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Real time liquid surface acoustical holography Pille, Peter 1972

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C '  REAL TIME LIQUID SURFACE ACOUSTICAL HOLOGRAPHY  by  Peter  Pille  B. Eng. C a r e l e t o n U n i v e r s i t y  1970  A THESIS SUMBITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  the Department of  Electrical  Engineering  We .accept t h i s t h e s i s as conforming required standard  to the  THE UNIVERSITY OF BRITISH COLUMBIA  October 1972  In presenting  this thesis in pa rt i al fu I f i lrnen t of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference  and  study.  I further agree that permission for extensive copying of this thesis for scholarly purposes may by his representatives.  be granted by the Head of my  It is understood that copying or publication  of this thesis f o r financial gain shall not written  permission.  Department of The University of B r i t i s h Columbia Vancouver 8, Canada  Date  Department or  be allowed without my  ABSTRACT  This t h e s i s provides  an a n a l y s i s o f an a c o u s t i c imaging  tech-  n i q u e u s i n g h o l o g r a p h y and the i n t e r a c t i o n o f a c o u s t i c and l i g h t waves at  a liquid-gas interface.  Real  time o p t i c a l images o f o b j e c t s  that  have been t r a n s i l l u m i n a t e d w i t h u l t r a s o n i c waves i n a l i q u i d medium can be o b t a i n e d . energy.  The l i q u i d s u r f a c e a c t s as a d e t e c t o r o f the u l t r a s o n i c  When coherent l i g h t  i s r e f l e c t e d o f f the l i q u i d  o p t i c a l image o f the o b j e c t i s o b t a i n e d . the mechanisms o f the l i q u i d  s u r f a c e an  An a n a l y s i s i s p r e s e n t e d  of  i n v o l v e d i n c l u d i n g an a n a l y s i s o f the t r a n s i e n t motion surface.  i  TABLE OF  CONTENTS  Page ABSTRACT  i  TABLE OF CONTENTS  i i  L I S T OF ILLUSTRATIONS  •  i i i  ACKNOWLEDGEMENT  v  1.  INTRODUCTION  1  2.  LINEAR SYSTEMS ANALYSIS OF THE IMAGING PROCESSES  2  2.1 2.2 2.3 2.4. 2.5 3.  The B a s i c System Surface-Light Interaction Plane Wave R e f e r e n c e . . . Focused Image Hologram L e n s l e s s F o u r i e r Transform Hologram  RADIATION PRESSURE AND THE FUNCTION g 3.1 3.2 3.3  4.  23  =  LIQUID SURFACE RESPONSE TO RADIATION PRESSURE  4.3  . . .  .  F o r m u l a t i o n o f t h e Problem Results f o r Single S p a t i a l Frequencies  23 28 31 39  Steady S t a t e Response T r a n s i e n t Response o f t h e L i q u i d S u r f a c e 4.2.1 4.2.2  39 41 . . . .  41 45  The S u r f a c e S p a t i a l Frequency Response t o P u l s e d Radiation Pressure  53  ACOUSTIC BEAM INTENSITIES, SOUND FIELD PARTICLE OSCILLATION, SURFACE BULGE  63  5.1 5.2 5.3 6.  . . . .'  . . . . . . . . .  R a d i a t i o n P r e s s u r e o f Sound Waves E f f e c t c f the M i n i t a n k c n the R a d i a t i o n P r e s s u r e The T r a n s f e r F u n c t i o n g^  4.1 4.2  5.  .  2 6 7 11 18  The U l t r a s o n i c Beam I n t e n s i t i e s The U l t r a s o u n d F i e l d P a r t i c l e O s c i l l a t i o n The S u r f a c e Bulge  63 65 66  CONCLUSIONS  70  APPENDIX A  F o u r i e r Transforms,  APPENDIX B  Numerical  APPENDIX C  Liquid  REFERENCES  Diffraction,  Imaging . . . . . .  S o l u t i o n of L i q u i d Surface Motion  S u r f a c e Response  . . . .  73 77 82  . .  83 ii  LIST OF ILLUSTRATIONS  Figure  Page  2.1  The b a s i c imaging  2.2  L i g h t r e f l e c t i o n o f f the s u r f a c e h ( x )  2.3  Focused image hologram  12  2.4  O p t i c s f o r f o c u s e d image hologram  13  2.5  Transform  15  2.6  L e n s l e s s F o u r i e r t r a n s f o r m hologram  3.1  Waves a t a boundary  25  3.2  Minitank  28  3.3  R e f l e c t i o n s i n the m i n i t a n k  29  3.4  |g-^(n,0)| v s . r\/ky  33  3.5  system  2 7  plane  /g^Cn.O) v s .  . . .  19  f o r v a l u e s o f d/A^ from 2 to 2 1/2 . . . f o r v a l u e s o f d/A  2  from 2 t o 2 1/2  . .  34  f o r v a l u e s of' d/A  2  from 9 to 9 1/2  . .  35  3.7  / g ^ Q l . O ) v s . n / ^ f o r v a l u e s o f d/A  2  from 9 t o 9 1/2  . .  36  4.1  L i q u i d s u r f a c e deformation  4.2  Response o f l i q u i d pressure  surface to step f u n c t i o n r a d i a t i o n  Response o f l i q u i d pressure  surface to a pulse of radiation  3.6  4.3  4.4  4.5  4.6  4.7  4.8  vjk  ±  |g (n,0)| v s . n/ky  . .  46  47  Overdamped l i q u i d s u r f a c e response r a d i a t i o n pressure Response o f l i q u i d pressure  41  to s t e p f u n c t i o n 48  s u r f a c e t o an impulse  of radiation 49  L i q u i d s u r f a c e ( f r e o n E-5) s p a t i a l frequency to s t e p f u n c t i o n r a d i a t i o n p r e s s u r e L i q u i d s u r f a c e (water) s p a t i a l frequency function radiation pressure L i q u i d s u r f a c e ( f r e o n E-5) s p a t i a l to a p u l s e o f r a d i a t i o n p r e s s u r e  iii  response 55  response  frequency  to step 56'  response 57  Figure  4.9  Page  L i q u i d s u r f a c e ( f r e o n E-5) s p a t i a l frequency response. to a s t e p f u n c t i o n r a d i a t i o n p r e s s u r e w i t h a f i n i t e f l u i d depth  58  L i q u i d s u r f a c e ( f r e o n E-5) s p a t i a l frequency response to a p u l s e o f r a d i a t i o n p r e s s u r e w i t h a f i n i t e depth o f f l u i d  59  5.1  Radiation pressure  68  5.2  Radiation pressure  69  A.l  Diffraction  74  A. 2  Lens  74  A.3  Imaging w i t h a l e n s  75  A. 4  F o u r i e r transform with  B. l  F l u i d d i v i d e d i n t o a r e c t a n g u l a r mesh o f c e l l s  B.2  P o s i t i o n of p r e s s u r e and v e l o c i t i e s  B.3  Deformation  4.10  a lens  of the l i q u i d  76  for c e l l i . j  surface . . . "  iv  77 . . . .  78 78  ACKNOWLEDGEMENT  I wish, t o express my g r a t i t u d e t o Dr. B.P. H i l d e b r a n d my o r i g i n a l s u p e r v i s o r and f o r s u g g e s t i n g  the t o p i c f o r t h e t h e s i s , and  f o r h i s c o n t i n u i n g i n t e r e s t i n my p r o g r e s s . Donaldson f o r h i s support  during  who was  A l s o , thanks t o Dr. R.W.  the l a t t e r stages  o f the work as my  supervisor.  I a l s o w i s h to acknowledge M i s s Norma Duggan f o r t y p i n g  the  A l s o , I am g r a t e f u l t o the N a t i o n a l Research C o u n c i l f o r  thesis.  financial  assistance.  V  I.  S i n c e 1966  INTRODUCTION  much r e s e a r c h has  taken p l a c e a p p l y i n g the  principles  o f h o l o g r a p h y to a c o u s t i c a l imaging methods, w i t h u l t r a s o u n d x>?aves used to i l l u m i n a t e o b j e c t s , and w i t h  images r e c o n s t r u c t e d u s i n g  Much o f t h i s work i s d e s c r i b e d i n r e f e r e n c e s the important  developments and  Of i n t e r e s t  results  [1-3,], and  a review o f  are d e s c r i b e d i n r e f e r e n c e  i s used to i l l u m i n a t e an o b j e c t . and  Sheridon  the t e c h n i q u e potentially  [4],  f o r t h i s t h e s i s i s the method of o b t a i n i n g images  o f o b j e c t s u s i n g a l i q u i d s u r f a c e to d e t e c t the u l t r a s o u n d  Mueller  lightwaves.  [ 5 ] , Green  E a r l y work i n t h i s method was [ 6 ] , and Brenden  are i n the n o n d e s t r u c t i v e  for medical  energy  [7].  that  done by  Applications of  testing of materials  [8],  and  d i a g n o s i s i n imaging human t i s s u e s t r u c t u r e  (e.g. d e t e c t i n g tumors). The  purpose of t h i s  t h e s i s i s to p r o v i d e  an a n a l y s i s o f  the  form of l i q u i d s u r f a c e a c o u s t i c a l h o l o g r a p h y t h a t g i v e s images i n r e a l time. in  The  treatment i s an e x t e n t i o n of the e a r l y work i n r e f e r e n c e s  t h a t the image o b t a i n e d  tions  that are  c h a r a c t e r i s t i c of the l i q u i d s u r f a c e and  the imaging method.  An  l i q u i d s u r f a c e used  and  performance and  i s r e l a t e d to the o b j e c t by  a n a l y s i s i s made of the i s shown t o be  understanding  Chapter 2 p r e s e n t s process  t r a n s f e r functhe geometry o f  t r a n s i e n t motion of  of c r i t i c a l  the  importance i n the  o f the imaging system. a t h e o r e t i c a l a n a l y s i s o f the  imaging  i n terms o f the t r a n s f e r f u n c t i o n s t h a t d e s c r i b e the  In Chapters 3 and  [5-7]  4 the p a r t i c u l a r form o f these  derived.  1  system.  transfer functions  are  2  II.  LINEAR SYSTEMS ANALYSIS OF THE  This 6,7]  chapter  which p r o v i d e s  o b t a i n e d by  an a n a l y s i s o f an imaging technique  u l t r a s o n i c waves.  u l t r a s o n i c waves, and  reflecting light  A hologram i s formed at a then an image of the o b j e c t i s  from the l i q u i d  surface.  In s e c t i o n s 2.1-2.3 the b a s i c r e l a t i o n s h i p s f o r the are d e r i v e d . analysed  and  In s e c t i o n s 2.4  and  space and  and  2.5  two  the o p t i c a l images o b t a i n e d  f i e l d which d e s c r i b e s is  d i f f e r e n t imaging methods are are r e l a t e d to the a c o u s t i c a l  the o r i g i n a l o b j e c t .  Since  time dependent, t h r e e d i m e n s i o n a l  c o n v o l u t i o n o p e r a t i o n s w i l l be  system  used, w i t h  this  Fourier  form o f imaging transforms  the n o t a t i o n d e f i n e d i n  Appendxx A. 2.1  The  B a s i c System We  s h a l l c o n s i d e r the f o r m a t i o n  shown i n f i g u r e  [5,  r e a l time o p t i c a l images o f o b j e c t s t h a t have been  t r a n s i l l u m i n a t e d with l i q u i d s u r f a c e by  presents  IMAGING PROCESS  of a c o u s t i c a l holograms as  2.1.  Water filled. tank  F i g u r e 2.1  The b a s i c imaging system  3  It  i s d e s i r e d to o b t a i n an o p t i c a l image of the i n t e r i o r  an o b j e c t which i s opaque to l i g h t . filled t . Q  tank and  The  t r a n s i l l u m i n a t e d by  a c o u s t i c x^aves t r a n s m i t t e d  The  The main tank by through but  t  a  to form a  from the water o f  Also, s i n c e the  the  u l t r a s o u n d waves to pass i n the main tank from  f l u i d p r o p e r t i e s are  of the m i n i t a n k can be  important  chosen x^ith  t h a t are more s u i t a b l e f o r imaging than i s x^ater. The  a c o u s t i c waves are r e f l e c t e d o f f the s u r f a c e and  a varying r a d i a t i o n pressure partially  transducer  any l a r g e s c a l e d i s t u r b a n c e s  fluid  transducer  minitank.  a t h i n membrane which a l l o w s  the imaging system, the  properties  produced by  f l u i d i n the m i n i t a n k i s s e p a r a t e d  prevents  from a  through the o b j e c t i n t e r f e r e w i t h  s u r f a c e of the  d i s r u p t i n g the s u r f a c e . in  o b j e c t i s p l a c e d i n a water  u l t r a s o n i c waves  r e f e r e n c e x^ave o f the same frequency hologram at the l i q u i d  of  determined by  p a t t e r n on  the s u r f a c e .  This pressure  the i n t e n s i t y o f the a c o u s t i c waves at  s u r f a c e , which causes the s u r f a c e to deform i n t o a p a t t e r n o f that represents  a hologram.  reflecting light The f l u i d xtfill be  An  ripples  image o f the o b j e c t i s then o b t a i n e d  from a l a s e r o f f the  represented  the v e l o c i t y p o t e n t i a l [15] § ( x , y , t ) .  satisfies  o f d i f f r a c t i o n apply  the s c a l a r wave e q u a t i o n ,  directly  a complex q u a n t i t y , and  to <J>.  A l s o , <j> w i l l be  t h e r e x ^ i l l be  an i m p l i e d time f a c t o r o f the form e  However, t h i s  time dependence w i l l not be  change  (x,y)  explicitly  i n the Since  the s c a l a r laws considered  s i n c e the u l t r a s o n i c f i e l d w i l l be  dependence of <(>(x,y,t) w i l l  by  surface.  d i s t r i b u t i o n i n a plane  by  is  the  acoustic f i e l d  if) i s a s c a l a r and  example a step  create  to  be  monochromatic,  (where j = / - l ) . shown, and  the  r e f e r to time changes i n amplitude,  for  time  4 <Kx,y,t) = <j>(x,y) u ( t ) where  • , ^ ( t )  =  Cl t > 0 (o t < 0  Consider face  the v e l o c i t y p o t e n t i a l  <t>^(x,y,t) i n c i d e n t  (x,y p l a n e ) to be the sum o f the v e l o c i t y p o t e n t i a l  a t the s u r -  due t o t h e  cj> ( x , y , t ) .  o b j e c t <{>^ (x,y, t) , and due t o the r e f e r e n c e Then ^ ( x . y . t ) = ^ ( x . y . t ) + <|>r(x,y,t) A radiation  pressure  However, t h i s p r e s s u r e  i s exerted  (2-1)  on the s u r f a c e due t o <j>^.  d i s t r i b u t i o n i s dependent on the a n g l e s  the v a r i o u s p l a n e wave components o f C|K s t r i k e the s u r f a c e .  a t which  I t w i l l be  shown i n Chapter 3 t h a t we can c o n s i d e r an e f f e c t i v e v a l u e o f the i n cident f i e l d  t o be g i v e n by  4> . ( x , y , t) = .^(x^y ,t) * g (x.,.y.,'t) ff  The  (2-2)  1  symbol * denotes a t h r e e d i m e n s i o n a l  p e n d i x A) o p e r a t i o n i n the x,y, and t dimensions. g^(x,y,t) incident  i s the e f f e c t i v e v e l o c i t y p o t e n t i a l velocity potential  o f g-^(x,y,t) inside  ( s e e Ap-  The complex f u n c t i o n  a t the s u r f a c e when the  i s an impulse 6 ( x , y , t ) .  The time dependence  i s due to the i n t e r n a l r e f l e c t i o n s o f the u l t r a s o u n d waves  the m i n i t a n k . The  is  convolution  radiation  pressure  d i s t r i b u t i o n p ^ on the s u r f a c e (x,y)  then g i v e n by  P (x,y,t) = c | * r  e f f  (x,y,t)|  T h i s r e s u l t i s d e r i v e d i n Chapter 3.  (2-3)  2  The c o n s t a n t  C depends on the  d e n s i t y o f the f l u i d i n the m i n i t a n k and the wavelength o f the u l t r a sound used. The h e i g h t o f the d e f o r m a t i o n ,  h, o f the l i q u i d s u r f a c e  from  5  the q u i e s c e n t l e v e l fluid,  the s p a t i a l  (z = 0 p l a n e ) i s dependent on the p r o p e r t i e s o f t h e f r e q u e n c i e s i n h e r e n t i n t h e f u n c t i o n p^. and time t .  I t w i l l be shown i n chapter 4 t h a t h(x,y,t) = p ( x , y , t ) * g (x,y,t) r  The  f u n c t i o n g ( x , y , t ) i s r e a l and r e p r e s e n t s the response o f the l i q u i d 2  s u r f a c e to an impulse  6(x,y,t) o f r a d i a t i o n pressure.  From e q u a t i o n s  (2-1) to (2-4) the s u r f a c e h e i g h t i s g i v e n by  h ( x , y , t ) = C|(cj> +'<j,) * g \  * g  2  r  We  (2-4)  2  x  1  (2-5)  2  can w r i t e (2-5) as h(x,y,t) = h ( x , y , t ) + h ( x , y , t ) r  Q  + h^x.y.t)  + h*(x,y,t)  (2-6)  where  *.g i  h (x.,y.,t) = C|(j> r  2  r  r  h (x,y,t) = C l ^ * g | o  2  x  h i C x . y . t ) = C{((|. * g-J* r  and  the symbol  denotes  *.g * g  (2-7)  2  (2-8)  2  (^  *  § 1  )}  * g  the complex c o n j u g a t e .  2  -  (2-9)  Note t h a t the h e i g h t  h ^ i s p r o p o r t i o n a l t o the square o f the amplitude o f the r e f e r e n c e beam, h  Q  i s p r o p o r t i o n a l t o the square  o f the o b j e c t beam a m p l i t u d e , w h i l e  h^ and h^ are p r o p o r t i o n a l t o t h e p r o d u c t o f the r e f e r e n c e and o b j e c t amplitudes. mation  In l a t e r s e c t i o n s i t w i l l be shown t h a t the u s e f u l l  infor-  from the o b j e c t b e i n g imaged i s c o n t a i n e d i n the f u n c t i o n s i n -  volving h (x,y,t). 1  Experiments  [7] have shown t h a t c o n s i d e r a b l y b e t t e r images  can be o b t a i n e d i f advantage i s taken o f the t r a n s i e n t motion liquid ^  surface.  Thus the t r a n s d u c e r s t  r  and t  o  a r e used  o f the  to produce  p u l s e s o f a c o u s t i c waves, and the f l u i d s u r f a c e i s sampled a t some  6  optimum  time by r e f l e c t i n g p u l s e d  coherent  o b t a i n a r e a l time image i n the o p t i c a l  light  field  o f f the s u r f a c e t o  after processing  through  suitable optics.  2.2  Surface-Light Interaction  The o p t i c a l  image o f the o b j e c t i n the water tank can be ob-  t a i n e d by r e f l e c t i n g  coherent  p a t t e r n on the s u r f a c e  light  o f f the l i q u i d  surface.  The  ripple  causes phase changes i n t h e r e f l e c t e d l i g h t which  w i l l be determined i n t h i s s e c t i o n . We  shall  c o n s i d e r a beam o f p l a n e  p a r a l l e l waves o f l i g h t t o  be r e f l e c t e d o f f the s u r f a c e h ( x ) .  The s u r f a c e h ( x ) ( f i g u r e 2.2) i s  considered  the r a d i u s o f c u r v a t u r e  t o be l o c a l l y  f l a t , with  p a r e d to the wavelength o f l i g h t .  l a r g e com-  The r e f l e c t e d wave a t the s u r f a c e  h(.a) i s U  = Be~  j k d  i [(a+c)sin6 ] k  e  z  £  j k [ ( x - a ) s i n ( e - 6 ) + ( z - h ) c o s (Q^S,,) ) z  s  s (2-10) where a,c and d are d e f i n e d as shown i n f i g u r e 2.2. The f i r s t the v a l u e value  exponential  o f the f i e l d  o f the f i e l d  light  a t (x,z) = (a+c,0).  a t (a+c,0).  q u i t e s m a l l so t h a t 8  f a c t o r accounts f o r the phase l a g from The second f a c t o r i s the  The s l o p e o f the l i q u i d  << 1 and, by s e t t i n g  U (x,y) = B e s  where  \  =  k  J  C O S l  6  x£  j 2  V  e  j( u  *  X  +  V>  be  z = 0 i n (2-10) then the  s d i s t r i b u t i o n U" becomes, f o r a two d i m e n s i o n a l s  surface w i l l  s u r f a c e h(x,y) ,  (2-11)  7  A  i s the wavelength of the l i g h t , and  cos  0  , cos  8  and  cos  6  the d i r e c t i o n c o s i n e s of the. wave r e f l e c t e d o f f the s u r f a c e z = We i n the plane if  assume t h a t (2-11) r e p r e s e n t s z = 0.  the l i g h t  T h i s i s an a p p r o x i m a t i o n which we  the r i p p l e p a t t e r n of the s u r f a c e  d e t a i l e d treatment of the r e f l e c t i o n of l i g h t used i n t h i s  F i g u r e 2.2  2.3  t h e s i s see  0.  distribution  can  say i s v a l i d  , o r the space v a r y i n g  o f h i s n o t l a r g e compared to a wavelength o f l i g h t .  than which w i l l be  field  component  For a much more  o f f rough s u r f a c e s  references  are  than  [9,10].  L i g h t r e f l e c t i o n o f f the s u r f a c e  h(x)  P l a n e Wave Reference  In t h i s s e c t i o n x^e s h a l l determine the r e l a t i o n s h i p between the l i g h t  d i s t r i b u t i o n at the l i q u i d s u r f a c e and  the a c o u s t i c  b u t i o n <j>, at the s u r f a c e when p l a n e waves o f l i g h t U and Js  distri-  acoustic  8  r e f e r e n c e d> a r e used, r The  reference transducer t  produces p l a n e waves (j)^, so t h a t  the s u r f a c e z = 0 we have * (x,y,t) = ^(x-y.t) e  j  (  r  u  r  x  V  +  °  (2-12)  where u v  k  = k  r  = k  r s  cos A  g  and  s s  cos 6 xs cos 6  .  =  2TT/A  2  2 2 8 + cos 8 + cos 8 xs ys zs  s =1  i s the wavelength o f the u l t r a s o n i c wave i n the f l u i d cos8  are the d i r e c t i o n  c i e s a r e 1 to 10 MHz. if  ys  c o s i n e s o f the p l a n e wave.  The amplitude  i n the m i n i t a n k Typical  frequen-  R^ i s shown as a f u n c t i o n o f time  the u l t r a s o u n d i s p u l s e d and as a f u n c t i o n o f x and y t o account f o r  d e v i a t i o n s from a p e r f e c t p l a n e wave. Using  (2-12), and w i t h some m a n i p u l a t i o n we f i n d  t h a t the s u r -  f a c e d e f o r m a t i o n h ( x , y , t ) g i v e n by (2-6) can be w r i t t e n as h(x,y,t) = h  + h  r  Q  + h  2  e"J  ( u  r  V>  X +  + h* e ^  ( u  r  X  +  V>  (2-13)  where h  ( x , y , t ) = (R* «fr) * ' { g e V j(  2  2  R (x,y,t) 2  = R  *' {  x  cb (x,y,t) = ^  *  2  The in  function R  2  g l  e"  gx  i s determined  +  j ( u  r  X  +  V  2  = R  1  ^  r  y  )  }  (2-14)  }  (2-15) (2-16)  by the amplitude  ( u , v , 0) r  0  f  .  (2-12), and i f R^ i s a c o n s t a n t then R R  v  2  2  R^ o f the r e f e r e n c e wave  i s a constant,  simply (2-17)  9  where g  (x,y,t) ^>  1  g  (u,v,(o)  (see Appendix A, F o u r i e r Transform) I f we w r i t e the f u n c t i o n h then  (x,y,t) = h  2  ( x , y , t ) i n (2-14) i n the form  (x,y,t) e  a  U  ^ '  V  >  t  (2-18)  )  (2-13) becomes h(x,y,t) = h  + h  r  + 2 h ( x , y , t ) cos (u x + v y - a ( x , y , t ) )  o  i  3.  j-  (2-19) Thus i n the t h i r d ject  f i e l d h^  term o f (2-19), the i n f o r m a t i o n from  ( x , y , t ) i s r e p r e s e n t e d as amplitude  c a r r i e r of frequency y u ^ +  h e i g h t o f the  , p r o d u c i n g v a r i a t i o n s i n the  fluid.  From (2-14),  (2-15) and (2-16)  •h  •{•R  Cx,y,t)  2  modulating  J  n a spatial  and phase  the ob-  =  *"{g  e  2  x 2  J ( u  (^  r  x +  * v  r  •g )} 1  y  V  (2-20)  * If R  2  i s a c o n s t a n t , g i v e n by  represents a f i l t e r e d of  the f l u i d .  R  constant,  2  h In  through  v e r s i o n of the f i e l d  Then i f we  (u,v,a>) = R *  ±  (x,y,t)  f u n c t i o n <j>^ at the s u r f a c e  (u,v,w) g  (U,V,CJ) g  2  (u-u  r >  v-v , u>)  can c o n s i d e r the f u n c t i o n <j>^(x,y,t) to have been  a f i l t e r with g  2  take the F o u r i e r t r a n s f o r m o f (2-20) w i t h  ^  2  (2-21) we  (2-17), then the f u n c t i o n h  2  .  2  ( u - u , v - v , to) r  If  r  i s n o t a c o n s t a n t then t h e r e i s some ad-  d i t i o n a l d i s t o r t i o n o f <J>^ as g i v e n by s l o w l y the amplitude  passed  transfer function  (u,v,w) g  and a m p l i f i e d by R  (2-21)  (2-15) and ( 2 - 2 0 ) .  R^ i n (2-12) v a r i e s i n space  The more  and time, the l e s s  10  cj> .  the d i s t o r t i o n o f the o b j e c t i n f o r m a t i o n When a coherent  plane wave o f l i g h t i s r e f l e c t e d o f f the s u r -  f a c e we o b t a i n a d i s t r i b u t i o n g i v e n by (2-11).  In order to evaluate  (2-11) we make use o f the f o l l o w i n g r e l a t i o n . .-jacosb e  =  °? , ..n _ . . - j n b Z (-j) J ( a ) e n=-°° n  where J i s a B e s s e l f u n c t i o n o f the f i r s t k i n d o f o r d e r n. n (2-11) and (2-19) the l i g h t  U (x,y,t) =  Then from  at the s u r f a c e becomes  ! U (x,y,t) e n=-°°  j  (  u  x n  +  V  n  y  (2-22)  )  where TT / ,_\ / . \ H T, ~ J 2 w ( h + h ) ina TJ ( x , y t ) = (-j) B e £ r o J (4w h ) e n ^ n Z a n  T  v  l  N  J  s  u v  n n  = u„ - n u I r = v. - n v I  As mentioned e a r l i e r  r  (2-22) i s v a l i d  o f the s u r f a c e deformation of l i g h t .  Any c o n s t a n t Equation  h(x,y,t)  component  i f the space v a r y i n g i s n o t l a r g e compared  i n h i s o f no  component t o a wavelength  concern.  (2-22) c o n s i s t s o f a s e r i e s o f sums o f t h r e e  distinct  The f a c t o r J (4w h ) e? c o n t a i n s t h e i n f o r m a t i o n o f the n £ a' a c o u s t i c a l f i e l d d i s t r i b u t i o n <j>^. A l o o k a t graphs o f B e s s e l f u n c t i o n s factors.  na  shows us t h a t J  o  i s o f no v a l u e  t o us.  near the o r i g i n t h e r e i s a p p r o x i m a t e l y J  n  (x) and x.  Since (0) = 1/2  then f o r 4w„h << 1 we have i a J  l  <  4 W  4 a> h  ~~  2  V a  However f o r n l > 0 we see that 1  1  a linear relationship  between  11 For l a r g e r v a l u e s o f 4w^h lationship  we s t i l l have an a p p r o x i m a t e l y  up to v a l u e s o f the argument o f about 1.  linear re-  Thus f o r n = +1,  from C2-18) J (4w h ) e 1  The light  Ji  j  factor e  distribution.  of the t r a n s d u c e r  ~ 2w^h  a  a  j  2  w  £^r  If h  (2-23)  ^o"* causes phase d i s t o r t i o n s  +  The term h  t . o  (x,y,t)  2  Q  can be made s m a l l b y r e d u c i n g  i s a constant r  wave.  i n practice  may be d i s t o r t e d . the l i g h t  source  (2-12))  a uniform  o f the f l u i d s u r f a c e due t o the r e f e r e n c e  Actually,  the power  ( i . e . R, a c o n s t a n t , 1  then i t i n t r o d u c e s no d i s t o r t i o n , and then h ^ r e p r e s e n t s h e i g h t o r bulge  i n the  ultrasonic  the f l u i d may not be u n i f o r m l y r a i s e d b u t  T h i s can be m i n i m i z e d by p u l s i n g the t r a n s d u c e r s and so t h a t the s u r f a c e i s sampled b e f o r e  become i n t o l e r a b l e .  A l s o , we can add a n o i s e  distortion  term h ^ to h ^ + h  Q  effects to  account f o r any n o i s e a t the s u r f a c e .  I f we c o n s i d e r the i n t e n s i t y o f  the l i g h t  W  a t the s u r f a c e the f a c t o r e ~ ^ ? , ^ r  +  ^o  * N^ drops o u t and  +  n  i n t r o d u c e s no d i s t o r t i o n as l o n g as the space v a r y i n g component o f h(x,y,t)  does n o t become l a r g e compared to a wavelength o f l i g h t . In  (2-22)  r e g i o n s o f space.  The f a c t o r exp ( j [ u  mines the d i r e c t i o n s ferent by  each term r e p r e s e n t s l i g h t 0  diffracted, into  - nu ) x + ( v  0  - nv ) y ] ) d e t e r -  i n t o which the d i f f e r e n t o r d e r s o r terms w i t h  values of n are d i f f r a c t e d .  Recall  that u  0  and v  the d i r e c t i o n c o s i n e s o f the l i g h t beam as i n d i c a t e d  larly,  different  dif-  a r e determined i n (2-11).  Simi-  and v^_ are determined by the a c o u s t i c r e f e r e n c e beam as g i v e n  i n (2-12). 2.4  Focused Image Hologram  We can now make use o f the p r e v i o u s  r e s u l t s t o c a r r y out an  12  a n a l y s i s of a s p e c i f i c form o f imaging. have been o b t a i n e d w i t h method i s due focus  the  distribution with  two  the method of the f o c u s e d  to Smith and Brenden  field  Some o f the b e s t  [11].  An  d i s t r i b u t i o n of the o b j e c t  <j>^(x,y) as i n f i g u r e 2.3.  results  image hologram.  This  a c o u s t i c l e n s i s used to  <j> (x,y) onto the o  The  [4]  surface  imaging c o u l d a l s o be  a c o u s t i c l e n s e s which p e r f o r m a double F o u r i e r t r a n s f o r m r  done of <J> . o  Image plane Laser Spatial  F i g u r e 2.3 Ignoring s u r f a c e , and  the  Focused image hologram time delay i n p r o p a g a t i n g  o  2  z /2a  2  o  k  where kw  w  from the o b j e c t to  the  u s i n g a s i n g l e l e n s , at the s u r f a c e the image cf>^(x,y) i s  r e l a t e d to the o b j e c t <j> (x,y) by e q u a t i o n  TT  filter  =  2IT/A  w  w  «  1  (A-7)  if  13  and  A i s t h e w a v e l e n g t h o f t h e u l t r a s o n i c wave i n water, w &  <j>^(x,y,t) appears as the f u n c t i o n '<)> (-mx,-my,t) p a s s e d a bandlimited  f i l t e r P(V(u + v ) z /ak. ) and w i t h o w v  The  d i s t o r t e d phase.  J  phase d i s t o r t i o n would n o t be p r e s e n t  through  r  i f two a c o u s t i c l e n s e s were  used, however, the phase d i s t o r t i o n i s o f no immediate concern s i n c e we a r e u l t i m a t e l y i n t e r e s t e d i n i n t e n s i t y o n l y .  The r e s o l u t i o n i s l i m i -  t e d by the s i z e o f t h e l e n s and the d i s t a n c e t o t h e l e n s , so the v a l u e of ak^/z s h o u l d be made as l a r g e as c o n v e n i e n t l y p o s s i b l e . In o r d e r  t o o b t a i n an image from e q u a t i o n  to i s o l a t e one o f the terms w i t h n = + 1. term i s d i f f r a c t e d i n t o a d i f f e r e n t  T h i s i s p o s s i b l e s i n c e each  r e g i o n o f space.  I f a l e n s i s used to F o u r i e r t r a n s f o r m at t h e s u r f a c e then a p i n h o l e i n t h e t r a n s f o r m o n l y t h e d e s i r e d o r d e r t o pass through. from performed by another l e n s w i l l  Surface  the l i g h t  plane  can be used t o a l l o w  Then a second F o u r i e r t r a n s -  plane  f o r focused  Image plane  image hologram.  I n f i g u r e 2.4 the f l u i d s u r f a c e i s shown v e r t i c a l venience. filter  distribution  r e s t o r e t h e d e s i r e d image.  Transform F i g u r e 2.4 O p t i c s  (2-22) i t i s n e c e s s a r y  f o r con-  The f o c a l l e n g t h s o f the l e n s e s a r e f ^ and f^-A- s p a t i a l  ( p i n h o l e ) i s p l a c e d i n the t r a n s f o r m p l a n e .  b u t i o n U.(x,y,t) r e p r e s e n t s  the image o f the o b j e c t  The f i e l d < j > (x,y,t).  distri-  14  An exact F o u r i e r t r a n s f o r m i s performed by l e n s i s of i n f i n i t e  s i z e , o r the hologram a r e a i s s m a l l and c o n t a i n s  s p a t i a l frequencies.  s h o u l d be o f a p p r o x i m a t e l y the  same s i z e o r l a r g e r than the hologram a r e a . the d i s t r i b u t i o n  frequency  limited  In o r d e r to reduce the e f f e c t s o f v i g n e t t i n g by  the f i n i t e s i z e o f the l e n s , the l e n s  in  only i f  The s p a t i a l  frequencies  u* (x,y,t) w i l l n o t d e v i a t e g r e a t l y from the s p a t i a l s  o f the l i g h t  used to r e f l e c t o f f the s u r f a c e s i n c e X « x,  X. . w  0  T h i s means t h a t t h e r e w i l l be no h i g h s p a t i a l f r e q u e n c i e s and no d e f l e c t e d f a r o f f a x i s f o r the l e n s to i n t e r c e p t .  light  In (2-22) t h e terms  f o r h i g h v a l u e s o f n a r e d e f l e c t e d f a r to e i t h e r s i d e o f the zero  order  term b u t these h i g h o r d e r terms a r e o f no i n t e r e s t . Taking lens  the s p a t i a l F o u r i e r t r a n s f o r m o f u" i n (2-22) w i t h s  i n f i g u r e 2.4 we o b t a i n i n the t r a n s f o r m p l a n e = d A ^ )  U (x,y,t) f  ? U n=-°°  n  (see (A-8))  (u-u , v - v , t ) n  n  (2-24)  where u = (k /f ) x £  x  v = (k^/^) y u  = u n  v  n  = v  - nu r  SL I  - nv  r  Thus we have a d i s p l a c e m e n t  o f the v a r i o u s o r d e r s i n the t r a n s f o r m .  p l a n e by an amount (u„ - nu ) f,/k„ I x 1 I  i n the x d i r e c t i o n  f-i/k.  i n the y d i r e c t i o n  (v  - nv ) Jo  TC  -i-  J6  /  The  transform plane with  the v a r i o u s o r d e r s appears as i n f i g u r e 2.5.  If there i s l i t t l e  o v e r l a p o f t h e d i s t r i b u t i o n U_^ o r  w i t h the o t h e r o r d e r s then a p i n h o l e can be used to a l l o w the d e s i r e d  15  A  F i g u r e 2.5  Transform  plane  o r d e r t o pass through w i t h the r e s t o f the o r d e r s  blocked.  From (2-12) and (2-24) and u s i n g the r e l a t i o n k  s  sin0 = k sin 8 zs w zw  the d i s t a n c e between the c e n t e r s o f a d j a c e n t o r d e r s i s r = f  (\/\ ) s i n 8  a  7  z  .(2-25)  w  where the angle o f the r e f e r e n c e beam t o the normal t o the s u r f a c e i s 8  i n the m i n i t a n k  zs  tial  frequency  transform  and 8 i n the main (water) tank. zw  The maximum s p a -  i n the o r i g i n a l o b j e c t , k , r e p r e s e n t s a d i s t a n c e i n the w  plane r = f  1  X  £  / A  (2-26)  w  However, t h e r e may be h i g h e r s p a t i a l  frequencies present  due to the e x t r a  phase f a c t o r i n (2-22) c o n t a i n i n g h +h +h. . r o N r  T  to  Thus from (2-25) and (2-26) we see t h a t t h e r e w i l l be some o v e r l a p o f the v a r i o u s terms i n (2-24) i n the t r a n s f o r m p l a n e . of  order  |n| > 1 w i l l be dim and may n o t cause any s e r i o u s o v e r l a p , b u t  the zero o r d e r term i s the b r i g h t e s t . 8  zw  The terms  To reduce the o v e r l a p the angle  t h a t the a c o u s t i c r e f e r e n c e beam makes w i t h  f a c e s h o u l d be made s u f f i c i e n t l y T y p i c a l values  the normal to the s u r -  large.  f o r the wavelengths may be  . 16 -3  A  - 0.15 mm and A = 0.6 ym, so that A /A = 4 x 10  Thus the dimensions  o f the v a r i o u s o r d e r s i n the t r a n s f o r m plane as  i n d i c a t e d by (2-24) and (2-25) would n o r m a l l y be q u i t e s m a l l . if  the f o c a l l e n g t h f  perform  However,  i s made l a r g e enough then i t may be p o s s i b l e t o  f i l t e r i n g w i t h s u i t a b l e t r a n s p a r e n c i e s i n s t e a d o f simply a  pinhole i n the transform plane. From (2-24) we have i n the t r a n s f o r m plane f o r the term o f o r d e r n = +1 U (x,y,t) = (l/A^f-p \  (u , v , t) i  f  f  f  • P ( u fj/k^P  , v  f  (u , v )  T  f  f  fj/k^P)  f  (2-27)  where  u  = u - u  f  V  f  p  (  -  x  r < k X  y  '  _  v  )  =  y ;  V ] L  = (k^/fp  x  = (k^/^)  x - u y - v  / l 'for + y { 0 otherwise  Z  £  £  + u  + v  r  r  < 1  P i s the r a d i u s o f the p u p i l i n the t r a n s f o r m p l a n e , and g^ ( u ^ , v^) r e p r e s e n t s the t r a n s m i t t a n c e o f the t r a n s p a r e n c y used.  I f the v a r i o u s  terms i n (2-24) do not o v e r l a p s i g n i f i c a n t l y and p i s made l a r g e enough we can s e t P = 1.  A l s o , i f no t r a n s p a r e n c y i s used  Applying light  g  = 1.  (A-8) to (2-27) and u s i n g (2-24) we o b t a i n f o r the  d i s t r i b u t i o n i n the image p l a n e , w i t h P = 1 U (x,y,t) = ( V V  [ U  ±  e  where - (f /f ) x x  2  y - - ( f ^ )y  1 J(  (x^'V U l  x + v  * i y  )  h  (x  '  y)]  (2T28)  17  I^Cx.y.t) i s d e f i n e d i n (2-22) g (x,yt)  g (u,v)  T  T  s  g  and  * and <—^ denote a s p a t i a l  convolution  and a s p a t i a l F o u r i e r t r a n s -  frora r e s p e c t i v e l y , as i n d i c a t e d i n Appendix A. The (2-28).  information  from the o b j e c t <j> (x,y,t) i s c o n t a i n e d i n o  To see t h i s we s e t g" = 1.  the i n t e n s i t y  i n the image p l a n e  U (x,y,t)  Then from (2-20),  becomes  = (B 2 w f / f )  2  ±  (2-22) and (2-23)  £  1  2  2  |{R*(x,y,t) [ ^ ( x - y - t ) *  g ; L  (x,y,t)]} *'{g (x,y,t) e  j(  2  u  r *  +  V °  (2-29) where, as i n (A-7), s e c t i o n A.5  ^(x.y) = m In o r d e r  j  (  x  2  +  y  2  )  m  k  /  e  2  f  w  [^(-n^-my) * q(m/(x +y )) ] 2  2  t o b r i n g the q u a d r a t i c phase f a c t o r caused by the use  o f a s i n g l e a c o u s t i c l e n s o u t o f the c o n v o l u t i o n o p e r a t i o n s  i n (2-29)  we r e q u i r e t h a t f o r some l e n g t h dimension d^[12]  § 1  (x,y,t)  =  0 and g ( x , y , t ) 2  e  j  (  u  r  X  +  V  y r  )  : 0  for \/x  2  + y  2  > d.  and (mk ./2f) d W  2  «  1  (2-30)  JL  Then we can w r i t e  |u.(x,y,t)| |{R  2  2  (x,y,t)  = (m B 2 w  £  f^fj)  2  [[* (-mx, -my, t ) * q ( m / ( x + y ) ) ] * o  Mg (x,y,t) e 9  j ( u  r ' X  V  r  y )  }|  2  g (x,y,t)]} 1  (2-31)  18  Thus the image we y  mf^/f^),  see appears as the s c a l e d o b j e c t f u n c t i o n  passed through a f i l t e r w i t h t r a n s f e r  (1/m ) P (/(u +v ) z./ak 2  2  ) |.  2  <f>(x mf^/f^, o  function  (u,v,co)  then m u l t i p l i e d by the f a c t o r R^Cx.y.t) and p a s s e d through a n o t h e r I2  ( -  In  the p r e c e e d i n g a n a l y s i s we  u  and the phase r  u  » "V v  r  r >  co)  d i s t o r t i o n s i n .U, due 1  h^)) were o f no consequence of  IL o n l y .  = 1 (no t r a n s p a r e n c y )  t o the f a c t o r exp  ( - i 2 w„  (h r  I  J  + h o  | -j2w (h { e  J i  2  r  = (B 2 w + h  Q  £  + h ) N  f /f ) x  ^  ( x , y , t ) d e f i n e d i n (2-20).  in  the c o n v o l u t i o n may  of  experiment  (2-23) and  (2-22)  2  2  (  ^  I  i t ) }  The  ^  (  f a c t o r exp  cause u n p r e d i c t a b l e r e s u l t s ,  X  )  5  0  |  2  (-j 2 ^ w  ( 2  (h  and i t may  + h  r  Q  _  3 2 )  + h ))  be a m a t t e r  t o see i f some form o f g^ can be used to improve  the image  quality.  2.5  L e n s l e s s F o u r i e r Transform Hologram We  shall  An  [26]  c o n s i d e r one more method o f image f o r m a t i o n by  ming a l e n s l e s s F o u r i e r t r a n s f o r m hologram  as i n f i g u r e  a c o u s t i c p o i n t s o u r c e a t (x^, y ) i n the f l u i d  cj) (x,y,t). o  R-^t)  i s used  The r e f e r e n c e i s denoted by an impulse <S( -x , x  r  y-y ) r  Thus at the s u r f a c e , by F r e s n e l d i f f r a c t i o n  for-  2.6.  as a r e f e r e n c e and i s the same d i s t a n c e z^ from the s u r f a c e as the object  +  s i n c e we were i n t e r e s t e d i n the i n t e n s i t y  have f o r the i n t e n s i t y i n (2-28) u s i n g  U^x.y.t)  2  set g  However i f use i s made o f a t r a n s p a r e n c y i n the t r a n s f o r m  p l a n e then we  with h  filter  (A-4)  the  19  Plane  wave  light  Image  plane  source  2  =0  0  F i g . 2.6  L e n s l e s s F o u r i e r t r a n s f o r m hologram  refe-renoe wave i s  (x,y,t)  4  r and  e^V *!^*"*/  V  (R.(t)A  =  2  +  ^ r  Y  (2-33)  w x  J.  from the o b j e c t j / . .., /, <f>, ( x , y , t ) = (1/X s  X  W  % z ) X  j (k / 2 z Y e w r  (x  J V  n  • "(xk /z , y\J\, w  2  2 + y3 )  t) P(x/d ,  1  2  y/d )  where s  • <j> ( u , v , t )  <Kx,y,t)  :(X  j./ _\ I t ^ \ j ( k /2z ) • (.x,y,t) = <j>o(x,y,t) e w V n  J V  T>  P P accounts  / J / i \ f l for (x2 ( x / d , y/d ) = |  + y ) 2  /  2  2  Q  o  t  h  e  r  w  i  s  e  2  2  + y ) y  < d 2  2  f o r the f i n i t e s i z e of the hologram. From (2-2)  and  (2-3)  the r a d i a t i o n p r e s s u r e i s  2  (2-34)  20  p (x,y,t) r  * g  = C|l* (x,y,t) + ^ ( x ^ t ) ] .  = (C/z^)  r  e~ r j ( u  |R (t)  2  2  X  V  +  (x,y,t)|  ±  (2-35)  + <j> (x,y,t)|  0  2  (2-36)  2  where u  <>K l> r  =  /z  r  v  r  =  ( k  w  / z  x  l>  y r  4> (x,y,t) = g ( x , y , t ) *'{*((k / z . ) x , (k /z )y , t ) 9  Z  X  W  X  W  X  •p ( / d , y / d ) } x  R (t) 2  2  = R^t)  2  * [ (x,y,t) e "  We have dropped a constant  0  ]  as c o n s t a n t  2  phase  factors  2  can do i f f o r some d [ 1 2 ] (see (3-33)) 3  (k /2z ) d lv  V  +  + y ) ] out o f the c o n v o l t u i o n o p e r a t i o n i n  w  (2-35) which we  X  A l s o , we have b r o u g h t a q u a d r a t i c phase  2 [j ( k / 2 z ^ ) ( x  r  phase f a c t o r i n R  h e r e are of no i m p o r t a n c e .  f a c t o r exp  j ( u  g l  1  2 3  «  1  (2-37)  and g  ±  (x,y,t) z 0  for (x  2  + y ) 2  < d  3  Thus we o b t a i n f o r p ( x , y , t ) r  i n (2-36) a form s i m i l a r to the  case o f a p l a n e wave r e f e r e n c e i n s e c t i o n at  the s u r f a c e i s then s i m i l a r U (x,y,t) = g  2.3  .  The l i g h t  distribution  to (2-22) and i s  Z U (x,y,t) e n=-°° n  j ( U  n  X  +  V  n  y  (2-38)  )  where TT / x.\ / -\ -n U ( x , y , t ) = (-j) B e n  n  -  J2w (h + h £ r o n  + h . J _ ,. , jna N J (4w h ) e x  II  &  a  J  21  u v  n  = u. + n u I r  n  = v. + nv it r  and h (x,y,t) = (C/ r  Z ; L  A )  h (x,y,t) = (C/z A ) o  1  h (x,y,t)  e  a  j R (t) |  2  w  w  i a ( X i V f t )  * g  2  2  2  | <j) (x,y,t)|  2  2  * g  (x,y,t)  2  = {R* ( t ) 4> (x,y,t)} 2  J  • r  - i ( u x + v y), e r r' }  /  * {g (x,y,t)  J  2  To  (x,y,t)  2  o b t a i n an image a l e n s i s used  face l i g h t d i s t r i b u t i o n .  V  y  t o F o u r i e r t r a n s f o r m the sur-  I n the t r a n s f o r m p l a n e o f the l e n s , o r the  image p l a n e , we have d i f f e r e n t  d i f f r a c t e d o r d e r s as i n f i g u r e 2.5.  As-  suming no s i g n i f i c a n t  o v e r l a p o f the o r d e r s , then a p p l y i n g ( A - 8 ) t o  (2—3-8) and-with 4w h  < 1 we o b t a i n f o r the o r d e r n = +1 the l i g h t d i s s.  tribution  U (x,y,t) f  [(4  = A D (u,v,t) * { e  Q  +  ^  )  (u + u , v + v r , t ) ] J  2  r  A = -j 2w  B e  u = ( k ^ )  x - u  v = ( k ^ )  y - v  2  ±  x  q (u)  = 2TT d  f^ =  lens focal length  2  (  o  with  2  c  J (d u)/u 1  /  2  * (<j, (-cfl, - c v , t ) * q  (t) [^(u.v.t)  i  j  2  (/("u + v ) ) ) ] ) 2  2  2  (2-39)  22  i  -j  4- h) N  2w„(h + h & r o  J  s — , . <r^> D ( u , v , t ) 2  We have taken the f a c t o r exp in  (2-28) and thus we  require  k ) (7r/d ) w z  (zJl  2  0  1  ( j c (u  «  2 + v )/2) out o f a c o n v o l u t i o n w i t h  [12]  1  r~ 2 2 s i n c e q^ ( v ( u + v )) i s n e g l i g i b l e f o r r  d V(u 2  In  2  2 + v ) > TT  (2-39) the o b j e c t f u n c t i o n <j> appears w i t h a l o s s o f o  resolu-  t i o n shoTm by the space c o n v o l u t i o n w i t h q^ due to the f i n i t e hologram s i g  Then i t i s passed through a temporal f i l t e r on the space c o o r d i n a t e s ( u , v ) . the  amplitude i n time.  g" which 2  (u,v,t) t h a t depends  1  The m u l t i p l i e r  ( t ) then changes  Then i t i s passed through another temporal  a l s o depends on the space c o r r d i n a t e s  (u,v) .  phase -factor then act-s as a m u l t i p l i e r over space.  The q u a d r a t i c  Finally  a d d i t i o n a l d i s t o r t i o n caused by the s p a t i a l c o n v o l u t i o n w i t h The  filter  there i s D.  d i s t o r t i o n due to D d i d n o t appear i n the f o c u s e d image  method d i s c u s s e d i n s e c t i o n 2.4 (2-32) the f a c t o r exp t h a t i n (2-32) and  (-j 2 w^  ( e q u a t i o n (2-31)). However i n e q u a t i o n (h  + h  Q  + h )) N  (2-39) t h i s d i s t o r t i o n due  does appear and i t seems to the phase f a c t o r  will  be s l i g h t o n l y i f the space v a r y i n g p a r t o f the s u r f a c e d e f o r m a t i o n h^ + h  + h^ <<  Q  A. £  Another d i s a d v a n t a g e o f the p r e s e n t method  from the d e m a g n i f i c a t i o n  of the image by  f,A /z A 1 x. 1 w Since A  w  >>  A^ a microscope must be used to see the image.  arises  23  III.  RADIATION PRESSURE AND THE FUNCTION g ( u , v , t i ) ) 1  In t h i s  chapter we s h a l l  d e r i v e t h e f u n c t i o n g^(u,v,co) which  was used i n c h a p t e r 2 to determine t h e e f f e c t i v e v a l u e o f the v e l o c i t y p o t e n t i a l a t the l i q u i d s u r f a c e o f the m i n i t a n k . e x e r t e d by a sound f i e l d  The r a d i a t i o n  on the l i q u i d s u r f a c e i s then  pressure  g i v e n by the e f -  f e c t i v e v a l u e o f the v e l o c i t y p o t e n t i a l as i n d i c a t e d by ( 2 - 3 ) . In the f i r s t s e c t i o n the f u n c t i o n g^ i s determined liquid  s u r f a c e i n the absence o f the m i n i t a n k .  s e c t i o n s the e f f e c t o f the m i n i t a n k  3.1  Radiation Pressure  fora  Then i n t h e f o l l o w i n g  on g^ i s determined.  o f Sound Waves on a Free L i q u i d  Surface  In t h i s s e c t i o n we s h a l l determine the r a d i a t i o n p r e s s u r e ted  by a sound f i e l d The  exer-  on the f i x e d s u r f a c e o f an o b j e c t .  l i n e a r i z e d equations  d e s c r i b i n g sound f i e l d s  i n ideal  f l u i d s a r e as f o l l o w s [15]  - c  dt  A<j> = 0  2  (3-1)  v = V <j> P  (3-2)  f£  = -P  1  (3-3)  c =|B-  (3-A)  2  P'  -  (J*-)  (3-5)  p'  o The  symbols v, <j> , p and p denote the f l u i d v e l o c i t y ,  p o t e n t i a l , p r e s s u r e and d e n s i t y r e s p e c t i v e l y . t i o n o f a sound wave i s g i v e n by c.  velocity  The v e l o c i t y o f p r o p a g a -  The primes denote v a r i a t i o n s about  24  the s t a t i c , v a l u e , v/here the s t a t i c v a l u e i s i n d i c a t e d by i.e.  p --- p* +  p  the s u b s c r i p t o.  o  P = P' + p o The  l i n e a r i z e d equations v <<  c  are v a l i d i f  or  p  << p  1  o In a sound f i e l d ,  the r a d i a t i o n p r e s s u r e p  r >  e x e r t e d on  f i x e d s u r f a c e of an i n t e r f a c e between d i f f e r e n t f l u i d s o r on the f a c e of an o b s t i c l e i n the f l u i d  i s g i v e n by  f l u x p e r u n i t a r e a of the s u r f a c e .  P  r  = p' n + p v  The b a r denotes a time  in  time  a first p' we  For an i d e a l f l u i d  average,  the sound f i e l d  momentum  t h i s becomes  order approximation, the e q u a t i o n s  [14]  (3-6) and n i s a u n i t v e c t o r normal to  the  fluid. c o n s i s t s of waves t h a t v a r y  then p' from l i n e a r theory i s z e r o .  must use  averaged  sur-  (v'n)  s u r f a c e , p o i n t i n g out of the If  the time  the  and  sinusoidally  However, t h i s i s o n l y  to o b t a i n a more a c c u r a t e v a l u e f o r  of n o n - l i n e a r a c o u s t i c s .  Then f o r an i d e a l f l u i d we  have  [14,15]  (3-7)  which i n c l u d e s terms up to second o r d e r .  The  (3-7)  equations.  can be o b t a i n e d from the l i n e a r i z e d If  the v e l o c i t i e s  the s u r f a c e (z=0)  then  normal, and i s , from  v a l u e s o f p' and v i n  at the s u r f a c e i n q u e s t i o n are normal to  the r a d i a t i o n p r e s s u r e p^ i s a l s o a l o n g  (3-6) and  the  (3-7) (3 8) T  25  We s h a l l now determine normal v e l o c i t y v • boundary  the v a l u e s o f the d e n s i t y p' and  When a t r a v e l l i n g p l a n e wave <JK s t r i k e s  (z=0) between two d i f f e r e n t  f l u i d s we o b t a i n  t r a n s m i t t e d waves < j > and d> as shown i n f i g u r e 2  Figure 3.1  (x,z) =  /  l  •  ^  (x,z)  r  j ( u , x + w,z)  A  x  R  1  e ^  u  l  1  " l  x  W  0> (x,z) = A J<V< * 2 +  2e  2  (3.1).  r e f l e c t e d and t r a n s m i t t e d waves a r e  A. e  = A  the r e f l e c t e d and  Waves at a boundary  I f the i n c i d e n t ,  4>.  a plane  then we have the f o l l o w i n g  z  '  (3-9)  )  Z )  relationships  Z . cos 9., - Z . cos6„ r _ _2 1 1 2 A^ Z cos0^ + Zy c o s 0  between the t h r e e waves [ 1 5 ]  A  2  (3-10)  2  R  12  2Z P  2  Z  1  2  cos0  cosS.^ 2  + Z  2  cos0  (3-11) 1  26  12 k. (3-12)  sin0.  (3-13)  PC  u. = k. s i n e , i i l w. = k. cos6. i i i k.  I  =  i  (3-14)  = 1, 2  2TT/A.  l  R.. i s the r e f l e c t i o n ij i n medium ( i ) and r e f l e c t e d d e f i n e d i n (3-10)!.  o f a p l a n e wave  travelling  from a p l a n e boundary w i t h medium ( j ) as  Similarly  a p l a n e wave t r a v e l l i n g  coefficient  T. . i s the t r a n s m i s s i o n c o e f f i c i e n t o f ij  from medium ( i ) i n t o medium ( j ) as d e f i n e d by  (3-11). As a p l a n e wave t r a v e l s quency u^ a l o n g the x d i r e c t i o n another.  through a boundary the s p a t i a l  fre-  does n o t change from one medium to  We s h a l l w r i t e , u. = k. s i n 0 . l l l  (3-15)  = n also cose. = /{i -  (3-16)  n /k. ) 2  2  Then we can w r i t e R.., T.., w. i n terms o f s p a t i a l ij i j l  f r e q u e n c y n.  I f medium (2) i s a gas and medium (1) i s a l i q u i d >> p  2  and R ^  ~ -1-  Then the v e l o c i t y  becomes the sum o f the i n c i d e n t tj>(x,z) = > (x,z) + T  i  i  J  potential  and r e f l e c t e d  then  (j) i n the f l u i d (1)  waves as  ^(x.z)  2 A. s i n w z i 1  e  1  (3-17)  27  Then from (3-2) the v e l o c i t i e s v v  =  0  -  J  and v  x  at the s u r f a c e z=0 a r e  z  z=0 j 2 w  z=0 = j 2  Thus the v e l o c i t y p o t e n t i a l by  1  V7  A. i  e  J  u  l  x  (3-18)  <(,.  1  T  l  z=0  a t the s u r f a c e i s r e l a t e d  (3-18), and the v e l o c i t y  to the i n c i d e n t  velocity  a t the s u r f a c e due t o a  single  p l a n e wave o f s p a t i a l frequency n becomes  ]J1 -  j 2 k  ±  n /^ 2  (3-19)  2  z=0 For s i o n s we and  z=0  an a r b i t r a r y v e l o c i t y p o t a n t i a l  can then a s s o c i a t e a t r a n s f e r  the i n c i d e n t  velocity  potential  v  z  (u,v) = j 2 k  v  z  (x,y) = j 2 k  g  function  r e l a t i n g the v e l o c i t y  a t the s u r f a c e as f o l l o w s  (u,v)  ±  (u,v)  (x,y): * ^  (x,y)  ±  x  d i s t r i b u t i o n i n 2 dimen-  J  (3-20)  or ±  g  ±  where g ( x , y ) Jl>  gy (u,v) =  1  n  Note t h a t  2  = u  yl - n  2  / ^  2  2 , 2 + v  g^(x,y) i s not a f u n c t i o n  o f time, a l s o , v h e r e i s a s p a t i a l  frequency and not v e l o c i t y . We  can now  determine the r a d i a t i o n  s u r f a c e due to an a r b i t r a r y  sound f i e l d .  p r e s s u r e on the l i q u i d  From  (3-17) <j>  n  =' 0 and-  z=U thus from (3-3) and (3-5) p ' l . = 0. z=u  Then from (3-8) and  (3-20)  28  with v  z  a complex  f u n c t i o n , the r a d i a t i o n p r e s s u r e on the s u r f a c e  is p (x,y) = p ^ r  2  | ( x , y ) * <J). g l  (x,y)|  (3-2.1)  2  2 T h i s i s the r e s u l t t h a t was used i n (2-2) w i t h C = p k . o 1 1  s i n c e p^_ i s a time average q u a n t i t y over s e v e r a l p e r i o d s o f the sound 3.2  Note  that  the time average s h o u l d be taken  field  oscillation.  E f f e c t of the M i n i t a n k on the R a d i a t i o n P r e s s u r e  In the f o r m a t i o n of a c o u s t i c a l holograms f a c e a m i n i t a n k i s used as shown i n f i g u r e  3.2  using a l i q u i d  .  sur-  A t h i n membrane  s e p a r a t e s the l i q u i d i n the m i n i t a n k from the water i n the main tank, to p r e v e n t the s u r f a c e from b e i n g d i s r u p t e d by any d i s t u r b a n c e s i n the water t h a t are not a s s o c i a t e d w i t h the sound  Air  ^  Water  Liquid surface  hologram  Minitank -Thin  Fig.  3.2  membrane  Minitank  P u l s e s o f u l t r a s o u n d waves t h a t the water  field.  form the hologram pass from  through the membrane and bounce around i n the m i n i t a n k .  t h i s s e c t i o n we  s h a l l determine the e f f e c t o f the waves t h a t bounce  around and i n c l u d e  these e f f e c t s i n the f u n c t i o n g ^ ( u , v ) .  be shown t h a t a time dependency We  In  i s introduced into  It w i l l  g^.  s h a l l assume t h a t the p r e s e n c e o f the membrane i n f i g u r e  3.2 has n e g l i g i b l e e f f e c t on the passage o f sound waves, which  requires  29  t h a t the membrane be much t h i n n e r than a wavelength o f sound. sider  the s i t u a t i o n i n f i g u r e  3.3 where a p l a n e wave  passes i n t o medium (2) (the m i n i t a n k ) and s t r i k e s  <j)^ i n medium (1)  the s u r f a c e a t time  t = 0 and i s r e f l e c t e d a t the s u r f a c e and i n t e r f a c e of  Con-  (2) - (1) a number  times. j(u +w(z+d)) ^(x.z.t) = e r x  J V  (3-2 2)  y(t)  0 for t < 0 1 for t > 0  y(t) =  Main tank  Fig. The  3.3  Reflections  i n the m i n i t a n k  velocity potential  incident  on the s u r f a c e i n medium  (2) i s = T  1 2  eJ  ( u  2  (3-23)  ^2 \(t) d)  z=0 =  and  the v e l o c i t y  multiple  T  12 * i  y(t) z=0  a t the s u r f a c e i s then, t a k i n g i n t o account the  reflections = j2k 2 = 0  2  /U-  2 n  /k  2 2  ) T  E  1 2  n  =  0  n^n j2nw d (-l) R^ e ^ 2 n  n  W  Q  y(t-t ) n  (3-24)  30  2 The  / ( l - n /k^ ) <(K i s s i m i l a r  factor j 2  when no m i n i t a n k (see  2  was  present while T^  to the r e s u l t i n (3-19)  i s the t r a n s m i s s i o n  2  (3-11)) i n p a s s i n g from medium (1) i n t o medium ( 2 ) .  the wave r e f l e c t s o f f the s u r f a c e , the i n t e r f a c e the s u r f a c e a g a i n , i t changes i n phase by z, where (-1)  and R  at the s u r f a c e and  interface  f a c e i s g i v e n by n + 1.  exp  time reaches  ( j 2 w^d) (see  and  along  (3-10))  d i s the  The number o f r e f l e c t i o n s o f f the  thicksur-  Also, /("l - n /k )  = n 2 d/c  n  R^  (2)-(l) respectively,  (2).  Each  (2) - ( 1 ) , and  are the r e f l e c t i o n c o e f f i c i e n t s  ness o f the f l u i d l a y e r  t  (-1)  coefficient  2  (3-25)  2  2  2 where c i s the v e l o c i t y of p r o p a g a t i o n  and  2d/c  2  / ( l - n /k  ) i s the  2  time between r e f l e c t i o n s o f f the s u r f a c e . Similarlyj  i f the wave <j>^ i s a s h o r t p u l s e or  ^(x.z.t) = then  the v e l o c i t y  e  J  (  u  l  x  +  (  z  +  d  )  -  J 2  k  2  impulse  6(t)  )  at the s u r f a c e w i l l be  time of d i f f e r e n t phase and  v J  l  w  (3-26)  a sum  o f impulses  d i m i n i s h i n g amplitude  /(W/k/)  T  I Z=0  1  2  *  !  t  n  as  delayed  in  follows  (-1)"  R  2 1  "  e^V  5  (t-t )  =-co  (3-27) We  can r e l a t e the v e l o c i t y  velocity potential  at the s u r f a c e to the  i n terms o f the t h r e e d i m e n s i o n a l  incident  Fourier transform  gj^ (u,v,w) as v (u,v,to)  = j 2 k  2  v (x,y,t) = j 2 k  2  z  g  1  (n,to)  ^(u.VjO))  (3-28)  or z  g l  (x,y,t)  * <j>..(x,y,t)  (3-29)  n  31  where  n  2  2 , 2 + v  = u  and g ^ ( x , y , t ) i s the e f f e c t i v e v a l u e o f the v e l o c i t y p o t e n t i a l when the i n c i d e n t v e l o c i t y p o t e n t i a l i s an impulse 6(x,y,t).  i n space and  time  Then w i t h g (x,y,t) ^» 1  we  (2-2)  g ; L  (u,v,w) = g  (n,w)  1  have  l  (n.ui) =  ±  A i - n /k ) 2  •Z n=0 where e ^  U t  2  2  (-D R n  n 9 i  T  1 2  (n)  (n)  j  2  n  d  w  e  2  (  n  )  e  j  U  t  n  (  n  (3-30)  )  n i s the F o u r i e r t r a n s f o r m i n time o f 6 ( t - t ) . n One  final  i t e m t h a t s h o u l d be mentioned i s t h a t  (3-10) and  (3-11) are v a l i d f o r 0 < n < (smaller of k , k ) 1 We  t h e r e f o r e d e s i r e the p r o p e r t i e s o f the f l u i d media (1) and  such  that k  2  > k  x  (or c . s 2  c^)  o t h e r w i s e no s p a t i a l f r e q u e n c i e s from the o b j e c t h i g h e r than k pass  through  minitank. interface  w i l l be  3.3  (2) to be  The  the i n t e r f a c e ,  2  will  from the water, i n t o the f l u i d i n the  In o t h e r words, i f k  2  < k^,  a l l p l a n e waves s t r i k i n g  the  (1) - (2) w i t h angles 6 g r e a t e r than the c r i t i c a l angle 0 = a r c s i n (k„/k.) c 2 1  t o t a l l y r e f l e c t e d a t the  Transfer Function  In  t h i s s e c t i o n we  interface.  (ri,w)  s h a l l c o n s i d e r the t r a n s f e r  function  0^  32  g^(u,v,w) i n more d e t a i l when a) = 0, i . e . when t h e r e i s no time t i o n o f the i n c i d e n t  ultrasound f i e l d .  at the time s t e p response The same.  g  as there would be no  \Jl - n / k 2  f o r v a r i o u s depths o f f l u i d 2  /.k. ")  are the same.  0  ).  (1) and (2) a r e the  i n t e r n a l r e f l e c t i o n s i n the m i n i t a n k , b u t the  ^ 1»  (see f i g u r e ^ ^  as g i v e n by  Some v a l u e s o f g^(n>0) a r e p l o t t e d  2  = c o  2  e f f e c t s of the i n t e r n a l r e f l e c t i o n s  -  (n  2  two l i q u i d s a r e d i f f e r e n t then T ^  r  state  and  h i g h s p a t i a l f r e q u e n c i e s would be a t t e n u a t e d  f o r V.(l - n  a steady  s i m p l e s t case o c c u r s when l i q u i d s  (n,oj) =  ±  T h i s i s e q u i v a l e n t to l o o k i n g  a f t e r i t has reached  Then T.^ = 1, R^^ = 0  varia-  d.  In f i g u r e s  .is .the v a l u e o f  I t i s apparent  =  ,g.  ^  w  e  m  u  s  t  u  m  t  n  e  (3-30).  i n figures 3-4  s  I f the  and  3.4  3.6  to  3.7  the curve  (.n.,.o) i f the f l u i d s (1) and (2)  i n figure  shown the m i n i t a n k has the e f f e c t  a n <  3,4).  3-4  t h a t f o r some depths d  of f l a t t e n i n g  the curves  for  g^(n>0)  f o r some depths, but not f o r o t h e r s , and t h a t the curves a r e f a i r l y s i t i v e t o depth.  F o r the g r e a t e r depths d i n f i g u r e  3.6  the  sen-  curves  become o s c i l l a t o r y f o r the h i g h e r s p a t i a l f r e q u e n c i e s . The phases o f g-^ruO) a r e p l o t t e d corresponding figure  3.5  to the a b s o l u t e v a l u e s i n f i g u r e s  3.4  3.5  and  and  3.6  the phase d i s t o r t i o n does not appear s i g n i f i c a n t  f o r the h i g h s p a t i a l f r e q u e n c i e s . quencies  i n figures  .  3.7 In  except  However, these h i g h s p a t i a l f r e -  might not be p r e s e n t a t the s u r f a c e anyway  f i n i t e appertures  used i n the imaging  d i n figure  the phase d i s t o r t i o n becomes o s c i l l a t o r y f o r the  3-7  higher s p a t i a l frequencies.  Again,  system.  because o f the  F o r the g r e a t e r depths  the curves i n f i g u r e s  3.5  and  33  34  35  36  9 to 9 1/2  37  3.7  are f a i r l y  s e n s i t i v e to the depth  Although  d.  the curves i n f i g u r e s  3.4  to  the v a l u e s o f g^(n-O) f o r n ~ 2 o r 3 are almost very high s p a t i a l frequencies. of for  3.7  were f o r n =  the same, except  That i s , i t takes o n l y a few  I^^Cu) the  coefficient  and the l e s s e f f e c t  the m i n i t a n k has on g^.  The  a steady  reason why  reflection state  the h i g h e r  s p a t i a l f r e q e u n c i e s take more • r e f l e c t i o n s , n, t o r e a c h a s t e a d y i s because  increases with s p a t i a l Thus we  see t h a t the e f f e c t  f o r the  state  The s m a l l e r the  more q u i c k l y g^(n,0) reaches  ,  reflections  the sound waves i n the m i n i t a n k i n order to r e a c h a steady the d e n s i t i e s and wavelengths i n d i c a t e d .  00  state  frequency. o f the f l u i d  tank on the imaging p r o c e s s e s i s to cause  l a y e r i n the m i n i -  amplitude  and phase  distor-  t i o n s i n the image o b t a i n e d , e s p e c i a l l y at h i g h s p a t i a l f r e q u e n c i e s . Although  t h e r e may  be some depths o f the m i n i t a n k f o r which the a m p l i -  tude v e r s u s s p a t i a l  frequency o f g^ i s q u i t e u n i f o r m ,  (figure  3.4),  t h e r e i s a phase d i s t o r t i o n a s s o c i a t e d w i t h t h i s a t h i g h s p a t i a l f r e quenices. finite  A g a i n , however, the h i g h f r e q u e n c i e s may  a p e r t u r e s o f the imaging A l s o , we  resonance  effect  can see t h a t  caused by  be  c u t o f f by  the  system. f o r the lower s p a t i a l  f r e q u e n c i e s the  c o n s t r u c t i v e i n t e r f e r e n c e o f the  r e f l e c t i o n s i n c r e a s e s the amplitude  o f g^.  a s p e c t does not appear s i g n i f i c a n t ,  as the more important c o n s i d e r a t i o n  is  the u n i f o r m i t y o f g^ w i t h s p a t i a l It  of  the f l u i d  does not appear obvious i n the m i n i t a n k .  curves i n f i g u r e s  3.4  and  3.5  However, t h i s  internal  positive  frequency. t h a t t h e r e i s any optimum  However, we  can at l e a s t  depth  say t h a t  the  are f a i r l y u n i f o r m o v e r a w i d e r range  s p a t i a l f r e q u e n c i e s than i n f i g u r e s 3.6  and  3.7.  Of  of  the curves shown  38  the b e s t may be f o r d/A^ = 2 3 / 8 . It  s h o u l d a l s o be mentioned t h a t i n d e r i v i n g  l a y e r was assumed t o be o f i n f i n i t e that  l a t e r a l extent.  g^ the l i q u i d  That  i s , we assumed  t h e r e were no r e f l e c t i o n s o f f the s i d e s o f the f i n i t e  minitank.  I t may be d e s i r a b l e t o have a sound a b s o r b i n g m a t e r i a l a l o n g s i d e s o f the m i n i t a n k  to absorb (2-30)  In i n e q u a l i t y some d^ o r d^ such condition in  and ( 2 - 2 7 )  that g^(x,y,t)  s p a t i a l frequency.  we assumed t h a t t h e r e was  ~ 0 f o r \/fx + y ) > d^. 2  2  This  F o r the case  = vfl - n / k ) 2  (3-31)  2  can o b t a i n g^(x,y) by t a k i n g the i n v e r s e Hankie t r a n s f o r m o f ( 3 - 3 1 )  and w i t h  [22],  then  g- (r) = — — £ (rk) L  f~2  where r = yx  2  + y .  (  rk - 4 . 5 , so i f we  "  S i  r k k  (3-32)  - cos rk}  r  2  At r = 0 we have g-^(O) = k / 3 , and f o r l a r g e r  v a l u e s o f r , g-^(r) decreases  can  the u l t r a s o n i c waves.  i s more e a s i l y met the more u n i f o r m l y g ^ ( n , t ) i s extended  ^(n) we  the i n n e r  and o s c i l l a t e s .  The f i r s t  z e r o i s at  take g ^ ( r ) t o be n e g l i g i b l e beyond r = 4 . 5 / k , we  take d  v  d  3  = 4.5/k  Finally, o f the l i q u i d  c h a p t e r we s h a l l  determine the  response  s u r f a c e to a pulse o f r a d i a t i o n p r e s s u r e , i . e . , the  function g (x,y,t). 2  i n the next  (3-33)  39  IV.  LIQUID SURFACE RESPONSE TO RADIATION PRESSURE  The  l i q u i d s u r f a c e a c t s as a d e t e c t o r o f the u l t r a s o n i c energy  t h a t i s produced by  the t r a n s d u c e r s  t  and  t  Q  of. f i g u r e 2.1.  When  sound waves are r e f l e c t e d o f f o f o r absorbed by  an o b j e c t , a r a d i a t i o n  pressure  case of the l i q u i d  [14]  i s e x e r t e d on the o b j e c t .  f a c e a deformation The we  of the s u r f a c e r e s u l t s which r e p r e s e n t s  deformation  shall i n i t i a l l y  t r a n s i e n t motion  4.1  s t a t e case and  sur-  the hologram.  of the s u r f a c e i s time dependent.  c o n s i d e r the steady  However,  l o o k at  the  later.  Steady S t a t e Response  In the steady surface height,  s t a t e there i s a balance  the r a d i a t i o n p r e s s u r e ,  between the  g r a v i t y and  surface  liquid tension  [6]  as f o l l o w s  P ( x , y ) - Pg h r  (x-y) + y  p^ i s the r a d i a t i o n p r e s s u r e , due  In the  A  h  (x,y) = 0  p the f l u i d d e n s i t y , g the a c c e l e r a t i o n  to g r a v i t y , h the h e i g h t o f the s u r f a c e above the q u i e s e n t  and Y the c o e f f i c i e n t I f we h  take  (u,v)  (4-1)  of s u r f a c e  tension.  the F o u r i e r t r a n s f o r m  = p (u,v)  level,  of the terms i n (4-1)  then  (u,v)  r  where g (u,v) = 2  g (n) 2  (4-3)  = l/(pg + yn ) 2  n The  2  = u  2 ^ 2 + v  t r a n s f e r f u n c t i o n g ( u , v ) was 2  used i n  (2-4)  40  Thus, as i n d i c a t e d by (4-2) and (4-3)  the h e i g h t h o f the  s u r f a c e d e f o r m a t i o n i s r e l a t e d to the r a d i a t i o n p r e s s u r e P pass t r a n s f e r f u n c t i o n g^i^).  r  by the low-  The amplitude o f the f u n c t i o n g2(n) i s  down by a f a c t o r of 1/2 when t h e s p a t i a l f r e q u e n c y n i s g i v e n by n = /("pg/Y)  (4-4)  For f r e o n E-5 (y = 16 dynes/cm, p = 1.8 gm/cm ) the h a l f frequency i s 1.7 cycles/cm.  I t appears  amplitude  then t h a t the r e s o l u t i o n we  can expect t o o b t a i n i n an image w i l l be l i m i t e d by the low pass response o f the l i q u i d s u r f a c e .  F o r f r e o n E-5 we may be s e v e r l y  l i m i t e d i n r e s o l u t i o n by the 1.7 c y c l e s / c m h a l f a m p l i t u d e , y e t we may is  be u s i n g u l t r a s o n i c waves w i t h wavelengths  o f around  0.03 cm which  33 c y c l e s / c m (5 MHz i n w a t e r ) . A l s o , we can see i n (4-3) t h a t t h e r e i s a zero frequency  •component to t h e s u r f a c e d e f o r m a t i o n .  In (4-3) t h i s r e p r e s e n t s a -uni-  form l e v e t a t i o n o r b u l g e o f the s u r f a c e , b u t i t may a c t u a l l y be a s o u r c e o f d i s t o r t i o n i n the image. t  r  and t D  I f the u l t r a s o n i c beams from t r a n s d u c e r s  o f f ibg u r e 2.1 cover o n l y p a r t o f t h e s u r f a c e o f the m i n i t a n k  then t h e r e w i l l be a d i s c o n t i n u i t y o f r a d i a t i o n p r e s s u r e a t t h e edges o f the area covered. be  Thus, r a t h e r than b e i n g u n i f o r m , the b u l g e may  curved, e s p e c i a l l y near the edges,  and d i s t o r t  t e r n c o n t a i n i n g the i n f o r m a t i o n o f the hologram  the s u r f a c e p a t -  o r v i o l a t e the c o n d i t i o n  t h a t the space v a r y i n g component o f h s h o u l d n o t be l a r g e to a wavelength  compared  of light.  I t s h o u l d be p o s s i b l e t o a v o i d the b u l g e i f the u l t r a s o n i c beams cover t h e e n t i r e s u r f a c e , s i n c e then the z e r o frequency component o f h cannot  r a i s e up t h e s u r f a c e .  s u r e i n the f l u i d w i l l s i m p l y decrease  In t h i s [20].  case, t h e average p r e s -  However, t h e r e may s t i l l  41  be  a d i s t o r t i o n or c u r v a t u r e  tank due  to s u r f a c e t e n s i o n .  of the s u r f a c e at the s i d e s o f the  mini-  I t remains a m a t t e r o f experiment to  see  i f b e t t e r images can be o b t a i n e d by i r r a d i a t i n g the e n t i r e s u r f a c e .  4.2  T r a n s i e n t Response of the L i q u i d  4.2.1  Formulation  I t was  suggested by B.B.  r a t h e r than to use  continuous  images were o b t a i n e d .  Surface  of the Problem Brendon to p u l s e  waves.  I t was  T h i s improvement was  i n c r e a s e i n the s p a t i a l frequency  the u l t r a s o n i c waves  then found t h a t much b e t t e r a t t r i b u t e d to a t r a n s i e n t  bandwidth o f the l i q u i d  surface  response to the r a d i a t i o n p r e s s u r e . An  a n a l y s i s of the t r a n s i e n t motion of the l i q u i d  response to the  r a d i a t i o n pressure  than f o r 'the steady  s t a t e case.  s o l v e t h i s problem by o b s e r v i n g of r a d i a t i o n p r e s s u r e  i s c o n s i d e r a b l y more i n v o l v e d  In the r e s t of t h i s s e c t i o n we  of s i n u s o i d a l space v a r i a t i o n . a finite  The  Z I.  L i q u i d surface  final solu-  d i f f e r e n c e scheme, and .  the r e s u l t s w i l l be shown to f i t an a n a l y t i c e x p r e s s i o n .  4.1  will  the response of the s u r f a c e to a p u l s e  t i o n w i l l be o b t a i n e d n u m e r i c a l l y , by  Fig.  surface  deformation.  42  As shown i n f i g u r e 4.1 we have a l i q u i d  l a y e r o f depth d  which i s assumed to be i n f i n i t e i n e x t e n t i n the x d i r e c t i o n . f a c e i s deformed i n t o a s i n u s o i d a l p a t t e r n n by  The  h(x,t) with s p a t i a l  the s i n u s o i d a l r a d i a t i o n p r e s s u r e p ^ ( x ) .  We wish  sur-  frequency  to determine  the  amplitude o f t h i s s i n u s o i d a l d e f o r m a t i o n h ( x , t ) which v a r i e s i n time. I t i s assumed t h a t the a i r above the l i q u i d the l i q u i d motion  i s d e s c r i b e d by  the f o l l o w i n g e q u a t i o n s  (4-5)  (4-12). Throughout  |J dt  the f l u i d :  ip  = v A v -  Vp + g  (4-5)  Vv = 0  (4-6)  Boundary At  the s u r f a c e  conditions:  (z=h) 8 h z "Y — K + 2 pv-rf; 3x 2  P =-p_ r  3z  +  T —  , z= -d  Initial h(x)|  t = Q  v(x,z)| The the  (4-7)  2  0  =  2  3x  the bottom (z = 1  9  9v X  v|  9 V  2  9v  At  on  and thus the a i r w i l l be r e p l a c e d by a vacuum.  The problem to  has n e g l i g i b l e e f f e c t  (4-8)  -d) =  0  (4-9)  c o n d i t i o n s ( t = 0): =0  t = ( )  (4-10) = 0  (4-11)  r a d i a t i o n p r e s s u r e on the l i q u i d  s u r f a c e (z=h) w i l l  take  form P (x,t) = P r  r  cos nx  [y(t) - y(t-At)]  (4-12)  43  P^ i s the c o n s t a n t pulse duration  r a d i a t i o n p r e s s u r e amplitude,  n = 2ir/X, At i s the  and 1 for t > 0 I 0 otherwise  A l s o , we v, t and and  g are  have the c o n d i t i o n t h a t h <<  Equation  (4-5)  The Equation  (4-6)  The  viscous  (4-7)  the v i s c o u s f o r c e s , and  i n t e r f a c e when h <<  c o n d i t i o n s are g i v e n by w i l l s o l v e (4-5) cell  for incompressible ultrasound be  time  equation satis-  i f h <<  A  (4-7)  and  to (4-9),  [18] from c o n s i d e r a t i o n s o f the b a l a n c e  A.  the s u r f a c e  Equation  states  Equation  (4-9)  (4-10) and  (4-11).  numerically using a s i m p l i f i e d  f i e l d w i l l not be  E f f e c t s due  form o f  to a c o u s t i c s t r e a m i n g  accounted f o r .  determined n u m e r i c a l l y , w h i l e  is  zero v e l o c i t y at a f i x e d w a l l .  f i n i t e d i f f e r e n c e method o f Harlow and Welch fluids.  of  tension  (4-8)  surface i s zero.  the c o n d i t i o n t h a t a v i s c o u s f l u i d has  the marker and  [16]  and must be  linearization i s valid  t h a t the shear s t r e s s at the l i q u i d  We  p,  i s the i n c o m p r e s s i b i l t i y c o n d i t i o n .  i s obtained  at the l i q u i d - v a c u u m  initial  fluid,  boundary c o n d i t i o n s are g i v e n by e q u a t i o n s  the r a d i a t i o n p r e s s u r e ,  The  viscosity,  i s the l i n e a r i z e d N a v i e r - S t o k e s  f i e d by a l l f l u i d p a r t i c l e s . Equation  symbols v,  to g r a v i t y , r e s p e c t i v e l y .  f o r the motion of an i n c o m p r e s s i b l e  d.  The  the f l u i d v e l o c i t y , p r e s s u r e , k i n e m a t i c  a c c e l e r a t i o n due  h <<  A.  i n the  Only the v e l o c i t i e s  the p r e s s u r e  can be  [21]  determined  will as  follows. I f we  take  the d i v e r g e n c e  of  (4-5)  and  use  (4-6)  Ap = 0 where A i s the L a p l a c i a n o p e r a t o r .  we  find  that  (4-13) From (4-5)  and  (4-9) we  obtain  the  44  boundary c o n d i t i o n 3 v 2  (ff " P  f)  v  = -Pg  (4-14)  3z Now, we s h a l l assume t h a t s i n c e the d r i v i n g f u n c t i o n p^, the r a d i a t i o n p r e s s u r e , v a r i e s s i n u s o i d a l l y a l o n g x, then the p r e s s u r e p and the v e l o c i t y v^ a l s o v a r y s i n u s o i d a l l y a l o n g x.  A l s o , s i n c e h <<X and h << d-  we assume t h a t we can s e t h = 0 i n d e t e r m i n i n g p(x,z,t).  the p r e s s u r e  distribution  Thus a t some time t we s e t p ( x , z , t ) = P ( z , t ) cos nx - Pgz  (4-15)  v ( x , z , t ) = V ( z , t ) cos nx z z  (4-16)  where P ( z , t ) and V ( z , t \ are the amplitudes  o f the p r e s s u r e and z  component o f v e l o c i t y r e s p e c t i v e l y a t some time t . Equations for at  (4-13) to (4-16) d e s c r i b e a boundary v a l u e problem  the p r e s s u r e p i n the f l u i d the time  l a y e r z = -d to z = 0.  The s o l u t i o n  ti s  p ( x , z , t ) = -pgz 3 V (-d,t) ^ s i n h nz} .2 2  +  C  ° f P ( 0 , t ) cosh n(z+d) + coshnd > ' n S  n  x  1  1  v  (4-17)  dZ  Equation  (4-17) r e p r e s e n t s the f l u i d p r e s s u r e p i n terms o f  the p r e s s u r e amplitude  P(0,t) at the f l u i d  s u r f a c e , and the second  d e r i v a t i v e o f the z component o f the v e l o c i t y a t the bottom. Now we can s o l v e f o r the v e l o c i t y d i s t r i b u t i o n i n the f l u i d s i n c e we know the i n i t i a l distribution. using  c o n d i t i o n s and the form o f the p r e s s u r e  I f (4-5) i s w r i t t e n i n f i n i t e  (4-17), the v e l o c i t i e s  using a d i g i t a l  computer.  d i f f e r e n c e form, and  can be s o l v e d f o r a sequence o f times  The method i s d e s c r i b e d i n Appendix B.  45  4.2.2  Results f o r Single S p a t i a l  Frequencies  A g e n e r a l s o l u t i o n of the problem d e s c r i b e d i n 4 . 2 . 1 quite complicated.  may  However, f o r the case of i n t e r e s t where the  s i p a t i o n of energy i n the f l u i d  due  to v i s c o s i t y i s s m a l l , we  be  dis-  can f i t  an a n a l y t i c e x p r e s s i o n to the r e s u l t s  t h a t were o b t a i n e d u s i n g the numer-  i c a l method d e s c r i b e d i n Appendix B.  We  shall  r e s u l t s by making use o f a n l a y t i c a l r e s u l t s propagation  of g r a v i t y  c a p i l l a r y waves on  I t i s known t h a t when a l i q u i d p r e s s u r e o f some s p a t i a l and may  o s c i l l a t e and  frequency  n>  decay i n time  comment on these  t h a t are a v a i l a b l e f o r the  the s u r f a c e o f a  fluid.  s u r f a c e i s s u b j e c t to  radiation  t h a t the s u r f a c e w i l l be  [22].  numerical  deformed  T h i s i s i n agreement w i t h  the r e s u l t s o b t a i n e d u s i n g the n u m e r i c a l method d e s c r i b e d i n Appendix B. F i g u r e s 4 . 2 ~to 4 . 5 of the amplitude H(t)  H ( t ) o f the s i n u s o i d a l s u r f a c e d e f o r m a t i o n  h(x,t) =  cos nx when s u b j e c t to r a d i a t i o n p r e s s u r e o f the form P^  f o r some d u r a t i o n At. by  are examples -of the v a r i a t i o n i n time  A number o f computer runs were made as  t a b l e B - l , f o r v a r i o u s v a l u e s o f the l i q u i d  d e n s i t y p,  t e n s i o n y> k i n e m a t i c v i s c o s i t y v, s p a t i a l frequency f l u i d depth g i v e n by The figures 4.2  the r a t i o o f depth to wavelength,  r e s u l t s o b t a i n e d by  to 4 . 5  n(n  cos  nx  indicated  surface = 2TT/A)  and  d/X.  the n u m e r i c a l method as shown by  are s i m i l a r to the f a m i l i a r t r a n s i e n t  response  o f a second o r d e r system as d e s c r i b e d i n any book on l i n e a r  systems.  These curves  The  damping r a t i o steady  are c h a r a c t e r i z e d by z, determines  two  parameters £ and u -  the o v e r s h o o t  n  of the response  s t a t e v a l u e , and co^, the n a t u r a l frequency  determines  the time s c a l e .  of  H(t)  above the  oscillation,  For a r a d i a t i o n p r e s s u r e o f d u r a t i o n At  2.0 EQUATION  (4-18)  NUMERICAL  1.5  ANAL  YSIS  <JO At - co n  3?  X=  1.0  0.10  cm  = 0.258  d/X  V = 15.9  0.5  g/crr>3  p=1.79  -  dyn/cm  V = 0.039  cm2/s  S = 0.207  C0 = 1427 n  0.0 0  rad/s  2  7 (sO t n  F i g . 4.2  L i q u i d s u r f a c e response h e i g h t H/H^,  v s . time w^t, t o a s t e p f u n c t i o n o f r a d i a t i o n  pressure.  2.0  •P = 0 r  aO &t = 3.86  x  =  0.20  cm  n  1.5  d/\ = 0.258 y = 15.9  dyn/cm  P = 1.79  9/cm  V = 0.039  1.0  3  cm /s 2  0.147 GO = 505 n  0.5  rad/s  Y 0.0  15 -0.5 EQUATION (4-18) ©  1.0  NUMERICAL ANALYSIS  ©  •1.5 4.3  Liquid  surface  wnAt =  response height  3.86.  H/H^, v s . t i m e  w^t, t o a pulse  of radiation  pressure  o f duration  48  0.6 A  d/X =  0.5  0.4  - 0.02  cm (jQ At=co  0.25  n  y = 5.0  -  dyn/cm  p = 3.0  g/cm3  V = 0.60  cm /s  —  2  1.65 6884  '  .  ^  A  /  /  y = 20.0  -  P  = 2.0  V  = 0.25  £  =2.32  (x) i  0.0  = 1.0  [  cm  d/\=0.08  0.2  O.J  0,4 NUMERICAL ANALYSIS  / rad/s  EQUATION (4-18)  i  n  = 33.9 i  dyn/cm g/cm3 cm /s 2  rad/s i  2  Fig. 4.4 Overdamped liquid surface response height H/H^, vs. time w^t, to a step function of radiation pressure.  0.8  0.6  -  0.4  -  .0.2 -  0.0  0.2  -  0.4  F i g . 4.5  Liquid surface response height H/K^, vs. time w^t to an impulse of r a d i a t i o n pressure. = P  /{k  1  + B )/ n^ 2  Y  (see ( C - l ) )  50  the response H ( t ) can be w r i t t e n as p H(t) =  X  —  Ul  - j  e^V  sin(a) t + 6)] y ( t ) d  pg+ny -[1 - j  e"  C u  n  ( t _ A t )  s i n (u> (t-A.t) + 0)] d  y(t-At)}  (4-18)  where  3 = /(i to  d  = gw  c ) 2  n  0 = tan  B/C  Equation  (4-18) can be w r i t t e n i n o t h e r forms i f g i s i m a g i n a r y  t > At.  These are g i v e n i n Appendix C. I t was  found  t h a t c l o s e agreement w i t h the n u m e r i c a l  or i f  results  .are o b t a i n e d i f  npv /Tn/(p(Pg  C = 2  <o  2  n  = (Pg + yn ) 2  / (  (4-18),  of  tanh  [tanh  (nd)]  E  (4-19)  ( d) .  (4-20)  n  curves i n f i g u r e s  4.2  to  .4.5  are obtained  (4-19) and (4-20), w h i l e the n u m e r i c a l  i n d i c a t e d by the c i r c l e s . sults  2  -2.3 i f d/X < 1/4 0 i f d/A > 1/4  The s o l i d equations  j  + Yn ))}  The d i f f e r e n c e s between  from  r e s u l t s are  the n u m e r i c a l r e -  and (4-18) to (4-20) are shown i n t a b l e B - l f o r the peak v a l u e s  the curves H ( t ) and the times a t which the peak o c c u r s .  The  differ-  ences are q u i t e s m a l l f o r the v a l u e s o f the parameters used. Figure  4.2  r a d i a t i o n pressure.  shows the response H ( t ) t o a s t e p f u n c t i o n Figure  4.3  i s s i m i l a r except  p r e s s u r e i s o f d u r a t i o n to^At = 3.86.  Two  t h a t the r a d i a t i o n  cases o f the response H ( t )  51  when £ i s g r e a t e r than one a r e shown i n f i g u r e  4.4  .  The response  H ( t ) to an impulse o f r a d i a t i o n p r e s s u r e ( a c t u a l l y a s h o r t p u l s e , w A t = 0.0121) i s shown i n f i g u r e The similar  4.5 .  response o f the l i q u i d s u r f a c e t o r a d i a t i o n p r e s s u r e i s  to the p r o p a g a t i o n o f g r a v i t y c a p i l l a r y waves on a l i q u i d  surface.  The s i m i l a r i t y  comes from the f a c t  that i n both  situations  t h e r e i s a s i n u s o i d a l d e f o r m a t i o n o f the s u r f a c e , the amplitude o f which decays is  i n time.  In (4-20), the n a t u r a l frequency o f o s c i l l a t i o n  the same as the r e s u l t  t h a t can be o b t a i n e d a n a l y t i c a l l y [17]  f o r the frequency o f o s c i l l a t i o n on the s u r f a c e o f an i d e a l A l s o , except coefficient  £10  o f c a p i l l a r y g r a v i t y waves p r o p a g a t i n g  f l u i d o f depth d.  f o r the dependence on depth ^ d , the damping  as g i v e n by (4-19) and (4-20) i s the same as t h a t  •which i s d e r i v e d i n .[17.] f o r the damping o f g r a v i t y waves p.rop.agating on the s u r f a c e o f a v i s c o u s f l u i d o f i n f i n i t e in  depth.  I t i s shown  [17] t h a t f o r g r a v i t y waves t h i s v a l u e o f damping c o e f f i c i e n t i s  v a l i d o n l y i f i t i s s m a l l , so t h a t the f l u i d motion t h a t o f an i d e a l f l u i d . g r e a t e r than about A/4 vn  <<  Then f o r f l u i d s o f i n f i n i t e this  i s approximately depth o r o f depth  condition i s  t^gn  (4-21) 2  I f we i n c l u d e the e f f e c t o f s u r f a c e t e n s i o n by r e p l a c i n g g by g + n y/p then (4-21) becomes vn  2  «  VTgT! + n  3  (4-22)  Y/P)  We would then expect t h a t the e q u a t i o n s if  (4-22) i s s a t i s f i e d .  easily satisfied interest,  (4-18) t o (4-20) to a p p l y  F o r f r e o n E-5 o r water,  for spatial  condition  (4-22) i s  f r e q u e n c i e s n from zero up to v a l u e s o f  f o r example 1000 rad/cm.  A t 10 MHz i n water the wavelength  52  is  0.015  for  the  cm,  f o r which the wave number k = 420  curves  i n f i g u r e 4.4  the  rad/cm.  Actually,  c o n d i t i o n (4-22) i s not  as b o t h s i d e s of the i n e q u a l i t y are a p p r o x i m a t e l y  satisfied,  the same.  This  seems to i n d i c a t e t h a t the c o n d i t i o n (4-22) i s too r e s t r i c t i v e when equations  (4-18) to (4-20) are  used.  For f l u i d s o f depth l e s s a p p l i e s as i n d i c a t e d i n (4-19). r e s u l t s w e l l f o r values  T h i s was  A/4  the s p a t i a l  analysis.  frequency  low v a l u e s point  o f n.  t h i s may  be  for  the  (4-20).  f o r most f l u i d s and  be  quite different  on a f l u i d  s p a t i a l frequency  that i s of  In the n u m e r i c a l  the n u m e r i c a l  f o r very  The  (4-19) and  f r e q u e n c i e s we  numerical  method used was  hot  that occur  given [14,17]  (4-20) seem to be may  use  in real  method i n Appendix B particular  fluid  interest.  a n a l y s i s d e s c r i b e d i n Appendix B,  to g r a v i t y , g, was  v a r i a t i o n s i n pressure  from t h a t  dis-  surface.  used to determine the s u r f a c e motion f o r any  c e l e r a t i o n due  the  the r e s u l t s i n r e f e r e n c e  (4-18),  spatial  time l i q u i d s u r f a c e h o l o g r a p h y .  and  and  f o r f l u i d s where the  T h i s i s shown by  However, the e q u a t i o n s  can be  frequencies  depth  reached.  decay o f s i n u s o i d a l deformation  sufficient  smaller  s t u d i e s were made to determine at what  s i p a t i o n of energy i s v e r y l a r g e may (4-19) and  but no  become d o u b t f u l  A l s o , the damping c o e f f i c i e n t  by  numerical  n v a r y i n g , the dependence on  (4-18) to (4-20) may  However no  as 0.02,  nd)  I f the depth d i s h e l d  becomes more s i g n i f i c a n t at the lower s p a t i a l v a l i d i t y of e q u a t i o n s  a f a c t o r (tanh  found to f i t the  o f d/A. at l e a s t as low  depths were used i n the n u m e r i c a l constant with  than about  s e t to z e r o .  T h i s was  s u i t a b l e f o r accounting a t the s u r f a c e due  the  ac-  done because f o r the  to g r a v i t y .  small  53  However, the e f f e c t  o f g r a v i t y i s n e g l i g i b l e f o r a l l b u t v e r y low  1  s p a t i a l f r e q u e n c i e s , and i n (4-19) and (4-20) g can be s e t t o zero f o r most s p a t i a l f r e q u e n c i e s o f i n t e r e s t . Finally,  f o r a s i n u s o i d a l deformation  we can w r i t e the response  o f the s u r f a c e amplitude  r a d i a t i o n p r e s s u r e o f amplitude  frequency n  H ( t ) t o an impulse o f  P^. 6 ( t ) as  UJ  P H(t) =  of s p a t i a l  ,  £  pg+yn  e  p  I f a s u r f a c e deformation  ?  V  sin(Bo) t ) y ( t ) n  h(x,y,t)  (4-23)  i s r e l a t e d t o an a r b i t r a r y  radiation  p r e s s u r e p ( x , y , t ) by (2-4), t h a t i s r  h(x,y,t) = p ( x , y , t )  * g  2  (x,y,t)  (u,v,w) g  0  (u,v,w)  r  or h  (u,v,w) = p  where g2(x,y,t)  t~  i s the response  of r a d i a t i o n pressure, of  r  (4-24)  o f the s u r f a c e t o an impulse  6(x,y,t)  then by t a k i n g the F o u r i e r t r a n s f o r m i n time  (4-23) we see t h a t 2 = g (u,v,w) = 2  R e c a l l that n  2  = u  2  1 pg + YH -  =  4.3  — ^ W  n  n  + 2j t,bi  2 03 -  n  2 + v , also, i t i s evident  g (u,v,to) = g 2  W  A 2  (4-25)  w  that  (-u,-v,-u)  The S u r f a c e S p a t i a l Frequency Response t o P u l s e d R a d i a t i o n  In l i q u i d f i g u r e 2.1 we s h a l l  s u r f a c e holography  the t r a n s d u c e r s  are used t o produce p u l s e s o f u l t r a s o u n d .  l o o k a t the response  o f the l i q u i d  t  and t  Pressure  in  In t h i s s e c t i o n  surface f o r various  spatial  54  f r e q u e n c i e s when the r a d i a t i o n p r e s s u r e i s a p u l s e o r s t e p i n time. Then i f P ( x , y , t ) i s g i v e n by r  p ( x , y , t ) = P^ 6 ( x , y ) r r i.e.,'p  i s unifjP^m i n s p a t i a l P ( x , y , t ) <4>  P  r  Then from  [u(t) - y ( t - A t ) ]  i  frequency  [y(t) - y(t-At)]  r  (4-18) the F o u r i e r t r a n s f o r m i n (x,y) o f the response o f t h e  surface h(x,y,t) f o r this _ _ h(u,v,t) = p  case i s  t _ (u,v,t) * g  r  2  (u,v,t)  = H(n,t)  Figures  (4-27)  (4-6) and (4-7) show the response H''(t) when At =°° (step  f o r f r e o n E-5 and water, quency range fluid  (4-26)  r e s p e c t i v e l y , a t v a r i o u s times t over a f r e -  from 1 to 2000 rad/cm.  depth d, o r a t l e a s t  These curves a r e f o r an i n f i n i t e  g r e a t e r than 1/4 o f the l a r g e s t  A = 2TT cm, so d > 1.5 cm a p p r o x i m a t e l y . H i s n o r m a l i z e d by  response)  wavelength  In a l l f i g u r e s o f t h i s  section  = P^/pg.  We can see i n f i g u r e s 4.6 and 4.7 t h a t t h e h i g h s p a t i a l  fre-  q u e n c i e s reach a steady s t a t e much sooner than t h e low f r e q u e n c i e s . The s p a t i a l of s p a t i a l it  frequency response r i s e s frequency  a t the r a t e o f 10 db p e r decade  (db = 10 l o g H/FL^) and 20 db p e r time decade u n t i l  i s c l o s e t o t h e steady s t a t e v a l u e .  Then the steady s t a t e  drops  at the r a t e o f 20 db p e r decade o f f r e q u e n c y a t the h i g h e r f r e q u e n c i e s . The o s c i l l a t i o n s o f t h e curves n e a r the steady s t a t e e s p e c i a l l y a t t h e lower s p a t i a l f r e q u e n c i e s c o r r e s p o n d to the o s c i l l a t i o n s once t h e steady s t a t e v a l u e i s passed As mentioned  i n t h e response  ( o v e r s h o o t ) as i n f i g u r e 4,2.  i n s e c t i o n 4.1 t h e steady s t a t e response i s  seen t o be v e r y poor, w i t h t h e h i g h e r s p a t i a l  frequencies dropping o f f  55  10  0  -10  -  •20  h  -30 3?  2r  -4 0  f  -50  60  h  70  h  80 F i g . 4.6  L i q u i d surface  ( f r e o n E-5)  s p a t i a l frequency  H(n,t) to r a d i a t i o n p r e s s u r e o f a s t e p f u n c t i o n  H„ = P /pg N r  response  i n time.  56 10  QQ  I F i g . 4.7  L i q u i d s u r f a c e (water) s p a t i a l f r e q u e n c y response  to r a d i a t i o n p r e s s u r e o f a s t e p f u n c t i o n i n time. H  N  "  VPS  H(n,t)  57  10  H  0  1—H-+-H-H-  H  1  I I I  10  -1  1  I I I Ij  100 7]  1000  rad/cm  -10 -20  -30  40  50  \0'  60  •70  y P  = 16.  dyn/cm  = 1.8  g/cm  V  = 0.039  cm /s  d = At  2  CD  - IO-  4  s  80 F i g . 4.8  L i q u i d s u r f a c e ( f r e o n E-5)  spatial  frequency  H(n-t) to r a d i a t i o n p r e s s u r e o f a p u l s e i n time.  response  3  58  70  -SO  LFig.  4.9  L i q u i d s u r f a c e ( f r e o n E-5)  s p a t i a l frequency  response  H(n,t) to r a d i a t i o n p r e s s u r e o f a step f u n c t i o n i n time, w i t h finite  fluid  depth.  59  10  0  H—i—i  i i 1111  1—i—i  M i n i  10  1  1—h-i i i n  100 T)  1000  rad/cm  10  -20  3  '30  -50  60  y = 16.  P  -70  dyn/cm  - 1.8  g/cm  V = 0.039  cm /s  3  2  d = 0.015 cm = 10~ s 4  -80 Fig.  4.10  L i q u i d s u r f a c e ( f r e o n E-5) s p a t i a l frequency  response  H(n,t) t o r a d i a t i o n p r e s s u r e o f a p u l s e i n time, w i t h f i n i t e depth.  H  N  = P /pg. r  fluid  60  at  20  db per  frequencies the  curves  decade.  However, the  is relatively  flat  t r a n s i e n t response a t the  over a r e s t r i c t e d  peak near the steady  s t a t e curve.  from the o b j e c t  <j> i s p a s s e d through the  where u  depend on the  beam as can be into  and  r  v  r  i n d i c a t e d i n (2-12). taken o f  s a m p l i n g the  Thus by  the  g r e a t e r the  Consider  ' ) u  r  >  reference  v^,  advantage  the o b j e c t  information flat  Evidently,  have an imaging system i n which the  maximum amount by which the o b j e c t is k  from the o b j e c t range from zero  , where k W  surface,  i n water.  any  t h a t has  Thus the sampling its flat  X  can "be  shifted  i s the wavelength w  time i s determined  r e g i o n between n  r\  and m  For example, f o r u l t r a s o u n d  spatial  to a maximum  information  = 2TT/X and w  W  of the u l t r a s o u n d curves  v  r e f e r e n c e beam to the  of n . m  of the  ~  v  r  time.  t h a t we  s p a t i a l frequency  u  the l a s e r a t t h i s moment.  a t the s u r f a c e  in  ~ »  u  2  information  response i s momentarily  frequencies The  g (  shifting  angle o f i n c i d e n c e of the  the e a r l i e r i s the sample  (2-31) the  a d j u s t i n g u^ and  the t r a n s i e n t motion by  surface with  range where  o r i e n t a t i o n of the a c o u s t i c  a r e g i o n where the s p a t i a l frequency  and  In  filter  Q  frequency  higher  a t 10 MHz  + k m  i n water k  by .  w = 420  rad/cm,  w but  say  t h a t the  f i g u r e 4.6 t = 0.5 if The to  r  y(u  with  x 10 2  r  angle  + v 0  -4 2 r  zw  the normal  0  zw  apertures f r e o n E-5  s has  the system l i m i t  i n the m i n i t a n k we  a flat  ) = 275 of  of  n  see  r e g i o n from about 275  rad/cm  then the  m  to 250  rad/cm.  t h a t the  curve  t o 525  / rad/cm.  sample time would be  the r e f e r e n c e beam i n water  0.5  In for Thus x 10  -4  s.  ( i . e . i n the main tank)  to the s u r f a c e i s then = a r c s i n v(u = 41.5°  2 r  + v  2 r  )/k  w (4-28)  61  Another p o s s i b l e sample time i s 0.1 9  rad/cm and  x 10  s with  /(u  ) =  r  150  =21°.  zw  With the t r a n s d u c e r s  t  r  and  t  producing pulses of °  o  sound, the p u l s e l e n g t h can be s e t so t h a t the p u l s e the s u r f a c e i s sampled. deformation  + v  Then a f t e r s u f f i c i e n t  to decay, another  of u l t r a s o u n d .  F i g u r e 4.8  ultra-  terminates  after  time to a l l o w the  image can be o b t a i n e d w i t h another  surface pulse  shows the response o f the s u r f a c e to a p u l s e  -4 of r a d i a t i o n pressure times  are t e r m i n a t e d  becomes n e g a t i v e , f i g u r e 4.3.  o f d u r a t i o n At = 10  s.  The  curves  f o r the  various  at those v a l u e s o f n where the response H ( n - t )  as the response decays and  The h i g h f r e q u e n c i e s b e g i n  oscillates  as shown i n  to decay r a p i d l y , b u t  the  -4 lower f r e q u e n c i e s to r i s e , and  to the l e f t o f the peak r e g i o n f o r t = 10  s u r f a c e f o r times If n  i n f i g u r e 4.8  t > At.  = 105  -that we  can s t i l l - s a m p l e "the  For example, at 5 MHz  i n water k  x 10  s (sample t i m e ) , and  the  curve  frequency  ) = 105  rad/cm, the angle  of the r e f e r e n c e beam  time between the p u l s e s o f u l t r a s o u n d i s determined by  the r a t e a t which the s u r f a c e decays a f t e r the p u l s e ends. at the b e g i n n i n g  of the p u l s e , then  a sufficient  t-^ at which the s u r f a c e at a g i v e n s p a t i a l  (C-l). e'^^l  j? f e ~  C a )  n s t  s i n (u,t  +6  freqeuncy  + 6)  If t = 0  c o n d i t i o n f o r the has  w i t h i n a f r a c t i o n f o f i t s h e i g h t at the sample time t from  210  2 +  The  is,  = w the  rad/cm, p u l s e l e n g t h At = 10 ^ s , w i t h -3  f o r t = 0.2  2 s h i f t /(u 6 =30°. zw  continue  then e v e n t u a l l y to decay.  I t i s -apparent 'in f i g u r e 4.8  rad/cm.  s  g  time  decayed to where t  ^ At,  (4-28)  62  Then the time a t which the next p u l s e Using  the v a l u e s  o f the p r e v i o u s  lowest s p a t i a l frequency w i t h i n  can b e g i n  can be taken as t ^ - t .  example and a p p l y i n g  (4-28) to the  the bandwidth o f i n t e r e s t , i . e . n  rad/cm, we o b t a i n f o r f = 0.05, t ^ 5 4.5 x 10  _3  =  100  s, o r the p e r i o d o f the  o  pulses r  t , - t >. 4.3 x 10 1 s  s.  In the f i g u r e s 4.6 to 4.8 t h e depth d o f the m i n i t a n k was a s sumed to be d > 1.5 cm so t h a t t h e r e was no e f f e c t o f the depth on t h e s u r f a c e motion f o r the s p a t i a l f r e q u e n c i e s  shown.  Figures  4.9 and 4.10  are s i m i l a r to f i g u r e s 4.6 and 4.8 r e s p e c t i v e l y except t h a t d = 0.015 cm. T h i s means t h a t f o r those s p a t i a l f r e q u e n c i e s  n < 2ir/4d = 100 rad/cm  the s u r f a c e motion i s more h e a v i l y damped i n f i g u r e s 4.9 and 4.10. The  f a c t t h a t a t h i n f l u i d l a y e r damps the motion o f low s p a t i a l  quencies i s a d e s i r a b l e feature s i n c e t h i s o f the f l u i d  surface.  contributes  fre-  to the s t a b i l i t y  63  V.  ACOUSTIC BEAM INTENSITIES, SOUND FIELD PARTICLE OSCILLATION, SURFACE BULGE  In t h i s  chapter we s h a l l  consider  the i n t e n s i t i e s o f the  a c o u s t i c beams r e q u i r e d , the amplitude o f the o s c i l l a t i o n o f f l u i d p a r t i c l e s i n the u l t r a s o n i c f i e l d ,  5.1  and the motion o f the s u r f a c e  bulge.  The U l t r a s o n i c Beam I n t e n s i t i e s  In o r d e r  to g a i n some i d e a o f the i n t e n s i t i e s  r e q u i r e d o f the -4  u l t r a s o n i c beams we s h a l l c o n s i d e r p u l s e s f i g u r e 4.8.  With the r e f e r e n c e  o f d u r a t i o n At = 10  <j> and o b j e c t  s as i n  <j>^ v e l o c i t y p o t e n t i a l s  b o t h p l a n e waves, then a t the s u r f a c e 4> = R e  J  ^  j (  (  u  r  = A e  r  ' l u  X  +  x  +  r  V  i  V  y  y  )  [u(t) - y ( t - A t ) ]  (5-1)  [u(t) - y ( t - A t ) ]  (5-2)  )  In f i g u r e s 3.4 and 3.5 we see t h a t a t l e a s t  for a thin  fluid  l a y e r o f f r e o n E-5 on water the e f f e c t o f m u l t i p l e r e f l e c t i o n s i n the m i n i t a n k f o r the lower s p a t i a l f r e q u e n c i e s  i s not s i g n i f i c a n t .  Then  the e f f e c t o f the t r a n s f e r f u n c t i o n g-^ (2-2) on the two beams §^ and <j>^ is  approximately  •  r  *  *1 *  =a*  8 l  l  g  * *1 a  where from f i g u r e 3.4 a pressure  r  s  1/2.  From (2-3) and (3-21) the r a d i a t i o n  on the s u r f a c e i s then p r r  = a  2  p k {R 2 2 2  0  2  + A  2  •[y(t) - y ( t - A t ) ]  + 2RA c o s [ ( u - u . ) x + (v - v . ) y ] } r 1 r 1 (5-3)  64  The t h r e e p r e s s u r e terms i n (5-3) c o r r e s p o n d t o the three h e i g h t terms i n (2-19) and we can use (4-18) o r f i g u r e 4.8 to d e t e r mine the amplitude o f the s u r f a c e d e f o r m a t i o n . In o r d e r t h a t the l i n e a r i z a t i o n the s u r f a c e h e i g h t h  (2-23) be v a l i d we must  due to the r e f e r e n c e and o b j e c t beams such  have that  3.  4 w. h < 1 £ a or h From  < 0.04 \ I cos 6  a  (see (2-11))  (5-3) we take the r a d i a t i o n p r e s s u r e from  P  r  =  a 2 p  2  k  2  and i f the s p a t i a l /T(u r  U l  2  the t h i r d  (5-4) term  M  (  5  _  5  )  f r e q u e n c y o f the o b j e c t wave i s s h i f t e d to )  2  + (v - v ) }  = 200 rad/cm  2  1  (5-6)  —3 then a t t = 0.2 x 10 s from f i g u r e 4.8 the n o r m a l i z e d s u r f a c e h e i g h t h  i s 0.275 x 1 0 ~ , and 2  n  h =h P /pg a  n  r  = h a k 2  n  RA/g  2 2  (5-7)  In o r d e r f o r (5-4) to be s a t i s f i e d  the p r o d u c t o f the beam  amplitudes RA must then be g i v e n by RA  < A„g/8TTh  Jo  n  a \  2  2  cos  6  The r e l a t i o n s h i p between the i n t e n s i t y amplitude  I and the v e l o c i t y  potential  < j > i n a p l a n e wave i s g i v e n by [24] I = pto  where  (5-8)  Z £  2  <}>/2c  (5-9)  2  w i s the f r e q u e n c y o f o s c i l l a t i o n  the v e l o c i t y o f p r o p a g a t i o n .  o f the u l t r a s o n i c wave and c  I f the r e f e r e n c e beam amplitude  i s larger  65  than the o b j e c t beam amplitude so t h a t say R = bA, then the r e f e r e n c e beam i n t e n s i t y becomes, w i t h I_ < p K.  0 = 0  c bA„g/l6fTa h  (5-10)  2  0  /  x,  n  I f we take b = 10, a = 1/2, h = 0.275 x 1 0 " , c = 0.7 x 10 n 3 -4 p = 1.8 gm/cm ( f r e o n E-5) w i t h A^=0.6xl0 cm then 2  T  5  cm/s and  < 200 mW/cm  (5-11)  <  (5-12)  2  R  and I  20 mW/cm  2  However, i f an a c t u a l o b j e c t i s used <j)^ w i l l  then the v e l o c i t y  potential  c o n s i s t o f a d i s t r i b u t i o n o f f r e q u e n c i e s , and the i n t e n s i t y  g i v e n by (5-12) may have to be h i g h e r .  A l s o , some energy  i n the u l t r a -  sound beams would be absorbed b y the o b j e c t and by the water,  so t h a t  (5-11) and (5-12) may be t o o s m a l l In water the a b s o r p t i o n c o e f f i c i e n t a causes an a t t e n u a t i o n of the amplitude o f a p r o p a g a t i n g wave by the f a c t o r e  . A t [25]  -3 -1 -2 -1 1 MHz a = 10 cm and a t 10 MHz a = 4 x 10 cm , so t h a t the i n t e n s i t y 2 is  down t o 1/e = 0.14 a t 1000 cm and 25 cm f o r 1 MHz and 10 MHz r e s -  pectively. 5.2  The U l t r a s o u n d F i e l d P a r t i c l e  So f a r no mention  Oscillation  has been made o f the amplitude  E, o f the  o s c i l l a t i o n o f the f l u i d p a r t i c l e s a t the frequency o f the u l t r a s o n i c beams.  T h i s motion  the motion  o f the f l u i d p a r t i c l e s a t the s u r f a c e , as opposed  o f the s u r f a c e h due t o t h e r a d i a t i o n p r e s s u r e , can be con-  s i d e r e d as a d d i t i o n a l n o i s e . I n a p l a n e wave the amplitude o f the f l u i d p a r t i c l e  oscillation  66  E i s r e l a t e d to t h e i n t e n s i t y I by [24] E At  2  = 21/pew  (5-13)  2  the f l u i d s u r f a c e the wave i s r e f l e c t e d and E would be twice  g i v e n by (5-13) i f the wave i s a p p r o x i m a t e l y surface.  W/cm  2  a t normal i n c i d e n c e  Note t h a t E i s i n v e r s l y p r o p o r t i o n a l t o the frequency  the u l t r a s o n i c wave.  that to the to o f  I f we c o n s i d e r the case w i t h to = 1 MHz, I = '0.2  , and f o r the f l u i d p r o p e r t i e s p = 1.8 gm/cm  3  and c = 0.7 x 10  5  cm/s  ( f r e o n E-5) then from (5-13) E = 0.9  x 10"  6  cm  (5-14) -4  If  the l i g h t wavelength ^ E/X  £  = 0.6 x 10  c  then  m  = 0.015  (5-15)  which i s comparable to h  i n (5-4)  Thus f o r h i g h i n t e n s i t y , low frequency amplitude o f the o s c i l l a t i o n o f the f l u i d may be comparable w i t h mation o f the image.  u l t r a s o n i c beams the  particles  a t the s u r f a c e  the s u r f a c e r i p p l e p a t t e r n c o n t a i n i n g the i n f o r However, i f t h e l i g h t  from the l a s e r t h a t i s  r e f l e c t e d o f f the s u r f a c e has a p u l s e l e n g t h which i s much l o n g e r  than  the p e r i o d o f o s c i l l a t i o n o f the u l t r a s o n i c beams, then we would expect t h a t the average e f f e c t o f the f l u i d p a r t i c l e o s c i l l a t i o n s on the image o b t a i n e d would be m i n i m i z e d . 5.3  The S u r f a c e  Bulge  In t h i s s e c t i o n we w i l l tude low s p a t i a l frequency  c o n s i d e r the e f f e c t o f a l a r g e  r a d i a t i o n pressure  term on t h e f l u i d  We can see i n (5-3) f o r the r a d i a t i o n p r e s s u r e frequency  terms a r e p r o p o r t i o n a l t o R  2  2 and A .  amplisurface.  t h a t the low s p a t i a l I f i n the r e f e r e n c e  67  beam the amplitude R i s s u f f i c i e n t l y l a r g e r than A i n the o b j e c t beam 2 then s i n c e we have R a l a r g e deformation parison with  i n the p r e s s u r e ,  the r e f e r e n c e beam may cause  o f the s u r f a c e at low s p a t i a l f r e q u e n c i e s  the h i g h s p a t i a l frequency  Initially,  i n com-  term p r o p o r t i o n a l t o RA.  t h i s would n o t seem to p r e s e n t  any problems s i n c e  the t r a n s i e n t response to a p u l s e i n f i g u r e 4.10 i n d i c a t e s t h a t the low  spatial  frequencies  thin f l u i d layer. i s much l e s s the  respond s l o w l y and a r e h i g h l y damped f o r a  A l s o , the s u r f a c e r i p p l e h e i g h t as g i v e n by (5-4)  than a wavelength o f l i g h t ,  so t h a t i t would appear t h a t  c o n d i t i o n mentioned i n s e c t i o n 2-«2, t h a t the v a r y i n g p a r t o f the  s u r f a c e h e i g h t s h o u l d n o t be much l a r g e r than a wavelength o f l i g h t , i s not v i o l a t e d .  But we must note t h a t f i g u r e 4.10 a p p l i e s f o r a  s i n g l e p u l s e , and i f a r a p i d s u c c e s s i o n the low - s p a t i a l f r e q u e n c i e s low  of pulses  a r e used, and s i n c e  decay -slower than the h i g h f r e q u e n c i e s , the  f r e q u e n c i e s may b u i l d up more than i n d i c a t e d by f i g u r e 4.10. A l s o , a l a r g e low s p a t i a l frequency  pressure  may c o n c e i v a b l y  cause an u n d e s i r a b l e  term i n the r a d i a t i o n d i s r u p t i o n o f the s u r -  f a c e , i f o n l y p a r t o f the s u r f a c e i s i r r a d i a t e d w i t h  u l t r a s o n i c waves.  Then a b u l g e i s formed on the s u r f a c e as mentioned i n s e c t i o n 4.1. At  zero s p a t i a l frequency  i n (4-18),  the s u r f a c e response has z e r o  h e i g h t , y e t i n (4-1) we o b t a i n f o r the steady P^/pg.  s t a t e the h e i g h t h =  T h i s d i f f e r e n c e i s due to t h e f a c t t h a t  case o f a f l u i d o f i n i f n i t e radiation  (4-18) a p p l i e s f o r the  s u r f a c e a r e a , a l l o f which i s s u b j e c t to  pressure. The  problem o f the s u r f a c e response t o r a d i a t i o n p r e s s u r e  c o v e r i n g o n l y a f i n i t e a r e a o f the s u r f a c e can be approached i n t h e f o l l o w i n g way.  Consider  the s i m p l e s t  case w i t h  the r a d i a t i o n p r e s s u r e  68  v a r y i n g i n the x d i r e c t i o n and surface.  i n the y d i r e c t i o n a l o n g  the  I f the hologram a r e a i s s m a l l compared to the a r e a o f the sur-  f a c e of the m i n i t a n k i n time  constant  and p^_ i s simply  a p u l s e a l o n g the s u r f a c e x  and  t then p  = P J .  G (x)  i d  [y(t) - u(t-At)]  (5-16)  where the gate f u n c t i o n G (x) = f 1 f o r -a < t < a I 0 otherwise and  2a i s the l e n g t h o f the hologram on the s u r f a c e ( f i g u r e 5.1  )  P fx) r  -3- X  -a F i g u r e 5.1  Radiation pressure  I f the l e n g t h of the m i n i t a n k  2b,  i s comparable to the  l e n g t h of the hologram 2a, then t h e r e are e x t r a c o n s t r a i n t s on f l u i d motion s i n c e the v e l o c i t i e s must be of  the m i n i t a n k .  the w a l l s by  We  can account  zero at the v e r t i c a l w a l l s  f o r the zero h o r i z o n t a l v e l o c i t y at  c o n s i d e r i n g p^ to be a p e r i o d i c , even f u n c t i o n a l o n g  as i n f i g u r e 5.2  x  .  Then w i t h p^ simply p u l s e s i n x and p (x,t) = P i  the  [y(t) - y ( t - A t ) ]  Z  JL  t G (x-2nb)  (5-17)  Cl  11=-oo  Thus from symmetry, the h o r i z o n t a l v e l o c i t i e s  i n the f l u i d a t x = + b  69  •b -a  •2b  F i g u r e 5.2 are z e r o .  a  2b  Radiation pressure  The v e r t i c a l v e l o c i t i e s  zero, but t h i s J o e s  b  a t x = +b, however, a r e n o t n e c e s s a r i l y  not appear to b e a s e r i o u s drawback u n l e s s the f l u i d  i s very v i s c o u s . Equation series  (5-17) f o r p^ can be w r i t t e n i n terms o f i t s F o u r i e r  i n x s o that P (x,t) = P r r  2a n  P\  [ y ( t ) - y ( t - A t ) ] {^+  CO  E n=l  cos rye}  sin (mra/b)  (5-19)  niTa/b  b  (5-18)  ri = nnx/b n Then, s i n c e each frequency  component  in  g i v e n by (4-18), the s u r f a c e response h(x,t) =  E a cos . n n=l  (5-18) would be a t t e n u a t e d as  h ( x , t ) can be w r i t t e n as  n x H(n , t ) n  n  (5-20)  (-b < x ^ b) where i n  (4-18) we r e p l a c e n by n . Thus the shape o f the s u r f a c e ,  the s p a t i a l spatial  frequency  frequency  (5-19) and (5-20).  components  components  p r e s e n t and the r a t e at which  oscillate  these  i n time a r e g i v e n by (4-18),  70  VI. . CONCLUSIONS  In chapter  2 we  have p r e s e n t e d  an a n a l y s i s o f a form o f  real  time imaging o f o b j e c t s u s i n g h o l o g r a p h y w i t h u l t r a s o u n d waves i l l u m i n a t i n g the o b j e c t i n water, and w i t h a l i q u i d surface. t i n g coherent  The  images o b t a i n e d  energy d e t e c t e d  i n the o p t i c a l domain by  by  reflec-  l i g h t o f f the s u r f a c e were r e l a t e d to the o b j e c t by  f u n c t i o n s g^ and  g^ which r e p r e s e n t  l a y e r i n the m i n i t a n k . the s u r f a c e has surface.  the u l t r a s o u n d  I t was  the impulse response o f the  liquid  shown that the l i g h t d i s t r i b u t i o n  a form s i m i l a r to t h e  a c o u s t i c d i s t r i b u t i o n at  the  at  the  However, s u i t a b l e o p t i c a l methods a r e r e q u i r e d to e x t r a c t  the useful/!' i n f o r m a t i o n The  focused  to form an o p t i c a l image o f the  image method was  shown to be  object.  insensitive  to n o i s e  on the l i q u i d s u r f a c e s i n c e the n o i s e merely causes phase changes i n the  light  t h a t forms the image.  T h i s i s t r u e as l o n g as  the n o i s e  not  cause s u r f a c e v a r i a t i o n s much l a r g e r than a wavelength o f  and  i f the v a r i o u s o r d e r s  form p l a n e . plane  o f l i g h t produced do not  Hie s e p a r a t i o n of the v a r i o u s o r d e r s  i s i n c r e a s e d by  i n c r e a s i n g the angle  i n the  light  i n the  the problem o f the image b e i n g  trans-  transform  of i n c i d e n c e o f the  r e f e r e n c e beam to the normal to the s u r f a c e . method a v o i d s  overlap  does  acoustic  A l s o , the f o c u s e d d e m a g n i f i e d by  the  image ratio  of the u l t r a s o n i c wavelength to the l i g h t wavelength. The by  the d e f o r m a t i o n  r e f e r e n c e and any  phase d i s t o r t i o n of the l i g h t o f the s u r f a c e due  to the r a d i a t i o n p r e s s u r e  o b j e c t a c o u s t i c beams a c t i n g alone  s u r f a c e deformation  minitank.  at the s u r f a c e i s caused  caused by  any  from  the  on the s u r f a c e , p l u s  external disturbances  reaching  Thus any method of imaging t h a t r e q u i r e s the l i g h t  to •  the  71  propagate over a d i s t a n c e b e f o r e to severe  distortion.  face v a r i a t i o n s  (h  + h o  T  to be  g e n e r a l l y o f low  o f the l i q u i d s u r f a c e phase d i s t o r t i o n due ultrasound. the low  The  subject  i n (2-22)) t h a t cause the phase  s p a t i a l frequency,  to low  S i n c e we and  sur-  distortion  to h^ + h  Q  + h ^ may  s u r f a c e s h o u l d be  s u f f i c i e n t l y long  + h  +  Q  response  i s slow, then  the  be m i n i m i z e d by u s i n g p u l s e s  sampled b e f o r e  to a l l o w  expect h  s i n c e the  s p a t i a l frequencies  s p a t i a l f r e q u e n c i e s becomes l a r g e , and  s h o u l d be  be  N  were much s m a l l e r than a wavelength o f l i g h t . h^  formed may  T h i s d i s t o r t i o n would be m i n i m i z e d i f the  + h r  an image can be  of  the response to  the  time between  pulses  the s u r f a c e to decay b e f o r e  the  next pulse. F i g u r e s 4.6 i s necessary  f o r any  the s p a t i a l  frequency  s u r e i n the steady i n t e r e s t , with  to 4.10  imaging system t h a t uses the l i q u i d response of 'the l i q u i d  ultrasound  t e r ) as d e r i v e d i n chapter  since  s u r f a c e 'to r a d i a t i o n  frequencies the 4,  above 1 MHz.  f u n c t i o n g^  pres-  I t i s the  of  liquid  (which a c t s as a bandpass  t h a t i s the most important  the method of o p e r a t i o n of the imaging system.  l i q u i d s u r f a c e response such as f i g u r e s 4.6 can be  surface  s t a t e i s v e r y poor f o r the s p a t i a l f r e q u e n c i e s  s u r f a c e response g i v e n by  in  show t h a t the p u l s e d method of imaging  to 4.10  determinant  Graphs o f and  the  equation  used to determine the angle o f i n c i d e n c e o f the a c o u s t i c  (4.18) refer-  ence, the p u l s e l e n g t h o f the a c o u s t i c beams, the sampling time o f l i g h t beam and The angle  fil-  the  the time between the p u l s e s o f the a c o u s t i c beams. f u n c t i o n g^,  d e r i v e d i n chapter  3, which accounts f o r the  o f i n c i d e n c e of the a c o u s t i c waves to the l i q u i d s u r f a c e and  p r e s e n c e o f the m i n i t a n k i s o f l e s s importance than the However, g.. shows t h a t at l a r g e angles  function  o f i n c i d e n c e o f the  the  g^.  ultrasound  72  a phase d i s t o r t i o n i s i n t r o d u c e d i n t o the a c o u s t i c f i e l d a t the s u r f a c e if  the f l u i d  tank.  i n the m i n i t a n k i s d i f f e r e n t  For thinner f l u i d s  from the f l u i d  i n the main  t h i s e f f e c t i s l e s s pronounced.  A l s o , a t h i n f l u i d l a y e r i n the m i n i t a n k i s d e s i r a b l e i n o r d e r to  damp the motion o f s u r f a c e r i p p l e s o f low s p a t i a l  frequency.  more, as i n d i c a t e d i n s e c t i o n 3.2 the v e l o c i t y o f p r o p a g a t i o n ultrasound  Further-  o f the  i n the m i n i t a n k s h o u l d be l e s s than o r n o t much g r e a t e r  than the v e l o c i t y i n the main tank. Finally,  a study  of this  t h e s i s can g i v e the r e a d e r  an under-  s t a n d i n g o f the mechanisms i n v o l v e d i n u s i n g a l i q u i d s u r f a c e i n an a c o u s t i c imaging system.  73  APPENDIX  FOURIER  A.l  TRANSFORMS,  A  DIFFRACTION,  IMAGING  F o u r i e r Transform  The  three dimensional  f ( x , y , t ) w i l l be  F o u r i e r transform  f(u,v,w) of a f u n c t i o n  i n d i c a t e d by  f ( x , y , t ) <-»  f(u,v,u)  where  f(u,v,a>) =  ///  f(x,y,t) e "  j ( u x  +  ^  +  (A-l)  dxdydt  W t )  —oo  A  two  dimensional  transform  i n the s p a t i a l  dimensions x,y w i l l  be  denoted by  f(x,y,t) <A» f(u,v,t)  A.2  Convolution The  x,y,t  c o n v o l u t i o n o f two  dimensions i s d e f i n e d f^Cx.y.t)  f u n c t i o n s f ^ ( x , y , t ) * f ( x , y , t ) i n the 2  as  * f (x,y,t) 2  OO  jjj  ^(e.n.x) f (x-c, 2  y-n,  t-x)  (A-2)  dedndT s  A  c o n v o l u t i o n i n x,y  i n t the A.3  o n l y w i l l be  c o n v o l u t i o n w i l l be  * f (x,y), 2  while  denoted f ^ ( t ) & f ( t ) . 2  Diffraction  Freshel D i f f r a c t i o n The  denoted f ^ ( x , y )  ( i g n o r i n g constant  f i e l d f ^ ( x , y ) at a d i s t a n c e Z  q  from f ( x , y ) Q  phase f a c t o r s ) i s g i v e n by  [13]  74  '  0  M  Fig. A . l  Diffraction  f ^ x . y ) = (1/XZ ) f (x,y) * e q  j ( k  0  2  / z  o  ) ( x 2  +  y 2 )  2 2 / i /-v -\ 1/ \ j(k/2z )(x + y ) = (1/Xz ) f ( u , v ) e o o J  J  "  (A-3) (A-4)  where t, N 4: / x i(k/2z ) ( x + y ) f(x,y) = f (x,y) e o 2  2  J  J  Q  u = (k/z ) x o v = (k/z ) y k  =  2TT/A  X i s the wavelength o f the r a d i a t i o n .  A.4  T h i n Lens  F i g . A.2  Lens  A t h i n l e n s i n t r o d u c e s a q u a d r a t i c phase change i n t h e f i e l d  75  I  f  (x,y) such t h a t  [13]  f x,y) - f (x,y) -J(k/2f)(x l (  o  2  y ) 2  e  +  p  (  x  /  a  >  y  /  a  (A-5)  )  where PU,y) a i s the l e n s  =  {I I  2  < 1  radius  f i s the l e n s f o c a l  A.5  2  1 for x + y ttherwise  length  Imaging With a Lens y A  4>,(x,y) F i g . A.3 Applying image  Imaging w i t h  a lens  (A-4) to $ ; (A-5) to <j> ; and (A-4) to 4>, then the  O  <f) o f the o b j e c t  function §  1  D  3L  is 2  ^(x.y) = m e  J a r  [(^(-mx.-my) e  J  m  r  2 s ) * q (mr) ]  (A-6)  where m = z /z, o 1 r = /("x + 2  a =  y ) 2  k/2z  q ( r ) = ( 1 / 2 T T ) 2aa J ( 2 a a r ) / r 1  1/z  o  ^  P(u/2aa, v/2aa)  + 1/z. = 1/f 1  a i s the l e n s r a d i u s ,  i s a Bessel  f u n c t i o n o f the f i r s t  76  kind  or o r d e r  1,  f i s the l e n s  Equation f o r x > TT.  I f the  (A-6)  focal  can be  length.  simplified  since J^(x)/x  is  negligible  geometry o f the imaging system i s such that  a(iT/2aa)  <<  [12]  1  then  ^(x.y) = m  A.6  j  (  m  k  /  2  f  )  [<j> (~ ,  r  e  mx  F o u r i e r T r a n s f o r m With a Lens  - my)  * q (mr) ]  (A-7)  [13]  y A  A V  •v (*,y)  'i  0  F i g . A.4 The the  light  l e n s becomes the V (x,y) A o  (x,y) =  with a  d i s t r i b u t i o n V ( x , y ) from the Q  lens  front  f o c a l plane  d i s t r i b u t i o n V^(x,y) i n the back f o c a l p l a n e .  of With  V (U,V) q  then i g n o r i n g a c o n s t a n t  V ; L  Fourier transform  phase f a c t o r  (1/Af) V ( x k / f , o  and  yk/f)  the  f i n i t e size  of the  lens (A-8)  77  APPENDIX B NUMERICAL SOLUTION OF LIQUID SURFACE MOTION  The n u m e r i c a l method t h a t was  used to determine the l i q u i d  s u r f a c e motion when s u b j e c t to r a d i a t i o n p r e s s u r e by Harlow in  and Welch.  i s based on a method  A d e t a i l e d d e s c r i p t i o n of the method i s g i v e n  [ 2 1 ] . With t h i s method the N a v i e r - S t o k e s e q u a t i o n  finite  d i f f e r e n c e form, and the v e l o c i t i e s o f the f l u i d  throughout the f l u i d are determined n u m e r i c a l l y steps  i s written i n particles  f o r a sequence o f time  and the subsequent motion of the s u r f a c e i s f o l l o w e d by  updating  the p o s i t i o n o f marker p a r t i c l e s a t the f l u i d s u r f a c e a f t e r each time step. The B . l with  liquid  i s d i v i d e d up i n t o r e c t a n g u l a r c e l l s  the f l u i d p r e s s u r e  as i n f i g u r e  d e f i n e d i n s i d e the c e n t e r o f each  cell  and the h o r i z o n t a l v e l o c i t i e s u and the v e r t i c a l v e l o c i t i e s v d e f i n e d on the c e l l b o u n d a r i e s as i n f i g u r e B.2. tially  The marker p a r t i c l e s a r e  s e t a t the c e n t e r o f the second from the top row o f c e l l s  the p o s i t i o n o f the s u r f a c e .  to mark  The s u r f a c e motion i s c o n s t r a i n e d t o s t a y  w i t h i n the second from the top row o f  cells.  \/4 J  1 /  Fluid surface  J  [  Bottom  ini-  JL  of  minitank Fig. B . l  F l u i d divided into a rectangular mesh o f c e l l s .  78  V. .  P; •  Fig.  B.2  P o s i t i o n o f p r e s s u r e and v e l o c i t i e s  for cell i , j .  The d e f o r m a t i o n of the s u r f a c e h i s assumed to' take the form H ( t ) cos nx when the r a d i a t i o n p r e s s u r e takes the form P^. cos nx [ u ( t ) u(t-At)].  S i n c e the s u r f a c e d e f o r m a t i o n h i s assumed t o be much l e s s  than the wavelength  A o r the depth o f t h e f l u i d  d, and the a c c e l e r a t i o n  due to g r a v i t y i s s e t to z e r o , then from symmetry c o n s i d e r a t i o n s i t i s o n l y n e c e s s a r y t o a p p l y the n u m e r i c a l method t o a l e n g t h o f f l u i d ( a l o n g x) o f A/4  ( f i g u r e B.3) o f depth d.  J_ Fig.  B.3  Deformation  o f the l i q u i d  As i n d i c a t e d i n s e c t i o n 4.2.1  surface  the l i n e a r i z e d N a v i e r - S t o k e s  e q u a t i o n can be used, so t h a t the n o n - l i n e a r terms i n the f i n i t e ference, e q u a t i o n i n [21] can be dropped. ytic to  dif-  A l s o , s i n c e we have an a n a l -  e x p r e s s i o n (4-17) f o r the p r e s s u r e , the i t e r a t i o n r e q u i r e d i n [21]  determine  the f l u i d pressure d i s t r i b u t i o n  can be  79  r e p l a c e d by  using  d e f o r m a t i o n and (4-7)  can be  time step  (4-17).  With the p r e s s u r e  the v e l o c i t y d i s t r i b u t i o n v a r y i n g  used to determine the p r e s s u r e  B . l considered  t o be  a f r e e - s l i p w a l l , and  n o - s l i p w a l l , the v e l o c i t i e s  r e q u r i e d by  the w a l l are  At  given by  [21].  c i t i e s on e i t h e r s i d e of and  the n e g a t i v e  pressures  on  nx,  f o r each  surface  The  the  top row  can be  Harlow and Welch's method. nx,  f o r s i n c e the s u r f a c e  The  velocities  of  can  fluid  at  the  shear  t h a t have been d i s c u s s e d ,  Since  the  tension  shape of the s u r f a c e forces  f o r t h i s purpose be  The  can be  system to a p u l s e d  as shown by  to 4.5.  The  motion  i s known to  accounted f o r , can be  assumed to be  r e s u l t s were t y p i c a l l y  ponse of a second o r d e r  the  u s i n g a s i m p l i f i e d form o f  A number of computer runs were made f o r v a r i o u s  f i g u r e s 4.2  a  cells.  normal s t r e s s e s a t the s u r f a c e  as l i s t e d i n t a b l e B-1.  be  the r i g h t  added c o n d i t i o n t h a t the  determined by  surface  X shear and  i n [21] , s i n c e  h o r i z o n t a l v e l o c i t i e s on  Thus, w i t h the m o d i f i c a t i o n s  s i n c e h <<  r i g h t w a l l are s e t to  ve-  z e r o , which determines the h o r i z o n t a l v e l o c i t i e s  i n the  o f the l i q u i d s u r f a c e  outside  Note t h a t the r i g h t ' w a l l ' i s n e i t h e r  the n u m e r i c a l method.  s t r e s s at the s u r f a c e be  form cos  the n u m e r i c a l method  a  the r i g h t ' w a l l ' the h o r i z o n t a l v e l o -  are s e t as i n [21], w i t h  the  the l e f t i n f i g u r e  the heavy h o r i z o n t a l l i n e  a n o - s l i p w a l l as d i s c u s s e d  w a l l are determined by  have the  x as cos  surface  f l u i d on  e i t h e r s i d e o f the  i s a l l o w e d to pass through i t .  outside  a t the  surface  the w a l l are s e t e q u a l , w h i l e the v e r t i c a l  of each o t h e r .  f r e e - s l i p w a l l nor  surface  along  the  o f the n u m e r i c a l method. With the heavy l i n e bounding the  locities  at the s u r f a c e ,  or step  and  accounted flat.  parameter  those o f the  values  res-  forcing function,  r e s u l t s were seen to f i t c l o s e l y  80  the e x p r e s s i o n s (4-18), and (4-19) and (4-20) and the percentage P.O. and the damped frequency o f o s c i l l a t i o n  taken from  overshoot  the n u m e r i c a l  r e s u l t s a r e l i s t e d i n t a b l e B-1. The peak v a l u e o f the response  H ( t ) as determined  by (4-19) was  compared w i t h the n u m e r i c a l r e s u l t s , and the d i f f e r e n c e i s shown i n t a b l e B-1 under column A as a p e r c e n t a g e percentage centage  indicates that  o f the n u m e r i c a l r e s u l t s .  (4-19) gave a h i g h e r v a l u e .  A l s o , the p e r -  d i f f e r e n c e i n the damped frequency OJ^ o b t a i n e d i s l i s t e d  column B, where the damped frequency i s determined H ( t ) reaches i t s peak v a l u e . from  A positive  (4-19).  The damping  under  by the time a t which  r a t i o £ l i s t e d was  determined  81  TABLE B - l  [A  Y  P  d/A  g./cm  cm  dyn/cm  . v xlO  2  P.O. d rad/s W  cm Is  A  B  %  %  1  0.20  0.258  1.79  15.9  3.90  65.3  491  -1.5  +1.6  0.147  2  0.10  0.258  1.79  15.9  3.90  53.3  1372  -1.2  +1.8  0.207  3  0.05  0.258  1.79  15.9  3.90  39.4  3880  -0.9  -0.5  0.293  4  0.40  0.258  1.79  15.9  3.90  76.0  178  -2.3  +1.7  0.104  5  0.05  0.258  1.79  15.9  1.95  64.5  3910  -1  +2  0.147  6  0.10  0.258  7.16  15.9  3.90  75.2  697  -1.8  +1.9  0.104  3.90  43  236  -3.5  +2.1  0.293  . 7  0.20 0.258  1.79  3.98  8  0.05  0.50  1.79  15.9  1.95  73  4110  -6.2  +0.9  0.147  9  0.05  0.129  1.79  15.9  1.95  30.9  3090  -1.6  +3.4  0.369  10  0.20  0.50  1.79  15.9  3.90  73.5  511  -6.2  +1.4  0.147  11  0.05  0.04  1.79  15.9  •0.0  103.5  2065  -1.7  +0-9  0.0  12  0.05  0.129  1.70  15.9  0.0  105.6  3270  -2.7  +4.9  0.0  .13  0.20  0..04  1.7.9  15.. 9  .0..3.90 . .30.4  224  -5 .4  +8.0  0.369  14  0.20  0.02  1.79  15.9  0.9075  19.6  151  +1.7 +10.6  0.438  15  1.0  0.07  1.00  72.0  1.00  85.9  +4.0  0.045  16  0.05  0.25  2.00  10.0  10.0  -  +1.7  0.25  17  0.02  0.25  3.00  18  1.0  0.08  2.00  20.0  19  0.05  1.00  1.79  15.9  5.00  Computer runs step f u n c t i o n in  the f i r s t  f i v e columns.  and UJ^ o b t a i n e d  and t o ^ .  1.95  0.0  -  -  -  1.65  0.0  -  -  -  2.32  73.4  -6.1  4110  r a d i a t i o n p r e s s u r e were made w i t h  A and B r e s p e c t i v e l y P.O.  25.0  -  2870  The r e s u l t i n g percentage of o s c i l l a t i o n oo^are  0.147  H ( t ) to a  the parameters overshoot  shown; a l s o ,  indicated (P.O.) the v a l u e s  from (4-19) and (4-20) a r e g i v e n i n columns  as a p e r c e n t a g e  o f the v a l u e s i n the columns  Numbers 1, 2 and 16 a r e shown i n f i g u r e s  4.5 r e s p e c t i v e l y ,  +0.9  t o determine the l i q u i d s u r f a c e response  o f H ( t ) and damped frequency o f P.O.  6.00  82.6 +0.5  and 17 and 18 i n f i g u r e  4.4.  4.3, 4.2 and  82  APPENDIX C LIQUID SURFACE RESPONSE  The  e q u a t i o n f o r the motion o f a l i q u i d s u r f a c e when s u b j e c t  to r a d i a t i o n p r e s s u r e ,  (4-18) can a l s o be w r i t t e n as  P  r-  _ V ( ( p g + Yn )  H(t) =  r  A  -Cu) t + B ) — s i n ( 3 u t + G + 6) 9  n  (C-l)  f o r t ^ A t , C < 1, and where  3 =yj{\..  A = -e  B = e  C ) 2  Cu At  C W  n  n  A t  .. s i n ($„ At) n  cos (B_ At) - 1 n "  t a n 0 = B/C tan 6 = A/B Equation following  (C-l) s t i l l holds i f 3 i s imaginary relations cos j x = cosh x sin j x = j sinh x tan j x = j tanh x  (c  > 1) as we can use the  83  REFERENCES  1.  A.F. M e t h e r e l l , H.M.A. El-Sum and Lewis Larmore, e d i t o r s , A c o u s t i c a l K o l o g f a p h y , Plenum P r e s s , Inc., New York, 1969, V o l . 1.  2.  A.F. M e t h e r e l l and-Lewis Larmore, e d i t o r s , A c o u s t i c a l Holography, Plenum P r e s s , I n c . , New York, 1970 , V o l . 2.  3.  A.F. M e t h e r e l l , e d i t o r , A c o u s t i c a l Holography, Plenum P r e s s , New York 1971, V o l . 3.  4.  R.K. M u e l l e r , " A c o u s t i c a l Holography", P r o c e e d i n g s o f the IEEE, V o l . 59, No. 9, p. 1319, Sept. 1971.  5.  R.K. M u e l l e r 1966.  6.  P.S. Green, " A c o u s t i c a l Holography w i t h the L i q u i d S u r f a c e R e l i e f C o n v e r s i o n Method", L o c k h e e l I n t e r v a l Report No. 6-77-67-42, Sept. 1972.  7.  B.B. Brenden, "A Comparison of A c o u s t i c a l Holography Methods", A c o u s t i c a l Holography, V o l . 1, Ch. 4, pp. 57-71.  8.  A c o u s t i c a l Holography, Plenum P r e s s ,  9.  Beckmann and S p i z z i c h i n o , The S c a t t e r i n g o f E l e c t r o - M a g n e t i c from Rough S u r f a c e s , M a c M i l l a n Co., 1963  and N.K.  S h e r i d a n , A p p l . Phys. L e t t e r s 9:  Inc.,  328-329,  V o l . 3, pp. 129-171. Waves  10.  B.P. H i l d e b r a n d , "An A n a l y s i s o f the L i q u i d S u r f a c e as a L i g h t S c a t t e r e r " , O p t i c s and Device Development T e c h n i c a l Report 68-15, B a t t e l l e Memorial I n s t i t u t e , R i c h l a n d , Wash., 1968.  11.  R.B. Smith, and B.B. Brenden, IEEE Symposium on Sonics and U l t r a s o n i c s , Sept. 25-27, New York, 1968.  12.  A. P o p o u l i s , McGraw H i l l ,  13.  J.W. Goodman, I n t r o d u c t i o n pages 60, 80.  14.  Z.A. G o l ' d b e r g , " A c o u s t i c R a d i a t i o n P r e s s u r e " , i n H i g h - I n t e n s i t y U l t r a s o n i c F i e l d s , e d i t e d by L. D. Rozenberg, Plenum P r e s s , New York, 1971.  15.  L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course o f T h e o r e t i c a l P h y s i c s , V o l . 6, Pergamon P r e s s , London, 1959, Chapter 8.  16.  L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course o f T h e o r e t i c a l P h y s i c s , V o l . 6, Pergamon P r e s s , London, s e c t i o n s 12, 15.  Systems and Transforms w i t h A p p l i c a t i o n s 1968, pp. 205, 418.  i n Optics,  to F o u r i e r O p t i c s , McGraw H i l l ,  1968,  84  17.  L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course o f T h e o r e t i c a l P h y s i c s , V o L 6, Pergamon P r e s s , London, 1959, s e c t i o n s 12, 25, 61.  18.  L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course o f T h e o r e t i c a l P h y s i c s , V o l . 6, Pergamon P r e s s , London, 1959, s e c t i o n s 15, 60.  19.  J.S. Gradshteyn and I.M. Ryzhik, T a b l e o f I n t e g r a l s , S e r i e s and P r o ducts , Academic P r e s s , New York 1965, page 688, formula 6.5671.  20.  R.F. Beyer, " R a d i a t i o n P r e s s u r e i n a Sound Wave", Am. J o u r . Phys., V o l . 18, No. 1, J a n . 1950, page 25.  21.  F.H. Harlow and J.E. Welch, "Numerical C a l c u l a t i o n o f Time-Dependent V i s c o u s I n c o m p r e s s i b l e Flow o f F l u i d w i t h F r e e S u r f a c e " , The P h y s i c s of F l u i d s , V o l . 8, No. 12, Dec. 1965, page 2182.  22.  B.P. H i l d e b r a n d and B.B. Brenden, An I n t r o d u c t i o n to A c o u s t i c a l Holography, Plenum P r e s s , New York, 1972, page 157.  23.  H.F. Budd, "Dynamical Theory o f T h e r m o p l a s t i c Deformation", Phys., V o l . 36, No. 5, May 1965, page 1613.  24.  J . B l i t z , Fundamentals o f U l t r a s o n i c s , B u t t e r w o r t h s , 1963, page 16.  25.  E.G. R i c h a r d s o n , U l t r a s o n i c P h y s i c s , E l s e v i e r P u b l i s h i n g Co., Amsterdam, New York. 1962. page 188.  26.  J.N. 227.  Goodman, I n t r o d u c t i o n t o F o u r i e r O p t i c s , McGraw-Hill  J-. Ap.  1968, page  

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