h e f i l a m e n t a t S l ^ T h e re fo re a h o u sin g w ith a s m a l l h o l e was p l a c e d a r o u n d t h e lamp L a n d t h e im ag e o f t h e h o l e was f o c u s e d o n -Sl-p The s e c o n d l e n s t h e n became n e c e s s a r y i n o r d e r to 56 Sc 1— CD A V C SI. F ig u re 6. L, Basi c- o p t i c a l SI BaTi 0 3 a r r a n g e m e n t u s e d f o r l i g h t r e f r a c t i o n m e th o d (M etho d I )• Yl p r o d u c e a s h a r p , n a r r o w , y e t w e l l - d e f i n e d beam b o t h t h r o u g h t h e c e l l , w e ll a s on th e s c r e e n . c ritic a l, The a d j u s t m e n t o f t h e w i d t h o f s l i t S I aa was v e r y s i n c e i t w as n e c e s s a r y t o k e e p t h e beam w i d t h t h r o u g h t h e c e l l s m a l l c o m p a r e d t o a wave l e n g t h o f s o u n d , a n d y e t i f S I w e r e m ade to o 2 n a r r o w , t h e s p r e a d i n g o u t due t o t h e d i f f r a c t i o n e f f e c t s w o u ld make t h e beam t o o w i d e i n t h e c e l l . V a r i a t i o n s o f t h i s b a s i c a r r a n g e m e n t w e re u s e d f o r p h o t o g r a p h i n g t h e s o u n d p r e s s u r e d i s t r i b u t i o n i n t h e s o u n d w av e. F o r l a t e r w o rk ( M e th o d I I ) w as u s e d . th is T h is arran g em en t i s a s o m ew h at d i f f e r e n t o p t i c a l a r r a n g e m e n t shown i n c a s e w as a G e n e ra l E l e c t r i c F ig u re 9- The l i g h t s o u r c e L i n t y p e AH- 4 1 0 0 - w a t t m e r c u r y v a p o r la m p . An im ag e o f t h e s o u r c e w as f o c u s e d on s l i t S l ^ by m e an s o f t h e c o n d e n s e r Q. The s l i t S l ^ t h e n b eca m e a s e c o n d a r y s o u r c e , a n d t h e l i g h t f ro m i t w as r e n d e r e d p a r a l l e l b y l e n s and p a s s e d th ro u g h th e sound c e l l a f t e r f i r s t b e i n g l i m i t e d i n w i d t h by t h e s l i t S l ^ , The f i n a l s l i t 3 1 ^ was a P o h l p r e c i s i o n t y p e s l i t m a n u f a c t u r e d by S p i n d l e r a n d H o y er i n G erm any. A G a e r t n e r t y p e L - 5 4 1 —E f i l t e r was e m p lo y e d b e h i n d t h e s l i t la te t h e m e r c u r y 5461 g r e e n l i n e . i B° — The l i g h t p a s s i n g t h r o u g h t h e f i n a l s l i t was m e a s u r e d by a n A m e ric a n I n s t r u m e n t Company No. 1 0 - 2 1 0 Aminco P h o to m u l ti p lie r M ic ro p h o to m e te r. The o u t p u t o f t h i s r a i c r o p h o t o m e t e r was f e d d i r e c t l y i n t o a M i n n e a p o l i s - H o n e y w e l l Brown E l e c t r o n i k r e c o r d e r , m o d e l No. 1 5 5 X 11 V -X -28. read r e la tiv e c a lib ra te S i n c e t h e m e t e r o n t h e m i c r o p h o t o m e t e r , w h ic h l i g h t i n t e n s i t y was n o t a l i n e a r s c a l e , i t was n e c e s s a r y to th e r e c o r d e r s c a l e a g a i n s t th e s c a l e on th e m ic ro p h o to m e te r• c a lib ra tio n c u r v e Bhowing t h e r e l a t i o n b e t w e e n t h e two s c a l e s i s Figure 10. 58 A shown i n 3 BoTIO Cell Photomultiplier F ig u re 9 . O p tic a l arrangem ent fo r o b ta in in g re c o rd e r cu rv es o f i n t e n s i t y c h a n g e i n u n d e f l e c t e d beam . (M e th o d I I ) * 59 90 - 70 60 - R ecover | R e ^ o iw g 80- 20 - lO o \o 20 30 40 50 M lC R O PH O T O r^eTE R F ig u re 10. C a lib ra tio n 70 SO lO O R eAD^G c u r v e . M i c r o p h o i o r - e 'e r r e a d i n g v s . read in g . ^0 90 reco rd er F re q u e n c y and C u r r e n t M e te rs M e a s u r e m e n t s o f f r e q u e n c y f o r u s e i n c a l c u l a t i n g s o u n d v e l o c i t y w ere m ade w i t h a U .S * Army S i g n a l C o rp s t y p e B C -2 2 1 -J f r e q u e n c y m e t e r f o r u s e i n th e ra n g e 0 .1 2 5 to 20 m e g a c y c le s /s e c o n d . N o m in a l v a l u e s o f f r e q u e n c y w e r e m ade w i t h a s i m i l a r b u t u n c a l i b r a t e d i n s t r u m e n t f o r t h e m e a s u r e m e n t s w here f r e q u e n c y d id n o t e n t e r d i r e c t l y i n t o th e c a l c u l a t i o n s . The v a l u e o f t h e R . F . c u r r e n t i n t o th e b ariu m t i t a n a t e tra n sd u c e r w as m e a s u r e d by a S im p s o n 0 —500 m a. t h e r m o c o u p l e t y p e R . F . mi H i am m eter • METHODOLOGY AND RESULTS P h o to g r a p h ic R ecord o f S p a t i a l P r e s s u r e D i s t r i b u t i o n i n a S t a t i o n a r y U l t r a s o n i c Wave P h o t o g r a p h i c r e c o r d b y l i g h t r e f r a c t i o n m e th o d (M e th o d I ) . g ra p h in g th e th e s p a tia l p ressu re d i s t r i b u t i o n i n a s t a t i o n a r y s o u n d wave by l i g h t r e f r a c t i o n m e t h o d (M e th o d I ) may be a c c o m p l i s h e d a s f o l l o w s (2 5 ): The b a s i c o p t i c a l a r r a n g e m e n t h a s b e e n d e s c r i b e d i n a p r e v i o u s shown i n F i g u r e 8 , p a g e 57* s e c t i o n and i s tic P h o to ­ A som ew h at s i m p l i f i e d s c h e m a ­ d i a g r a m o f t h e a r r a n g e m e n t s h o w in g t h e m o d i f i c a t i o n s u s e d i n t h e b a s ic se t-u p is shown i n F i g u r e 11# L i g h t , a f t e r p a s s i n g t h r o u g h t h e s l i t S l^ was c o n f i n e d t o a v e r y n a r r o w beam* I t is t h e n p a s s e d t h r o u g h l e n s L2 w h ic h f o rm e a s h a r p im ag e o f t h e e d g e s o f S l ^ u p o n t h e r o t a t i n g c y l i n d e r w h ic h h a s r e p l a c e d t h e screen . T h i s r o t a t i n g c y l i n d e r h a d a p h o t o g r a p h i c f i l m m o u n te d on i t . T he c e l l c o n t a i n i n g t h e s t a t i o n a r y s o u n d wave w as t h e n m oved b y m e a n s o f a m ic ro m e te r screw d riv e n by a v a r i a b l e speed m o to r, and s im u lta n e o u s ly , t h e c y l i n d e r w as r o t a t e d s l o w l y by m e a n s o f a s e c o n d s i m i l a r m o t o r . l i g h t r e a c h i n g t h e p h o t o g r a p h i c f i l m was l i m i t e d i n t h e v e r t i c a l The d ire c ­ t i o n by a f a i r l y w id e h o r i z o n t a l s l i t p l a c e d i m m e d i a t e l y i n f r o n t o f t h e c y lin d e r. The a l t e r n a t e in c r e a s e and d e c re a s e i n th e b ro a d e n in g o f th e beam a s i t t r a v e r s e d t h e v a r i o u s p o r t i o n s o f t h e s t a t i o n a r y s o u n d wave r e s u l t e d i n t h e t y p e o f p h o t o g r a p h i c r e c o r d shown i n F i g u r e 1 2 . As h a s b e e n p o i n t e d o u t e a r l i e r , 42 t h e b r o a d e n i n g o f t h e beam i s -icrzO Q ) P >* P p P O cd oc Cm *H p P «S C P C D CD 0 < D cri bO C § *H P P c CO o * •ft < — 1 p t— c GO 3 O p T3 •H •H O P P X P. P P O CD < D •ft s r~H ori o d> •H P P o cd cj a & P< *H •r-4 p o Cc. bO co o ari p p o p X p Ph 3 F i g u r e 12. P h o t o g r a p h i c r e c o r d , of s p a t i a l p r e s s u r e d i s t r i b u t i o n in a s t a t i o n a r y u l t r a s o n i c w ave o b ta i n e d b y lig h t r e f r a c t i o n m e th o d (M ethod I). 44 g r e a t e s t a t t h e p r e s s u r e n o d e s an d l e a s t a t t h e p r e s s u r e a n t i n o d e s . T h ere fo re in t h e p h o t o g r a p h o f F i g u r e 12 t h e b r o a d p o r t i o n s r e p r e s e n t th e p r e s s u r e n o d e s , w h ile th e narrow p o r ti o n s r e p r e s e n t th e p r e s s u r e lo o p s* P h o t o g r a p h i c r e c o r d b y i n t e n s i t y v a r i a t i o n i n u n d e f l e c t e d beam ( M e th o d I I ) . I f i n s t e a d o f p h o t o g r a p h i n g t h e b r o a d e n i n g o f t h e beam d i r e c t l y a s d e s c r i b e d a b o v e , u s e i s made o f t h e v a r i a t i o n i n i n t e n s i t y o f t h e u n d e f l e c t e d beam (M e th o d I I ) i s o b t a i n e d , w h ic h a l s o show s t h e s p a t i a l p r e s s u r e d i s t r i b u t i o n s t a t i o n a r y u l t r a s o n i c wave ( 2 4 ) . is still ( 1 1 ) , a d i f f e r e n t ty p e o f p h o to g ra p h The o p t i c a l a r r a n g e m e n t f o r t h i s c a s e b a s i c a l l y t h a t show n i n F i g u r e 8 , p a g e 37# m o d i f i e d a s 3hown i n t h e s c h e m a t i c d i a g r a m o f F i g u r e 13* p la c e d in to film * in a I n t h i s c a s e , a narrow s l i t S l^ i s t h e p a t h o f t h e l i g h t beam b e t w e e n t h e s o u n d wave a n d t h e As t h e c e l l c o n t a i n i n g t h e s o u n d wave i s m oved i n t h e d i r e c t i o n in d ic a te d , t h e l i g h t r e a d i n g t h e f i l m w i l l be v a r i e d ; m o re l i g h t p a s s i n g t h r o u g h t h e s l i t S l ^ when t h e beam i s n a r r o w a n d l e s s when t h e beam i s broad. T h u s, th e g r e a t e r th e v a r i a t i o n i n r e f r a c t i v e in d e x i n th e t h i n l a y e r w h i c h t h e beam t r a v e r s e s t h r o u g h t h e s o u n d f i e l d , th e g r e a t e r w ill b e t h e b r o a d e n i n g , a n d t h e s m a l l e r w i l l be t h e e x p o s u r e o n t h e f i l m . T h erefo re, i f th e film c o n ta in in g th e s t a t i o n a r y i s moved i n t h e same d i r e c t i o n a s t h e c e l l s o u n d w av e, an im age o f t h e s o u n d wave i s p r o ­ d u c e d i n w h ic h p r e s s u r e g r a d i e n t v a r i a t i o n s i n t h e s o u n d f i e l d a r e c o n ­ v e rte d in to d e n s i t y v a r i a t i o n s o n t h e p h o t o g ram . p h o to g ra p h o b ta in e d i n t h i s m anner. (n e g a tiv e ) F i g u r e 14 shows a S i n c e on t h e p h o t o g r a p h i c t h e g r e a t e r e x p o s u r e s w i l l o c c u r w h ere b r o a d e n i n g i s 45 film le a st, a i o to -H © p P <0 P, -H P r— t CO aj > •H P >» cd P P i *r-f (Q CO © c © p p c P bO C «M •H r> id P O a .8 © p p © i~H © I 1P aJ -P o •H > » SiP. P bo cd o & -p o p p o p p c © © _ o •H C o © © p p >> !— t P bO cS t ) £ Q o si •H p P © cd 35 P CO © © P P P ■H 13 £ © o P •H O K\ P © 3* P P a ■H T3 CJ c P P © •H © © p p 3 P in o 'P. F i g u r e 14. P h o t o g r a p h i c r e c o r d of s p a t i a l p r e s s u r e d i s t r i b u t i o n in a s t a t i o n a r y u l t r a s o n i c w ave o b t a in e d by the m e th o d of i n t e n s i t y v a r i a t i o n of the u n d e f l e c t e d b e a m (M ethod II). 47 i.e ., a t p re s s u re lo o p s, th e d a rk p o r t i o n s o f th e p o s i t i v e r e p r o d u c tio n o f F i g u r e 14 r e p r e s e n t t h e p r e s s u r e n o d e s , a n d t h e l i g h t e r p o r t i o n s p r e s ­ su re lo o p s, o r th e d a r k p o r t i o n s may be c o n sid e re d to r e p r e s e n tp re s s s u re g r a d i e n t lo o p s and th e l i g h t e r p o rtio n s nodes o f p re s s u re g r a d ie n t. I t may b e o f i n t e r e s t t o p o i n t o u t by c o n s id e r in g th e t h a t t h i s m e th o d c a n be e x p la in e d s t a t i o n a r y u l t r a s o n i c f i e l d i n a t r a n s p a r e n t l i q u i d to be o p t i c a l l y e q u i v a l e n t to a s e r i e s o f c y l i n d r i c a l l e n s e s ; a l t e r n a t e l y d i v e r g e n t and co n v e rg e n t* l a t i o n to th e The l o c a t i o n o f t h e s e l e n s e s i s f i x e d i n r e ­ s o u n d f i e l d b u t t h e l e n s e s v a r y c o n t i n u o u s l y fro m a m a x i ­ mum p o s i t i v e p o w e r t o a maximum n e g a t i v e p o w e r . At th e nodes o f th e sound p r e s s u r e g r a d i e n t t h e r e i s n o th in g to cau se a d iv e rg e n c e o r con­ vergen ce o f th e l i g h t . A l i g h t beam p a s s i n g t h r o u g h t h e s e p o s i t i o n s w i l l t h e r e f o r e show no d e v i a t i o n a n d p a s s t h r o u g h s l i t S l ^ . s itio n s in B etw een t h e s e p o ­ t h e s o u n d w a v e , l i g h t w i l l b e d e f l e c t e d aw ay f r o m t h e s l i t . The g r e a t e r t h e v a r i a t i o n i n p r e s s u r e g r a d i e n t , t h e m o re l i g h t w i l l be d e f l e c t e d and th e s m a lle r th e exposure o f th e f ilm . M easu rem en t o f Sound V e lo c ity The m e t h o d o f d e c r e a s e i n i n t e n s i t y o f t h e u n d e f l e c t e d beam was f o u n d t o b e r a t h e r e a s i l y a d a p t a b l e to t h e f a i r l y r a p i d m e a s u r e m e n t o f sound v e l o c i t y i n t r a n s p a r e n t l i q u i d s . t h a t shown i n F i g u r e 8 , p a g e 57* as fo llo w s: The s e t — u p u s e d was b a s i c a l l y The m e th o d o f v e l o c i t y m e a s u r e m e n t was The c e l l i n w h ic h t h e p o s i t i o n o f t h e t r a n s d u c e r h a d b e e n a d j u s t e d f o r o p tim u m s t a n d i n g w av es was m o u n te d o n a m i c r o m e t e r s c re w t r a v e r s e m o u n t w h ic h c o u l d be o p e r a t e d m a n u a l l y , c a u s i n g t h e c e l l 48 t o move a t r i g h t a n g l e s t o t h e l i g h t beam . T he s o u n d was t u r n e d o n a n d t h e m i­ c r o m e t e r s c r e w a d j u s t e d t o g i v e a minimum d e f l e c t i o n on t h e r e c o r d e r . The f r e q u e n c y m e t e r w as a d j u s t e d f o r z e r o b e a t # A p o s itio n read in g fo r t h e c e l l w a s t h e n t a k e n fr o m t h e m i c r o m e t e r s c r e w . The c e l l w as t h e n m o v e d b y m e a n s o f t h e m i c r o m e t e r s c r e w a t r i g h t a n g l e s t o t h e l i g h t beam . The d i s t a n c e t r a v e l e d i n m o s t o f t h e m e a s u r e m e n t s was s l i g h t l y l e s s t h a n fiv e c e n tim e te rs. As t h e c e l l m oved a c r o s s t h e l i g h t beam a s e r i e s o f m a x im a a n d m in im a w e r e drawn o n t h e r e c o r d e r c h a r t . th e end o f i t s As t h e s c r e w n e a r e d t r a v e l t h e p h o t o m e t e r n e e d l e was w a tc h e d c l o s e l y a n d t h e e f f o r t w as m ade t o minimum r e a d i n g , s t o p t h e m o t i o n a s c l o s e l y a s p o s s i b l e a t an e x a c t so t h a t i n s o f a r a s was p o s s i b l e a d i s t a n c e e q u a l t o a n i n t e g r a l n u m b e r o f h a l f - w a v e l e n g t h s w o u ld be t r a v e r s e d . The m o t i o n o f t h e s c r e w w as t h e n s t o p p e d a n d a s e c o n d p o s i t i o n r e a d i n g o f t h e c e l l w as ta k e n . T he d i f f e r e n c e b e t w e e n i n i t i a l a n d f i n a l p o s i t i o n r e a d i n g s g a v e th e t o t a l d i s t a n c e t r a v e r s e d by th e c e l l . th e fre q u e n c y m e te r d i a l , The f r e q u e n c y was r e a d fro m and t h e num ber o f h a l f - w a v e l e n g t h s t r a v e r s e d c o u l d e a s i l y b e c o u n t e d fro m t h e r e c o r d e r c h a r t . From t h e num ber o f h a l f —wave l e n g t h s c o v e r e d a n d t h e d i s t a n c e t r a v e r s e d t h e wave l e n g t h , c o u ld be d e te rm in e d . , By u s e o f t h e f r e q u e n c y r e a d i n g , f , t h e v e l o c i t y c w as d e t e r m i n e d f r o m t h e e q u a t i o n C = f K . ( 62) A t e m p e r a t u r e r e a d i n g o f t h e medium was t a k e n b e f o r e a n d a f t e r a r u n w i t h a m ercu ry in g la ss t h e r m o m e t e r , and t h e mean v a l u e t a k e n a s t h e t e m p e r a ­ tu re . 49 F ig u re 15. T y p i c a l r e c o r d e r "brace o b t a i n e d i n m a k in g s o u n d v e l o c i t y m e a s u r e m e n t s by t h e m e th o d o f i n t e n s i t y v a r i a t i o n o f t h e u n d e f l e c t e d beam . TABLE I R e s u lts o f Sound V e lo c it y Measurements S u b stan ce Sound V e lo c it y ( m ./s e c .) A cetone* (5 rune) mean v a lu e Carbon T e tr a c h lo r ­ id e* * (4 runs) mean v a lu e 0 -X y len e ( 4 runs) mean v a lu e Dow-Coming 200 f l u i d ( l c e n t ! s t o k e ) ( 5 runs) 1155 1165 1169 1165 916.1 9 1 8 .9 9 1 5 .5 9 1 7 .9 917 .I 1555 1542 1551 1546 1548 9 8 1 .6 9 8 7 .1 9 8 5 .1 9 8 4 .6 Temperature (D eg. C) at at at at 2 7 .0 2 7 .8 2 7 .6 2 7 .5 at at at at at 2 8 .1 2 8 .2 2 8 .5 2 8 .4 2 8 .? at at at at at 2 6 .5 2 6 .4 2 6 .5 2 6 .6 2 6 .5 at at at at 2 6 .1 2 6 .4 2 6 .5 2 6 .5 * F reyer (2 6 ) g iv e s 1190 m ./s e c . a t 20° C fo r a ceto n e w ith A v /& T * - 5 .6 m ./ s e c . d eg. C. This y ie ld s 1148 m ./s e c . a t 2 7 .5 ° 0 . ** F reyer (2 6 ) g iv e s 9 0 4 .0 m ./s e c . a t 5Q0 0 fo r carbon te tr a c h lo r id e w ith ZSiv/^T * 5 .1 m ./s e c . deg. 0 . T his y ie ld s 9 0 9 .5 m#/sec» 2 8 .5 ° C. 51 A t y p i c a l r e c o r d e r re c o rd o f a t y p i c a l run i s shown i n F i g u r e 1 5 . I n d i v i d u a l r u n s i n a g i v e n s u b s t a n c e w e re f o u n d t o a g r e e g e n e r a l l y w i t h i n l e s s t h a n 1.59& b e t w e e n e x t r e m e v a l u e s , a n d a l s o w e ll w ith v a lu e s g iv e n in th e l i t e r a t u r e . su red v alu es i s s eem e d t o a g r e e A l i s t i n g o f some o f t h e m e a ­ shown i n T a b l e I . A lth o u g h o t h e r m e th o d s e x i s t f o r th e m e asu re m en t o f sound v e l o c i t y i n l i q u i d s w h i c h a r e c e r t a i n l y m o re p r e c i s e t h a n t h i s m e t h o d , e v e n a t th e se fre q u e n c ie s (2 5 ), n e v e rth e le ss i t i s f e l t t h a t th e r a p i d i t y w ith w h i c h a r u n c a n b e made (10 m i n u t e s o r l e s s i n c l u d i n g c a l c u l a t i o n s ) s h o u l d m ake t h i s m e th o d o f some u s e , e v e n w i t h t h e r e l a t i v e l y low p r e ­ c isio n * T h e m e th o d c o u l d b e made a l m o s t f u l l y a u t o m a t i c a n d p r o b a b l y fa ste r i f th is sh o u ld be th o u g h t d e s i r a b l e . A p p r o a c h t o Wave Form D e t e r m i n a t i o n An a t t e m p t w as m ade t o a p p l y t h e m e th o d o f d e c r e a s e i n i n t e n s i t y o f t h e u n d e f l e c t e d beam t o t h e s t u d y o f wave fo rm o f t h e s t a n d i n g w av e. m e t h o d w as f i r s t a p p l i e d t o g l y c e r i n e . (F ig u re 16) i n t h i s The The e x p e r i m e n t a l c u r v e s o b t a i n e d l i q u i d seem e d t o f o l l o w t h e g e n e r a l s h a p e o f t h e t h e o r e t i c a l c u r v e s p r e d i c t e d from t h e t h e o r y ( F i g u r e 4 , p ag e 2 6 ) and th u s i n d i c a t e d t h e e x i s t e n c e o f w aves i n t h i s l i q u i d a t l e a s t a p p ro x im a te ly sin u so id a l in n a tu re . F i g u r e 16 shows t h e e x p e r i m e n t a l r e c o r d e r t r a c e o f th e " r e c o r d e r r e l a t i v e in te n sity " f o r t h r e e d i f f e r i n g sound i n t e n s i t i e s , i n d i c a t e d by t r a n s d u c e r c u r r e n t s o f 100 , 200 , a n d 500 mi H i a m p e r e s r e ­ sp e c tiv e ly . T h e s e c u r v e s w e re o b t a i n e d f o r t h e l i g h t beam p a s s i n g a p ­ p r o x i m a t e l y 2 . k cm. f r o m t h e t r a n s d u c e r . 52 T h is p la c e d i t a p p ro x im a te ly /3 F ig u r e 1 6 . R e c o rd e r t r a c e o f ” r e c o r d e r r e l a t i v e i n ­ t e n s i t y ” o f l i g h t th r o u g h f i n a l s l i t v e r s u s p o s i t i o n i n th e wave f o r a s t a t i o n a r y wave i n g l y c e r i n e . T ra n s d u c e r c u r r e n t a ) 100 m a ., b) 200 tn a ., c ) 500 m a. 55 14 cm. f r o m t h e r e f l e c t i n g e n d o f t h e s o u n d c e l l . S i m i l a r c u r v e s , h o w e v e r, w ere n o t o b ta in e d f o r o t h e r l i q u i d s i n ­ v e s t i g a t e d , e x c e p t p e r h a p s a t v e r y low i n t e n s i t i e s , w h e r e i t w as d i f f i ­ c u l t to d e te rm in e th e sh ap es o f th e c u rv e . R a t h e r c u r v e s o f a n u n sy m — m e t r i c a l n a t u r e w e r e o b t a i n e d a n d t h e c u r v e s w e re n o t n e c e s s a r i l y t h e same s h a p e f r o m o n e wave t o t h e n e x t . As a n e x a m p le o f t h i s a s y m m e tr y F i g u r e 17 o f a c u r v e f o r b e n z e n e t a k e n a p p r o x i m a t e l y 2 . 5 cm. fro m t h e t r a n s d u c e r , a n d F i g u r e 18 o f a c u r v e f o r a c e t o n e , t a k e n a t a b o u t t h e same p o s i t i o n r e l a t i v e t o t h e s o u n d s o u r c e a r e show n. I t was t h e c o n - s i s t a n c y w i t h w h ic h t h e s e a s y m m e t r i c a l c u r v e s a p p e a r e d w h ic h l e d to t h e q u a lita tiv e ta in i f i n v e s t i g a t i o n o f t h e t y p e o f c u r v e o n e w o u ld e x p e c t t o o b ­ a s i n u s o i d a l wave i n t e r f e r e d w i t h a s a w t o o t h t y p e w av e. In th i s c o n n e c t i o n t h e c u r v e o f F i g u r e 19 i s i n c l u d e d b e c a u s e o f t h e i n t e r e s t i n g s i m i l a r i t y i n s h a p e b e t w e e n i t a n d t h e c u r v e o b t a i n e d fro m t h e a n a l y s i s r e f e r r e d t o a b o v e , a n d shown i n F i g u r e 7, p a g e 55* The c u r v e i n F i g u r e 19 w as o b t a i n e d f o r c a r b o n t e t r a c h l o r i d e a t a d i s t a n c e a p p r o x i m a t e l y 8 . 5 cm. fro m t h e t r a n s d u c e r . The c o m p a r i s o n b e t w e e n t h e c u r v e s i s m e n t i o n e d m a i n l y b e c a u s e o f t h e i n t e r e s t i n g s i m i l a r i t y , a n d no c l a i m i s made t h a t t h e a s y m m e tr y i s n e c e s s a r i l y p r o o f o f t h e e x i s t a n c e o f a s a w t o o t h wave form i n th e l i q u i d . T here a r e o th e r p o s s i b i l i t i e s , such as r e f l e c t i o n s f ro m t h e b o u n d a r i e s o f t h e c e l l w h ic h m i g h t p o s s i b l y g i v e r i s e to such a s y m m e tr ic a l c u r v e s and f u r t h e r i n v e s t i g a t i o n , b o th t h e o r e t i c a l p e rim e n ta l, in th is m a tte r i s p ro b a b ly c a l l e d fo r b e fo re a d e f i n i t e s t a t e m e n t c a n be made a b o u t t h e wave s h a p e i n t h e s e l i q u i d s . is an d e x ­ H o w e v e r, i t h o p e d t h a t t h i s m e th o d p r o v i d e s a n a p p r o a c h to t h i s p r o b I s m w h ic h c a n 54 F i g u r e 17* * T ^ j -J d — 4 ----"S!\^ J f / i f / f R e c o r d e r t r a c e f o r s t a n d i n g wave i n benzene. ' " r ■■■ ! ■ i ! 1 ! 1 f ' ; : i ^ . F i g u r e 18• ___ _ --— |.—J--T j I ! j...... ... ”--- — “1 I --1 —— |--■ . — i— J j!t_ --- . I V y ===]f=zj=== O : z z t i z jV ? | \ OL I ’ ;\ f • 1 / \ • / t.V ...... \ s 1 / .. .1 I . f/ y -i f y i \; ..i, .■ i V \ Q ■-i : 1 V i "CD f ' ~ TF.. V 3 7 1 ... - H rV / -1 -- 7 }Y — ; — I ■l ;■ 1... 1 \ r \ .. J i 1 V r— j.. |... : ’ I i > 1 I Ti ! \ ..- L L_;„,_._J— ___ L . R e c o r d e r t r a c e f o r s t a n d i n g wave i n ac e to n e . 55 F i g u r e 19« R e c o r d e r t r a c e f o r s t a n d i n g wave i n c a rb o n te tra c h lo rid e . 56 be im p ro v ed and i n v e s t i g a t e d f u r t h e r i n th e f u t u r e . A p p r o a c h "to S o u n d P r e s s u r e M e a s u r e m e n t G en era1. As h a s b e e n m e n t i o n e d e a r l i e r , th e m a th e m a tic a l th e o ry d e v e l o p e d p r e v i o u s l y s h o u l d make i t p o s s i b l e t o u s e t h e m e th o d o f d e ­ c r e a s e i n i n t e n s i t y o f t h e u n d e f l e c t e d beam (M e th o d I I ) fo r th e e s t i ­ m a t i o n o f t h e sound p r e s s u r e a m p litu d e in th e s t a n d i n g w ave. R e c a ll t h a t S r e p r e s e n t s th e l i g h t en erg y p a s s in g th ro u g h th e f i n a l s l i t SI . 5 I f t h e l i g h t beam p a s s e s t h r o u g h a p r e s s u r e n o d e S w i l l h a v e a minimum So v a lu e , r e p r e s e n t s t h e l i g h t e n e r g y p a s s e d by s l i t S I o f no s o u n d . The r e l a t i v e l a t i o n b etw een S energy p a s s e d , S R is 5 f o r th e case th e r a t i o S /S ^ The r e - and t h e p a r a m e te r A i s g iv e n by e q u a tio n 6 1 , page ^ n in 2 5 , w h ic h i s w r i t t e n a g a i n h e r e . SW - [ '- f + U + - L ) ( 2 P (64) y- w here a - th e w id th o f th e s l i t S l ^ » g — t h e d i s t a n c e fro m t h e c e n t e r o f t h e s o u n d c e l l t o t h e slit ^ - t h e p a t h l e n g t h o f t h e l i g h t beam i n t h e s o u n d f i e l d , 57 K s trh© c o m p r e s s i b i l i t y o f t h e m e d iu m , n s t h e r e f r a c t i v e i n d e x o f t h e m edium , th e w ave l e n g t h o f l i g h t i n t h e l i g h t b eam , th e wave l e n g t h o f t h e s o u n d i n t h e medium u s e d , r = th e d is ta n c e from s l i t S I 2 to s l i t 5 1 , and 5 2 P = t h e p r e s s u r e a m p l i t u d e o f t h e s t a t i o n a r y s o u n d w av e. N o t c o n s i d e r i n g t h e p r e s s u r e a m p l i t u d e 2P f o r t h e m om ent, a l l o f t h e a bove q u a n t i t i e s a r e c e p tio n cf e i t h e r known o r m e a s u r a b l e , w i t h t h e p o s s i b l e e x ­ th e q u a n t i t y ^ , th e p a th le n g th f i e l d , w h ic h h o w e v e r, can be e s t i m a t e d . tie s o f th e l i g h t i n t h e sound A ss u m in g t h e n t h a t t h e s e q u a n t i ­ a r e k n o w n , t h e p r e s s u r e a m p l i t u d e c a n be o b t a i n e d p a r a m e t e r A. The r e l a t i o n s h i p b e t w e e n t h i s p a r a m e t e r a n d 65) can be p l o t t e d as a c u rv e . Such a c u rv e i s i n te rm s o f th e S (e q u a tio n m in shown i n F i g u r e 2 0 . S in c e S c a n b e o b t a i n e d fr o m t h e minimum v a l u e o f t h e c u r v e on r e ^ m in c o r d e r t r a c e ( F i g u r e 16, page 55) and th e r e c o r d e r c a l i b r a t i o n c u rv e ( F i g u r e 1 0 , p a g e 4 0 ) , t h e c u r v e o f F i g u r e 2 0 e n a b l e s on e t o d e t e r m i n e th e v alu e o f A f o r a g iv e n m easured v a lu e o f S^. T hen by t h e u s e o f e q u a t i o n 6 4 , t h e p r e s s u r e a m p lit u d e c a n be c a l c u l a t e d . R e l a t i o n b e t w e e n t r a n s d u c e r c u r r e n t an d t h e p a r a m e t e r A. The v a l u e s o f A o b t a i n e d fro m F i g u r e 2 0 f o r t h e t h r e e w a v e - f o m i c u r v e s f o r g l y c e r i n e (see F i g u r e 1 6 , p a g e 5 5 ) , a r e shown p l o t t e d a g a i n s t t r a n s d u c e r c u r r e n t i n F ig u re 2 1 . The r e l a t i o n s h i p a p p e a r s t o be f a i r l y lin e a r in th is c a se , a l t h o u g h t h e a m o u n t o f d a t a on w h ic h t h i s s t a t e m e n t i s b a s e d i s ra th e r sm all • A l t h o u g h t h e r e i s some d o u b t a s t o w h e t h e r t h e 58 t h e o r y c a n be a p p l i e d Figure 20. V ariatio n of 'S^ min with - ^ £ j 1 0 5 , ( I 895 ) 14. S . W o l i n , J . O p t . S o c . Am., 4 ^ , 5 7 5 , (1955) 15. G. S t o k e s , P h i l . M a g . , 1848 16. B. R ie m a n n , A b h a n d l u n g e n , V I I I , 17. S . E arnshaw , P h i l . 18. L o rd R a y l e i g h , The T h e o r y o f S o u n d , S e c o n d E d i t i o n ; M a c m i l l a n 1896 V o l . I I p p . 5 2 f f * ( R e p r i n t e d 192 9) 19 . R. D. F a y , J . 20* P . B i q u a r d , S u r L ffA b s o r p t i o n Des O n des U l t r a s o n o r e B P a r Lea L i q u i d s , T h e s e s , L " U n i v e r s i t e De P a r i s ( 1 9 5 5 ) T rans. 1 9 9 . 1107, (1 9 5 4 ) 549, 2 4 5 , (1680) 150, 155, (I8 6 0 ) A c o u s t . S o c . Am., 67 2 2 2 , (1951) 21. J. C. H u b b a r d , J . A, F i t z p a t r i c , B. T. K a u k o v s k y , a n d W. J . T h a l e r , P h y s . R e v . , j[4 2 n d S e r i e s , 107* (1 9 ^ 8 ) 22. G. D. 25* J . K o lb a n d A. P . L o e b e r , J . 24. A. P . L o e b e r a n d E. A. H ie d e m a n n , J . (1954) 25. G. M. G ra h a m , J . 26. E . B. E . B. F r e y e r , J . Am. Ohem. S e e . , 5 1 , 759# (1 9 2 9 ) an d F r e y e r , J . Am. Chem. S o c . , 5 ^ , 1515* (1 9 5 1 ) M i k h a i l o v , D o k la d y A k a d e m ii Nauk SSSR, 8 9 , 6 5 5 , (1 9 5 5 ) A c o u st. Soc. A c o u s t . S o c . Am., 2 6 , 2 4 9 , ( 1 9 5 4 ) A c o u s t . S o c . Am., 2 6 , 257# Am., 2 5 , 1 1 2 4 , ( 1 9 5 5 ) 68