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A study of electrically driven standing waves on fluid surfaces Ionides, George Nicos 1972

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A STUDY OF ELECTRICALLY DRIVEN STANDING WAVES ON FLUID SURFACES by GEORGE B.Sc, M.Sc.,  NICOS  ION I D E S  A.R.C.S., U n i v e r s i t y o f L o n d o n , 1968 U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1970  A THESIS THE  SUBMITTED  IN P A R T I A L F U L F I L M E N T OF  R E Q U I R E M E N T S FOR THE DEGREE OF DOCTOR OF P H I L O S O P H Y  in  the Department of  PHYSICS  We  accept  required  THE  this  thesis  as c o n f o r m i n g  to the  standard  UNIVERSITY i  OF B R I T I S H  J u n e 19 7 2  COLUMBIA  In p r e s e n t i n g t h i s t h e s i s  in p a r t i a l  f u l f i l m e n t o f the r e q u i r e m e n t s  an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree the L i b r a r y  s h a l l make i t  freely  avai1 able for  r e f e r e n c e and  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s  It  i s u n d e r s t o o d that c o p y i n g o r  thesis  Department  of  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada  or  publication  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written permission.  that  study.  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department by h i s r e p r e s e n t a t i v e s .  for  ABSTRACT  The ducting  properties  liquids  conditions applied  (mercury  where  the  normally  to  the  are  spatially  time.  The  amplitudes  shift  of  containing water.  the  the  Both  an a c c u r a c y  of  small  free  on  mercury;  amplitude  surface  10  (^0.005  cm.  The  cm),  linearized  of  the  strength  of  results  are  in  agreement  of (up at  surface to the  excited  waves  55 k V / c m ) fluid by  field show do  applying  theory. that  not  surface.  an  affect Single  electric  i i  by  be of  waves  has as  wave at  been  a field.  appropriately on  boundary  stresses  measured  electrostatic an  for  frequency  the  electrostatic  surface  resonator  used  container,  with  in  observing  was to  Observations  the  periodic  decrease  applied  strong  The  a microwave  amplitudes  function  uniform  and  in  fields  fluids.  technique  a cylindrical  close  the  con-  studied  measured  of  on  electric  wave  for  modified  in  were  examined  The  mercury  of  waves  been  by  static  optical  permit  5 x  have  frequency  an  surface  excited  mercury  resonant  of  are  non-uniform,  methods  to  standing  and w a t e r )  waves  fields  the  of  layer  damping  fields structure  modes- h a v e  the  been  displacement  antinodes  of  a desired  the  applied  stress  the  surface  mode  distance  from  surface,  the  and  is  New  field  driving  modes  on s h a l l o w  change  of  strated. limiting  with  Viscous  theory.  the  force The  water  criteria  on  dissipation the  are  been modes  was  the  results  are  to  to wave  amplitude,  formulated, mixing and  conclusively  in  as  the  good  mean  fluid  an  studied,  When  the  the  produces  taken  of  mode p e r i o d .  Non-linear  also  period  comparable  non-linearity  has  the  electrode  depends  among s u r f a c e  and  natural  becomes  observations.  energy  factor,  and s e t t i n g  applying  stability  well  to  amplitude  the  agree  with  equal  non-linear.  surface.  mode,  overstable which of  surface  the  ex-  demon-  amplitude  agreement  TABLE OF CONTENTS  Page ABSTRACT  Ii  L I S T OF T A B L E S  viii  L I S T OF F I G U R E S .  ix  ACKNOWLEDGEMENTS  xii 1  Chapter 1  INTRODUCTION  1  Chapter 2  EQUIPMENT AND E X P E R I M E N T A L T E C H N I Q U E S . . . .  8  2.1  Microwave E l e c t r o n i c s  9  2.2  Microwave C a v i t y and B a s i c A c c e s s o r i e s .  13  2.3  H i g h V o l t a g e Wave G e n e r a t o r  17  •2.4  R e c t a n g u l a r W a t e r Wave T a n k  19  2.5  T i m e a n d Wave A m p l i t u d e  Recording  Systems 2.5.1  20 M i c r o w a v e C a v i t y Wave R e c o r d i n g .  2.5.2  Chapter 3  Optical Wave-Monitoring Technique C A L I B R A T I O N OF T H E MICROWAVE RESONATOR TO S T A T I C S U R F A C E DEFORMATIONS, AND I N V E S T I GATION OF E L E C T R O S T A T I C F I E L D E F F E C T S ON L I N E A R S U R F A C E WAVES  20  iv  22  27  Pag_e 3.1  Microwave Resonator S e n s i t i v i t y S t a t i o n a r y Deformations of i t s Boundaries 3.2.1  3.2.2  3.2.3  3.3  Chapter  4  to 28  Dispersion Relation for Surface Waves on a C o n d u c t i n g Field Under the A c t i o n of a Normal Electrostatic Field-Theory . . . Dispersion Relation with a S p a t i a l l y Non-Uniform F i e l d .  Experiment  that  4.2  4.2.2  50  56  . . .  61  S u r f a c e Mode E x c i t a t i o n by a S p a t i a l l y N o n - U n i f o r m , Time Varying E l e c t r i c Field - Theory.  62  S u r f a c e Wave Containers  70  Experimental with Theory 4.2.1  42  did  E X C I T A T I O N OF PURE S U R F A C E MODES BY S P A T I A L L Y NON-UNIFORM TIME PERIODIC E L E C T R I C F I E L D S - S U R F A C E MODE DAMPING  4.1.2  .  E x p e r i m e n t a l R e s u l t s of S u r f a c e Mode F r e q u e n c y R e d u c t i o n by Spatially Non-Uniform, Electrostatic Fields  An I n s t a b i l i t y not Succeed  4.1.1  .  33  Results  Damping  -  in  Comparison 76  S u r f a c e Mode E x c i t a t i o n by S p a t i a l l y N o n - U n i f o r m , Time Dependent E l e c t r i c F i e l d s Experimental Results  76  S e l e c t i v e E x c i t a t i o n of Pure S u r f a c e M o d e s by T i m e Dependent, S p a t i a l l y Non-Uniform Electric Fields  87  v  Page 4.2.3  Chapter  5  Dependence o f Damping F r e q u e n c y o f S u r f a c e Modes on an Applied Electrostatic Field. . .  D I S C U S S I O N OF THE IMPORTANCE OF NON-UNIFORM E L E C T R I C F I E L D S 5.1  The  Importance  of  SPATIALLY  Chapter 6  7  Measurements  101 103 104  with Theory  109  Di s c u s s i on  113  NON-LINEAR I N T E R A C T I O N OF WAVES  ON  SHALLOW WATER  118  7.1  Theory  119  Experimental Results C r i t i c a l Discussion of Experimental Results of the Non-Linear I n t e r a c t i o n b e t w e e n S h a l l o w W a t e r S u r f a c e Modes . .  128  7.2 • 7.3 Chapter 8  95  NON-LINEAR D R I V I N G OF SURFACE WAVES BY ELECTRIC FIELDS 6.1 Theory of Non-Linear D r i v i n g of S u r f a c e Modes by E l e c t r i c F i e l d s . . . . 6.2 Experimental R e s u l t s - Comparison  6.3 Chapter  Damping F r e q u e n c y  95  Non-Uniform  Electric Fields 5.2  92  CONCLUSIONS - FUTURE WORK  132 139  BIBLIOGRAPHY  178  APPENDICES A R a d i a l D i s t r i b u t i o n of Non-Uniform E l e c t r i c F i e l d on t h e M e r c u r y S u r f a c e . . . . B E x p e r i m e n t a l D i f f i c u l t i e s and P r e c a u t i o n s i n M e a s u r i n g S u r f a c e Wave D a m p i n g  182  vi  191  Appendices C  D  E  F  G  Page A d d i t i o n a l C a l c u l a t i o n s on t h e Non-Linear D r i v i n g o f S u r f a c e Waves by S p a t i a l l y N o n Uniform P e r i o d i c E l e c t r i c Fields  195  Additional Effects Wave I n t e r a c t i o n  208  in  Second  Order  Wave-  D a m p i n g on End W a l l s o f a R e c t a n g u l a r C o n t a i n e r of Width S m a l l Compared to i t s Length  222  E x c i t a t i o n o f S u r f a c e Modes and Time P e r i o d i c F i e l d s  224  by  Space  D a m p i n g o f a P r i m a r y Mode by t h e o f a S e c o n d a r y Mode - a P o s s i b l e zation Technique  vii  Presence Stabili227  LIST OF TABLES  Table  Page  2.1  Some P a r a m e t e r s  of Experimental  Equipment ...  2.2  Summary o f S y m b o l s F r e q u e n t l y U s e d i n t h e Thesis  15  25  3.1  Parameters  Used i n the E x p e r i m e n t s  51  3.2  Zero of D e r i v a t i v e s of Bessel Functions  of  the F i r s t Kind  53  3.3  Results  57  4.1  Experimental  Parameters  80  4.2  S u r f a c e Modes E x c i t e d w i t h D i f f e r e n t Electrodes. . . I d e n t i f i c a t i o n o f S u r f a c e Modes by  89  t h e i r Damping F r e q u e n c i e s  91  4.3  6.1  Experimental  Parameters  112  7.1  Experimental  Parameters  130  7.2  Experimental Theory  R e s u l t s - Comparison  with 133  vi i i  LIST OF FIGURES  Figure  Page  1  E l e c t r o n i c microwave system  2  D i s p l a y of microwave resonance  148 on  oscilloscope  149  3  M i c r o w a v e c a v i t y and a c c e s s o r i e s  150  4  Gas t i g h t m i c r o w a v e c a v i t y f i l l e d w i t h SFs  5  S t a t o r o f h i g h v o l t a g e wave g e n e r a t o r  152  6  Rotor  152  7 8  O u t p u t o f h i g h v o l t a g e wave g e n e r a t o r 153 M o d i f i e d s t a t o r o f h i g h v o l t a g e wave g e n e r a t o r , p r o d u c i n g " s i n u s o i d a l " type output. . . . 154  9  R e c t a n g u l a r w a t e r c o n t a i n e r a n d wave excitation electrode  10  o f h i g h v o l t a g e wave g e n e r a t o r  . . 151  155  S u r f a c e wave a n d t i m e m a r k e r r e c o r d i n g on f i l m  156  11  Time m a r k e r p r o d u c t i o n and d i s p l a y  157  12  O p t i c a l wave m o n i t o r i n g waves  158  ix  method f o r w a t e r  Figure  Page  13  Photoresistor c i r c u i t f o r monitoring s u r f a c e w a v e s on w a t e r  159  14  Calibration of photoresistor to yield a b s o l u t e m e a s u r e m e n t s o f s u r f a c e wave amplitude  160  Graph f o r c a l i b r a t i n g t h e s e n s i t i v i t y o f t h e microwave resonator to s t a t i c boundary deformations  161  16  G e o m e t r y o f s u r f a c e wave i n f l u e n c e d by a u n i f o r m , normal e l e c t r o s t a t i c f i e l d  162  17  G e o m e t r y o f s u r f a c e wave i n f l u e n c e d by a non-uniform e l e c t r o s t a t i c f i e l d  163  18  P l o t o f s q u a r e o f s u r f a c e mode f r e q u e n c y v s . the square o f the v o l t a g e a p p l i e d t o the electrode  164  19  High v o l t a g e g e n e r a t o r used f o r a p p l y i n g s u p e r c r i t i c a l f i e l d s to the f l u i d surface.  165  20  Unstable growth  21  Power r e s o n a n c e  22  E s t i m a t i o n o f phase d i f f e r e n c e between t h e d r i v i n g f i e l d a n d s u r f a c e wave  168  23  S u r f a c e wave a m p l i t u d e as a f u n c t i o n o f t h e -amplitude o f t h e e l e c t r o d e v o l t a g e squared . .  169  24  D i r e c t m e a s u r e m e n t o f s u r f a c e wave a m p l i t u d e  .  170  25  Various  .  171  15  . .  s u r f a c e wave a n d i t s e x p o n e n t i a l curve  166  f o r ( 0 , 3 ) s u r f a c e mode .  s h a p e s o f wave e x c i t a t i o n e l e c t r o d e s  x  167  Fi gure  Page  26  Measurement o f damping f r e q u e n c y s u r f a c e modes.  of  27  D a m p i n g f r e q u e n c y o f s u r f a c e modes as a function of the square root of the o s c i l l a t i o n f r e q u e n c y ; t h e l a t t e r i s r e d u c e d by an applied electrostatic field  172  d r i v e n s u r f a c e wave a m p l i t u d e  173  28  Non-linearly  29  Comparison o f p r e d i c t e d n o n - l i n e a r l y driven amplitude with experimental r e s u l t s  30  S e c o n d a r y mode a n d d r i v i n g v o l t a g e w a v e form i n r e c t a n g u l a r water container  176  31  P r i m a r y a n d s e c o n d a r y mode a m p l i t u d e s functions of the driving voltage  177  Al  Analog experiment f o r obtaining radial p r o f i l e of applied e l e c t r i c f i e l d at the f l u i d surface  183  A2  Radial  174  . .  175  185  A3  Graph demonstrating experiment  A4  D e p e n d e n c e o f a t t r a c t i o n f o r c e on v e r t i c a l p o s i t i o n f o r brass probe used i n analog experiment  188  CI  N o n - l i n e a r l y d r i v e n s u r f a c e wave i n rectangular container  197  D2  N o n - l i n e a r l y d r i v e n s u r f a c e wave in rectangular container  199  as  p r o f i l e of e l e c t r i c f i e l d squared  . . .  f e a s i b i l i t y of analog  xi  . .  amplitude  186  Fi gure C3  C4  C4  Page Dependence of c r i t i c a l v o l t a g e dimensions of e l e c t r o d e i n the of a r e c t a n g u l a r c o n t a i n e r  on case 200  D e p e n d e n c e o f p h a s e d i f f e r e n c e on s u r f a c e wave a m p l i t u d e f o r non-linearly d r i v e n waves  203  E l e c t r o d e f o r a p p l y i n g both time periodic electric fields . .  226  xii  and  space  ACKNOWLEDGEMENTS I wish t o thank Dr. F.L. Curzon f o r h i s e x c e l l e n t supervision of t h i s i n v e s t i g a t i o n . t h a n k t h e members o f my  Ph.D.  I a l s o would  committee,  D r . R.W.  B u r l i n g and D r . C F .  gestions  f o r improving the p r e s e n t a t i o n  like  D r . R.A..  to  Nodwell,  Schwerdtfeger f o r t h e i r sugof this t h e s i s .  T h e a s s i s t a n c e o f Mr. R. H a i n e s i n my e a r l y  days  i n t h e s t u d e n t w o r k s h o p , and o f Mr. A. F r a s e r a n d h i s t e c h n i c a l s t a f f i n the c o n s t r u c t i o n of parts of the apparatus is g r a t e f u l l y acknowledged.  My t h a n k s a r e a l s o due  to  Mr. D. S i e b e r g , Mr. J . A . Z a n g a n e h and Mr. R. D a C o s t a f o r t h e i r a s s i s t a n c e i n t h e e l e c t r o n i c s and t h e g e n e r a l tenance of the  equipment.  I would  l i k e to express s p e c i a l a p p r e c i a t i o n  Mr. T. M a t h e w s f o r h i s e x p e r t during  t h e summers o f 1970  to  assistance in computations  and  1971.  I am g r a t e f u l t o t h e g r a d u a t e s c h o o l o f B.C.  main-  of the U n i v e r s i t y  f o r f i n a n c i a l a s s i s t a n c e t h r o u g h the award of  graduate fellowships during T h i s w o r k was Energy Control  Board of  the c o u r s e o f t h i s work.  s u p p o r t e d by a g r a n t Canada. xtii  from the  Atomic  Chapter  1  INTRODUCTION  The in  the Plasma  part  of  work  Physics  a program  incompressible  reported  in this  group  oriented  fluid  thesis  was c a r r i e d  at the U n i v e r s i t y towards  analogues  to  o f B . C . , as  developing certain  out  and a p p l y i n g  dynamical  plasma  systems. The subject, ing  a)  which  of plasma dynamics  is  a very  broad  c a n be s i m p l i f i e d by one o f t h e t h r e e  follow-  approximations.  Transport This  the  theory  and is  kinetics: the branch  of plasma dynamics  Maxwel1-Boltzmann p a r t i c l e  space  is  used  a plasma under particle collision  to describe steady  interactions dominated,  distribution  the evolution  state  in  in  which  velocity  and p r o p e r t i e s  conditions.  The  charged  i n t h e p l a s m a a r e assumed t o be in order  mathemati c s .  1  to s i m p l i f y  the  complex  of  2  b)  Orbit  theory: Systems  which  collisions  perturbations by  the  play  of  and  Lorentz  by  a minor  the  theory  The  electric  orbits,  only  orbit  role.  "prescribed"  plasma p a r t i c l e  consistent, the  describable  force  are  those  dynamics  in  involve  and  magnetic  fields  the  model  self-  so  that  on  a charged  is  particle  q  is  force  £ = q(JE + y_ x B_) where  E_,v_  velocity  and  and  c)  B_  the  are  the  magnetic  electric  induction  field,  the  particle  respectively.  Magnetohydrodynamics: In  involve scopic  contrast  a microscopic continuum  ducting  cases  is  much  enter  The  in  the  of  which  smaller  previous  approach,  basic  field  equations  in  the  interactions  fluids.  electromagnetic Stokes  to  fluid  equations  mean the  of  free  typical  motion.  are  path  of  has  macro-  with  the  model  the  physical It  fields  with  This  which  combines  Maxwell's  together  dynamics.  domains,  model  magnetic  equations  equations  the  than  of  this  two  con-  classical Navier-  applies  plasma  particles  dimensions  that  proved  very  useful  3  in  the  description  collective  of  plasma  area  of  may in  have,  analogous  research  importance  because e.g.  in  atmospheric  that of  the  the  electricity,  fields  fluid  continua.  dent  of  the  electric  demonstrate actions  on  importance  limits  troublesome  on  from the  where  acquired  of  of  conducting  renewed  unstable is  surfaces,  between  static fluid  applications  it  "electrohydro-  particular  effects  is  applications  This  Our  and in  plasma  macroscopic  instabilities,  by  general  electric  interest  was  and  depen-  time  surfaces, that  and  these  to  inter-  have.  utmost  impose  etc.  practical  Instabilities the  recently  interactions  dynamic  fields  some  and  magnetohydrodynamics  stabilization  involves  study  dynamics,  man/possible  which  the  to  has  dynamics," and  discharge  effects.  Exactly an  pinch  for  a certain  the  way  they  plasma p h y s i c s , confinement  the  are  displacement  equilibrium  since  times.  instabilities  which  develop  The  are  of  they most  exchange  type  perturbation  configuration  is  governed  equation  _r i s  analogous  a radius to  the  vector  equation  and V  t 4  is  V ~ 0  time. for  This  is  completely  incompressible  4  fluid  d y n a m i c s [\f_the  fluid  easy h a n d l i n g , the ready simplicity  velocity).  a v a i l a b i l i t y o f f l u i d s , and  of the necessary  due  The  results  r a t h e r than with a c t u a l  can t h e n be g e n e r a l i z e d ( e x e r c i s i n g  c a u t i o n ) to analogous The  situations in a  plasma.  i n t e r a c t i o n o f f l u i d s u r f a c e waves w i t h  s t a t i c and t i m e p e r i o d i c e l e c t r i c f i e l d s to the f r e e f l u i d  applied normally  s u r f a c e has b e e n t h e m a i n o b j e c t o f  investigations.  We  unstable system,  i n which  originally  so t h a t the e x p e r i m e n t a l theoretical  the waves were s u f f i c i e n t l y results  c o u l d be c o m p a r e d  p r e d i c t i o n s g i v e n by a n a l y t i c a l Two  Besides  non-linear  a l l o w i n g the i n v e s t i g a t i o n of  the added  r e c o r d i n g s p e c i f i c modal r e s p o n s e s ,  g r a p h i c and p r o b e  (which  the theory.  linear  s e t - u p , and i t s wave m o n i -  t o r i n g and e x c i t a t i o n t e c h n i q u e s , h a d  r a t h e r t h a n £ ( r, Z )  problems  non-  the r e s u l t s of l i n e a r  s u r f a c e waves, our experimental  of  with  i s i n t r o d u c e d i n a c o n t r o l l e d manner, w i t h of extending  small,  s o l u t i o n of  are a l s o i n v e s t i g a t e d i n t h i s work; i n t h e s e , the  s p e c i f i c purpose  our  intended studying a simple  the r e l e v a n t , l i n e a r i z e d e q u a t i o n s .  linearity  the  d i a g n o s t i c equipment, i t i s  much e a s i e r t o w o r k w i t h f l u i d s plasmas.  Because of the  advantage  i . e . £"(CJ,/c)  i s g i v e n by t h e common  diagnostic techniques  used i n  ,  photo-  plasmas);  5  where the  (J i s  the  wavevector  means  that  characteristic  angular  associated with  the  normal  labour  the  of  frequency  fluid  Fourier  a n d k_  motion.  This  analysis  is  rendered  unneces s a r y . A main stress ities  the of  of  using  less  overriding  an  stability  applied  of  uniform  on  it  field  surface the  on  for  the  dispersion  a conducting  reduction  of  the  the  surface  surface  surface  and  of  spatial  proceeded  to  through  investigate  fields  of  excitation  wave  effect  of  a strong  for  is  much  misleading  electro-  linearized  (mercury)  by  observing  frequency.  For  unstable  being  instability  by is  disturbances  these  inhomogeneity  non-uniform the  simplicity  approach  large pulled  produced set  up  much by  non-uniformities.  We r e a l i s e d portance  and  common  some  becomes  transient  field  dynamics The  of  to  non-uniform-  to  mode  more  electric  been  lead  fluid  This  the  the  relation  upwards.  by  on  has  spatial  waves.  to  uncontrollably easily  the  sake  the  on  fields  work  criteria.  waves  enough  of  field  shown  examined  the  this  surface  is  stability We f i r s t  static  electric  fields  and  of  importance  interacting  realistic  results  contribution  on  a fluid  the  in  investigations the  effect  surface.  electrode  and  the  electric  field,  of  time  By  proper  setting  of  im-  periodic tailoring  the  applied  6  field  frequency,  surface  mode  surface.  can  This  we  demonstrated  be  excited  is  that  virtually  individually  a powerful  on  experimental  any  the  tool  fluid (see  below).  When t h e a m p l i t u d e o f t h e d r i v e n s u r f a c e was  between This the a  the  led  to  of  criterion.  is  becomes  in  Navier-Stokes  the  tions,  then  surface ing  cations  in  lead  to  the  an  water  surface  non-linear waves  periodic  electric  modes  controlled Chapter  equipment  in  of  the  in in  the the  used  to  arises  when  are  wave  (_y_.gradvj  boundary spectrum  condiof  mechanism f o r  spatially  were  mechanism.  terms  and has  of arises  hydrodynamic  oceanography, conditions  by  instability  non-linear and  surface.  formulation  non-linearity  that  distance  waves  of  spectrum,  fluid distribut-  direct  appli-  non-linear  optics,  met  shallow  in  the  non-uniform, excite  very  time pure  amplitude.  2 contains and  the  driving  important  which  fields  type  terms  wave  plasma p h y s i c s ,  Resonant  mental  is  fluid  surface  to  waves  the  the  The  equation,  to  in  so  forcing  This  through  etc.  of  type  non-linear.  modes.  energy  effects  the  and  This  increased,  system  of  field,  independent  amplitude  comparable  equilibrium  driving  electric  non-linear An  became  and t h e  non-linear  stability  because  soon  electrode  non-uniform  new  the  this  increased,  desired  relevant  a description  of  the  experi-  techniques.  In  Chapter  3 we  7  present of  theoretical  strong,  mercury of  the  we  effects  results  relation  properties Chapters  surface  by  The  modes  the  driver  and  the  surface  to  6 and  problems.  the  in  modes  basic  non-uniform 4 is  spatially  an  mercury  some  detail  the  the  the  devoted  to  deals  with  former of  amplitude  deals on  in  the  the  comparable  to  equilibrium  with  the  dynamic  conclusions,  a n d make  the  In  the  on  dynamic  field. non-linear  driving  of  distance  8 we for  of  surface,  interaction  suggestions  5  experi-  the  fluid  Chapter  a  especially  distinct  the  water.  of  Chapter  electric  from  shallow  In  non-linear  on  periodic  exerted  two  effects  investigations  chapters,  influence  the fields  time  surface.  two  non-uniformities  of  of  implications  previous  dominant  7 are  account  non-uniform,  a free  of  results  electrostatic  on  electrode  latter  experimental  Chapter  of  fields  discuss  mental in  spatially  surface.  electric  and  between summarize  future  work.  Chapter 2  EQUIPMENT AND EXPERIMENTAL TECHNIQUES  In t h i s  c h a p t e r we d e s c r i b e t h e e s s e n t i a l  niques i n v o l v e d i n our e x p e r i m e n t a l work.  tech-  S i n c e one g o a l  was t h e s t u d y o f l i n e a r , s t a n d i n g s u r f a c e w a v e s , t h e wave amplitude  i n o u r c a s e h a d t o be o f t h e o r d e r o f IO""  cm,  5  in order to s a t i s f y the s t r i n g e n t l i n e a r i z a t i o n c o n d i t i o n $/k H « Z  depth.  i  3  (Stokes ( 1 8 4 7 ) ) , where H i s the f l u i d  S t a n d i n g s u r f a c e waves i n r e s o n a t o r s were  throughout, excitation  a n d o u r wave e x c i t a t i o n of single  p u r e modes  we o n l y had' t o d e a l w i t h a f i x e d i n g wavenumber  techniques  at a time.  used  guaranteed  In t h i s  f r e q u e n c y and  way  correspond-  a t a t i m e , and p r o b l e m s o f d i s p e r s i o n  a s s o c i a t e d w i t h t r a v e l l i n g waves were e l i m i n a t e d . Two w o r k i n g f l u i d s w e r e e m p l o y e d , m e r c u r y a n d w a t e r ( w a t e r i s d e s i g n a t e d h e r e as a c o n d u c t i n g f l u i d to i t s high d i e l e c t r i c constant r e l a t i v e  to a i r ) .  due  Amplitude  a n d f r e q u e n c y m e a s u r e m e n t s o f w a v e s on m e r c u r y w e r e o b t a i n e d by m o n i t o r i n g t h e r e s o n a n t f r e q u e n c y s h i f t o f a m i c r o w a v e  8  9  resonator lower  caused  boundary.  by s u r f a c e w a v e s on m e r c u r y f o r m i n g  the  Corresponding  sur-  m e a s u r e m e n t s on w a t e r  f a c e s w e r e c a r r i e d o u t by an o p t i c a l t e c h n i q u e a microscope spatial  resolution We  system.  (see F i g . 12).  Both methods a f f o r d e d a  o f about 5 x 10"  shall first  cm.  d e s c r i b e the e l e c t r o n i c  T h i s i s f o l l o w e d by a d e s c r i p t i o n  c o n t a i n e r , and  the e l e c t r o d e used  of time  voltage generator,  and  o f t h e m i c r o w a v e and  Microwave  this chapter  used  A low i n the  the theory  concludes  frequency, application  with  o p t i c a l wave m o n i t o r i n g  proceeding  t r o n i c m i c r o w a v e s y s t e m , we behind  t i o n s f o r i t t o be  details  techniques.  to a d e s c r i p t i o n  of the  elec-  s h a l l give a b r i e f account  the resonator technique,  and t h e  of  condi-  useful.  When t h e b o u n d a r i e s  of a microwave  are s l i g h t l y d i s t o r t e d , the resonant ^F.  mercury  Electronics  Before  by  of the  dependent f i e l d s to e x c i t e s u r f a c e waves, i s  described then,  2.1  microwave  for applying e l e c t r i c  f i e l d s to the f r e e s u r f a c e of the f l u i d . high amplitude,  employing  T h i s f a c t c a n be u s e d  frequency  resonator F changes  to d e t e c t s u r f a c e waves  on m e r c u r y by m a k i n g t h e f r e e m e r c u r y s u r f a c e one  of  the  10  resonator AF  can  boundaries.  be  calculated  distortion, of  the  and  if  the  compared w i t h is  in  is  accordance  with  is  used  A F  terms  of  the  an e x a c t  giving the  F as  cavity,  microwave  analytic  electromagnetic  in  the  these  in  our  theorem  resonant  conditions system,  (Slater  are  s o we  (1950))  (see  is  Thi  start  equations fields  without  Waldron  satisfied  for  we  boundaries  eigenvalue,  eigenfunction  cavity.  That  the  the  small  expressions  of  of  is  are  perturbation  distortion.  perturbation  spatial function  amplitude  of  when  conditions  a function  All  F,  these  characteristic  the  the  a linear  the  corresponding  in  obtain  of  of  ing  Slater's  the  useful  solution  and  degree  is  Both  of  only  a function  assumptions  boundary  change  high  the to  a small the  is  perturbation  dimensions  which  from  as  amplitude.  theory in  exactly  boundary  the  technique  s i m p l e s t when A F  perturbation  satisfied  The  to  are  justified  for  the  in  causes destroy  ( 1969)) . a  in  fractional  very using shift  namely  (2  o where V  £  E  and B  M  a n d )f  B  are  M  are  the  constants  electric depending  and on  magnetic the  fields,  resonator  and  and  11  microwave f i e l d s Sc  geometries  i s the volume element  ( s e e C u r z o n and P i k e ( 1 9 6 8 ) ) . "removed" from the r e s o n a t o r  by t h e b o u n d a r y p e r t u r b a t i o n . In a normal  our case, f o r a p e r t u r b a t i o n of the form  f l u i d s u r f a c e mode o f a m p l i t u d e  cTr = i>lf(r,e)]  where f g i v e s the r a d i a l of  v o l u m e g i v e s A F <=< 7? In  ,  (2  .  2)  dependence  I n t e g r a t i o n over the r e s o n a t o r  (see Curzon  our experiments  and P i k e ( 1 9 6 8 ) ) .  on s u r f a c e w a v e s on  a m i c r o w a v e r e s o n a t o r t e c h n i q u e was many a d v a n t a g e s  ,  ( r ) a n d a z i m u t h a l {6)  the s u r f a c e d i s p l a c e m e n t .  of  used because  mercury, of i t s  o v e r c o n v e n t i o n a l wave m o n i t o r i n g t e c h n i q u e s .  N a m e l y , t h e f a c t t h a t i t does n o t i n t e r f e r e w i t h t h e w a v e s , i t s s i m p l i c i t y , the ease of data a n a l y s i s , i t s F o u r i e r a n a l y z i n g p r o p e r t i e s (see Curzon finally  because  and P i k e ( 1 9 6 8 ) ) , and  the monitored amplitude i s e s s e n t i a l l y  an a v e r a g e d d i s p l a c e m e n t o v e r t h e e n t i r e f r e e f l u i d s u r f a c e , so t h a t i t i s r e l a t i v e l y i n s e n s i t i v e to s m a l l ripples.  The  a p p l i c a t i o n s of the r e s o n a t o r are however  l i m i t e d only to h i g h l y conducting f l u i d s metals), of dimensions the  amplitude  microwaves.  (i.e. liquid  d e t e r m i n e d by t h e w a v e l e n g t h  of  12  The 1.  It  consists  reflex  of  klystron,  stability, power  electronic  and  heat  currents  frequency  F of  is  voltage scope  the  a function  of  transmitted  isolator the  prevents  klystron.  factor TEon  stray  The  calibrated mode.  It  of  1 MHz.  is  in  two;  other  half  is  At  half  our  cavity  The  In on  by  this the  rates the  from  basically  of  it  goes  dissipated  at  the  tee into  The to  output  9.6  repeller oscillo-  sawtooth  also  way,  klys-  the  of  1 kHz. are  through  Fig.  as  a  1).  The  returning  into  high  Q  operating  in  the  frequencies  to  an  resonator  magic  ± 325  klystron  (see  microwave  output  oscilloscope  resonator  reflections is  8.6  Fig.  regulated  display  beam).  accessories  the  of  ± 200 t o  klystron  of  produced  measure  28  range  modulation  wavemeter  can  the  base.  cylindrical  accuracy split  and  purpose  and  the  displayed  a microwave  waveguides  in  output  typical  microwaves  the  in  Corp.)  100 mA a v a i l a b l e ) .  time  be  shown  (Raytheon  V AC,  5 5 1 , dual  can  F at  to  6.5  sweeping  type  power  The  of  by  for  is  Lambda model  klystron,  oscilloscope  output  system  of  sawtooth  (Tektronix  drives tron  this  the  two  from 0 to  modulated with  by  system  723A/b  shielded  (outputs  V DC w i t h  kMHz,  a standard  powered  supplies  microwave  the the  a very  microwave cavity,  matched power  power and  the  terminator.  13  The  power r e f l e c t e d from  crystal  t h e c a v i t y i s p i c k e d up by a  d e t e c t o r a n d d i s p l a y e d on t h e o s c i l l o s c o p e as a  f u n c t i o n o f microwave frequency is the tuning post.  (Fig. 2).  A last  feature  T h i s i s a small metal  probe,  moveable  a l o n g and p e r p e n d i c u l a r used  t o t h e waveguide a x i s , and i t i s  t o e f f e c t t h e most e f f i c i e n t  coupling of the micro-  waves t o t h e r e s o n a t o r .  2.2  Microwave C a v i t y and B a s i c The  3.  Accessories  m i c r o w a v e c a v i t y i s shown i n d e t a i l  I t i s e s s e n t i a l l y a brass  l e v e l l i n g p l a t f o r m to ensure is horizontal.  in Fig.  c y l i n d e r m o u n t e d on a that the base o f the c y l i n d e r  The whole arrangement, i n c l u d i n g  micro-  wave c o m p o n e n t s , i s m o u n t e d on a h e a v y , r i g i d t a b l e t o minimize  extraneous  vibrations.  the c a v i t y i s e f f e c t e d through  Microwave coupling to a small hole  (diameter  R-j/5, w h e r e R-j = 3.64 cm i s t h e c y l i n d e r r a d i u s ) i n t h e c a v i t y l i d ; t h e h o l e i s a t a d i s t a n c e R^/1.5 f r o m A f l e x i b l e tube  (Tygon) connects  the microwave  the axis.  resonator  to a mercury r e s e r v o i r through  an a d j u s t a b l e f i n e t a p .  The  a c c e s s o r i e s l i a b l e t o come  c a v i t y and any o t h e r metal  i n t o contact with mercury are n i c k e l p l a t e d with in thick layer of nickel i n order to avoid  a 0.003  corrosion  14  from mercury.  Table  2.1  gives relevant  experimental  parameters. A s t r o n g e l e c t r o s t a t i c or a l t e r n a t i n g f i e l d c o u l d be a p p l i e d t o t h e m e r c u r y s u r f a c e by a d i s c s h a p e d brass  e l e c t r o d e m o u n t e d a few  m i l l i m e t e r s above the  The  e l e c t r o d e , of radius R (<R-j),  and  i t s edges were rounded  to m i n i m i z e  inch diameter  b a k e l i t e rod.  r o d was  found  The  presence  5/6  of the e l e c t r o d e  t o h a v e v e r y l i t t l e e f f e c t on t h e q u a l i t y The  p e r i n c h ) and p a s s e d  r o d was  through  threaded  a threaded  (24  hole  the c e n t r e of the c a v i t y l i d . A t h i n w i r e , running a n a r r o w (0.1  cm d i a m e t e r )  axis connected potential  trol  DC p o w e r s u p p l y , t y p e  with  used  blowing  1020-30 ,  potential.  The  The  power  a regulated current overload  con-  damage t o t h e e l e c t r o d e i n c a s e o f s p a r k i n g  from the e l e c t r o d e to the mercury. was  of high  maximum c u r r e n t 30 mA).  kept at ground  equipped  to minimize  hole along the b a k e l i t e rod  v o l t a g e 20 kV,  m e r c u r y i t s e l f was  at down  the e l e c t r o d e to a r e g u l a t e d source  (S.orensen,  maximum o u t p u t  s u p p l y was  mm  f a s t e n e d t o the l o w e r end o f a  the microwave resonances.  threads  2  the p o s s i b i l i t y of d i e l e c t r i c breakdown at I t was  and  very h i g h l y p o l i s h e d ,  to a c u r v a t u r e of about  high f i e l d s .  of  was  2  surface.  i n our e a r l y experiments, p e r i o d i c a i r pulses through  In t h i s s y s t e m ,  which  w a v e s w e r e e x c i t e d by a hole i n the  centre  T a b l e 2.1 SOME PARAMETERS  OF E X P E R I M E N T A L E Q U I P M E N T  •k  R.|  = Microwave resonator  R  = E l e c t r o d e r a d i u s = 2.0 cm  2  r a d i u s = 3.64 cm  L  = L e n g t h o f r e c t a n g u l a r w a t e r c o n t a i n e r = 97.8 cm  b  = W i d t h o f r e c t a n g u l a r w a t e r c o n t a i n e r = 5.0 cm  R-| h a s t o b e r e d u c e d t o e f f e c t i v e R-j b y s u b t r a c t i n g ~" 0.2 cm, t h e r a d i u s o f c u r v a t u r e o f t h e r i g i d m e r c u r y m e n i s c u s , when u s e d i n c o n n e c t i o n w i t h s u r f a c e waves on c l e a n m e r c u r y ( s e e P i k e and C u r z o n ( 1 9 6 8 ) ) .  16  of  the  electrode  Curzon  (1968).  kV/cm down  could  field  in  seal  on  cavity,  to  3)  in  the  fashion  of  Pike  arrangement  fields  up  to  the  surface  mercury  about 4.  By  using  raising  90 k V / c m .  The  eliminating  0-ring  on  the  resonator  was  gas  To  remove  were  this  surface  on  and  35  before  gases  a rotary  of  break-  of  the  sealed  waves  by  surface  system  air  hole  the  rim  and  the  modifi-  limiting  the  lid so  in a  of  the  that  pump  George and  it  a powerful that  and  Richards  would  of  passed  drying  could  (SFg),  a bottle  S F g was  the  mylar  hexafluoride  (see  moisture  the  tight,  The  certain  modified  in  sulfur  (PgOg),  m e t h o d was the  with  employed.  traces  strength  alternative  end,  pentoxide  In  made  insulating  this  any  breakdown  refilled  best  phosphorous  SFg  over  agent,  reduce  the  gas. system,  blowing  it air  employed. by  stresses  onto  selected  This  most  conveniently  was  seals  hole  and  fields  cavity,  coupling  pressure  excited  the  higher  microwave  the  excite  attain  the  (1969)). under  to  Fig.  and  flushed of  to  made t o  strength  electrode  to  this  applied  order  were  shown  one  Fig.  occurred.  cations  be  With  be  In  is  (see  done  of by  not  possible  pulses.  Pure  applying  regions  was  surface  strong, the  The  free  following  modes  periodic fluid  modulating  the  to  were electric  surface. voltage  1.7  on  a  with  geometrically  tailored  a square  voltage  variable The  wave  controlled  modulation  generator,  2 .3  was  which  High  easily  Wave  6.  which two  In the  its  variable,  high  plates  on  the  plates  1,  3 and  6-of and  the the  stator. slots  precautions and  also  passage  The steel a  ball,  lite the  shaft stator.  next  it  is  is  One  power  supply  the  second  edges  be the  risk  10  Hz.  wave  with  of  the  motor  ball  is  by  5 in to  brass  connected  plates  2, are  resin. between  to  4  and  rounded,  These plates, the  them.  rod,  from  through  facilitating  consists  shown  Fig.  switch,  plates  epoxy  over  a bakelite as  to  sparking  bounce  contact  one  in  connected  is  adjacent  contact  DC m o t o r ,  steel  to  and  section.  alternately  of  filled  contact  insulates  0.3  a rotary  supplies  on  4 kV,  given  power  mounted  to  cavity  a square  voltage  rotating  The  the  generator  form,  rotating  1 / 2 0 HP B o d i n e ,  with  is  could  the  in  range  contact  The  reduce  effected  the  and  eliminate of  of  stator. 5,  the  up  the  Generator  simplest  rotating  inside  amplitude in  described  A diagram and  of  frequency  is  Voltage  electrode  of  a spring  loaded  which  is  driven  in  Fig.  6.  The  the  high  voltages  electriclly  connected  by  bakeon to  18  t h e wave e x c i t a t i o n e l e c t r o d e i n s i d e t h e c a v i t y by means of a s o f t metal  brush  and h i g h v o l t a g e  transistor potentiometer trol  c i r c u i t enables  frequency  A  simple  i n c o r p o r a t e d i n t o the motor con-  very f i n e adjustments  of the  generator  t o be made. The  Hz.  cable.  maximum f r e q u e n c y  of the generator  was  10  T h i s c a n be i n c r e a s e d r e a d i l y by u s i n g a f a s t e r m o t o r  and more s e c t i o n s on t h e s t a t o r . generator  i s shown i n F i g . 7 a .  modes, s q u a r e  waves w i t h  up t o 4 kV w e r e u s e d . h i g h e r peak v o l t a g e s  A typical  amplitude  of the  In o r d e r t o e x c i t e  surface  a g r o u n d e d base and peak  In a n o t h e r  voltages  s e t of experiments  much  ( u p t o 20 kV) w e r e u s e d , w i t h t h e  b a s e l i n e a few h u n d r e d v o l t s l o w e r . small  output  square  This superposes  a  wave on a l a r g e s t a t i c f i e l d , s o  t h a t w a v e s c a n be e x c i t e d a n d t h e i r p r o p e r t i e s s t u d i e d i n the presence  of large static  fields.  In o t h e r a p p l i c a t i o n s i t was c o n v e n i e n t a more c o n t i n u o u s was a c c o m p l i s h e d  v a r i a t i o n of voltage with by c o n n e c t i n g  time.  t o use This  a s e t of high voltage re-  s i s t o r s t o a s t a t o r w i t h e i g h t p l a t e s as shown i n F i g . 8. The  values  o f t h e r e s i s t o r s were chosen t o give a v o l t a g e  proportional  t o (1 + c o s i i t )  makes t h e e l e c t r i c a l at a f l u i d s u r f a c e  2  on t h e e l e c t r o d e .  This  s t r e s s e s p r o p o r t i o n a l t o (1 + c o s J " l t ) ( X L i s the rotor frequency).  An RC  19  i n t e g r a t o r r e s u l t e d i n f u r t h e r smoothing of the output,  a typical  p i c t u r e o f w h i c h i s shown i n F i g .  I t s h o u l d be n o t e d pulse generator  was  table supporting any  2.4  that in a l l experiments  kept mechanically  the f l u i d  Rectangular  i s o l a t e d from  c o n t a i n e r i n order to  the the  avoid  W a t e r Wave T a n k o f n o n - l i n e a r wave-wave i n t e r a c t i o n s o f  s u r f a c e w a v e s on s h a l l o w  w a t e r were a l s o undertaken.  o r d e r to s a t i s f y the c o n d i t i o n wavelength), dimensions  and s t i l l  X » H  (where A  have a w o r k a b l e f l u i d  of the f l u i d  depth,  technique.  an o p t i c a l wave  r e c t a n g u l a r c o n t a i n e r , of was  constructed  from  e x c i t e o n l y modes w i t h  The  case.  p l e x i g l a s s and  placed  Waves w e r e  e l e c t r o d e s h a p e was  a displacement  con-  monitoring  e x c i t e d by an e l e c t r o d e on w h i c h a s i n u s o i d a l t y p e applied.  as  dimensions  on a l e v e l l i n g p l a t f o r m as s h o w n i n F i g . 9.  v o l t a g e was  (about  A transparent water  method ( s e e S e c t i o n 2.5.2) were e m p l o y e d i n t h i s  x 5 x 5 cm,  the  c o n t a i n e r were l a r g e enough  t a i n e r , in conjunction with  The  In  i s the  t o p r o h i b i t t h e use o f t h e m i c r o w a v e r e s o n a t o r  t h e wave m o n i t o r i n g  97.8  7b.  external vibration noise.  Studies  1 m)  voltage  antinode  output  chosen at the  to centre  20  of  the  tank.  surface under  2.5  These  transients  Time  a n d Wave  measuring  on w a t e r  2.5.1  and  resonances 2,  iodic  surface  quency  dip  on is  Chapter  made by  mask  minimization  the  second  of  order  effects  running the  the  deals  amplitudes  Cavity  adjusting the  with and  obtained wave  the  Wave the  swept  the  Systems techniques  periods  of  involved  surface  modes  in  level  the  the  of  oscilloscope  a periodic  oscilloscope  A time a narrow  to  (0.5  movie  Relative  of  screen.  the  of  slit  as  film  amplitude  the  shown  in  the  as  the  the The  surface  mm)  face  in  cavity,  shown  screen.  perturbs  shift  record  oscilloscope  mercury  frequency,  mercury  proportional 3).  Recording  microwave  on  on  a continuous slit.  Recording  mercury.  putting  covering  to  the  resulting  shift  (see  in  were  fields,  the  would  Amplitude  section  Microwave By  Fig.  which  ensure  investigation.  This in  precautions  microwave fre-  amplitude  an  wave  opaque  Fig.  a direction  measurements  of  amplitude  surface  in  per-  resonant  wave  in  A  in  10,  was screen and  perpendicular can  be  readily  21  obtained  from  i s t h e time of from  t h e f i l m r e c o r d , a n d t h e s u r f a c e wave p e r i o d  between c o n s e c u t i v e  100 t o 200 ms w e r e u s e d  peaks.  Typical  periods  i n the  experiments.  In o r d e r t o a v o i d c o u n t i n g d o t s on t h e f i l m are produced simple  at the t r i g g e r rate of the o s c i l l o s c o p e ) , a  and e f f i c i e n t t e c h n i q u e  p e r i o d s t o an a c c u r a c y nique  that enables  t i m i n g wave  o f 0.2 ms was d e v i s e d .  The t e c h -  i n v o l v e s d i r e c t r e c o r d i n g o f time markers onto t h e  movie f i l m , and Fig.11 Three  (dots  square  channel  shows t h e e l e c t r o n i c b l o c k  wave p u l s e s  diagram.  a r e added and f e d i n t o t h e second  of the recording oscilloscope.  These are produced  a t i n t e r v a l s o f 1, 10 a n d 100 ms r e s p e c t i v e l y by t r i g g e r i n g t h r e e T e k t r o n i x t y p e 163 p u l s e g e n e r a t o r s sponding The  pulses from  with  the corre-  a Dumont t y p e 781A t i m e mark  generator.  1 ms t i m e m a r k e r i s a l s o u s e d t o t r i g g e r t h e o s c i l l o -  scope.  A l l t h r e e p u l s e s h a v e t h e same a m p l i t u d e ,  but the  d u r a t i o n o f t h e 100 ms one i s t w i c e t h e d u r a t i o n o f t h e o t h e r two ( s e e F i g . 1 0 ) . When t h e 1 ms p u l s e a r r i v e s a t the s l i t ,  a horizontal line i s recorded  t h e 1 a n d 10 ms p u l s e s  on t h e f i l m .  arrive simultaneously,  added and t h e r e c o r d i s b l a n k e d ;  When  they are  a n d when a l l t h r e e  pulses  a r r i v e a t t h e same t i m e , t h e t r a i n o f 1 ms p u l s e s i s d i s p l a c e d h o r i z o n t a l l y as a r e s u l t o f t h e l o n g e r d u r a t i o n o f t h e 100 ms p u l s e .  22  By t h e f i l m and accurate made.  r e c o r d i n g a number ( a r o u n d u s i n g the time  readily yield  instabilities,  2.5.2  periods  onto  markers, a q u i c k , easy  and  m e a s u r e m e n t o f t h e s u r f a c e wave p e r i o d c o u l d  R e l a t i v e amplitude  f i l m can  20)  and  be  measurements d i r e c t l y from  information  damping  the  about growth r a t e s  of  frequencies.  Optical Wave-Monitoring  Technique  This simple, but h i g h l y r e s o l v i n g technique, used  t o m e a s u r e wave c h a r a c t e r i s t i c s o f s h a l l o w  waves i n a l o n g ( a b o u t 2.4)  1 m)  r e c t a n g u l a r tank  where the microwave t e c h n i q u e  was  e m p l o y e d by Z r n i c * and H e n d r i c k s  12a  and  focused  m a g n i f i c a t i o n 60, 0.25  cm).  opposite the f i e l d  inapplicable.  and  It  technique  s e t up as shown i n  p o i n t of the  focal  o f f o c u s 0.09  s i d e of the tank,  by t h e m i c r o s c o p e , light  depth  An o r d i n a r y r e a d i n g  of view  was  on t h e l o w e s t  of the w a t e r s u r f a c e (microscope  Section  ( 1970).  A t r a v e l l i n g microscope Fig.  water  (see  i s b a s e d on a m o d i f i c a t i o n o f a wave m o n i t o r i n g  was  meniscus  l e n g t h 2.6 cm,  lamp was  field  arranged  cm, width on  the  the r e s u l t i n g p i c t u r e i n  i s s h o w n i n F i g . 12b.  It is inverted  and  ( s h a d e d ) and  c o n s i s t s of a dark  (unshaded) region.  These areas  correspond  to  a  light  23  totally  i n t e r n a l l y r e f l e c t e d by t h e w a t e r m e n i s c u s on  s i d e o f t h e l a m p , and  to l i g h t p a s s i n g  straight  the  through  t h e w a t e r as shown i n F i g . 1 2 c , w h i c h i s a c r o s s - s e c t i o n of the tank  perpendicular  The  presence  to i t s l e n g t h .  of small  amplitude  w a v e s on  w a t e r s u r f a c e changes the r e l a t i v e p r o p o r t i o n s and  dark  areas  linearly.  t o r i n g wave a m p l i t u d e s (Phillips  T h i s p r o p e r t y was  down t o 5 x I O ' -2  of b r i g h t  used  cm.  , p o w e r r a t i n g 0.2  j u s t o u t s i d e the eyepiece  of the microscope.  c l o s e d i n a l i g h t t i g h t j a c k e t to prevent f a l l i n g on t h e p h o t o r e s i s t o r .  The  i n moni-  A photores i s tor  8 7 3 1 0 3 , maximum r e s i s t a n c e i n t h e d a r k  r e s i s t a n c e i n l i g h t 300/1  10 M i l ,  W) was  was  on an o s c i l l o s c o p e .  change of v o l t a g e  A typical  d a m p i n g w a v e f o r m i s s h o w n i n F i g . 13b. be e a s i l y and  of water i n the tank. F i g . 14.  by p o u r i n g  AH  across was and  device  can measure-  known a m o u n t s  A c a l i b r a t i o n g r a p h i s shown i n  An a d d e d v o l u m e o f w a t e r AD"  of f l u i d depth  The  from  p i c t u r e of a  a c c u r a t e l y c a l i b r a t e d to give absolute  m e n t s o f s u r f a c e wave a m p l i t u d e s  en-  stray light  p r o p o r t i o n a l t o t h e s u r f a c e wave d i s p l a c e m e n t , recorded  mounted  I t was  t h e p h o t o r e s i s t o r i n t h e c i r c u i t shown i n F i g . 13a then  the  given  by  produces a change  24  where  A is  duces  a corresponding  the  the  photoresistor The  basically monitoring of  area  5 ms .  by  the  Fig.  In  r  surface.  in  the  This  voltage  pro-  across  14).  time  our  water  A V p  resolution  response  circuit.  free  change  (see  time the  of  of of  case,  the  system  the this  is  limited  photoresistor was  of  the  and order  25  T a b l e 2.2 SUMMARY O F S Y M B O L S  ^=  Fluid surface displacement  7?- A m p l i t u d e (S,U)  F R E Q U E N T L Y USED  o fsurface  from  IN THE THESIS  equilibrium  wave  = D e s i g n a t i o n o f a s u r f a c e mode i n a c y l i n d r i c a l container (Uth p o s i t i v e s o l u t i o n o f Bessel f u n c t i o n o f order S - see equation (3.40))  J^(kr) = Bessel  function o fthe f i r s t kind o forder S  D = Distance o fe l e c t r i c f i e l d applying from e q u i l i b r i u m f l u i d s u r f a c e J  electrode  ?/°  =  X = W a v e l e n g t h o f s u r f a c e wave k=  2TT/A.  =  £J = A n g u l a r  f  =  ^/'2JF  W a v e n u m b e r o f s u r f a c e wave frequency  o fsurface  = F r e q u e n c y o f s u r f a c e wave  JQ = A n g u l a r f r e q u e n c y e l e c t r i c f i el d  o f a p p l i e d , time  F = Resonant frequency <5> = D a m p i n g f r e q u e n c y Q =  0)/2ry  wave  o f microwave o fsurface  oscillation periodic  resonator  modes  = Quality factor o fdriven surface  "° = F l u i d k i n e m a t i c  viscosity  v_ = F l u i d v e l o c i t y <f = F l u i d  v e l o c i t y p o t e n t i a l (y_ = g r a d <f)  modes  26  Table  p  = Fluid  depth  g = Acceleration = Fluid  = Permittivity  V  = Voltage  Q  E  = V /D 0  y  0  of  surface  t  Q  (Continued)  density  H = Fluid  T  2.2  gravity  tension of  applied  free to  space  the  field  applying  electrode  = E l e c t r i c f i e l d p r o d u c e d by t h e f i e l d e l e c t r o d e n o r m a l l y to the f r e e f l u i d  = Disturbance e l e c t r o s t a t i c by t h e s u r f a c e w a v e  potential  R-j = C y l i n d r i c a l  microwave  resonator  R2 = D i s c  electrode  radius  shaped  L  = Length  b  = Width  of of  rectangular rectangular  water water  produced  radius  container container  applying surface in  E  Q  Chapter 3  CALIBRATION OF THE MICROWAVE RESONATOR SENSITIVITY TO STATIC SURFACE DEFORMATIONS, AND INVESTIGATION OF ELECTROSTATIC FIELD EFFECTS ON LINEAR SURFACE WAVES  We s t a r t t h i s c h a p t e r by d e m o n s t r a t i n g sensitivity  o f a m i c r o w a v e r e s o n a t o r c a n be c a l i b r a t e d  observing the s t a t i c surface deformation electrostatic field inside  how t h e  on t h e s u r f a c e o f m e r c u r y  the resonator.  of experimental  produced  The  T h i s i s f o l l o w e d by t h e r e s u l t s  and t h e o r e t i c a l  investigations  r e l a t i o n of l i n e a r hydrodynamic  dispersion  r e l a t i o n which  with the wavevector  on t h e  equations  connects  on t h e  s u r f a c e waves.  t h e wave  k_ c o n t a i n s a l l t h e n e c e s s a r y  t o d e s c r i b e t h e wave m o t i o n . out t h i s chapter.  by an  contained  effect of a strong, applied electrostatic field dispersion  by  frequency information  L i n e a r t h e o r y i s used  through-  This allows exact manipulation of the  by o r d i n a r y a n a l y t i c a l  27  t e c h n i q u e s , and t h e r e s u l t s  28  are  generally  applicable  Non-linear  problems  cases,  and  usually  to  specific  3.1  the  Microwave  static  field  mercury R  4  by  in  surface,  at  Resonator  Q  is  its  is  held  and  a shift  it  is  caused in  the  is  shown  normal  by  a few  special  applicable  to  at  the  in to  microwave of  a distance  held  in  only  Stationary  Boundaries  electrode  at  are  situations.  hand.  applied  shaped  tackled  Sensitivity  a cylindrical  deformation as  E  be  similar  solutions  arrangement  a disc  electrode  their  of  other  only  problem  Deformations The  can  to  Fig. the  radius  surface  electrostatic resonant  of  2  Q  .  of  radius The  4  above  2  V  electro-  R (< R ) .  R )  a potential  microwave  free  resonator  D ( D «  An  3.  the  The  grounded  surface  stress  is  registered  frequency  on  the  os c i 1 1 o g r a m . With corresponding of  the  field  siderations of the  the  d and h t h e parts  (see and  surface  resonator  of  Fig.  the  depression mercury  3 ) , the  assumptions deformation sensitivity.  surface  following  permit to  and e l e v a t i o n  be  the  and  the  presence  justifiable  a direct  made,  in  of  con-  calculation used  to  calibrate  29  1. This allows  D  «  R  2  us t o t r e a t t h e n o r m a l e l e c t r o s t a t i c f i e l d  distribution  on t h e s u r f a c e as a s t e p r  o <  obtained  (3.1)  justified  from  the radial  field distribution  i n A p p e n d i x A. 2.  fluid  r < R  < r < R<  0  This i s f u l l y  function  T h e mass o f m e r c u r y i s c o n s e r v e d ,  and t h e  i s incompressible. 3.  The e l e c t r o s t a t i c s t r e s s i s b a l a n c e d  hydrostatic pressure depressed  by t h e  d i f f e r e n c e between t h e e l e v a t e d and  m e r c u r y l e v e l s , u n d e r n e a t h t h e e l e c t r o d e and  outside i t respectively. 4.  The d e f o r m a t i o n s  a r e s m a l l enough f o r s u r -  f a c e t e n s i o n t o be n e g l i g i b l e . . Conservation  o f mass  gives  (3.2)  30  Balance  of stresses V*  6*  From e q u a t i o n  (2.1),  the f r a c t i o n a l  frequency  s h i f t of  the microwave r e s o n a t o r i s  o  o o  (3.4)  where r , Q , z are c y l i n d r i c a l  polars with o r i g i n at the  c e n t r e o f t h e u n d i s t u r b e d f l u i d s u r f a c e , and t h e z a x i s vertically  upwards. Since B  n  and E  n  do n o t v a r y s i g n i f i c a n t l y  v a l u e s o f z ~ > ( h , d ) , we c a n i n t e r g a t e d i r e c t l y  over  over z to  obtain  A F -  Ah  + / - KcC c L  (3.5)  31  where X  a n d j*.  are constants  d e p e n d e n t on t h e g e o m e t r y  o f t h e m i c r o w a v e c a v i t y and f i e l d s . (3.5)  by u s i n g  Eliminating h  from  (3.2) gives  d=  (3.6)  /3AF  w h e r e y(3 i s t h e s e n s i t i v i t y S u b s t i t u t i n g from  constant. ( 3 . 2 ) i n ( 3 . 3 ) f o r h and re-  arranging  =  J)-  (3.7)  wi t h E l i m i n a t i n g d b e t w e e n ( 3 . 6 ) a n d ( 3 . 7 ) we  obtain  (3.8)  AF  is arbitrarily  d e f i n e d from our measurement  as t h e d e f l e c t i o n i n cm o f t h e m i c r o w a v e r e s o n a n t the o s c i l l o g r a m .  Absolute  values  of AF  technique p e a k on  were found  t h e c a l i b r a t e d w a v e m e t e r ( s e e F i g . 2) as AF  = 1.8 MHz p e r cm s h i f t on t h e o s c i l l o g r a m ,  using  32  From e q u a t i o n V /i/AF  ( 3 . 8 ) we s e e t h a t a p l o t o f  v s . A F s h o u l d be a s t r a i g h t l i n e .  Q  Such a p l o t  i s made i n F i g . 15 w h e r e t h e r e s u l t i n g s t r a i g h t l i n e that the assumptions justified  shows  made i n t h e d e r i v a t i o n o f ( 3 . 8 ) w e r e  i n t h e range  of fields  used.  For higher  values, the resulting points deviated considerably  field from  the s t r a i g h t l i n e , i n d i c a t i n g t h a t t h e e x c l u s i o n o f s u r f a c e t e n s i o n f r o m t h e s t r e s s e q u a t i o n i s no l o n g e r  valid.  From F i g . 15, t h e s l o p e o f t h e s t r a i g h t 3  is  n  5  -V  = 285 x 10  V/m *• , a n d t h e i n t e r c e p t on t h e A F  a x i s i s c * 0.14 m. p  line  U s i n g t h e s e , two i n d e p e n d e n t  constants  c a n be c a l c u l a t e d as /3  f  = 5.35 x 1 0 ~  3  ( s e e T a b l e 3.1 f o r values o f the parameters ) ^  and  The  = 5.31 x 1 0 "  3  v e r y g o o d a g r e e m e n t o f t h e s e two  estimates of the s e n s i t i v i t y  independent  f u r t h e r confirms the v a l i d i t y  o f t h e c a l c u l a t i o n s . T h u s a d e f l e c t i o n o f 1 cm on t h e oscillogram corresponds  t o a d e p r e s s i o n o f about  5.3 x 10'  33  at  the mercury s u r f a c e .  T h i s i s of the o r d e r  from the microwave resonator  3.2.1  ( s e e C u r z o n and  Dispersion  Relation for Surface  Conducting  F l u i d Under the A c t i o n  Electrostatic  (1968)).  Waves on a of a Normal  is applied  to the o s c i l l a t i n g s u r f a c e of a conducting f a c e waves d i s t o r t the f i e l d , Since  Pike  Field-Theory  When an e l e c t r o s t a t i c f i e l d  in i t .  expected  and  fluid,  consequently  the e l e c t r o s t a t i c f i e l d  normally the  the  energy  supplies energy  the waves, the d i s p e r s i o n i s a l s o a f u n c t i o n of the strength. shall and  employ a v a r i a t i o n a l technique  field  a p p l i e d to k i n e t i c  p o t e n t i a l energy changes a s s o c i a t e d with The  virtual  following  dis-  assumptions  made:  1. normally charged the  to  In o r d e r t o d e r i v e t h e d i s p e r s i o n r e l a t i o n we  placements of a moving system. are  sur-  A uniform  to the  undisturbed  to a p o t e n t i a l V  undisturbed  electrostatic field  surface,  0  fluid and  E  0  is applied  s u r f a c e by an  h e l d at a d i s t a n c e  as s h o w n i n F i g .  16.  electrode D above  34  The f l u i d  2. the  motion  tion,  is  so t h a t  it  remains  function  3. sidered velocity,  container  write  and  circula-  throughout. i n terms  where  linearized where  inviscid,  without  the v e l o c i t y  amplitude,  of a  V <f= 0.  waves  are  the  fluid  y_ i s  con-  neglected).  A last  with  rest  irrotational  t h e t e r m ,v.grady_,  c a n be  4.  from  y> a s v_= g r a d ^ ? ,  Small  (i.e.  incompressible,  assumed t o s t a r t  We c a n t h e n potential  is  rigid  assumption walls,  is  that  and t h a t  it  the f l u i d is  held  is  at  in a  ground  potenti al .  According displacements state  to  to Hamilton's  that  another  transfer  in a given  (TL  principle  a moving time  system  for  virtual  from  one  final  T,  - o  (3.9)  where  L=j '{Y-?)dt,  (3.40)  T  0 in the  which  K and P a r e t h e k i n e t i c  system  respectively.  and p o t e n t i a l  energy  of  35  Equation we  shall  use.  is  used,  with  face, The  the  xy  z  fluid  has  a finite  .  the  p  axis  is  variational  cartesian  on  the  vertically  the  upwards  H.  fluid  density,  performed  co-ordinate  undisturbed  depth  is  equation,  (see  system  fluid Fig.  which  sur-  16).  then  over  the  fluid  volume  Then  [ { IvSvelr]  T,  motion  the  handed  integration  c f K r J L - J  which  is  plane  the  where f  A right  and  If  V  (3.9)  for we  the  incompressible,  assumed becomes  theorem  (3.12)  f  irrotational ,  have  From G r e e n ' s  V  dt  linearized  36  .  where surface  is  the  free  with  the  rigid  UM (jX i s the  the  linearized  normal  (where from  J  its  since  on  (3.14)  any  contact  container  to  kinematic  V r f L L - V ?  surface,  *«*  ;  Z0  unit  fluid  tf*  the  ? V  contact  boundary  = *  (  surface).  condition  at  By z  the  equilibrium)  vertical we  displacement  '  1  v  = 0  (3-16)  of  the  fluid  JI.JJ  (3.i7)  0  upon  H  integration  ;  )  get  £k=p( ( *S£I1 f j ff s, This,  5  using  r i f - r i J L .  denotes  3  yields  J J L sicji.ois  (3.18)  37  The tional  energy  surface  potential P,  ,  energy  the  electrostatic  surface  define  the  P^  + P  the  energy  gravitaP^  and  the  .  + P  z  surface  of  tension  energy  P = P,  We f i r s t  consists  wave  (3.19)  3  displacement  as  4 tot  where  is  C|  of  the  x  and y  the  wave,  wave  and  such  F (x,y) 0  that  Jx in  which  k is  of  rigidity  it  + 1  the  and  g the  satisfies  u  }y-  angular  of  the  Laplace's  •• -fr- R * i H>  defined  impermeability F  the  a function  eigenvalue  ^° Then w i t h  amplitude,^  at  the  -  space  variables  equation  (3.20)  u  by  the  conditions  container  — 0  gravitational  frequency  walls  (3.21)  acceleration,  38  P, _= f 7  1  (3.22)  gi ves  T  "71 (3.23)  '  $  From e q u a t i o n  $  0  (3.20), i f T i s the s u r f a c e t e n s i o n of the  fluid,  f  s  o  (3.24)  So  JC  Thus,  (3.25)  Fi n a l l y ,  39  if  cy~  i s the s u r f a c e  charge density  b a n c e e l e c t r o s t a t i c p o t e n t i a l (due w o r k i n g i n t h e MKSQ s y s t e m  ^±.U\E. 4J s  s  (  f  -  ( 3 . 2 6 )  f  ^  equation  and  surface; n e  x  the  P  r  e  s  s  i  o  ^  n  o r  boundary  is  where A i s a c o n s t a n t .  A c a n be d e t e r m i n e d f r o m  the  condition  V(  at  a  s  f i e l d at the  that s a t i s f i e s Laplace's  boundary  wave),  = - ^ ^ J s  E = - g r a d y = ~ ~J~^—~~~^~z~ * conditions  to the s u r f a c e c  where E i s the p e r t u r b a t i o n  Y  the d i s t u r -  o f u n i t s we h a v e ^ e r - E . £ )  Z-kj^-Y.Jj  V  and  the grounded f l u i d  i n a Taylor  series  T)  +  y  surface,  = 0  Expanding to f i r s t  (3.28)  order  40  vm  =v(o)*rl^l  :  o j  i.e.  V(O) = - T E  0  w h i c h i n ( 3 . 2 2 ) g i v e s , a t z = 0;  A -y^Eo  S u b s t i t u t i n g from we  COi*ck(Kt>)  (3.27) and (3.29) i n e q u a t i o n  (3.26)  have  P, = - - £ s L  and  (3.29)  f KE C0tU(KD)  T oU  2 o  (3.30)  %  consequently  <fP --NICE'S 3  cotUktij  S u b s t i t u t i n g - f o r K, P, , P ^ , P  J rSTJsofi  ?  from (3.18),  (3.23), (3.25) and (3.31) i n the v a r i a t i o n a l  (3.31)  (3.19), equation  ( 3 . 9 ) we g e t  ' j^^pUTK T'-iofiCcoiL(KP)TjiT^dit ?  r  =o  (3.32)  41  S i n c e t h e v a r i a t i o n i s a r b i t r a r y , the i n t e g r a n d can be i d e n t i c a l l y s e t equal t o zero, i . e .  pi£ + eqX + T^T-M  E^CotU (KD)T  A g a i n , an e x p r e s s i o n f o r Cf t h a t s a t i s f i e s equation  = 0  (3.33)  Laplace's  and the boundary c o n d i t i o n s i s  Bcoslo k (? + H) Kfay)  y -  Using this  and t h e e x p r e s s i o n  for 7  (3.34)  i n the kinematic  b o u n d a r y c o n d i t i o n ( 3 . 1 6 ) we o b t a i n  B  -  l  U  /  J  ;  J ^ ~ ~ ^ 1 L  >  Substituting in equation  (3.33)  coiU(Url)  r e s u l t s , upon  (3.35)  simplifica-  t i o n and rearrangement, i n  X f  K  £  0  E?cotk(Kt»  (3.36)  42  This i s the f i n a l was  first  v e r s i o n of the d i s p e r s i o n r e l a t i o n .  d e r i v e d by L a r m o r ( 1 8 9 0 ) f o r an i n f i n i t e l y  f l u i d , and more r e c e n t l y by M e l c h e r ( 1 9 6 3 ) , McEwan ( 1 9 6 5 ) a n d M i c h a e l  <<t/k and t h e f r e q u e n c y  waves, u n t i l , f o r high  enough v a l u e s  r i g h t hand s i d e o f e q u a t i o n gives  imaginary  values  deep  and  (1968).  We s e e t h a t t h e a p p l i c a t i o n o f E phase v e l o c i t y  Taylor  It  f =  reduces the  G  of the  CA//2TT  of the f i e l d ,  the  (3.36) becomes n e g a t i v e .  t o t h e p h a s e v e l o c i t y , a n d we  This now  h a v e u n s t a b 1e w a v e s . In p r a c t i c e i t i s v i r t u a l l y conditions  of complete uniformity  to have  of the e l e c t r o s t a t i c  f i e l d , so t h a t the d i s p e r s i o n e q u a t i o n be m o d i f i e d .  impossible  (3.36) w i l l  have to  The m o d i f i c a t i o n w i 1 1 d e p e n d on t h e p a r t i c u l a r  e l e c t r o d e g e o m e t r y , and i n t h e n e x t  s e c t i o n we  t h e c h a n g e s b r o u g h t a b o u t by t h e e x p e r i m e n t a l  derive conditions  in our arrangement.  3.2.2  Dispersion  Relation with  In g e n e r a l , f o r the case  a S p a t i a l l y Non-Uniform  i n order to modify equation  o f a n o n - u n i f o r m a p p l i e d f i e l d , we  (3.36) simply  m u l t i p l y the e l e c t r o s t a t i c f i e l d term i n the equation constant  defined  by  Field  by a  43  (3.37)  in  which  static and d a  (? ) 3  and  0  energy  produced  a non-uniform for  our  field  is  R^(R <R,) a  placement given  applied  distant  the  perturbations  the s u r f a c e  applied  field  of  by  D from  which  radius  a disc  waves  shaped  for  the  uniform  fluid  3 and  The  electro-  We c a l c u l a t e  electrode  the s u r f a c e .  the e q u i l i b r i u m  a  has t h e  (Fig.  Rj  in  for  respectively.  arrangement  container  from  are  |  by  particular  cylindrical  The  (||)  17). of  radius  vertical  fluid  in  dis-  surface  is  by  t  (where order with the  J}S  )  the z  is in  the  designated  at  the  (S,U)  The wave  dition  v^ = 0 at  fluid  the  vertically  below.  of  function  a cylindrical  origin  axis  Bessel  polar  upwards. mode;  the  k is  r  , where  = R,  fluid  The  first  single  v^  by is  kind  of  system  surface,  nomenclature  defined  Therefore,  the  co-ordinate  undisturbed  number  particles.  of  (3.38)  ( r , $ ,  and  mode w i l l is  explained  the boundary the  be  radial  con-  velocity  z)  44  ( which  0  )  —  gives  KR,  is  the  =  Uth  j '  (3.40)  positive  denotes  differentiation  respect  to  its Using  and  non-uniform  electric tials  (3.39)  fields  where,  root  of  the  of  J5 i)sv)  Bessel  ~ ®  '  function  with  refer  the  argument. subscripts  0 and  field  cases  (? ,  can  0  since  there  1 to  respectively,  be are  expressed no  free  in  to  the  perturbation  terms  charges  uniform  in  of the  potencavity,  (3.41)  and  The  the  disturbance  V*  V|/  latter  subject  to  electrostatic  = 0  (l  equation, the  i.e.  appropriate  potential  =  O  or  satisfies  1)  (3.42)  Laplace s,gives'\j> when ,  boundary  |  conditions  that  solved all  45  solid  metal  surfaces  equipotentials. (3.28),  At  in the  the  fluid  where  of  surface,  the  cavity  using  are  equations  (3.29)  r-TE;  y  interior  ((:  is  surface.  the  From  applied the  o  „  l)  electrostatic  definition  of  <=<  (3.43)  field  and  at  (3.41)  the we  fluid have  (3.44)  which  by  Green's  theorem,  (3.41)  and  (3.43)  becomes  (3.45)  1,  fy'&E.T^rJt, O  For to  a first  surface  by  •»  the  n •»  evaluation  approximation, the  non-uniform  of  that  I  0  the  applied  and  l  x  we  distortion field  E^  assume, of  the  leaves  fluid the  46  surface  mode e i g e n f u n c t i o n s  reduction  of  the  oscillation  Again, Laplace's  unaffected,  a solution  equation  and t h e  Eo  frequency for  except  the  f.  that  boundary  for  satisfies  conditions  is  n($.\J)Ts(^)cos(S8)si*Ufrlz-8»]  (3.46)  S im U (f where  E  0  =  V /D. 0  Substituting  I  0  =  for  and  JL ott(KW  evaluate  in  I  gives  Q  f  ViS^Vof  C  To  T  j/(K*Ore/r  R>  (3.47)  D 1  (  we  note  that  (3.48)  d where  F (r,9 ) 0  corporating  the  2  is  a function  appropriate  that  c a n be  boundary  evaluated  conditions  by  into  inthe  47  solution the  radial  E(r) so  of  is,  Laplace's field  to  equation.  distribution  a very  good  that  in  this  region  Jto  „  M  ^  77"  ' T T  For  R^ <  r <  E,  c a u s e d by  ing  quantity  R,  the for  E Then,  from  <  where  q  and q  A2 , A p p e n d i x uniform  for  r  that A) <  = 1)  (3.49)  ^  assume t h a t wave  the  fractional  is  less  field,  i.e.  than  the  change  in  correspond-  _  (3.50)  (3.45),  * (3.46),  — — ^  = kR,  however  Q  E,  equations  clear  (;. e. F (r)  a uniform  ^  0  Fig.  ^  surface  >  (see  is  approximation,  ° , we  It  = kR,^  (3.49)  and  (3.50),  (3.51)  48  Since E /E (  i s very small f o r  0  < r < R, ( s e e F i g . A 2 )  ( 3 . 5 1 ) c a n b e w r i t t e n as  ^  Mdl)  (3.52)  where  The i n t e g r a l s can be e v a l u a t e d a n a l y t i c a l l y by using equations  11.3.33  a n d 11.3.34, p. 484, a n d e q u a t i o n  9 . 1 . 2 8 , p . 361 o f A b r a m o w i t z expressions  andStegun (1965).  The  f o r M are:  For S odd:  M^J = -~ [ Jo* (l>+J,*«0] - <J J> (<\) I, (<{) (3.53)  and f o r S even  Mtyrifa^+J.V]-  ^  (1)  (3.54)  49  These hold f o r a l l S except the e x p r e s s i o n s  M(?)  actual  and  ?  values f  was  also evaluated  of E /E (  = 0 for  values  from  0  < r < R,  of E / E (  0  F i g . A2, .  instead of  Equation  the c h a r a c t e r i s t i c equation  to  we  this experimentally,  and  360/67 c o m p u t e r .  is s t i l l Our  field.  of the  proportional  a i m was  to check  to see what a d d i t i o n a l e f f e c t s might  be i n t r o d u c e d by t h e s t a t i c s u r f a c e d e f o r m a t i o n non-uniform  integration  (3.36) i s u n a l t e r e d , so  o f the e l e c t r i c f i e l d .  the  used,  see t h a t the form  o f t h e s u r f a c e mode f r e q u e n c y  the square  (3.56)  assuming  ( 3 . 5 1 ) was  were f e d i n t o a numerical  From t h i s e x a m i n a t i o n  {Sz\)  more e x a c t l y by u s i n g  p r o g r a m u s i n g t h e U n i v e r s i t y o f B . C . ' s IBM  square  cases  simply  po^C^) + ^ ( ^ y i l ^ ^ T j ^ )  r-1-  I  that E  are  S = 0 and S = 1, i n w h i c h  due  to  the  50  3.2.3  Experimental  Results  Reduction  Spatially  by  of  Surface  Mode  Frequency  Non-Uniform,  Electrostatic  Fields We s t a r t surface  modes  of  their  of  the  were  reduction  of  the  surface  predictions  excited  spatially setting  since see  in  their  to  fluid  mental  the  driving both  4.2).  1 (the depth  of  the  Section  time  factors For  all  was  parameters).  about In  by  a n d we  surface of  modes  mercury  either  in  equal modes  to  blowing  the  periodic  fluid  10 cm ( s e e  be  approximation)  instance,  3.1 the  for  By  mode  very  (about  could  Table  using  fields.  were  high  tanh(kH)  or  natural  excited  quite  were the  electric  first  compare  3.2.2.  periodic  deep  the  applied,  experiments),  were  results  Modes  pure  waves  infinitely  by  (earlier  frequency the  measurement  present  fields,  surface  surface  individual  accurate  T h e n we  of Surface  cases,  quality  an  how  mode f r e q u e n c y  resonator  non-uniform,  Section  equal the  onto  the  frequency  on  microwave  pulses  from  experiments,  individually  cylindrical air  these  showing  electrostatic  Identification In  by  frequencies.  non-uniform  with  section  identified  oscillation  spatially these  this  pure  150  -  set since experi-  electrostatic  51  Table 3 . 1  P A R A M E T E R S  USED  I N  T H E  E X P E R I M E N T S  R.J = R a d i u s o f m i c r o w a v e r e s o n a t o r  R  2  = 3 . 6 4 cm  r a d i u s = 2 . 0 cm  = Electrode Electrode  thickness  = 0 . 5 8 cm  o f mercury = 4 7 0 dyne.cm"  T  = Surface  tension  H  = D e p t h o f m e r c u r y = 8 t o 1 2 cm  p  = Density  g  = A c c e l e r a t i o n o f g r a v i t y = 9 8 1 crn.s"  o f m e r c u r y = 1 3 . 6 g.cm~*  EM mode r e s o n a n t  y  =  ( R , / R  A  f  )  %  -  1  =  frequency  2  = 8 7 1 8 MHz  2 . 3 1  £o= P e r m i t t i v i t y of f r e e space = 8 . 8 5 x 10  F.m  52  field sion  was  set  equal  to  zero  (i.e.  V  = 0),  0  and  the  disper-  r e l a t i on  cv*  5-  =  I *  H +  3  f was  solved  of  c*j ,  in  an  by  using  of  Section  were  k,  having  the  appropriate  accuracy in  for  2.5.1.  lated  Values  =KRI  values  produced  surface  \'  and S t e g u n  to  get  the of  measured  iteration  program.*/was time  value  technique  measured  marker  g and T  Table  3.2  known  were  against  taken  detergent Minute  and  20% ( s e e  identified, k satisfy  modes  of  to  of  to  scheme  given  small  T  U.  The  here.  slight  with  in  an  described  Table  3.1  used  dispersion  investigated  and  surface  since  to  value  clean tend  no  the  to  T  from it  is  special from  cavity  of  periodically as  S and  much U were  computed  exactly.  given  Table  the  not  precautions  T by  after  re-  was  traces  relation  experimentally  tabu-  partly  tension  made t h e  in  with  by  Abramowitz  reduce  Therefore, to  identified  (3.40))  contamination  (1949)).  the  be  table  used, The  adjustments  to  then  equation  contaminants  Burdon  the  could  411 was  start  solvents  traces  value  S and p.  as  modes (see  (1965),  accurately  the  using  the  used.  comparing  of  Newton-Raphson  computer  0 . 2 % by  The  as  substituted  This  3.1.  change  in  values set  For k for  the the  53  T a b l e 3.2  ZEROS  OF D E R I V A T I V E S OF THE  OF BESS-EL F U N C T I O N S  F I R S T KIND  u  J  4,«  1  0.00  1 .84  3.05  4.20  5.32  2  3.83  5.33  6.71  8.02  9 .28  3  7.02  8.54  9.97  11 .35  12.68  4  10.17  11.71  13.17  14.59  15.96  5  13.32  14.86  16.35  17.79  19 .20  6  16.47  18.02  19.51  20.97  22.40  1  54  largest to  reasonable  produce  change  ambiguous  Decrease  with the  (S,U)  of Surface  was  the  voltage  the  electrode  measured of  the  the  to  cavity  an  of  (about  the  equation  of  surface  theory  (3.45)).  measured f  first  vs.  V„  to  a disc  to  the  of  of The  an for  by  all  about  surface  the  the  which  exceeds section  ranges  0  the  3.2.2  of  f  of  kept  Fig.  be  below  mm, i n  the  of  read  employed.  order  surface  it  to  below modes  18 s h o w s  employed;  the  distortion  (assumptions  0.2%.  modes  0.1  rim  the  equilibrium  about  frequency  was  depths  upper  could  0  of  was  the  With V  the  altering  2 mm, a n d  from  V /D  in  distance  measuring  used,  cases  by  The  microscope.  supply  Application  three  varied  electrode.  instance,  three  modes.  electrode  a n d was  50 V i n  accuracy the  shaped  was  enough  experimentally,  In  0 . 0 5 mm b y  power  at  the  large  Field  (0,3).  surface  about  25 kV/cm)  mercury  satisfy  the  voltage  the  and  a travelling  of  of  1%, not  investigated  and m e r c u r y  using  about  Mode Frequency  3 and 4 ,  accuracy  accuracy  the  was  Figs.  from  high  by  applied  Q  an  In limit  of  electrode  Sorensen to  V  (2,2)  applied  configuration  is  Electrostatic  modes w e r e  = (4,2),  field  T  identification  of a Normal Three  in  plots  verifies  s  55  the c o n c l u s i o n s f  that  of Sections  3.2.1 a n d 3 . 2 . 2 , s i n c e i t shows  i s p r o p o r t i o n a l to the square  electrostatic field p r e d i c t e d from  of the applied  (proportional to V  equation  ) , e x a c t l y as  Q  (3.36)  lhTT  x  modified  by e q u a t i o n  non-uniformity The  (3.51) which accounts  of the f i e l d . curved  p a r t o f g r a p h 18b c o r r e s p o n d s  higher e l e c t r o s t a t i c f i e l d strengths for  the r e s t of the graphs.  rate of reduction of f  the theory  a  than  tends  the values  used  to accelerate the  , hence i t w i l l  f o r smaller field o f S e c t i o n 3.2.2.  values  lead to a surface  than  expected  from  H o w e v e r , p u r e modes w e r e  n o t r e a d i l y e x c i t a b l e on t h e s u r f a c e f o r f i e l d g r e a t e r than  to  T h i s g r a p h shows t h a t t h e  stationary surface deformation  instability  f o r the spatial  values  a b o u t 50 t o 55 kV/cm, p r e s u m a b l y due t o t h e  large surface deformation characteristics  at these  higher fields.  i n t h i s r e g i o n were n o t pursued  Wave any  f u r t h e r , s i n c e i t was n o t o u r p u r p o s e t o i n v e s t i g a t e i n detail  t h e e f f e c t o f s u r f a c e t e n s i o n on t h e s t a t i o n a r y  surface deformation, fluid  n o r t h e mode s t r u c t u r e on t h e d i s t o r t e d  s u r f a c e when t h e mode e i g e n f u n c t i o n s  are modified.  56  F o r t h e l i n e a r p a r t s o f t h e g r a p h s i n F i g . 18 the d i s p e r s i o n r e l a t i o n quoted above, m o d i f i e d f o r the field  n o n - u n i f o r m i t y p r e d i c t the  f  All  D  slope  a  t h e v a r i a b l e s i n t h i s e x p r e s s i o n a r e known, o r m e a s u r e -  a b l e d i r e c t l y , e x c e p t cC , s o we and  can use t h i s  expression  t h e s l o p e s e s t i m a t e d f r o m F i g . 18 t o g i v e v a l u e s f o r .  T h e s e a r e c o m p a r e d i n T a b l e 3.3 w i t h t h e v a l u e s  d i c t e d by e q u a t i o n s  ( 3 . 5 2 ) t o ( 3 . 5 4 ) , and e q u a t i o n  in the m o d i f i e d e l e c t r o s t a t i c f i e l d (3.2.2).  The  tions  (3.51)  theory of S e c t i o n  very good agreement between  r e s u l t s and t h e o r e t i c a l  pre-  experimental  p r e d i c t i o n s shows t h a t t h e  assump-  ( l i n e a r i z a t i o n , e i g e n f u n c t i o n i n v a r i a n c e , e t c . ) made  in the m o d i f i c a t i o n of the d i s p e r s i o n r e l a t i o n f o r nonuniform f i e l d s  3.3  valid.  An I n s t a b i l i t y E x p e r i m e n t t h a t d i d n o t One  ing  are  of our o r i g i n a l  Succeed  research goals, besides  t h e e f f e c t o f s t r o n g e l e c t r o s t a t i c f i e l d s on t h e  persion equation ensuing  ( 3 . 3 6 ) , was  to i n v e s t i g a t e f u l l y  the  i n s t a b i l i t y when t h e r i g h t h a n d s i d e o f t h e  studydis-  Table  3.3  RESULTS  Mode  (S,U)  D (cm)  k (cm"' )  a  a  a  kPq  ( A n a l y t i c) (eq. (3.54)  (Numeri c a l ) (eq. (3.51)  Observed  (4,2)  0.285  2.55  9 .29  0.378  0 . 385  0.396  (0,3)  0.220  1.93  7.02  0.553  0.557  0 .576  (0,3)  0.166  1 .93  7.02  0.553  0.557  0.555  (2,2)  0.T70  1 .85  6.73  0.528  0.533  0.524  58  dispersion  equation  +  K  —  K  f  x  f  b e c o m e s n e g a t i v e ; i n p a r t i c u l a r , we w a n t e d t o m e a s u r e t h e growth r a t e o f t h i s i n s t a b i l i t y . expected  t o be f c p k l  v e l o c i t y CJ/H  , where c  p  This growth r a t e i s i s the imaginary  g i v e n by t h e d i s p e r s i o n e q u a t i o n  s t a b l e regime.  Having  ceeded  larger electric fields  to apply  phase  i n t h e un-  o b t a i n e d t h e a b o v e r e s u l t s , we  kV/cm) t o make t h e s u r f a c e  pro-  ( b e t w e e n 70 a n d 90  unstable.  A n o n - d e s t r u c t i v e t e c h n i q u e was d e v i s e d t o a v o i d damaging t h e h i g h l y p o l i s h e d e l e c t r o d e and a l s o the f o r m a t i o n  of oxides  An e l e c t r i c f i e l d  surface. with  a t t h e m e r c u r y s u r f a c e by s p a r k i n g .  o f large amplitude  short duration (about  prevent  ( a b o u t 80 kV/cm) a n d  10 ms) was a p p l i e d t o t h e m e r c u r y  T h e s u r f a c e was i n i t i a l l y d r i v e n t o o s c i l l a t e  a small amplitude.  The f i e l d  caused  the surface to  become u n s t a b l e , and p e r m i t t e d o b s e r v a t i o n o f t h e growth rate.  However, the s h o r t d u r a t i o n o f t h e f i e l d  t h e u n s t a b l e wave f r o m to  cause  cavity.  prevented  coming c l o s e enough t o t h e e l e c t r o d e  d i e l e c t r i c breakdown i n t h e SFg atmosphere o f t h e The e x p e r i m e n t a l  a r r a n g e m e n t was as f o l l o w s .  59  A second the r o t o r (see corresponding  spring loaded  F i g . 1 9 ) , and t o i t was  voltage generator  electric field  the  T h i s s e c t i o n was  (where s u p e r c r i t i c a l  on  a d d i t i o n a l brass  high enough to apply  cause i n s t a b i l i t y ) . c o u l d be u s e d  a small  mounted  f i x e d on t h e s t a t o r o f t h e  o f F i g . 6.  p o t e n t i a l w h i c h was  c o n t a c t was  section  high  h e l d at a  a supercritical  means s t r o n g e n o u g h  A f a s t , two-way r e l a y s w i t c h  to  (Fig.  to t r a n s f e r c o n t a c t to the e l e c t r o d e i n s i d e  c a v i t y from  the f i r s t to the second  When t h e s w i t c h was  a c t i v a t e d , and  rotor  contact.  a supercritical  voltage  a p p l i e d to the e l e c t r o d e , a m i c r o s w i t c h  automatically  the power to the r o t o r .  a p p l i c a t i o n of a  supercritical now  19)  voltage  Hence a second  i n the next  cause s p a r k i n g from  cut  r e v o l u t i o n , that would  the e l e c t r o d e to the s u r f a c e ,  was  avoided. Two F i g . 20d with  modes w e r e s t u d i e d , t h e ( 0 , 2 )  shows a s c h e m a t i c  and  the  (0,3).  diagram of a r e s u l t i n g f i l m ,  the s u r f a c e i n s t a b i l i t y  clearly depicted.  The  dots,  o c c u r r i n g at the t r i g g e r r a t e of the o s c i l l o s c o p e , t y p i c a l l y 0.5  ms,  served  as t i m e m a r k e r s i n t h i s c a s e .  t h e l o g a r i t h m o f t h e wave a m p l i t u d e  from  obtained  A typical  very good s t r a i g h t l i n e s .  i n F i g . 20b,  and  the e x p o n e n t i a l  is c l e a r l y demonstrated.  By p l o t t i n g  the f i l m ,  growth of the  we  p l o t i s shown instability  However, the r e s u l t i n g e - f o l d i n g  60  times  differed  operating 5 ms)  by  than  as  30% u n d e r  and  in  addition  expected  initiation  for  theoretically  with  subcritical  of  the  from  Section  instability  subcritical fields  especially  this  in  we w e r e  boundary  stage,  the  case  dealing  value  uniformities  with  problem,  played  of  occur  values.  was  and  even  Instability  an  initially  a dominant  Chapter  after  presentation  importance have  of  a time  spatial varying  electric  role  of of  showed  that,  non-uniform  a difficult  that  discussion  5,  thought  spatially  A full  fields  shorter  Furthermore,  to  for  identical  much  3.2.1.  even  a little  unstable.  the  were  seemed  field  occurred  apparently  surface. At  what  much  conditions,  the  static  as  in  this  is  initial field  driving  non-uniformities  and non-  the  deferred  some m o r e  component.  fields,  surface until  evidence  of  when  applied  the  Chapter  4  EXCITATION OF PURE SURFACE MODES BY SPATIALLY NON-UNIFORM TIME PERIODIC ELECTRIC FIELDS - SURFACE MODE DAMPING  tigations periodic  In  this  of  the  in  the  used to  previous  as  behave  damping of  the  viscous  cluded  in  viscosity large  fields  on  the  theory  chapter,  the (i.e.  the  electrostatic  theory)  at  with  of the  energy fluid  rate  the  is  surfaces,  61  surface  the  of  the  energy  energy in  is  being shown the  theory also  in-  dissipation the  examined. boundary  a strong  used  is  of  a  and  resonator  outline wave  in  one  which  also in  the  in  frequency)  occurs  as  investime  mercury  system,  surface  damping fields  an  of  throughout,  exactly  driven  The  used  Since  role, of  of  a microwave  tool.  a crucial  theory.  is  results  non-uniform,  surface  again  arrangement  dissipation  dissipation  is  a resonantly  plays  the  spatially  diagnostic  as  present  of  Linear  experimental  we  effects  electric  container. the  chapter  by  presence Since layer  of  viscous (see  electrostatic  field  62  might a f f e c t the p h y s i c a l the boundary  4.1.1  processes  i n v o l v e d by  modifying  layers.  Surface  Mode E x c i t a t i o n by a S p a t i a l l y N o n - U n i f o r m ,  Time V a r y i n g  E l e c t r i c F i e l d - Theory  When a p e r i o d i c e l e c t r i c s t r e s s i s a p p l i e d mally will  to a f r e e f l u i d  surface,  the r e s p o n s e of the  norsurface  r e s u l t i n t h e s e t t i n g up o f s u r f a c e w a v e s , i n a  s i m i l a r m a n n e r t o any  other  the d r i v e n  a d i s c r e t e s e t o f modes i n i t s f r e -  s y s t e m has  quency s p e c t r u m (as i n our fluid  in a container),  driven  case of standing  temporal  t h e d r i v i n g m e c h a n i s m can be  since  This  i t eliminates  theory  w a v e s on  a  of  used to e x c i t e pure, prede-  is a great  experimental  convenience, of  by v i r t u e o f t h e f a c t t h a t  of free o s c i l l a t i o n s gives  f u n c t i o n of the wavenumber  system  F o u r i e r components  the need of F o u r i e r a n a l y s i s  arbitrary disturbances,  If  the s e l e c t i v e response of the  t o p a r t i c u l a r s p a t i a l and  t e r m i n e d modes.  harmonic system.  the frequency  OJ  the as  a  k.  In t h i s s e c t i o n , s t a r t i n g by t h e i n c l u s i o n o f a time v a r y i n g a fluid  e l e c t r i c s t r e s s i n the equation  s u r f a c e , we  derive  the necessary  of motion  conditions  for  of the  63  d r i v i n g o f pure tainer.  s u r f a c e modes on a f l u i d s u r f a c e i n a c o n -  We s h a l l  u s e t h e same g e o m e t r y as t h e one e m p l o y e d  i n t h e p r e v i o u s s e c t i o n ( i . e . F i g s . 3, 1 7 ) , w i t h t i o n t h a t we a r e now d e a l i n g w i t h We a g a i n u s e l i n e a r The  a general  the excep-  e l e c t r o d e shape  theory.  vertical  displacement  J  o f an i n v i s c i d  fluid  s u r f a c e i n t h e a b s e n c e o f an e l e c t r i c f i e l d s a t i s f i e s t h e e q u a t i on  (4.1)  where a l l t h e symbols have been d e f i n e d i n S e c t i o n This i s simply together with  3.2.1.  the l i n e a r i z e d form o f B e r n o u l l i ' s e q u a t i o n , the c o n d i t i o n that the f l u i d  the s u r f a c e remain  p a r t i c l e s on  on t h e s u r f a c e t h r o u g h o u t .  I t expresses  the c o n s e r v a t i o n o f s t r e s s across the s u r f a c e . I n ' o r d e r t o k e e p t h e a l g e b r a s i m p l e we a s s u m e a t t h e o u t s e t t h a t we a r e d e a l i n g w i t h deep f l u i d  (i.e. tanh(kH)s  in the experiments).  an  shall infinitely  1, w h i c h was a c t u a l l y t h e c a s e  T h e n we c a n w r i t e  (J> as ( s e e e q u a t i o n  (3.34))  r . r » ( j > w : i f  V K ^ K ^ W ^  (4.2)  64  where  y  is  (S,U);  the  manner  as  mode  given  notation for  the  (equation  often  omit  fusion. at  the  we  get  in  general (S,U)  (S.U),  when  the  surface  is  For this  sum o f in  normal  exactly  does  give  henceforth  rise  to  any  k i n e m a t i c boundary  (3.16))  in  (4.1)  to  same  a normal  we s h a l l  not  modes  the  associated with  brevity  linearized  (equation  the  defined  displacement  (3.40)).  Using  as  con-  condition  eliminate  with  (4.3)  From t h e  orthogonality  give  dispersion  the  In will  still  except  in  boundary Parkinson  the  the  of  (1957)),  of  of  modes, (cf.  in  the  the  V  fluid, body  boundaries,  thickness where  the  last  equation  a viscous  everywhere  vicinity  layers  the  relation  case  vanish  of  two  (3.36)).  the of  vorticity  the  the  fluid,  specifically see  is  equations  kinematic  Case  the  and  viscosity  of  65  the f l u i d .  In o u r c a s e , t h e s e v i s c o u s e f f e c t s w i l l  duce a d d i t i o n a l motion  (4.1);  factor.  t e r m s o f o r d e r ^/R  t  intro-  into the equation of  k i s a l s o m o d i f i e d by a s i m i l a r  correction  These c o r r e c t i o n s a r i s e because the d i s p e r s i o n  equation will  i n v o l v e t h e non v a n i s h i n g c o m p o n e n t o f t h e  vorticity  \/~ .  H e n c e we c a n now w r i t e e q u a t i o n ( 4 . 3 )  in the form  _i L.£l'K'  ie -  xfff + T/f' ) T'  (4.5)  1  w h e r e t h e p r i m e s i n d i c a t e t h a t c o r r e c t i o n s o f o r d e r ^/R, ( i . e . a b o u t 1 0 ~ * * i n o u r s y s t e m ) m u s t be made t o t h e a p p r o priate  parameters. In o r d e r t o f i n d t h e e f f e c t o f a p e r i o d i c  electric  f i e l d o f f r e q u e n c y j f l on t h e s u r f a c e , we a d d an e x t r a s t r e s s term t o the e q u a t i o n o f motion, to o b t a i n  (4.6)  where E  0  i s a time i n d e p e n d e n t f i e l d , and E  0  a n d E, h a v e  66  t h e same s p a t i a l d e p e n d e n c e ( i ^ B ) .  The f o l l o w i n g assump-  t i o n s a r e made:  E, I E .  1}  as i s t h e a c t u a l c a s e of second  harmonic  6 1  <-> 4  a t hand, and t o reduce  caused  8  approximation,  by t h e s u r f a c e w a v e s  T h e l a s t c o n d i t i o n t h a t h a s t o be s a t i s i f e d  t h e e x c i t a t i o n o f p u r e modes i s t h a t t h e g r a d i e n t o f  the normal f i e l d s m a l l e r than  (where and  field  4  neglected.  3) for  ( - )  r < < D  the fact that, to a f i r s t  changes i n the e l e c t r i c c a n be  the p o s s i b i l i t y  generation.  2) f, » F o — These express  7  V  a n y w h e r e on t h e f l u i d s u r f a c e be much  E /D, i . e . t  i s the two-dimensional  i and j a r e u n i t v e c t o r s  operator  i n t h e r and  ~ + ~  ""^Q"^ *  9 direction  67  respectively). comparable not  The  to  E,/D  sufficiently  (transients) electric modes  and  for  set  is  up  the  for  large  on  the  is  near  that  electrode  travelling surface  These  if  when  mix w i t h  7£T,  ,  is  edges  disturbances a time the  periodic  basic  pure  them.  equation  driving  this  happens  applied.  contaminate Using  sion  (this  rounded),  are  field  reason  (4.7)  we  can  linearize  the  expres-  field  (4.10)  Expanding variables, JTL  we  E  in and  obtain  terms  equating from  of  orthonormal  terms  equation  v  with  (4.6)  modes  the by  the  same S a n d  space U,  and  o r t h o n o r m a l i z a t i on  No ti  in  (4.11)  - J l *  No  68  where  (4.12)  in equation  ( Cu^  u  (4.11)  c~  the damping f r e q u e n c y V|'  (o)  surface  and  su  (.fl)  i s c o m p l e x , and i t i n c l u d e s o f t h e w a v e s , i . e . CJ^ are r e s p e c t i v e l y the  d e f o r m a t i o n c a u s e d by E  t h e (S,U)  =  +C  a  J ^ l K (G  w h e r e G, and G ^ a r e i n t e g e r s both  ^^(o)  Cr^yTj  therefore  and that  e  (4.13)  a  c o n s t a n t s , then  b e z e r o u n l e s s S = G, a n d  have t o s a t i s f y the c o n d i t i o n  p» 0, s i n c e E i s a l w a y s We  (4.11)  i s chosen such  and C, and C  and ^ ^ ( X l ) w i l l  U = G ^ , C, a n d E  { >  static  .  from equations  t h a t i f the geometry o f E  i^ ^ '  , and t h e a m p l i t u d e o f  mode e x c i t e d a t f r e q u e n c y I t i s now o b v i o u s  (4.12)  0  +  that  real.  a r r i v e a t the important r e s u l t  by t a i l o r i n g t h e g e o m e t r y o f t h e d r i v i n g f i e l d ,  that,  selected,  69  pure  surface  ually.  The  when  the  i.e.  at  modes  can  amplitude  driving  of  preferentially the  frequency  resonance.  amplitude  be  Then,  excited  is  as  excited  wave  close  is  as  from equation  individ-  maximized  possible  (4.11),  to  the  (  *  J  S x )  wave  is  f  1  E: / liojyfi  A/  0  e  (4.14)  it) where  Q  driven phase and  is  s  mode.  In applied  to  analysis  between  wave  coefficients  I  r  to-—fe,» where  E^  i  s  the  the  =  I ( /  )  resonance, driving  experiments  a square  excitation  a square  r  at  (Q  there  force  of  is  the  a fr/2  (electric  stress)  wave.  our  the  of  factor  Furthermore,  driven  f\/o  1  quality  difference  the  priate  the  in'  wave E  0  in  and  .  F  .  amplitude  electrode.  time,  E,  we  can  voltage Then, get  was  by  the  Fourier  appro-  as  -  1  ' - rr of  wave  the  Ed  (4.15)  7= applied  square  wave  field.  70  4.1.2  Surface  Have  From t h e surface  wave  portional ing  field. we  to  tainer  boundary  depth,  damping  layers  then  of  is  the  of  frequency  general  formula  (Landau  of  E-j-is  E$ , E g the  and  viscous  the Eyvu  of  the  electric  theory  of  containers. by  energy  dissi-  f~ and  (y/cv) the  compared  oscillation, theory  and  the  can  be  calculated  ,/x  thick-  to conused,  using  the  (1959))  »  «1 +  where  linear  small  Lifshitz  damp-  discussion  amplitude  be  the  pro-  the  applied  thickness  both  cr c a n  and  Also  caused  a linearized  damping  .  the  the  wave  are  inversely  theoretical  layers  If  the  •  the  by  that  is  a modification  rectangular  wavelength  dimensions,  •  surface  and  surfaces.*  4 . 1 . 1 we s e e  mode  by  predictions  boundary  the  fluid  the  complete  the  Section  affected  viscous  fluid  and  to  of  Containers  frequency  wave  at  of  in  in  a driven  Surface in  the  be  cylindrical  pation  ness  might  order  in  of  damping  summarize  damping  the  the  processes In  now  results  amplitude  frequency  physical  Damping  total are  boundary  energy the  stored  rates  layers  on  of the  (4.16)  WL  in  the  energy fluid  waves,  and  dissipation surface,  the  in  71  container and  base  and  the  side  walls  cr^  u  are  the  corresponding  tively.  In  our  case  ness  surface  and  entering  the  the  wave  approximations  were  damping  ratio  of  amplitude  equations  was  truly  respectively;  about valid  boundary  any  other  10"^", (see  ,  frequencies  the  to  OJ  so  Case  respec-  layer  thick-  typical  length  the and  linear Parkinson  (1957)). Expressions containers drical  have  containers  expressions fluid  been  are  surface  by  is  results,  Pike  and  assuming  that  a thin,  immobile.  (4.16)  its  For results  publication  in  for  given  by  Hunt  (1952),  and  Case  and  with  of  the  (1968)  In and  replaces oj  s  in  film  the  cylin-  These  to  precise  the  for  that  order  modified  surface  rectangular  (1957).  assumption  theory  rigid, This  Parkinson  mobile.  this  counterpart  the  free  account  for  experimental  this  theory  renders  o~} t e r m  laterally  by  the  in  surface  equation  immobile  damping.  a cylindrical  (corrected  transcription  a n d E^t,  Curzon  laterally  model  , Eg  laterally  between  surface  Ej-  derived  discrepancies  by  for  from  for  typographical  Pike's  1968)  container,  are  Ph.D.  Pike  errors  dissertation  and  Curzon's  during (1967)  the to  the  72  The  o n l y e f f e c t o f an e l e c t r o s t a t i c f i e l d  reduce the frequency e f f e c t on  Cv»  su  o f t h e w a v e s , so t h a t t h e  ('Vrv)  be t r u e as l o n g as t h e i m m o b i l e s u r f a c e f i l m i s  n o t d i s r u p t e d by t h e e l e c t r o s t a t i c f i e l d , and the d i s t o r t i o n  of the f l u i d  as l o n g  dissipation.  For a r e c t a n g u l a r c o n t a i n e r , equation becomes ( i n the l a t e r a l l y immobile s u r f a c e  a £  -  E  +  +E  B  + Eir  as  s u r f a c e by t h e s p a t i a l l y n o n -  u n i f o r m f i e l d does not i n t r o d u c e e x t r a  -  only  w o u l d be a r e d u c t i o n , p r o p o r t i o n a l t o  This will  CT  i s to  (4.16)  theory)  ^  (4.18)  T  where E  w  and  Ep-  are the r a t e s of energy d i s s i p a t i o n  s i d e w a l l and e n d w a l l o f t h e c o n t a i n e r .  The  total  energy  i n t h e w a v e s e q u a l s t h e maximum k i n e t i c e n e r g y o f t h e i.e.  at each  fluid,  73  (4.19)  in the usual n o t a t i o n .  By u s i n g G r e e n ' s  theorem.  (4.20)  A s s u m i n g t h e waves a r e p r o p a g a t i n g a l o n g the l e n g t h o f t h e c o n t a i n e r ( s e e F i g . 9 f o r geometry) whose w i d t h i s b, t h e e n e r g y c a n be c a l c u l a t e d  from a v e l o c i t y p o t e n t i a l  of the  form  Cf =  Cj3  COf  Ii (Kz) C Of  [Ujt)  rof  (Hie)  (4.21)  to give ( s e t t i n g t = 0 to maximize the k i n e t i c energy)  (4.22) -  ?  <o  %  b cosk(kH)  S^liKH).  where L i s the l e n g t h o f the tank. the  c a s e , a n d s i n c e no g e n e r a l i t y  negligible  (HL)  If L>^b  ( a s was  i s l o s t ) then  (see Appendix E f o r j u s t i f i c a t i o n ) .  is Hunt's  »  expressions f o r the d i s s i p a t i o n  terms  IP  actually  EI^J(A^ a n d  *  ItlE^/w  74  are  derived  velocity  by  potential  cp =  In  <?  our case,  Hunt's  assuming of  in  the f i n a l  equation  (19))  (Hunt,  equation  (13))  from  E  Pike  I ;  Substituting  Kx)  cosk  of  expressions.  and Curzon  described  by a  (hrzj  the form  are a p p l i c a b l e  (Hunt,  And  -  wave  form  a potential  calculations  by  the  c*s (cut  with  a travelling  (4.23)  of  provided  equation we r e p l a c e  The r e s u l t s  H)  0ih  (JL  ) ' \ y L b (KO> )  2  —  (4.24)  -  (1968)  r Eg co/^UW)  from  <f  are  — { ^ ) Ojl\(<f^ i U(H  Ep - 1  (4.21),  equations  (4.22),  (4.25)  (4.24)  and ( 4 . 2 5 )  for  75  E  T  ,E  , £g  w  final  E,is  and  expressions  w  —  for  in GT  equation , <TL  and  (4.18)  we o b t a i n  the  (TT XX  Cost»*(kH)n'hU{KM (4.26)  From in  cylindrical  the  expressions  for  the  damping  and  rectangular  containers  frequencies  (equations  and  (4.26)  respectively)  the  corresponding  quality  for  driven  surface  can  be  from  modes  calculated  (4.17) factors  the  general  equation  (4.27)  We s h a l l mation the  give  ( k H »  shallow  container, periments We t h e n  here 1)  expressions  and w i t h  fluid since  (see  the  S = 0 for  approximation these  Chapter  were 7 for  the the  in  the  the  fluid  approxi-  c y l i n d r i c a l ^ and  (kH « 1 ) cases  deep  for  the  investigated  rectangular  in  rectangular in  ex-  container).  have:  (4.28)  76  (for  a cylindrical  container)  (for  a rectangular  container).  4.2  Experimental In  results modes  of  section  we  shall  investigations  on  the  surface. strong  to  the  is  also  of  30 k V / c m  order on  hexafluoride  the  experimental  time  of  investigation the  method  modes of  the  damping  surface  periodic  this  surface  of  of  on  of  the  effect surface  fluid of modes  described.  Excitation  to  study  was at  by  Electric  surface  resonator  pure  on  Theory  characteristics  non-uniform,  fields  Dependent  In  microwave  Mode  with  present  application  generation  Surface  sulfur  and  electrostatic  Time  than  spatially  An e x p e r i m e n t a l  mercury  4.2.1  by  fields,  excitation  of  this  driven  electric  Results-Comparison  made  Fields  the  wave  Spatially -  effect  Experimental of  parameters gas  tight,  atmospheric  Non-Uniform,  fields in  and  pressure,  Results  stronger  mercury, filled and  the with  the  air  hole  77  in  the  electrode  technique  for  which  to  ing,  led  eliminated  surface the  spatially  wave  as  shown  in  excitation  investigation  non-uniform  of  was  the  electric  Fig.  4.  A  then  effect  fields  developed,  of  on  new  time  the  vary-  mercury  surface. The by  tailoring  applied the  to  the  the  the  of  surface,  conveniently  wave  excitation voltage,  the  stress  The  voltage  the  high  Section  at  shift  was  frequency  to  the  and (see  again  dynamic  that of  of  surface  low the  is  with  resonator  as  the  by  as  applying  generator  microwave  a surface  using could  this be  discussed  in  the  This on  the  wave to  is  in  maximize  antinodes. output  described by  resonator  the frequency  amplitude  excitation for  calibration Section  3.1.  of in  changing  calibrated The  at  possible  mode.  the  wave  wave  as  shape  done  The  mode;  (square  displacement  varied  stress  voltage  electrode  is  effected  maximized  close  a periodic  the  is  surface  particular  disturbances. was  set  is  mode  6).  By  stress  modulating  frequency  employed  deformation  was  modes  electric  a desired  frequency  Fig.  surface  the  the  tailoring  monitor.  microwave  tionary  and  modulation  2.3,  speed  by  electrode  voltage,  rotor  the  done  surface  a periodic  field  frequency  most  time)  so  applied  pure  of  antinodes  the  resonant  of  geometry  displacement  frequency to  excitation  and  method, sensitivity for  sta-  78  Dr-tven  Surface  Before measurements modes, faces  "fresh" obtained the  of  an  by  the  the  the  A fresh  mercury  from  the  a hole  particles cavity,  the  diffusion  lasted  an  of  to  elastic  properties  possible  for  "old"  about  surface (see  indirectly  know w i t h  to  by  higher in  the  what  all  type  of  are  a  into  rush  in  the  fluid  was  back centre  flow  pushed  towards  the  clean  surface.  minutes.  After  that,  changed  the  the  surface the  surface  B for  a full  "old"  surface  than  visco-  discussion damping  the "fresh" s u r f ace  experiments  damping  sur-  a "fresh"  changing  The  of  surface  it  impurities  three  Appendix  essential,  rapid  producing  considerably  is  or  thus  other  implications).  is  it  and  The  contaminants  surface  frequency  resonator.  surface  resonator  from  the  with  the  mercury  rapidly  of  of  These  very  surface  to  surface.  respect.  deals  types  resonator  This  modes,  this  distinct  letting  the  directly  in  two  then  walls  so  that  frequency  3 ) , and  dust  one,  damping  which  Fig.  adsorbed  and  out  section,  (see  base  of  "old"  draining  reservoir  into  point  this  the  investigated  and  Characteristics  starting  involving  we m u s t were  Mode  frequency  surface  we  involving of are  either  surface dealing.  79  (a)  Quality For  electric voltage  steps the  by  using  amplitude  disc  driving motor  response  section  was  The  about  50 r e v o l u t i o n s  to  an  accuracy  in  Table  Q  *l  {  The  From of  1/4  was  altered  circuit  surface shift  on  of  Fig.  in 6,  fine and  the  oscillogram.  one  and  the  resonant  rotor  the  measured  the  (0,3)  frequency  of  baseline  to  the  driving  was  used  mode  was  with  in  found  a stop  Experimental  s.  21 i s  this  the  (0,3)  a plot  power mode  theory  of  gives  a driven,  of  as  in  the A  the  frequency the by  cylintiming  watch  parameters  ( ^ ^ v s .  resonance can  'I half-power bandwidth)  (4.27)  frequency  to  the  amplitude the  employed,  of  the  and  identical  driving  the  of  reading  are  given  4.1. Fig.  mode.  of  to  the  corresponded  cavity.  with  control  of  amplitude  2 kV  frequency  electrode  = 7.82 s"'  drical  the  equal  modes.  the  contant,  set  the  resonant  shaped  previous  set  wave  Then  of driven  experiment,  was  square  microwave  f  this  field  grounded.  factor  plot,  be  obtained  Q  jo ^  damped  _  f for the  the  quality  (0,3) factor  as  | a  harmonic  • ,  oscillator  (  4  <  3  (equation  0  )  80  Table  4.1  E X P E R I M E N T A L PARAMETERS  = Resonator  r a d i u s = 3.42 cm ( c o r r e c t e d f o r s u r f a c e  tension)  R  = E l e c t r o d e r a d i u s = 2.0 cm  H  = D e p t h o f m e r c u r y = 10 cm  D  = Distance  x  o f e l e c t r o d e from  EM mode r e s o n a n t  f  = Density  y = Kinematic  frequency  f l u i d s u r f a c e = 0.39 cm = 8 7 1 8 MHz  o f m e r c u r y = 13.6 g.cm"^  v i s c o s i t y o f m e r c u r y = 0.11 c P . c m  3  .g"'  81  ^03 ~  (4.31)  F o r t h e ( 0 , 3 ) mode, t h e v a l u e s o f t h e o s c i l l a t i o n 6^03  and damping f r e q u e n c y  <r  o3  face theory model, i . e .equation approximation)  <^o  which  give  = 48.98 s '  the t h e o r e t i c a l  Q  <T ^ 0  was.also  ment.  0 3  and  (4.28) i n t h e deep  fluid  < £ = 0.19 s " ' 3  value of Q  ,  as  = 129  s u r f a c e as  4.2.2)  = 0.195 s ~ ' .  two v a l u e s a r e s e e n t o be i n e x c e l l e n t a g r e e -  This lends experimental  the damped, d r i v e n , harmonic used.  sur-  measured e x p e r i m e n t a l l y (see S e c t i o n  f o r a " f r e s h " mercury The  (using the immobile  a r e c a l c u l a t e d t o be  -  3  frequency  support to the v a l i d i t y of  oscillator  m o d e l t h a t we  have  82  (b)  Phase at  between  driving  field  and  surface  mode  resonance.  A second  important c h a r a c t e r i s t i c o f thedriven  modes i s t h e p h a s e d i f f e r e n c e b e t w e e n t h e d r i v i n g f o r c e and t h e s u r f a c e mode d i s p l a c e m e n t .  Again from t h e d r i v e n  o s c i l l a t o r t h e o r y , t h i s p h a s e d i f f e r e n c e s h o u l d be rr/2 at resonance.  T h i s was v e r i f i e d f o r t h e ( 0 , 2 ) a n d ( 0 , 3 )  modes a s f o l l o w s . One  channel  o f o u r d u a l beam o s c i l l o s c o p e m o n i -  t o r e d t h e s u r f a c e wave i n t h e u s u a l m i c r o w a v e  resonant  f r e q u e n c y s h i f t m e t h o d , a s shown i n F i g . 2 2 . T h e s e c o n d channel monitored  t h e v o l t a g e V ( t ) o n t h e wave  e l e c t r o d e , r e d u c e d by a h i g h v o l t a g e p o t e n t i a l ( T e k t r o n i x P6013) which produces  22).  divider  a 1000:1 r e d u c t i o n . T h e  g r o u n d e d b a s e l i n e o f V ( t ) was b r o u g h t with the viewing s l i t  excitation  into coincidence  i n t h e o s c i l l o s c o p e screen (see F i g .  T h e o s c i l l o s c o p e was t r i g g e r e d e v e r y 5 ms.  a l s o shows s c h e m a t i c a l l y a r e s u l t i n g f i l m . sinusoidal  The dotted  t r a c e c o r r e s p o n d s t o t h e s u r f a c e mode, w i t h t h e  d o t r e p e t i t i o n r a t e e q u a l t o 200 H z . T h e c l o s e l y horizontal  F i g . 22  spaced  l i n e s on t h e f i l m c o r r e s p o n d t o t h e b a s e l i n e o f  the e x c i t a t i o n v o l t a g e , and t h e blank r e g i o n s between s e t s of these l i n e s corespond t u d e o f t h e s q u a r e wave.  to theduration of V  0  , the ampli-  From t h e f i l m we r e a d i l y  verified  83  t h a t t h e p h a s e d i f f e r e n c e b e t w e e n t h e two wave f o r m s was T T / 2 . Small  indeed  random d e v i a t i o n s  o f t h e phase  angle  f r o m r r / 2 a r e due t o t h e f a c t t h a t t h i s r e l a t i v e l y h i g h quality factor (see previous drifts  section)  mechanical  system  i n and o u t o f r e s o n a n c e , because our d r i v i n g  mechanism i s n o t s u f f i c i e n t l y s t a b l e . agreement between our simple mental  S o , a g a i n we h a v e  t h e o r e t i c a l model and e x p e r i -  results.  (c)  Dependence on  the  of  driving  A final  driven  surface  field  important  mode  amplitude  strength.  characteristic of driven,  damped  h a r m o n i c o s c i l l a t o r s i s t h e d e p e n d e n c e o f t h e a m p l i t u d e on the d r i v i n g f o r c e .  Again the harmonic o s c i l l a t o r theory  predicts that the amplitude should to the f o r c e . tude should electric  In o u r case,  be p r o p o r t i o n a l  tation  t h e d r i v e n s u r f a c e mode  pulses  to the square of the a p p l i e d t o t h e wave e x c i -  electrode. O n c e m o r e we u s e d t h e same d i s c s h a p e d  (see and  ampli-  to the square of the d r i v i n g  f i e l d , i.e. proportional  amplitude of the voltage  be d i r e c t l y p r o p o r t i o n a l  electrode  F i g . 3) a n d two modes w e r e i n v e s t i g a t e d ; t h e ( 0 , 2 ) t h e ( 0 , 3 ) mode.  T h e s u r f a c e mode a m p l i t u d e was m e a s u r e d  i n a r b i t r a r y u n i t s as t h e s h i f t o f t h e m i c r o w a v e r e s o n a n c e on  84  the  oscillogram.  small  reduction  D was  0.39  the  natural  in  of  the  surface  modes  on  the  surface  was  the  driving  out.  by  The  to  mercury  results  average by  ensure  the  higher of  static  was  plotting  a  oscillation field  reduction  in  conditions  again  the  fields,  electric  a slight  resonant  surface  of  At  frequency  compensated  frequency  A "fresh"  the  cm.  through-  used.  observed  amplitudes  o on  the  shown the  oscilloscope in  Fig.  23.  theoretical  perimental  (d)  of  the  In  this  model  we  are  see  in  each that  of  the  the  excellent  two  modes  predictions  agreement  of microwave  resonator  boundary  In  3 . 1 we d e m o n s t r a t e d  Section of  the  its  are  of  with  ex-  surface  section, surface the  using  wave  the  This  is  driver  using  an  the  the  results of  useful  electrode  sensitivity  stationary  static  defor-  distortion  electrostatic of  the  a driven  sensitivity a very  the  to  applied  amplitude  resonator  deformations. of  by  resonator  by  with  sensitivity  deformations.  microwave  boundaries,  mercury  calibrate  for  0  to dynamic  of  presence  Again  Calibration  observed  V  results.  calibration mations  vs.  to  inside  predicted m o d e , we  dynamic  option, the  field.  shall  surface  since  the  microwave  and  85  cavity  distorts  lations no  of  longer  the  of  wave  radius  The  the  field  Rj  is  D « R ^ »  normal  given  by  Curzon  and  theory  of  Section  a surface of  field  the over  E = E  wave  driven  amplitude  given  ( 0 , 2 ) mode was  since  fields  tically Stegun  calcu-  (1968)  are  integral by  using  the  by  E^ i n  calculated  a spatially  non-uniform,  a cylindrical  resonator  by  investigated  experimentally,  and  again,  electrode  radius,  we  can  approximate  the  surface  by  the  step  fluid  r <  A  = 0  the  Pike  that  (4.1.1),  R  R  in  a  < r <  (1965), t o g e t h e r  Appendi x  (4.32) can  11.3.20, p.  with  function  A)  R,  equation  equation  the  x  (see  Then  so  possible.  amplitude  of  electromagnetic  type  From  square  the  our  be  484 o f  equation  performed  analy-  Abramowitz  (3.55),  i.e.  and  86  H e n c e , by c a l c u l a t i n g \qj the frequency value  of V  t h i s method, and  , t h e p r e d i c t e d s e n s i t i v i t y c a n be  e  obtained  to v e r i f y the v a l i d i t y of the c a l i b r a -  t i o n , the experiment  was r e p e a t e d  t i o n s , and t h e a b s o l u t e measurement.  observing  4 F on t h e o s c i l l o g r a m f o r a c e r t a i n  shift  In o r d e r  under i d e n t i c a l  v a l u e o f \n'  was f o u n d  condiby d i r e c t  T h i s was d o n e f o r t h e ( 0 , 2 ) mode as f o l l o w s .  A vertical  n e e d l e was m o u n t e d on t h e c a r r i a g e  of a t r a v e l l i n g microscope introduced  with  ( s e e F i g . 2 4 ) . The n e e d l e  i n t o the microwave c a v i t y through  in the l i d of the cavity at a radial the r e s o n a t o r  axis.  D = 0.39 cm f r o m  a small  square  lowered  t h e m e r c u r y s u r f a c e , and u s i n g a " f r e s h "  to j u s t touch  the needle  The needle  was  the o s c i l l a t i n g surface.  ampli-  gradually The e x c i -  r e m o v e d , a n d a f t e r t h e s u r f a c e damped o u t  was l o w e r e d  equilibrium surface.  to r e e s t a b l i s h contact with the T h e e x a c t moment o f c o n t a c t was i n -  d i c a t e d by a lamp s h o w i n g c o m p l e t i o n o f F i g . 24.  from  With the e l e c t r o d e at a d i s t a n c e  wave v o l t a g e p u l s e s .  t a t i o n was t h e n  hole  d i s t a n c e o f 3 cm  s u r f a c e , a s u r f a c e wave was e x c i t e d by u s i n g 1.5 kV tude  was  of the simple  T h e d i s t a n c e b e t w e e n t h e two v e r t i c a l  of the microscope mode a t r = 3 cm.  carriage gives the amplitude  circuit positions  of the surface  T h i s d i s t a n c e was 0.2 ± 0.02 mm.  Therefore  87  0.2  ± 0 . 0 2 mm, w h i c h  V\' = 0 . 6 3 ± 0 . 0 6 mm a t The (equation  theoretically  (4.32))  with  the  This  demonstrates  tivity to  linearly  turbation  quency the  in  = 0 . 6 6 mm, i n amplitude,  that  assumptions  are  the  correct.  a resonator fields  dependent  (see  observed  the  above  sensitivity  for  (as on  Section  The  long  the  shift  We s h a l l of  the  wave  Spatially now  excitation  as  the  error in  bounds.  the  can  sensi-  be  resonator of  0)  agreement  geometries  amplitude  of  experiment  Excitation  Dependent,  method  =  of t ^ ^  full  made  (r  used of  the  response  a boundary  per-  2.1)).  constant  Selective  The  centre  within  arbitrary  the was  microwave  resonant  16 mm, w h i c h  gives  frefor  J3  B " O O Ut H> "~ ' ^  4.2.2  the  value  measured  calibration  electromagnetic is  was  directly  calibrate  predicted  gives  ^ mm s h i f t screen) mm  of  Pure  o  f  w  how,  electrode,  v  e  on  Surface  Non-Uniform  describe  a  by  pure  amplitude per oscilloscope  Modes  Electric tailoring surface  by  Time  Fields the  modes  shape of  88  controlled surface. the  amplitude Again  microwave  diagnostic. the  the  Square in  all  setting  up  order  surface  shock  was  again  quency  the  for  amplitude  effected  the  already  used  in  these  voltage  reduction  was  negligible.  was  employed  frequency.  to  symmetric resonant  generator. (Fig. the  25b) (1,U)  and and  respectively.  the (2,U) Full  transient  fre-  solving  the  dispersion  few  modes  (0,2)  using  butterfly in  results  are  a grounded  base-  are  25a)  static  the  degenerate  range  electrode  given  in  of  rotawith  our  electrode  (Fig.  range  Table  the  modes  semicircular  frequency  in  below.  (0,U) the  field  scheme  excited  (0,5).  shaped our  volts  further  beyond  For  hundred  average  which  (Fig. to  3.2.3.  identification  discussed  frequencies  modes  natural  and  mode  or modes  the  small  the  surface  finding  the  were  of  (a  by  25  avoid  experiments  is  by  Fig.  disturbances  with  a  modes.  to  by  electrode  Similarly,  rounded  generator,  modes  to  Section  f  scheme  applied  in  distinguish  This  U 2> 6 h a v e  in  in  and as  surface  shown  fluid  employed  discussed  A different  A circular tionally  mode  as  the  the  used,  were  identification  k,  of  was  pulses  carefully  The  a particular  fields  line),  waves.  F i g . 3 was  shapes  frequency  basically  of  equation  high  were  a conducting  shift  excite  the  on  of  voltage  to  of  edges  of  excited  frequency  wave  Electrodes and  be  arrangement  resonator  electrode  used,  can  25c),  were  4.2.  excited  89  Table  4.2  S U R F A C E MODES E X C I T E D WITH D I F F E R E N T  .  ELECTRODES  f (Hz)  k (cm"')  Mode (S,U)  El e c t r o d e  5.35 7.82 9.63 12.18*  1.11 2.06 2.96 3.88  (0,2) (0,3) (0,4) (0,5)  F i g . 25a  3.44 6.46 8.70  0.507 1 .55 2. 50  (1,1) (1 ,2) (1 ,3)  F i g . 25b  4.76 7.44 9.50  0.89 1 .96 2.91  (2,1) (2,2) (2,3)  F i g . 25c  6.46  1 .55  (4,1)  F i g . 25d  T h i s mode e x c i t e d w i t h e i g h t p l a t e s on t h e s t a t o r (see F i g . 8 ) .  90  These the  shape  mode  of  Section the  of  the  4.2.1  could  (c))  lay  in  its  ability  excite  kR,  modes  in 3.2.  the  to  (1,2)  (4,1)  screws  mounted  in  used. both  the  Pure  cases.  typically their  on  in  Since  the  excitation  fact  that  our  range  high  level,  required  This  is  following.  the  The cylindrical on  the  largest of  azimuthal  electrode near  as  by  method type  range  of  were the  of  identification  of  on  the  the  brass  Fig.  modes  to  are (a)),  with  such  modes  through  a  technique.  wavenumber  modes  25d  excited  coupled  precise  surface  to  used.  4.2.1  not  of  (4,1),  order  four in  Section This,  the  was  is  both  configuration  of  less  container  shown  factors  overlap.  depend  the  heads  200 ( s e e  damping f r e q u e n c i e s  container  follows.  frequency  the  the  cross  quality  a new  degenerate  as  same f r e q u e n c y  100 t o  timing  highly  In  the  bandwidths  excitation  5.32 r e s p e c t i v e l y .  is  see However,  and  a plexiglass of  -  0  excited. wave  our  are  V  (1,2)  consisting  modes  the  ones  desired  k differed  a semicircular  mode, which  an e l e c t r o d e  in  tailoring  any to  demonstrated  selected  5 . 3 3 and  mode,  this  wavenumbers  Convenient  equal  to  was  whose  500 w e r e  walls,  was  be  of  1 part  the  principle  worthiness  from Table  For  (proportional  the  This  suitably  electrode,  of  individually.  excite  in  by  test  Two  with  that  excitation  amplitude  technique  than  wave  showed  controlled  true  modes  results  in  a  k, the  and  91  i n t e g e r S (see S e c t i o n quencies  4.1.2).  Therefore,  the damping  o f t h e modes e x c i t e d a t t h e same r e s o n a n t  by e l e c t r o d e s  25b  and  recording  of the damping o s c i l l a t i o n  modes ( s e e  F i g . 26)  and  compared with  in Section  the damping f r e q u e n c i e s  The  2.5.1.  vs. time f o r each of  the  were f o u n d ,  the t h e o r e t i c a l v a l u e s , i n order  i d e n t i f y e a c h mode.  the  a b o u t 50 o s c i l l a t i o n s  as d i s c u s s e d  From t h e p l o t s o f 1og(amp 1 i t u d e ) two  frequency  25d w e r e m e a s u r e d by r e m o v i n g  e x c i t a t i o n m e c h a n i s m , and  fre-  to  r e s u l t s a r e shown i n T a b l e  4.3.  T a b l e 4.3 I D E N T I F I C A T I O N OF S U R F A C E MODES BY T H E I R DAMPING  FREQUENCIES  (s- ) 1  The  Immobi1e surface theory  Electrode  Mode (S,U)  Fig.  25b  (1,2)  0.146  0.143  Fig.  25d  (4,1)  0.195  0.198  damping frequency  theory  was  Experiment  c a l c u l a t e d from the immobile  model of the damping (see S e c t i o n  4 . 1 . 2 ) , and  the  surface close  92  agreement support Table  of  to  this  4.2)  are  separation timing  the  is  experimental theory.  also  pure  mode  that  the  i.e-.  amplitude an  voltage  face  two  between  insulator.  the  substantial  (2,3)  modes  frequency,  but  (see their  identification  and  can  be  as  that  they  have  an A p p l i e d  damping  investigate  an  by  <r  of is  applied and  exciting added  our  of  the  one  at  of  Frequency  Section  of  the  then  dielectric  propor-  generator.  them  insulators,  any  parameter,  pulse  waves  method  advantage  directly  controlled  as  different  theory  of  is  excite  be  excitation  the  wave  Electrostatic  frequency, this.  has  long  to  Damping  frequency  Hence  It  to  wave  capable  squared  used  happen  the  and  accurately  liquids,  of  this  driven  both  damping  ditions.  of  If  From  the  in  that  amplitude  Dependence on  seen  surface.  easily  it  the  and  enable  versatile  a fluid  Furthermore,  necessary  now  cheap, on  to  the  4.2.3  to  lends  method.  simple,  tional  (0,4)  degenerate  sufficient  We h a v e is  The  results  inter-  is  an  it  is  permittivities.  Surface  Modes  Field 4.1.2  a function  of  electrostatic experiments  it the  field  were  is  obvious  boundary might  undertaken  that con-  affect to  93  The they is  could  also  be  used  A typical  (0,2)  and  excited to  run  (0,3)  with  produce  of  square  wave  voltage  equal  having  the  amplitude  at  by  appropriate  (HV2)  of  mercury, damp The to  Fig. the  under  5.  the  quency, cr, log  as  shift  the as  surface  static  field,  surface shown vs.  The 27,  the  in  the  of  the  power  and  microwave  in  a loss  were  no  longer  results  of  plotting  together  equation  with (see  the  26)  by of  and  easily  done  and  excited  f,  on  allowed  and  plot.  the the  was  the to  damping  obtained  higher  from  Fields fields,  non-uniform surface  fre-  mode  a as  the electropurity,  possible. |cr| v s . t/T a r e  theoretical  Section  volts,  used  was  For  the  resonator  10.  Fig.  of  field.  Fig.  (T  field.  (HV1)  wave  which  electrostatic  frequency,  caused  was  the  a strong  (see  is  supplies  wave  employed.  resulted  This  mode  time  baseline  thousand  300 V.  stopped,  deformation  and m e a s u r e m e n t s  by  of  55 k V / c m w e r e  static  Fig.  was  several  a surface  influence  (amplitude)  high  of  the  since  electrode,  electrostatic  setting to  employed,  shaped  strong  about  After  rotor  frequency measure  setting  a disc  the  consisted  modes w e r e  4.1.2)  lines  shown  predicted  in  94  This  i s the expression  f o r t h e damping  a (0,U) mode i n t h e i n f i n i t e l y From t h e p l o t we in  cr d e p e n d s  through  experimental  on t h e a p p l i e d e l e c t r i c  o""J f o r u  approximation. theory i s  observations. field  solely  t h e change i n f , i t i s c l e a r t h a t t h e p h y s i c a l  mechanism  o f wave damping  of the e l e c t r o s t a t i c surface  deep f l u i d  see t h a t the immobile s u r f a c e  e x c e l l e n t agreement w i t h  Since  frequency  film  i s n o t a f f e c t e d by t h e p r e s e n c e  field,  that i s the l a t e r a l l y  i s not d i s r u p t e d  immobile  by t h e e l e c t r o s t a t i c s t r e s s .  Chapter  5  DISCUSSION OF THE IMPORTANCE OF SPATIALLY NON-UNIFORM ELECTRIC FIELDS  tigations 6 and  7,  cations In  Before  proceeding  of  distinct  we of  two shall  the  particular  spatially  5.1  to  is  attack  mathematical of  and  results  electric  of on  fields  frequencies,  results  situations  some d e t a i l  concentrate  damping  Importance It  tion  in  on  the  the the  of in  and  the  impli-  two  effects the  Chapters  basic  last  on  inves-  chapters.  of  excitation, stability  of  modes.  The  search  we s h a l l  report  non-linear  experimental  non-uniform  oscillation surface  discuss  to  the  of  Non-Uniform  a custom new  a necessity  problems  simplicity physical  and  with  using as  processes  95  Electric in  models  faithful involved.  Fields scientific that  as  re-  combine  possible  Sometimes  descripcomplete  96  satisfaction to  solve  of  with In  actions, and of  the  are  case  non-uniform  of  waves  and  used  has  work  and  the  research,  the  with  uniformity.  Two  and  an  fields  the  notable  on  the  stabilizing  excep-  who  used  of  studied  incompressible  for  assumption  stabilization  fields  inter-  theoretical  (1966),  ( 1 9 6 5 , 1 9 6 7 ) who  non-uniform  non-uniform  Warren  feedback  of  impossible  surface  both  out  Melcher  surface  problem  techniques.  carried  Crowley  spatially on  past  field  for  the  electrohydrodynamic  been  of  fields  instabilities,  of all  electric  the  renders  analytical  practically  complete  effect  latter  known  experimental,  tions  of  the  surface  the  .  excitation fluid  the  jet,  jet  by  feedback. In uniformity tainer  cannot  will  especially with  the  practice,  always near  container  frequency  electrostatic cation This  model  maybe  is is  of  complete  A finite  non-uniform  discontinuity  fluid  applied  at  the  field in  a  electric  common  confields,  interface  walls. seen  that  surface  fields  (Section  deformation (and  of  condition  satisfied.  have  the  We h a v e tion  be  the  is  3.2.2)  modes  well of  non-linear)  enough  reduction by  linear,  accurate to  effects  of  the  spatially  represented  the  sufficiently large  the  bring  by  non-uniform, a small  uniform  until in  (Section  oscilla-  the  field static  additional 3.2.3).  modifitheory. surface linear  Beyond  this  97  point,  the  context to  concept  used  pursue  so  this  of  far,  a single, breaks  problem  pure  down.  further,  surface  It  but  was  it  mode,  not  in  our  certainly  the  purpose merits  more  attention. Another spatially  important  non-uniform  electric  surface  unstable  much m o r e  linear,  modified  theory  buted  to  the  turbances stress  profile  bances  have  mode,  and  travel This  the time  the  account electric the  electric  energy  field  field.  uniform by  supplied  distortions  m e c h a n i s m we  field  This  the  by is  the attri-  travelling  These  than  the  unstable  dis-  in  the  distur-  normal when  surface  they  applying  electrode.  times  of  the  unstable  shorter  the  wavelength,  The  hence  observations  theoretically  the  predicted  edges.  e-folding  k,  drive  discontinuities  become  expected.  applied,  the  shorter  the  e-folding  (3.36)).  a perfectly  discussed  the  the  wavenumber  equation  periodic  assuming  why than  Similar time  under  to  3.3).  by  electrode  to  that  amplitude,  nature)  surface  region  smaller  larger (see  the  explains  are  the  than  Section  large  is  tend  a much s m a l l e r w a v e l e n g t h  the  also  waves  under  cause  to  of  a transient  fields  readily  (see  generation  (of  observation  assume  were  made f o r  the  An i n s t a b i l i t y alternating  Yih to due  (1968).  an o s c i 11 a t i n g to  the  unimportant  in  wave. our  of  a  mechanism  field This  case  has  been  takes  into  surface This  is  treatment  by  the  precisely of  the  98  analogous 6),  problem  and A p p e n d i x  culations This  lead  with  a spatially  C justifies  to  a Mathieu  electrohydrodynamic instability  fluid  when  ically  in  (1954)),  the and  to  stability in  century been  of  Benjamin  by  and  perfectly  Wolf  (1970)  to  a radial,  there  to  surface.  A surface  the  the  to  the  incompressible  and  periodUrsell of  an  applied  parameteric  is  no  the  which  had  been  19th  and  20th  It  has  recently the  gravitational was  in-  difficulty  dynamically  method  stabilize  used  field  by  Zrnic  R a y l e i g h - T a y 1 or  in-  feedback.  perfect  impossible  in  stabilize  diagram.  (1965)).  (1954)).  since  cal-  instability  instability,  Ursell  Yih's  oscillated  Benjamin  Reynolds  A similar  magnetic  the  is  an  gravitational  type  instability,  virtually  in  by  Ursell,  and  (1969)  -However,  develop  (see  experimentalists  uniform.  by  the  and  Benjamin  Hendricks  stability  of  Mathieu  several  Ray!eigh-Tay1 or is  case  (see  (Chapter  analogous on  container  stressed  field  the  (see  is  develops  direction  jet  In  by  used  fluid  the  that  field  instability  electrohydrodynamic  electric  observing  observed  the  equation  fluid  vertical  incompressible alternating  the  assumptions.  instability  gravitational surface  our  non-uniform  to  electric  field  uniformity  achieve  on  a bounded,  instability  is  then  fashion  discussed  in  free  much m o r e  Chapter  6 by  is fluid  likely  to  non-linear  99  driving ties,  of  the  rather  surface  than  From of  spatially  terion  for  predicted be  of  1  time  taken  field the  the  foregoing  non-uniform  by  the  uniform  instability for  surface  non-uniformities  electrode,  L is  (e.g.  radius  of  the  this the  of  condition  the  we  can  draw  to  hold  if  importance  a rough an  approximation  is  going  must  be  very  much s m a l l e r  transients to  travel  set to  up  the  e-folding  by  cri-  instability  the  V  L  )  in  the  to  time than  the  electric  unstable  wave  under  and  disturbances.  of  the  investigations  of  individually  The  the  non our  in  on  the  for  phase  attempt  Section  of  to  This  of observe  3.3.  Fields  spatially  (Section  exciting  surface.  velocity  Electric  effect  fields  arrangement  satisfaction  Periodic  electric  a fluid  is  of  waves  a technique on  experimental V£  failure  unstable  periodic  "  the  of Non-uniform^  development  modes  the  Yih.  that  a container)  rates  time  of  by  is  length  caused  Application  uniform,  discussed  This  (  a typical  Our  has  non-uniformi-  i.e.  transient  growth  as  field  discussion  field  «  where  electric  fields  that  observable.  an  the  parametrically  a condition  clearly  (AJ"  by  pure  4.2)  led  surface  method  has  nonto  100  several  advantages over conventional  methods.  Common m e t h o d s f o r e x c i t a t i o n o f s t a n d i n g waves i n c o n t a i n e r s t i o n of the f l u i d support.  surface  have been the p e r i o d i c v e r t i c a l  vibra-  c o n t a i n e r , or the r o c k i n g of the  These methods are f u l l y  container  d i s c u s s e d i n Abramson  (1966).  H o w e v e r , due  to the d i f f i c u l t y  inertial  d r i v i n g mechanism, l a r g e t r a n s i e n t s were s e t  when t h e d r i v e r was  switched  c a u t i o n s were taken  to prevent  tude  is difficult  modes c a n n o t reason  this.  t o c o n t r o l , and  expensive  the  pre-  A l s o t h e wave  finally,  higher  mode m i x i n g .  up  ampli-  order  The  last  i s the main cause f o r the f a c t t h a t p r a c t i c a l l y a l l w o r k on g r a v i t a t i o n a l s u r f a c e w a v e s on a  i n a c o n t a i n e r has  been performed  s t a n d i n g wave m o d e s . able without  mixing  The by P i k e and  on t h e f i r s t two  by u s i n g i n e r t i a l  C u r z o n ( 1 9 6 8 ) by b l o w i n g  the f r e e f l u i d  early experiments.  only f a i r l y  low  obtain-  eliminated  p e r i o d i c a i r pulses frequency  have a l s o used  I t g i v e s p u r e modes o f s m a l l  H o w e v e r , t h e mode a m p l i t u d e  three  d r i v i n g systems.  s u r f a c e at the resonant  T h i s i s a m e t h o d we  fluid  or  T h e s e modes a r e t h e o n l y o n e s  p r o b l e m o f l a r g e t r a n s i e n t s was  d e s i r e d mode.  and  o f f , unless  be o b t a i n e d w i t h o u t  experimental  onto  in decoupling  in  of a our  amplitude.  i s not e a s i l y c o n t r o l l a b l e ,  o r d e r modes c a n be  excited.  101  By fields,  we  using  have  spatially  been  able  non-uniform,  to  excite  surface  purity  and  controlled  amplitude  method  can  be  excite  surface  can  be  used  that  it  excites  between by  two  virtue  removes (since for  the  the  the  of  low  the  5.2  Damping  fluid  stressed  in  by  surprising, surface  ment w i t h  of  both  high  order.  the  The  interface  system,  modes,  of  modes  of  the  gives  and  it  results, results  the  feasibility  extremely  surface and  pure,  modes  space  on  even  of in  shallow  periodic  fields  surface.  damping  Measurements frequency  a cylindrical  that  the i.e.  it  a model  assuming  high  as  measurements  container,  fields  up  electric  surface.  as  single  analysis  are  time  the  fields  at  a sealed  demonstrate  factor  electrostatic in  in  surface  which  quality  viscosity,  properties  Even  modes  waves  of  electric  modes).  F we  Frequency  The mercury  normal  modes  irrespective  Fourier  single  applying  onto  of  Appendix  surface  by  of  of  exciting  of  fact  theory  In  fluids,  it  necessity  a spectrum  case  to  fluids,  of  the  used  periodic  had The  field  no  the  had  no  surface  surface  on  effect  the  were  to  have  rather on  the  physical  in  immobility  seemed  the  55 k V / c m w e r e  effect  results  lateral  55 k V / c m  to  with  on  close  of no  the  agreesurface  effect  in  102  removing giving model  the  results of  a fluid  fields  in  in  damping In  on  lateral  surface  agreement could  Appendix  surface, partial  be B we  and  immobility, with  clean,  no  measurements  mobile  surface  taken. discuss  possible  removal  the  but  of  the  effect  of  impurities  applications  of  electric  surface  contaminants.  Chapter 6  NON-LINEAR DRIVING OF SURFACE WAVES BY ELECTRIC FIELDS  In C h a p t e r 4 we c o n s i d e r e d s u r f a c e modes o n m e r c u r y  the e x c i t a t i o n o f pure  i na cylindrical  c o n t a i n e r by  using s p a t i a l l y non-uniform, time p e r i o d i c e l e c t r i c The s u r f a c e w a v e a m p l i t u d e ^ distance  fields.  was much s m a l l e r t h a n t h e  D o f t h e wave e x c i t a t i o n e l e c t r o d e f r o m t h e  librium fluid  surface.  Then  equi-  t h e s u r f a c e c o u l d be t r e a t e d  as a d a m p e d , h a r m o n i c o s c i l l a t o r , d r i v e n b y a c o n s t a n t amplitude  (V /D) 0  ( where V  0  i s the amplitude o f the  a p p l i e d t o the electrode) f o r c e .  voltage  The analogous problem f o r  a f l u i d j e t h a s been i n v e s t i g a t e d by C r o w l e y  (1965).  In t h i s c h a p t e r we i n v e s t i g a t e t h e d r i v i n g o f s u r f a c e w a v e s b y p e r i o d i c e l e c t r i c f i e l d s when t h e a m p l i t u d e Vji  o f t h e d r i v e n s u r f a c e mode b e c o m e s c o m p a r a b l e  This modifies  t o D.  t h e d r i v i n g f o r c e a m p l i t u d e t o (^/(D-^))  i . e . the f o r c e depends  n o n - l i n e a r l y on t h e s u r f a c e  ment T . T h e n o n - l i n e a r d r i v i n g l e a d s , f o r l a r g e  103  ,  displaceenough  104  amplitudes  o f t h e s u r f a c e d i s p l a c e m e n t , t o a new  type of  i n s t a b i l i t y , w h i c h i s n o t p r e d i c t a b l e by u n i f o r m theory  (see Yih (1968)).  i n t h e c a s e o f two  A s i m i l a r treatment  neighbouring drops.  field  c a n be  Ackerberg  (1969)  has a l r e a d y d i s c u s s e d n o n - l i n e a r e f f e c t s c o n c e r n i n g neighbouring drops  held at d i f f e r e n t  given  two  electrostatic  potenti a l s . We  first  g i v e t h e r e l e v a n t t h e o r y , and f o l l o w i t  up w i t h p r e s e n t a t i o n o f e x p e r i m e n t a l  r e s u l t s , and a f u l l  discussion.  6.1  T h e o r y o f N o n - L i n e a r D r i v i n g o f S u r f a c e Modes Electric We  by  Fields again t r e a t the s p e c i f i c problem i n v e s t i g a t e d  e x p e r i m e n t a l l y ; t h i s i s e f f e c t i v e l y an i n f i n i t e l y d e e p occupying the r e g i o n r < R ,  , z<0  c o - o r d i n a t e system (see F i g . 17).  of a c y l i n d r i c a l Axisymmetric  is  to the s u r f a c e .  The  vertical  °U  frequency  displacement of the  then  ( *),u  polar  surface  w a v e s a r e e x c i t e d by a p p l y i n g p e r i o d i c s t r e s s e s o f XL  fluid  J  fluid  105  where  all  the  symbols  sections,  and  n is  tion can  of be  normal now  have  been  an i n t e g e r .  stress  at  the  defined The  fluid  in  the  previous  condition surface  of  conserva-  (equation  (4.6))  written  (6.2)  where  E(r,_n.  surface.  ,  t,J  E(r,-fl  )  ,  is  t, J  the )  electric  can  be w r i t t e n  $(i)h(T)  where  (see  Fig.  f  (  r  )  g  (t)  These fied  to  the  as  Vo  (6.3)  %  r<  = 0  = i  Appendix  R*  (  i  ' " e  a  s  S  U  m  e  D  K  <  R  *>  <rtT/a  TT/1  = pa^',  assumptions in  applied  17)  = 0  h(D  field  for C.  < \ n.-t \ < ir  x/P)  the  ^  x  form  of  f  E(r,I^,  r  o  m  t,  e  q  u  )  a  t  i  o  are  n  ( - ^ 6  1  justi-  106  The tated then the  for resonantly replaced  resonant  mode  modes  harmonics surface  (Q  o y  To j u s t i f y  since  T  the displacement this  we n o t e  that  driven  if  of the applied  field,  i.e.  GV  could  be d r i v e n  o f SL .  However,  ensures  that  facili-  i n h(J") i s  i s resonantly  OJ )  i.e. i t  (6.2) is greatly  by s t r e s s e s  only  its  spectrum  one r e s o n a n t  i s not possible  the Uth frequency  oscillating  the frequency  to find  mode  of  at f o r the  is  an i n t e g e r  driven n > 1  that  ~  where  150  Q  o i  see Chapter  ,is  Then,  matches  i n the d r i v i n g  0  small  4)  the quality  Cu> y .  For  modes,  JofKr)  mode.  modes  a time;  such  driven  the frequency  Other  of equation  by  (frequency  equals  at  solution  even  amplitudes  the resonant  if  (6.5)  v  factor  o f a mode  non-linear  stresses  frequencies  f o r a l l modes  will  except  of ei genf requency are large,  result  in  mis-  negligibly  the resonant one.  mode  J l - (^Vljf' )r 1 1  \)nSl~% \ < J ^ L  i 0J  0iJ  <r~  ou  (6.6)  107  where  expressions  rived  in Section  for  0~"  4.1.2. equation  ( 6 . 2 ) by J ( k r ) ,  assumptions  leads  the f o l l o w i n g  above  for  v\  the  surface  are de-  0{J  Multiplying the  a n d Cf> ,  0(j  the damping frequency  the phase  to  and u s i n g  0  between  expression  the d r i v i n g  field  and  mode.  (6.7)  The  non-resonant  moved  by i n t e g r a t i n g  plying  by  period,  in  section. Abramowitz The  IT  terms,  final  sineut  and t i m e  over  the f l u i d  and coso^ t  exactly  and S t e g u n  0,<r 7 <os<f z»  over  are then  surface,  and i n t e g r a t i n g  t h e same m a n n e r  The i n t e g r a l  results  dependence  as i n  the surface  (1965),  p.  and, over  the is  re-  multithe  wave  previous  given  484, equation  by 11.3.34.  are  • J,V,)] M  £  (  fl  2  J ' f c » ' ; » » ' * ^ 6 . 8 a )  108  J (z,)+J, 0  (6.8b)  TT  °  -77  where  I ^  f  l  / 0  U  Z,  ;  and t h e s u f f i c e s  = KR,;, Z ^ r : KK*,  0~^t  (6.8c)  on w a n d <r a r e now o m i t t e d f o r t h e s a k e  of bre vi ty . In o r d e r t o s i m p l i f y to a f i r s t (6.8a)  t h e a p p r o a c h , we n o t e  a p p r o x i m a t i o n , we c a n s e t ip-ft/i  and ( 6 . 8 b )  the i n t e g r a l  over  (see Appendix  C).  C  C  =  I(T)  k i <  -  i n equations  T h e n we c a n p e r f o r m  B i n (6.8b) to obtain  Y/I(T) ~ _ ^°  where  that,  f^*  Jo(z)Zofz  (6.9)  109  Equation V  0  (6.9) connects  , the amplitude  1(7)  the dimensionless  T  amplitude  with  of the voltage applied to the e l e c t r o d e .  c a n be e v a l u a t e d  of J  f o r various values  I n A p p e n d i x C we  by a n u m e r -  ical  i n t e g r a t i o n technique.  consider  from  t h e t h e o r e t i c a l p o i n t o f v i e w n o n - l i n e a r wave e x c i -  t a t i o n i n r e c t a n g u l a r c o n t a i n e r s , w h e r e i t i s shown t h a t to I ( T )  the i n t e g r a l corresponding  c a n be  performed  analyti cally.  6.2  Experimental  Resu1ts-Comparison with  We s h a l l of the previous  now a n a l y z e  the predictions of the theory  s e c t i o n , and compare them w i t h  r e s u l t s f o r a s p e c i f i c s u r f a c e mode. the c y l i n d r i c a l  Theory  experimental  T h e ( 0 , 2 ) mode i n  m i c r o w a v e r e s o n a t o r was i n v e s t i g a t e d e x -  perimentally.  I t was e x c i t e d by a p p l y i n g s q u a r e  voltage pulses  t o a d i s c s h a p e d e l e c t r o d e , e x a c t l y a s shown  in  F i g . 17.  dip  The s h i f t o f t h e m i c r o w a v e r e s o n a n t  on t h e o s c i l l o g r a m was a g a i n u s e d  amplitude  wave  on t h e m e r c u r y  frequency  as a m o n i t o r  o f wave  surface.  We f i r s t v e r i f i e d  that our assumption,  that the  phase d i f f e r e n c e between t h e d r i v i n g e l e c t r i c f i e l d and the  r e s o n a n t l y d r i v e n s u r f a c e wave was  f/2  i n t h e n o n - l i n e a r r e g i o n , was v a l i d .  approximately T h i s was d o n e  i n e x a c t l y t h e same m a n n e r as f o r t h e l i n e a r c a s e .  The  no  technique films the of  the  type  such  in  and  our  voltage  the  out  for  the  long  time  from  on  the  0.1. Fig.  IBM  of  integrals predicted  linear.  For  increase  in  was  )  .  time.  in  of  phases due  equation  the  case. due  the  Any  to  drift-  fact  that  enough  results  setting  dependence  to  a stable  These  fre-  confirmed  <f = rr/Z  in  evalu-  (6.8).  of  evaluated for  T  by  T  on  V  for  values  up  7  numerical  from  ( i . e . £ V<? / D )  that  higher 7  keep  latter  of  regime  was  0  ob-  C v?  l [ J)  We s e e  not  ratio  non-linear  the  was  the  in  electrode  deviation  random  This  the  the  in  is  both  (6.9)  .  7 /I ( J  the  assumption  360 c o m p u t e r  Then 28.  could  equation  integral  found  resonance.  our  J  The  was  period  of  The tained  of  in  of  Several  taken  T =Y\/D  measurable  m a s k e d by  generator  validity  ating  was  where  and  4.2.1.  22 w e r e  distance  surface)  f r o m rr/2  <f  Section  Fig.  0,  the  No s y s t e m a t i c  deviation  ing  to  in  in  (i.e.7—^  equilibrium  angle  quency  depicted  amplitude  -»-' 0 . 5 ) .  phase  described  regime  wave  the  fully  the  linear  from (J  of  is  of  Furthermore,  0.1  to  integration  0.9  3  was  plotted  to  0.3  the  we  J it  is  in  steps  vs . 7  dependence  observe obvious  a  of in is  non-linear from  the  graph  in  t h a t t h e r e a r e two T ^  0.7  t h e two  the c u r v e .  values of  J  f o r e a c h v a l u e o f Vo  A t any h i g h e r v o l t a g e s t h e r e a r e no r e a l s o l u unstable.  The  characteristic  f e a t u r e s o f g r a p h 28 a r e f u l l y d i s c u s s e d b e l o w . values to V  0  t h i s g r a p h we 0  At  v a l u e s come t o g e t h e r a t t h e maximum on  t i o n s , i . e . t h e w a v e s a r e now  V  .  and  f i n d i n g the c o r r e s p o n d i n g  amplitudes  p l o t t e d the t h e o r e t i c a l l y p r e d i c t e d J  o f F i g . 29.  The  ( s e e S e c t i o n 4.2.1  from vs.  c u r v e i n F i g . 29a c o r r e s p o n d s  " f r e s h , " and t h a t i n F i g . 29b  The  By g i v i n g  to a  t o an " o l d " m e r c u r y s u r f a c e  for definitions).  dependence of ?  t a l l y by m e a s u r i n g T  on V  a a  was  found  experimen-  i n a r b i t r a r y u n i t s of the s h i f t  AF  o f t h e m i c r o w a v e r e s o n a n t f r e q u e n c y on t h e o s c i l l o g r a m . T h e s e m e a s u r e m e n t s w e r e t h e n m u l t i p l i e d by a s c a l e f a c t o r , so. as t o b r i n g t h e l i n e a r p a r t o f t h e e x p e r i m e n t a l  curve  i n t o c o i n c i d e n c e with the t h e o r e t i c a l l y p r e d i c t e d l i n e F i g . 29) and  t h e n p l o t t e d i n F i g . 29 f o r d i r e c t  with t h e o r y i n the n o n - l i n e a r r e g i o n .  A g a i n two  (see  comparison sets of  r e s u l t s w e r e t a k e n , f o r a " f r e s h " and f o r an " o l d " s u r f a c e . Experimental  p a r a m e t e r s a r e g i v e n i n T a b l e 6.1.  varies i n v e r s e l y with  cr ( s e e e q u a t i o n  ( 6 . 9 ) ) , the  f o r t h e " o l d " s u r f a c e a r e b e l o w t h o s e f o r t h e new r a t i o o f t h e s e s l o p e s was frequency  Since  used to c a l c u l a t e the  amplitudes one.  as  <T  = 0.095 s " ' .  The  damping  f o r the " o l d " s u r f a c e ; f o r a f r e s h s u r f a c e  m e a s u r e d i n S e c t i o n 4.2.1  J  was  112  Table  6.1  E X P E R I M E N T A L PARAMETERS  p  = D e n s i t y o f m e r c u r y = 13.6 g.cm"  T  = Surface  g  = A c c e l e r a t i o n o f g r a v i t y = 981 cm.s  R,  = Radius o f microwave c a v i t y ( c o r r e c t e d f o r s u r f a c e t e n s i o n ) = 3.42 cm  R^  = E l e c t r o d e r a d i u s = 2.0 cm  D  = Distance o f e l e c t r o d e from e q u i l i b r i u m s u r f a c e = 0.172 cm  t e n s i o n o f m e r c u r y = 470 d y n e . c m ' -  °~  = D a m p i n g f r e q u e n c y o f ( 0 , 2 ) mode = 0.095 s - ' ( " f r e s h " s u r f a c e ) = 0.136 s - ' ( " o l d " s u r f a c e )  k  = W a v e n u m b e r o f ( 0 , 2 ) mode = 1.11 cm"  ^  = Angular frequency  Q  = Q u a l i t y f a c t o r o f ( 0 , 2 ) mode = 177 ( " f r e s h " s u r f a c e ) = 124 ( " o l d " s u r f a c e )  1  o f s u r f a c e mode = 3 3 . 6 2 s " ' .  k D = 0.19 C  = 3.07 x IO- " n T . V 15  =  3  2.14 x 10"' " m-2.V 5  7,  X  ("fresh" ("old"  mercury)  mercury)  113  S i n c e t h e v i s c o s i t y c o u l d n o t be c o m p l e t e l y trolled  and kept c o n s t a n t  i n e i t h e r s e t (a) or s e t (b) o f  the r e s u l t s , and f u r t h e r m o r e ,  s i n c e i t was n o t a l w a y s  s i b l e to p i n p o i n t the surface resonance o f 10 m e a s u r e m e n t s was t a k e n o f Fig. 29.  con-  These were then  exactly, a series  f o r each o f the data averaged  pos-  points  to give the p l o t t e d  poi n t s . From F i g . 29 we c a n s e e t h a t t h e p o i n t s agree  well with  experimental  the t h e o r e t i c a l l y predicted  values,  i n d i c a t i n g that our assumptions f o r the a n a l y t i c a l estimation of F T from  were w e l l founded.  Experimental  points  beyond  ~ 0.6 w e r e n o t r e a d i l y o b t a i n a b l e , s i n c e s p a r k i n g  occurred  t h e e l e c t r o d e t o t h e s u r f a c e when t h e d r i v i n g v o l t a g e  was i n c r e a s e d f r o m  t h e F ~ 0.6 v a l u e .  This i s a t t r i b u t e d  t o t h e s u r f a c e b e c o m i n g o v e r s t a b l e , and i s f u l l y  discussed  below.  6.3  Di s c u s s i on The most i n t e r e s t i n g f e a t u r e o f t h e r e s u l t s o f t h e  theory  and e x p e r i m e n t s  two v a l u e s  o f t h e wave a m p l i t u d e  given amplitude F i g . 28.  i n this chapter  V  Q  i s the existence of  ^ (or  J 7/D) =|  fora  o f t h e d r i v i n g v o l t a g e , as shown i n  Unfortunately  i t i s not easy  to observe  the solutions  114  for 7^ these  0.7.  We s h a l l  now show by an e n e r g y a r g u m e n t  solutions are unstable. S u p p o s e we s e t t h e s u r f a c e  amplitude  ^  ( p o i n t P,  , with  (  f  When t h e v o l t a g e  i s z e r o , and t h e s u r f a c e t e m p o r a r i l y  to D  correspond  , s u c h t h a t v£  %  t o t h e p o i n t P^  e r r o r h a s b e e n made i n V w h e r e Vy >  .  z  <  0.7  on t h e e l e c t r o d e  /D|  to  , and t h e  = Vj^/Df  (Fig. 28).  and  S u p p o s e now t h a t an  s o t h a t we a r r i v e i n s t e a d a t P  The q u e s t i o n  then  ,  3  a r i s e s whether or not F  the o s c i l l a t i o n s o f i n i t i a l amplitude steady  , and  f l a t , we c h a n g e t h e  o f t h e s q u a r e wave g e n e r a t o r  distance  with  t h e e l e c t r o d e a t a p o t e n t i a l V,  i n F i g . 28).  amplitude  into oscillation  f r o m t h e s u r f a c e , s u c h t h a t ty /0, = F,  d i s t a n t D,  a  will  decay to the  s t a t e p o i n t P^- . The  termined field  amplitude  o f the surface o s c i l l a t i o n s i s de-  when t h e w o r k d o n e p e r c y c l e by t h e a p p l i e d  electric  c o m p e n s a t e s t h e e n e r g y l o s s p e r c y c l e c a u s e d by  viscous occurs  dissipation. for V  Therefore  0  =  .  F o r a wave o f a m p l i t u d e But V  (at P  3  more work i s b e i n g  wave a m p l i t u d e  increases, i n order  t h e wave i s u n s t a b l e .  above the surve  3  this  ) i s g r e a t e r than  .  d o n e on t h e s u r f a c e t h a n c a n  be d i s s i p a t e d by a wave o f a m p l i t u d e  i.e.  that  J%.  Hence a t P  the  to increase d i s s i p a t i o n ,  This argument holds  f o r a l l points  ( F i g . 28) a n d t o t h e r i g h t o f ? ^  S i m i l a r l y , i f we a r r i v e a t P ^  3  instead of P  z  (\<  0.7. \ ) * the  115  v o l t a g e i s t o o s m a l l t o do t h e w o r k n e c e s s a r y t o m a i n t a i n t h e v a l u e o f 7 = 7% , h e n c e 7  decreases.  However,  l a r g e r v o l t a g e s a r e needed t o m a i n t a i n waves w i t h ^6 ^ ^ ^ ^ " i  *  T n u s  even amplitudes  > p o i n t s b e l o w t h e c u r v e ( F i g . 28)  move t o t h e l e f t u n t i l  t h e y e n d up on t h e c u r v e a t some  point with  S i m i l a r a r g u m e n t s show t h a t p o i n t s  J <  with  F<  0.7.  0.7 a n d a b o v e t h e c u r v e , move o n t o i t , i . e .  the o s c i l l a t i o n s are s t a b l e , i n agreement with the observations.  Finally, i f  unstable.  ? / I ( y ) > 0.43, t h e waves a r e a l s o  T h i s occurs because  t h e maximum v i s c o u s  energy  t h a t c a n be d i s s i p a t e d p e r c y c l e o c c u r s f o r w a v e s w i t h 7 = 1 ( i . e . when t h e f l u i d s u r f a c e t o u c h e s more e l e c t r i c a l  energy  the e l e c t r o d e ) . I f  i s pumped i n p e r c y c l e t h a n  l i m i t , t h e n c l e a r l y t h e wave a m p l i t u d e w i l l  this  grow u n t i l  waves t o u c h t h e e l e c t r o d e ( o r , i n p r a c t i c e , u n t i l  the  sparking  occurs). Although  the theoretical  d e l i n e a t e d as T / I ( T ) (Fig.  >  instability  region i s  0 . 4 3 , "Y > 0.7 a n d a b o v e t h e c u r v e  2 8 ) , we s h o u l d p o i n t o u t t h a t i t i s v e r y d i f f i c u l t  to o b s e r v e s t a b l e waves near t h e peak o f t h e c u r v e i n F i g . 28.  S i n c e t h e peak i s f a i r l y f l a t , even s m a l l s p o r a d i c  fluctuations in V  0  can k i c k the o p e r a t i n g p o i n t i n t o the  u n s t a b l e r e g i o n , which p o i n t s f o r which J s t a b i l i t y would  explains our i n a b i l i t y  to observe  was g r e a t e r t h a n a b o u t 0.6.  The i n -  a l s o d e v e l o p q u i c k e r due t o t h e p r e s e n c e  of  116  short wavelength t r a n s i e n t s or small are non-resonantly  higher  e x c i t e d , a n d r i d e on t h e r e s o n a n t  In o u r e x p e r i m e n t s we u s e d t h e o c c u r r e n c e criterion  f o r the existence  s e t s i n when t h e e l e c t r o d e is reduced  o r d e r modes  of unstable  to mercury surface  t o a b o u t 0.35D by s u r f a c e w a v e s .  ing  of the S F  we o b s e r v e d  g  as a  Instability gap l e n g t h  Since  was i n a l l e x p e r i m e n t s w e l l b e l o w t h e d i e l e c t r i c strength  mode.  of sparking waves.  that  V /(0.35D) o  breakdown  gas i n t h e microwave c a v i t y , t h e s p a r k -  i n o u r e x p e r i m e n t s was n o t s p u r i o u s , b u t  r e s u l t e d from the growth o f u n s t a b l e  waves.  E n h a n c e m e n t o f t h e d r i v i n g f o r c e by t h e e l e c t r i c field  d i s t o r t i o n has been n e g l e c t e d  i n the theory.  It is  shown i n A p p e n d i x C t h a t f o r t h e mode we i n v e s t i g a t e d , f o r c i n g t e r m s d e p e n d i n g on t h e c u r v a t u r e surface  are n e g l i g i b l y small  electric field  compared with  f o r c e u s e d i n S e c t i o n 6.1.  the e f f e c t o f these  and s l o p e  of the  the dominant In o r d e r  t e r m s on t h e s u r f a c e wave  to study  amplitude,  s u r f a c e modes w i t h much s m a l l e r w a v e l e n g t h s h a v e t o be employed. and  T  The t h r e s h o l d  i s , i n general, expected  e f f e c t s supply surface  f o r i n s t a b i l i t y i n terms o f V t o be r e d u c e d , s i n c e  additional electric field  0  these  energy to the  waves. An e f f e c t t h a t c e r t a i n l y n e e d s f u r t h e r i n v e s t i g a -  t i o n i s t h e phase d i f f e r e n c e between t h e d r i v i n g f i e l d and  117  the s u r f a c e d i s p l a c e m e n t .  In o u r c a s e i t was  accurate to s e t t h i s phase, c a l c u l a t i o n s of Appendix system  y> , as <jf>  sufficiently  = r r / 2 , and  C c o u l d n o t be c h e c k e d  due t o t h e h i g h Q - f a c t o r ( a b o u t 150)  i n our  of the s u r f a c e  mode i n v e s t i g a t e d , and t h e i n a b i l i t y o f o u r s i m p l e to keep a c o n s t a n t f r e q u e n c y o v e r s u f f i c i e n t l y of time.  Suggestions  associated effects  the  driver  long periods  f o r i n v e s t i g a t i n g the phase  and  (e.g. the frequency response of the  s u r f a c e mode i n t h e n o n - l i n e a r r e g i m e ) Experiments  are given i n Chapter  on d i f f e r e n t g e o m e t r i e s , e s p e c i a l l y  the case of r e c t a n g u l a r geometry, are very d e s i r a b l e a l s o . T h e r e a s o n i s t h a t t h e r e s u l t s c a n be c h e c k e d lytical  e x p r e s s i o n s (see Appendix  against  ana-  C for relevant calcula-  t i o n s ) and, i n a d d i t i o n , the r e s u l t s are a p p l i c a b l e to the analogous  case of a l i q u i d j e t .  given i n Chapter  8.  Again suggestions  are  8.  Chapter 7  NON-LINEAR INTERACTION OF WAVES ON SHALLOW WATER  A very important non-linear problem i s considered in this chapter. (Kadomtsev  T h e r e has been a t r e n d i n r e c e n t y e a r s  a n d Karpman (1971)) towards i n v e s t i g a t i n g  entire  c l a s s e s o f n o n - l i n e a r problems, r a t h e r than towards attempting t o solve individual system.  problems, p a r t i c u l a r to a certain  An i m p o r t a n t c l a s s , w i t h w i d e a p p l i c a t i o n s i n a  v a r i e t y o f f i e l d s , on w h i c h i n t e n s e emphasis h a s been i n t h e l a s t few y e a r s , i s the n o n - l i n e a r , dynamic a c t i o n b e t w e e n modes o f o s c i l l a t i o n spectrum.  placed  inter-  i n a normal f r e q u e n c y  This type o f i n t e r a c t i o n i s capable o f producing  a m u t u a l e x c h a n g e o f e n e r g y among t h e e i g e n m o d e s , a n d i t has v e r y i m p o r t a n t a p p l i c a t i o n s i n f i e l d s such as plasma physics  (Tsytovich  (1970)), non-linear optics  (1965)) and oceanography ( P h i l l i p s  (I960)).  (Bloembergen Experimental  i n v e s t i g a t i o n o f t h i s i n t e r a c t i o n was made p o s s i b l e b y t h e flexibility  a f f o r d e d us b y t h e e x c i t a t i o n o f p u r e  modes b y s p a t i a l l y n o n - u n i f o r m , t i m e p e r i o d i c  118  surface  applied  119  e l e c t r i c s t r e s s e s (see Chapter 4).  This type o f n o n - l i n e a r i t y  however i s c l e a r l y a c h a r a c t e r i s t i c f e a t u r e o f a moving f l u i d , and, u n l i k e the n o n - l i n e a r s i t u a t i o n  investigated  i n C h a p t e r 6, h a s n o t h i n g t o do w i t h t h e d r i v i n g (i.e. the e l e c t r i c  7.1  mechanism  stresses).  Theory In l i n e a r s y s t e m s  t h e r e s p o n s e t o an a r b i t r a r y  s t i m u l u s c a n be e x p r e s s e d as a l i n e a r s u p e r p o s i t i o n o f waves o f t h e form  (7.1)  where  A^  i s t h e a m p l i t u d e , k_ t h e w a v e - v e c t o r  a n g u l a r f r e q u e n c y o f t h e k mode. governing the system are weakly terms w i l l  a n d UJ^ t h e  I f the dynamic equations n o n - l i n e a r , the non-linear  g e n e r a t e q u a d r a t i c terms  o f the form  A_K,A,_e in the equations of motion. i n t e r a c t i o n o f modes k,  These  and k  a  .  a r e due t o t h e d y n a m i c The n o n - l i n e a r i t y  i n t r o d u c e s a f o r c e on t h e s y s t e m o f f r e q u e n c y a/,t(V^  thus and  1  wave-vector  k^ + k_ ^ .  20  I f i n the frequency spectrum of the  o s c i l l a t i n g s y s t e m t h e r e e x i s t s a t h i r d mode k_^ , s u c h  that  (7.3)  and i n a d d i t i o n i f A^  t  i s not perpendicular to  t h e n t h i s t h i r d mode w i l l  >  be r e s o n a n t l y d r i v e n b y t h e  dynamic i n t e r a c t i o n o f t h e o t h e r two. L i n e a r i z e d s h a l l o w water s u r f a c e waves, propagati n g i n one d i r e c t i o n , s a t i s f y a l l c o n d i t i o n s f o r r e s o n a n c e . We s h a l l  now d e v e l o p t h e n e c e s s a r y t h e o r y f o r t h e n o n -  l i n e a r i n t e r a c t i o n o f s t a n d i n g s u r f a c e w a v e s on s h a l l o w water.  S i n c e i n o u r e x p e r i m e n t s we s t u d i e d t h e s i m p l e s t  c a s e o f t h e s e l f i n t e r a c t i o n o f a wave t o g e n e r a t e a wave of d o u b l e t h e f r e q u e n c y and w a v e - v e c t o r  (see equation  i n o r d e r t o s i m p l i f y t h e e x p o s i t i o n , we s h a l l s p e c i f i c problem. mode w i l l  treat  this  The n o m e n c l a t u r e p r i m a r y and s e c o n d a r y  be e m p l o y e d t o r e f e r t o t h e mode w h i c h  i n t e r a c t s a n d t o t h e mode d r i v e n by t h i s s e l f respectively.  ( 7 . 3 ) ) ,  self  interaction  T h e p r i m a r y mode was e x c i t e d by a p p l y i n g  s p a t i a l l y non-uniform, periodic e l e c t r i c stresses  onto  the f r e e s u r f a c e o f water i n a r e c t a n g u l a r c o n t a i n e r o f length L (see Fig. 9 ) .  T h e p r i m a r y mode h a s a w a v e l e n g t h  L a n d t h e s e c o n d a r y mode L / 2 .  1 21  We s h a l l  u s e a p e r t u r b a t i o n m e t h o d , w i t h an 1  expansion  parameter  w h e r e A,  i s the amplitude  displacement  f = A,/k, H  (see Stokes  of the primary  (1847))  mode.  The f l u i d  a n d v e l o c i t y p o t e n t i a l c a n t h e n be w r i t t e n as  l-.if\et  X)  ;  c^ey^V  w h e r e t h e s u p e r s c r i p t s on J the terms.  3  Neglecting  and  '  (7 4)  0  denote the order of  s u r f a c e t e n s i o n ( s e e A p p e n d i x D)  we c a n w r i t e t h e e q u a t i o n  of stress conservation  at the  s u r f a c e as  w h e r e C , ( t ) i s an a r b i t r a r y c o n s t a n t  of integration, f(x,t)  i s a f u n c t i o n t h a t d e s c r i b e s t h e s p a t i a l and temporal ation of the applied e l e c t r i c f i e l d that only the primary  vari-  ( f ( x , t ) i s chosen so  mode i s r e s o n a n t l y e x c i t e d - s e e  A p p e n d i x D) , a n d D i s t h e m i n i m u m d i s t a n c e o f t h e e l e c t r o d e from on  the e q u i l i b r i u m f l u i d ¥  will  be u s e d  surface  (see F i g . 9).  t o denote the components o f  Suffices displacement  122  in the c a r t e s i a n c o - o r d i n a t e system suffices  shown i n F i g . 9,  on t h e p a r t i a l d i f f e r e n t i a l c o e f f i c i e n t s w i l l  the p o i n t at which the c o e f f i c i e n t s The is  (Moiseev  kinematic ( 1 9 5 8 ) , p.  s u b s t i t u t e from e q u a t i o n (7.6)  about  evaluated.  -  surface  (7.4)  (v(f)  (7.6)  into equations  (7.5)  and e x p a n d a p p r o p r i a t e t e r m s i n a T a y l o r  z = H to  denote  861)  +  and  a r e t o be  boundary c o n d i t i o n at the  i £ i J L . Uk We  and  series  get:  (7.7)  H  J  (from equation  (7.5))  where a l l the p a r t i a l c o e f f i c i e n t s at z =  are from  now  on  evaluated  H.  )r *  f }  ir  n)  - c i t - » r -Lz~ + e  z  ;r  0>  )r  co  +t  r  (7.8) if  1 23  ( f r o m e q u a t i on  (7.6)).  Collecting first  and s e c o n d  the f o l l o w i n g equations  J ">  ~7T~ 9  5  j">  '  z  o r d e r terms  of motion  obtain  and b o u n d a r y c o n d i t i o n s ;  V, fM)  U  ( f i r s t order equation of  s e p a r a t e l y we  v>  ( .9) 7  1  motion);  (7.10)  (second order equation of motion) w h e r e we terms  have n e g l e c t e d f o r the time b e i n g second  i n the s u r f a c e s t r e s s (see Appendix  order  D) ;  ( f i r s t order kinematic boundary c o n d i t i o n ) ;  tl£  }  11*11  *  7  *  -J*  ^  V  "  ,  >*  M  (7.12)  124  (second order kinematic boundary  condition).  T h e s e f o u r e q u a t i o n s a r e t o be s o l v e d s u b j e c t t o the boundary  condition  displacement  normal  = 0, w h e r e  10  T-J°  T  w h e r e /!,  U) X  -  >  Ai  ^  coth(H,T)  A>  UJ  ct>i(c^,i)  t  (7.13)  t)  S>'*(t«,t)  i s p r o p o r t i o n a l to V  e  S<h(K,*)  (see Section (4.2.1)),  co, , t h e r e s o n a n t f r e q u e n c y o f t h e p r i m a r y  mode ( f r o m t h e d i s p e r s i o n r e l a t i o n fluid  ( 7 . 9 ) and (7.11) a r e  co (k,x)  sin (UJ,  cos(k,\)  ~ -/), caikiK.U)  k, = 2 T / L , a n d  i s the f l u i d  H  to the s u r f a c e of the c o n t a i n i n g vessel  The s o l u t i o n s o f e q u a t i o n s  Co  T  approximation  (3.36) f o r t h e s h a l l o w  ( i . e . tanh(kH) = kH), n e g l e c t i n g s u r f a c e  t e n s i o n and t h e n e g l i g i b l e e f f e c t o f t h e s m a l l s t a t i c ponent of the e l e c t r i c  com-  f i e l d ) i s g i v e n by  * L M  V  (7.14)  l  We c a n s u b s t i t u t e t h e s e e x p r e s s i o n s i n t o t h e equations o f the second  o r d e r m o t i o n (7.10) and  w h i c h c a n t h e n be s o l v e d f o r Cf  —  and  J  (7.12),  (1.) 2  . Eliminating  125  C  from e q u a t i o n  (7.10) u s i n g e q u a t i o n  an e q u a t i o n c o n t a i n i n g p r o d u c t s  (7.12) leads to  o f the terms  a to e below:  a)  b)  c)  The  products  ae, af g i v e r i s e to s m a l l s t a t i c  deformations  o f t h e s u r f a c e , and s i n c e t h e s e a r e n o n - r e s o n a n t  they  s m a l l e n o u g h t o be n e g l i g i b l e  The  (see Appendix  d u c t s bd, cd c o n t r i b u t e o n l y time terms  D).  (but not space)  are cross products r i s e to resonant  (7.10).  pro-  dependent  and c a n be c a n c e l l e d by p r o p e r c h o i c e o f t h e  constant C (t) in equation  are  arbitrary  The o n l y r e m a i n i n g  o f b and c w i t h e o r f , and t h e y  terms  give  e x c i t a t i o n o f modes w i t h  (7.15)  K-x - 2- K  {  The  r e s u l t a n t e q u a t i o n f o r Cp  is  0  (7.16)  1 26  (%)  Since Co  ^  i s a solution o f Laplace's equation with k  az  2k, ,  %= 2.<*s, , we c a n w r i t e i t i n t h e f o r m  ( X )  = 0L«COsl>(K<>7i)  where the amplitude Using equation equation  cos ( K*X) tw(bs  %  (7.17)  t)  a ^ can be f o u n d from e q u a t i o n  (7.17) t o e l i m i n a t e  (7.16) from  ^ y ' ^ / d z  (7.16) leads t o  (7.18) --2.  ou  a  (  A i ^ coi( Kryk) s>* ju>% t) £l - ( Ki H)*]  which f o r shallow f l u i d s  (Jl +  Since  (k^H « 1 )  ti)<f --IwJ^Apo^x) (%)  OJ^ r J K% H  f°  r  becomes  s»(u*0[HKH)*j  shallow f l u i d s , this  d e s c r i b e s resonant d r i v i n g o f the secondary resonance,  t h e l e f t hand s i d e o f e u q a t i o n  equation  mode. A t (7.19) can be  r e p l a c e d by the u s u a l e x p r e s s i o n i n v o l v i n g the  damping  (7.19)  127  frequency  0\  _  J  of  the secondary  (see Section  .  With  this  substitution  cot(^£)cosl(KiV  Substituting order with  V  Q  a  of  A^  is  leads  (7.12)  to the f i n a l  r  (ft tf)]'** (  into  the  7 , 2  °)  second  and i n t e g r a t i n g  result  for  S  x  ,  i  the amplitude (cf.  the secondary  water  equation  mode,  in Section Thus  shallow  condition  Cf>^  obtain  h  =&i/'kc\  derived  for  we  (^,x)[(l^iHf)sin(2O,t)+i(l-C0 (iu),fc))  C 0 S  *  expression  boundary  to time  ^  =  ~  where  now t h i s  kinematic respect  4.1.1)  a>(V  CVJ_ —  i.e.  mode  of the secondary (4.27))  expressions  is  mode, and  the quality  f o r which  have  factor been  4.1.2.  the s e l f will  interaction  generate  "5  A  a second  of  a surface  order  mode  mode o n of  amplitude  128  The t h e o r y a l s o p o i n t s o u t t h a t t h e S t o k e s ' o r d e r £  f o r t r a v e l l i n g waves £, -  by t h e p a r a m e t e r  (Stokes  Q<iA,Itt  ( 1 8 4 7 ) ) ,  s h o u l d be r e p l a c e d  f o r resonantly driven standing  waves i n o r d e r f o r t h e p e r t u r b a t i o n e x p a n s i o n s vergent. £  Our e x p e r i m e n t a l  and  parameter  t o be c o n -  c o n d i t i o n s were such t h a t both  £, w e r e a b o u t 0 . 2 , i . e . o u r p e r t u r b a t i o n e x p a n s i o n s  were c o n v e r g e n t  ( s e e f o l l o w i n g s e c t i o n on e x p e r i m e n t a l  resu1ts) . F i n a l l y , even  though  we t r e a t e d t h e p a r t i c u l a r  c a s e o f t h e s e l f i n t e r a c t i o n o f a m o d e , an e x a c t l y treatment  analogous  c a n be e m p l o y e d when d e a l i n g w i t h t h e i n t e r a c t i o n  o f more t h a n o n e mode.  7.2  Experimental  Results  A r e c t a n g u l a r water The e x p e r i m e n t a l  arrangement  p r i m a r y mode ( w a v e l e n g t h soidal  c o n t a i n e r o f l e n g t h L was i s shown i n F i g . 9.  The  L ) was e x c i t e d by a p p l y i n g a s i n u -  t y p e v o l t a g e ( s e e F i g . 7b) t o t h e e l e c t r o d e E.  wave a m p l i t u d e s  used.  o f both t h e p r i m a r y and t h e s e c o n d a r y  were measured w i t h t h e o p t i c a l  The mode  method u s i n g a m i c r o s c o p e i n  c o n j u n c t i o n w i t h a p h o t o r e s i s t o r as d e s c r i b e d i n S e c t i o n 2.5.2.  We f i r s t  show t h a t A,  i s p r o p o r t i o n a l t o V©  1  , and A  ( t h e p r i m a r y mode a  amplitude)  ( t h e s e c o n d a r y mode  amplitude)  129  is proportional to V quality factor Q  0  .  We t h e n d e r i v e a v a l u e f o r t h e  o f t h e s e c o n d a r y mode, and u s e i t t o  a  show t h a t , t o a g o o d a p p r o x i m a t i o n , as p r e d i c t e d by e q u a t i o n ( 7 . 2 1 ) .  = (3/8)Q^(A, / H ) ,  Experimental  parameters  are g i v e n i n T a b l e 7.1. A typical  p i c t u r e o f t h e o u t p u t a t x = L/4 ( s e e  F i g . 9) w h i c h i s a n o d e f o r t h e p r i m a r y a n d an a n t i n o d e f o r t h e s e c o n d a r y mode i s shown i n F i g . 3 0 .  T h i s was t a k e n by  e x c i t i n g t h e p r i m a r y mode a t a b o u t i t s r e s o n a n t  frequency,  and t h e n c a r e f u l l y t u n i n g t h e r o t o r f r e q u e n c y t o g e t t h e maximum a m p l i t u d e  o f t h e s e c o n d a r y mode.  It i s evident  t h a t t h e wave c a n be r e p r e s e n t e d by t h e e x p r e s s i o n  ... R  h  f a )  /...a I  where t h e o r i g i n o f t h e component o f f r e q u e n c y amplitude  and  B i s e x p l a i n e d i n Appendix D and d i s c u s s e d i n  S e c t i o n 7.3. obtained  (7.22)  T h e s e c o n d a r y mode a m p l i t u d e A ^  c a n be  from  A, = [(V(p)-V(q)) + (V(p)-V(r))]/4  where V ( j ) denotes  the p h o t o r e s i s t o r voltage at the point j ,  130  Table  E X P E R I M E N T A L  7.1  P A R A M E T E R S  H = W a t e r d e p t h = 2.96 cm L = L e n g t h o f t a n k = 9 7 . 8 cm D = Minimum d i s t a n c e o f e l e c t r o d e f r o m e q u i l i b r i u m s u r f a c e = 0.76 cm g = A c c e l e r a t i o n o f g r a v i t y = 981 j  3  = D e n s i t y o f w a t e r = 1.0  g.cm  cm.sec"*  - 3  T = S u r f a c e t e n s i o n o f w a t e r = 73.0  dyne.cm"'  y = K i n e m a t i c v i s c o s i t y o f w a t e r = 0.01 b = W i d t h o f t a n k = 5.0 cm  Poise,cm  .g  131  and t h e p o i n t s p , q a n d r a r e d e f i n e d i n F i g . 3 0 .  A^  was  measured f o r v a r i o u s values o f V . o The  amplitude  o f t h e p r i m a r y mode, A , , was a l s o  m e a s u r e d as a f u n c t i o n o f V t a n k ; a t x = L/8 a l s o as A,  a t two p o s i t i o n s a l o n g t h e  0  ( n o d e o f s e c o n d a r y mode) as A  / |/~2, a n d  (  a t x = 0 ( a n t i n o d e f o r both modes).  Slight  asymmetries i n the s i n u s o i d a l waveform a t the l a t t e r p o i n t w e r e c a u s e d by t h e p r e s e n c e o f t h e s e c o n d a r y mode ( o f amplitude V  0  about  1 0 % t h a t o f t h e p r i m a r y mode a m p l i t u d e ) .  m e a s u r e d t o an a c c u r a c y o f 2% by m o n i t o r i n g i t on t h e  o s c i l l o s c o p e with a high v o l t a g e probe see F i g . 30.  A b s o l u t e v a l u e s o f A,  ( T e k t r o n i x P6013)  and A  2  were o b t a i n e d  by c a l i b r a t i n g t h e p h o t o r e s i s t o r o u t p u t ( s e e F i g . 1 4 ) , by varying the water level The  i n the tank.  r e s u l t s a r e shown i n F i g . 3 1 . i s p r o p o r t i o n a l t o \l °"  mode a m p l i t u d e A,  0  The primary and t h e s e c o n d a r y  u  mode a m p l i t u d e  A  with theoretical  proportional to V  a  0  , i n agreement  p r e d i c t i o n s (equation. (7.21)).  The p r i m a r y mode was a l s o e x c i t e d , a n d t h e d r i v i n g v o l t a g e s w i t c h e d o f f , t o o b t a i n a damping waveform o f t h e p r i m a r y mode ( s e e F i g . 1 3 b ) . From t h i s p h o t o g r a p h  logA, v s .  t was p l o t t e d t o g i v e t h e d a m p i n g f r e q u e n c y o f t h e mode. T h e n t h e q u a l i t y f a c t o r o f t h e p r i m a r y mode was o b t a i n e d f r o m Q, = «*>, /2 ay .  F r o m  t n i s  >  b  ^ u s i n g Q /Q, =  (°\/OJ,  ) *= V  fl  ,  132  f o r t h e s h a l l o w f l u i d modes we e m p l o y e d i n t h i s s t u d y , a value of  c o u l d be o b t a i n e d .  c o u l d n o t be m e a s u r e d  d i r e c t l y due t o t h e p r e s e n c e o f t h e c o m p o n e n t at f r e q u e n c y  cv, ( s e e e q u a t i o n  oscillating  (7.22)).  T a b l e 7.2 shows t h e s l o p e s o f t h e g r a p h s i n F i g . 21 t o g e t h e r w i t h t h e s l o p e p r e d i c t e d f o r g r a p h " 3 1 b by equation the Aj  , Y"*,,-  (7.21), i . e . j o . vs. V  graph ( F i g . 31a).  0  The t a b l e a l s o  a comparison between the e x p e r i m e n t a l l y o b t a i n e d for Q  (  and Q  , where m i s the s l o p e presents  values  w i t h t h e v a l u e s p r e d i c t e d by e q u a t i o n  %  (4.29)  i.e.  Hb H + b  and  the f a c t that  = Q ^2~}  1 J  The agreement between  theory  and e x p e r i m e n t i s s e e n t o be q u i t e s a t i s f a c t o r y  considering  t h e e r r o r s i n t r o d u c e d by t h e m e a s u r e m e n t o f \I  and Q  and  ,  (  the a d d i t i o n a l complications discussed i n Appendix D  and S e c t i o n  7.3.  Q  7.3.  Critical  Discussion of Experimental  R e s u l t s o f t h e Non-  L i n e a r I n t e r a c t i o n Between S h a l l o w Water S u r f a c e  Modes  We s t a r t t h i s s e c t i o n by c o n s i d e r i n g how w e l l t h e experimental  r e s u l t s of the n o n - l i n e a r i n t e r a c t i o n of shallow  Table  EXPERIMENTAL  ^  RESULTS -  7.2  COMPARISON WITH THEORY  = p r i m a r y mode f r e q u e n c y = 3.48 s e c " ' = s e c o n d a r y mode f r e q u e n c y = 6.96 s e c " '  k, = p r i m a r y mode w a v e n u m b e r = 0.064 cm"  1  k ^ = s e c o n d a r y mode w a v e n u m b e r = 0.128 m, = s l o p e o f A,  cm"  1  v s . V© g r a p h ( F i g . 3 1 a ) = 0.163 mm/(kV)  = s l o p e o f A ^ v s . Vp g r a p h ( F i g . 3 1 b )  THEORY 0.071 s e c " ' (equation (26))  EXPERIMENT 0.078 ± 0.004 s e c - 1  24.4 (equation (26))  Q  1 ma  34.5 (equation (26)) 0 . 0 1 0 5 + 0.005 mm/(kV)** ( e q u a t i o n ( 7 . 2 1 ) , m, a n d Q*)  * The  22.1  ±1.0  31.3 ± 1.4 ( u s i n g Q^= fz Q, a n d Q ) (  0 . 0 1 1 8 ± 0 . 0 0 0 6 mm/UV) *  experimental value of Q  4  n  i s used  here.  134  water  s u r f a c e modes a r e d e s c r i b e d by t h e o r y , a n d t h e n we  compare and c o n t r a s t o u r r e s u l t s w i t h s i m i l a r i n v e s t i g a t i o n s of other  workers.  Experimental  Results  As S t o k e s p o i n t o u t ( 1 8 4 7 ) f o r t r a v e l l i n g the l i n e a r i z a t i o n parameter A /k  H  waves,  f o r s h a l l o w w a t e r waves i s  , w h e r e A i s t h e s u r f a c e wave a m p l i t u d e , k i t s  wavenumber,  and H t h e f l u i d  depth.  For driven, standing  s u r f a c e waves, r e s o n a n t e f f e c t s a r e i m p o r t a n t , and t h e parameter  becomes  Q^A,/ H.  In o u r e x p e r i m e n t s  this  par-  a m e t e r was .— 0 . 2 , a n d i s t h e r e f o r e s m a l l e n o u g h f o r t h e c a l c u l a t i o n s t o be v a l i d . Since A  l  i s p r o p o r t i o n a l t o A,  i t i s clear  ( f i g . 3 1 b ) t h a t t h e s e c o n d a r y wave i s i n d e e d r e s o n a n t l y d r i v e n by m i x i n g o f p r i m a r y m o d e s .  However, A ^  i s larger  t h a n t h e t h e o r e t i c a l l y p r e d i c t e d v a l u e o f ( 3 / 8 ) 0 ^ , /H g i v e n by e q u a t i o n ( 7 . 2 1 ) .  T h i s d i s c r e p a n c y (—> 1 2 % ) i s  s l i g h t l y l a r g e r than the estimated measuring  errors (see  T a b l e 7 . 2 ) . A l s o A«j_ c o n t a i n s a c o m p o n e n t w h i c h at frequency  Cu{ .  In Appendix  oscillates  D.VI we show t h a t t h e p r i m a r y  mode d i s t o r t s t h e a p p l i e d e l e c t r i c f i e l d , s o as t o d r i v e a c o m p o n e n t o f t h e s e c o n d a r y mode  J  {2,2,2)  ( T"(2,2,2)  i s the  135  c o n v e n t i o n a l s e c o n d a r y mode).  T h e two c o m p o n e n t s d i f f e r i n A  phase  by  TT/2 u n d e r i d e a l  c o n d i t i o n s , and  a b o u t 20% o f  1(2,2,2).  the magnitude  of the secondary  which  I f the phase  17(2,2,2)1  due t o mode  T h i s i s n o t l a r g e e n o u g h t o e x p l a i n t h e 12%  crepancy cited  above.  l a r g e phase  produce  s h i f t s with s m a l l changes  d i f f e r e n c e of  dis-  However, i n r e s o n a n t systems,  s h i f t s i n f r e q u e n c y ( l e s s t h a n <~7Q)  phase  d i f f e r e n c e i s fT/2,  components i s  i s o n l y 2% l a r g e r t h a n  mixing.  7(2,2,2) i s  TT/3, i n s t e a d o f  small  comparatively  in amplitude. tr/2  A  could easily  a c c o u n t f o r a 10% i n c r e a s e i n t h e c o m p u t e d a m p l i t u d e the secondary The  mode. component at f r e q u e n c y  c a n be d r i v e n i n two w a y s .  <v, ( a m p l i t u d e A  s u r f a c e wave ( A p p e n d i x  D.V).  3  d e c r e a s e s as A ^  say)  primary  T h i s mechanism generates a  component whose a m p l i t u d e i s 0 . 0 2 ( A ^ ) . m e n t s show t h a t A / A ^  5  The f i r s t o f t h e s e i n v o l v e s  mixing of a s t a t i c s u r f a c e deformation with the  mechanism does  of  However, the e x p e r i -  i s t y p i c a l l y b e t w e e n 5 and 1 5 % ,  is increased.  and  Hence the n o n - l i n e a r d r i v i n g  not account f o r A 3  Another mechanism f o r d r i v i n g A j d r i v i n g by t h e a p p l i e d e l e c t r i c f i e l d T h i s g i v e s an a m p l i t u d e w h i c h  is simply  (Appendix  D.VII).  i s p r o p o r t i o n a l t o A|  c o n s t a n t of p r o p o r t i o n a l i t y l e s s than 0.03).  direct  (with  Even a s m a l l  136  driving stress will  t h e r e f o r e g i v e an A  significant fraction of A as A  .  a  i s increased since A /A^  a  3  w h i c h may be a  ?  Furthermore, A^/A^ i s proportional  decreases  t o H/A, .  A t v e r y s m a l l a m p l i t u d e s o f t h e p r i m a r y mode t h e c o m p o n e n t A^  i s t h e main  o b s e r v a b l e f e a t u r e o f t h e s e c o n d a r y mode.  Hence t h e e x p e r i m e n t a l r e s u l t s c o n f i r m t h a t t h e component o f f r e q u e n c y &4 field.  in I  i s d r i v e n d i r e c t l y by t h e a p p l i e d  F o r t u n a t e l y t h e p r e s e n c e o f t h i s component does n o t  i n f l u e n c e t h e measurement o f A^ .  I f desired, A^  g r e a t l y r e d u c e d by a d j u s t i n g t h e g e o m e t r y electrode.  Appendix  F demonstrates  c a n be  of the driving  the f e a s i b i l i t y of  d r i v i n g a p u r e s u r f a c e mode by a s p a t i a l l y a n d t e m p o r a l l y periodic  Other  field.  works  The p r o b l e m o f wave-wave i n t e r a c t i o n i n w a t e r waves i s t r e a t e d f r o m two p o i n t s o f v i e w i n t h e l i t e r a t u r e . group o f a u t h o r s ( P h i l l i p s and t h e i r c o - w o r k e r s H i g g i n s and Smith  (1960,1967),  One  Longuet-Higgins  ( M c G o l d r i c k et al. ( 1 9 6 6 ) ,  (1962)  Longuet-  (1966)) d i s c u s s e d the t r a n s f e r o f energy  by t h i r d o r d e r n o n - l i n e a r m i x i n g o f t r a v e l l i n g w a v e s on d e e p water  ( k H » 1 ) . A l a r g e t a n k ( a b o u t 10 f t . l o n g a n d o f  comparable  width) w i t h s h e l v e d edges t o p r e v e n t  reflection  o f w a v e s was u s e d i n t h e i r e x p e r i m e n t s , a n d t h e y m e a s u r e d t h e  137  spatial  growth  r a t e o f t h e n o n - l i n e a r l y d r i v e n wave  the boundaries.  Although  t i g a t i n g the second w a v e s , we mental  Phillips  from  (1960) s u g g e s t e d  inves-  order mixing process f o r shallow  water  h a v e n o t b e e n a b l e t o f i n d any p u b l i s h e d e x p e r i -  results. The  o t h e r group  o f a u t h o r s has s t u d i e d w a v e s o f  f i n i t e a m p l i t u d e on f l u i d s i n c o n t a i n e r s . i s g i v e n by A b r a m s o n , Chu  A useful survey  and Dodge ( A b r a m s o n ( 1 9 6 6 ) ) ,  who  c i t e i n v e s t i g a t i o n s by P e n n e y and P r i c e ( 1 9 5 2 ) , T a y l o r ( 1 9 5 3 ) and M o i s e e v  (1958).  More r e c e n t l y , C h e s t e r ,  and  C h e s t e r and B o n e s ( 1 9 6 8 ) h a v e p r e s e n t e d r e s u l t s on w a v e s of large amplitude  e x c i t e d on a s h a l l o w f l u i d  in a rectan-  gular container. P h i l l i p s , Longuet-Higgins  and t h e i r  co-workers  do n o t i n c l u d e v i s c o u s d i s s i p a t i o n e f f e c t s i n t h e i r w o r k because  they examine amplitude  i s n o t l i m i t e d by d i s s i p a t i o n .  growth  i n r e g i o n s where i t  The s e c o n d  group  of  w o r k i n a r e g i m e w h e r e A , /H i s l a r g e ( i . e . Q A / H » (  (  H e n c e t h e r e s o n a n t f r e q u e n c i e s d e p e n d on t h e wave  authors 1).  amplitude,  s o t h a t n o n - l i n e a r i n t e r a c t i o n s o f t h e t y p e we h a v e c o n s i d e r e d become n o n - r e s o n a n t .  In t h i s c a s e , t h e n o n - l i n e a r l y d r i v e n  wave has f i n i t e a m p l i t u d e b e c a u s e  the d r i v i n g frequency i s  d i f f e r e n t f r o m t h e r e s o n a n t f r e q u e n c y o f t h e mode, i . e . t h e amplitude  i s n o t l i m i t e d by v i s c o u s d i s s i p a t i o n .  Viscous  138  damping and  effects  Chester  excite which  and  gravity leads  to  The work  is  mixing  first of  amplitude  are  only  Bones  taken  (1968). by  waves  propagating  main d i f f e r e n c e that  waves,  and  secondly  in  it  is  the  we  Finally,  because  A,/H  resonant  interaction  is by  to  have  of  less  and  we  our  than  take  Chester,  of  the  and  second of  channel.  earlier  order  such  account  of  small  viscous  mechanisms.  2%, d e t u n i n g dispersion  container,  along  excitation  they  the  work  used waves  the  amplitude  fro  studied  to  by  experiments  sloshing  between  essential  analysis  account  their  horizontal  all  dissipation  In  waves  of  that  into  of  does  the not  occur.  Chapter 8  CONCLUSIONS - FUTURE WORK  Conclusions  In t h i s c h a p t e r the b a s i c c o n c l u s i o n s suggestions future.  we s h a l l f i r s t g i v e a summary o f  d r a w n f r o m o u r w o r k , a n d t h e n make  for extending  these  These suggestions  investigations i n the  a r e made i n a s e r i o u s  spirit,  and  i t i s o u r b e l i e f t h a t the p r o j e c t s would l e a d t o f r u i t f u l  and  i n t e r e s t i n g r e s u l t s , which have a p p l i c a t i o n s i n a v a r i e t y  of f i el ds. The  r e s u l t s (Chapter  3) o f t h e  the square o f the o s c i l l a t i o n frequency of the  l i n e a r decrease i n f  applied electrostatic field strength,  with  the  (V /D) 0  d e m o n s t r a t e d t h a t a l i n e a r t h e o r e t i c a l model g i v e s d e s c r i p t i o n o f the p h y s i c a l  processes  the s p a t i a l n o n - u n i f o r m i t y  agrees well with  o f the  an e x c e l l e n t wave  The c o r r e c t i o n  applied e l e c t r i c stress  e x p e r i m e n t , as l o n g as t h e s u r f a c e  139  , have  i n v o l v e d when t h e  a m p l i t u d e o v e r t h e w a v e l e n g t h i s Q~ ~ IO"**. for  square  mode  140  eigenfunctions  are not modified  tion of the f l u i d The  by t h e e q u i l i b r i u m d i s t o r -  s u r f a c e caused  by t h e n o n - u n i f o r m  field.  e q u i l i b r i u m d i s t o r t i o n o f t h e s u r f a c e c a n be u s e d t o  c a l i b r a t e the microwave resonator monitoring waves. theory  Again  the surface  t h e r e s u l t s a r e w e l l d e s c r i b e d by l i n e a r  n e g l e c t i n g s u r f a c e t e n s i o n , up t o t h e p o i n t when  the s u r f a c e d i s t o r t i o n i s l a r g e enough ( l a r g e r than ^ to  c a u s e s u r f a c e t e n s i o n e f f e c t s t o become We h a v e shown ( C h a p t e r  uniform,  time  4) t h a t s p a t i a l l y  non-  excitation  s u r f a c e mode on a f l u i d i n  a container.  T h e mode a m p l i t u d e  to  o f t h e a p p l i e d e l e c t r i c , f i e l d , and hence can  the square  is directly  be e a s i l y a n d a c c u r a t e l y c o n t r o l l e d . uniform  proportional  The s p a t i a l l y  e l e c t r i c stresses are applied to the f l u i d  so a s t o m a x i m i z e t h e s t r e s s a t t h e d i s p l a c e m e n t o f a d e s i r e d s u r f a c e mode. s t r e s s e s i s s e t equal  The frequency  as l o n g as t h e s u r f a c e wave a m p l i t u d e  remai n  smal1.  surface antinodes  of the applied  t h e s u r f a c e b e h a v e s as a d a m p e d , d r i v e n h a r m o n i c  equilibrium surface  non-  t o t h e n a t u r a l mode f r e q u e n c y ,  the d i s t a n c e o f t h e f i e l d  mm)  important.  p e r i o d i c e l e c t r i c f i e l d s enable  o f v i r t u a l l y any pure s t a n d i n g  0.1  i s small  and  oscillator,  compared t o  a p p l y i n g e l e c t r o d e from the  ( d i s t a n c e D) , a n d w a v e - w a v e i n t e r a c t i o n s  141  When t h e s u r f a c e wave a m p l i t u d e t o D, n o n - l i n e a r  d r i v i n g r e s u l t s (Chapter  which the dominant non-linear spatial  non-uniformity  becomes c o m p a r a b l e 6).  A model i n  d r i v i n g f o r c e i s due t o t h e  of the applied e l e c t r i c f i e l d  q u i t e good agreement with  experimental  results.  gives  The r e s u l t s  a l s o p o i n t o u t a new i n s t a b i l i t y c r i t e r i o n f o r p e r i o d i c , applied electric field  theory.  mode a m p l i t u d e  f i e l d s , w h i c h i s n o t p r e d i c t e d by  The c r i t e r i o n i s t h a t t h e d r i v e n becomes s u f f i c i e n t l y  viscous  by t h e e l e c t r i c f i e l d e f f e c t s i n the f l u i d We h a v e o b s e r v e d  frequency  than  surface  large f o r overstabi 1i t y  t o r e s u l t , due t o t h e f a c t t h a t more e l e c t r i c a l supplied  uniform  energy i s  c a n be d i s s i p a t e d by  wave. ( S e c t i o n 4.2.3) t h a t t h e damping  o f s u r f a c e modes i n t h e p r e s e n c e o f s t r o n g ,  s p a t i a l l y non-uniform e l e c t r o s t a t i c f i e l d s i s s o l e l y determined i.e.  by t h e r e d u c t i o n  the non-uniform f i e l d  boundary l a y e r s t r u c t u r e .  i n t h e s u r f a c e mode f r e q u e n c y  h a s no s i g n i f i c a n t e f f e c t on t h e The r e s u l t s o f a l l damping  quency measurements a r e i n v e r y good agreement w i t h t h a t a s s u m e s an i m m o b i l e s u r f a c e f i l m on t h e f r e e s u r f a c e , and t h i s f i l m i s n o t d i s r u p t e d fields of strength The  f,  fre-  a theory fluid  by e l e c t r o s t a t i c  up t o 55 kV/cm.  g r o w t h r a t e s o f t h e i n s t a b i l i t y c a u s e d when a  strong e l e c t r o s t a t i c field  i s a p p l i e d t o an o s c i l l a t i n g  fluid  142  surface  (Chapter  Larmor theory,  3) a r e n o t c o n s i s t e n t w i t h appropriately modified  to account  u n i f o r m i t i e s i n the applied e l e c t r i c f i e l d . that the modification length  instabilities  change r a p i d l y with container).  Again  electric field  ignores  ( i n t h e case  non-uniformities  water (Chapter  dynamic equations generate  by t h e m i x i n g  quite  e f f e c t s i n the  and t h e s u r f a c e b o u n d a r y c o n d i t i o n s i n s e c o n d a r y modes w h i c h a r e r e s o n a n t l y  driven  The advantages o f a fixed  (effects of dispersion  t r a v e l l i n g waves, i n which t h e wavenumber a n d t h a t much m o r e  s y s t e m s may be u s e d i n t h e e x p e r i m e n t s ;  the geometry of the primary chosen to optimize  compact also  a n d s e c o n d a r y modes c a n be  the observations,  s e c o n d a r y mode d i s p l a c e m e n t displacement  s u r f a c e w a v e s on  non-linear  is f r e e to change, are avoided), simple  stability.  w a v e s a r e t h a t we a r e d e a l i n g w i t h  associated with  by t h e  7 ) , we h a v e b e e n a b l e t o show  wavenumber'for a c e r t a i n frequency  and  of a c y l i n d r i c a l  on t h e s u r f a c e  o f f i r s t o r d e r modes.  using standing  wave-  stresses  we s e e t h e d o m i n a n t r o l e p l a y e d  c o n c l u s i v e l y that second order  a fluid  The reason i s  where the s u r f a c e  By e x p e r i m e n t s on s t a n d i n g shallow  f o r non-  the e x c i t a t i o n of short  i n regions radius  the o r i g i n a l  antinode  node o f t h e p r i m a r y  by a r r a n g i n g  the  to coincide with the  mode.  By u s i n g  spatially  n o n - u n i f o r m e l e c t r i c f i e l d s f o r t h e s u r f a c e mode e x c i t a t i o n ,  143  v e r y p u r e modes c o u l d be e x c i t e d .  This i s not easy  vibrators or "sloshing" of the f l u i d In a d d i t i o n , t h e u s e o f s t a n d i n g small  amplitude  ^  5 x IO"**" cm  mersion  technique,  by  T h e u s e o f an o p t i c a l  with a s p a t i a l r e s o l u t i o n of  e l i m i n a t e s problems a s s o c i a t e d with im-  probes (see f o r example d i f f i c u l t i e s  by C h e s t e r  Future  s u r f a c e waves o f s u f f i c i e n t l y  i n the theory, since detuning  dispersion i s avoided.  wave m o n i t o r i n g  container are used.  necessitates taking viscous d i s s i p a t i o n of  wave e n e r g y i n t o a c c o u n t amplitude  when  and Bones  encountered  (1968)).  Work  Future  i n v e s t i g a t i o n s of the e f f e c t s of s p a t i a l l y  n o . n - u n i f o r m , e l e c t r o s t a t i c f i e l d s on f l u i d s u r f a c e s concentrate  on t h e s t a t i c s u r f a c e d e f o r m a t i o n  should  for fields  l a r g e enough to d i s t o r t the s u r f a c e s u f f i c i e n t l y f o r s u r f a c e t e n s i o n e f f e c t s t o become s i g n i f i c a n t . O p t i c a l and wave t e c h n i q u e s ,  with  micro-  t h e i r high s p a t i a l r e s o l u t i o n s , are  ideally suited for this.  Also the e f f e c t of the surface  d i s t o r t i o n on t h e p u r i t y o f t h e s u r f a c e modes s h o u l d  be  looked a t . In a n y f u t u r e w o r k on a l t e r n a t i n g a p p l i e d fields  t o e x c i t e s u r f a c e modes, a d r i v e r t h a t  electric  supplies  144  periodic voltages  of constant  period  over long  o f t i m e ( a b o u t 100 wave o s c i l l a t i o n s ) c o u l d employed. period  t o b e t t e r than 1 p a r t i n 1000. to important  linear driving of surface the q u e s t i o n  f l u i d , with  (i.e. smaller  Such a d r i v e r  will  i n v e s t i g a t i o n s on t h e n o n -  modes by p e r i o d i c f i e l d s ,  wave a t r e s o n a n c e .  comparatively  Q-factors),  r e l a t i v e l y accurate can  constant  namely  o f phase d i f f e r e n c e between the d r i v i n g  and t h e s u r f a c e  shallow  u s e f u l l y be  T h i s means a d r i v e r t h a t c a n k e e p a  o p e n up t h e r o a d  field  intervals  By e m p l o y i n g a  broader resonance  and a c o n s t a n t  electric  period  measurements o f the phase  curves  motor, difference  be made. Non-linear  should  a l s o be i n v e s t i g a t e d  relevant tical  d r i v i n g by p e r i o d i c e l e c t r i c i n rectangular  vessels.  i n t e g r a l s i n t h i s c a s e c a n be e x p r e s s e d  expressions,  and t h e y  pressible fluid jets. preferable  The  by  analy-  to the case o f incom-  The e x p e r i m e n t s i n v e s s e l s a r e  ( t o t h o s e on j e t s ) b e c a u s e t h e f l u i d  at r e s t , and s e c o n d l y , tude small  also apply  fields  is initially  i t i s e a s i e r t o k e e p t h e wave  enough f o r t h e dynamics o f t h e f l u i d  ampli-  i t s e l f to  remain l i n e a r ( i . e . only the d r i v i n g mechanism i s n o n - l i n e a r ) . It i s also d i f f i c u l t  t o make q u i e s c e n t  e x c e e d a few m i l l i m e t e r s .  j e t s whose  radii  W i t h wave a m p l i t u d e s o f c o m p a r a b l e  145  m a g n i t u d e , t h e l i n e a r i z e d form of the N a v i e r - S t o k e s will  equation  n o t p r o v i d e a good d e s c r i p t i o n o f j e t dynamics. Another  important aspect of the n o n - l i n e a r l y driven  waves i s t h e u n s t a b l e r e g i o n ( F i g . 2 8 ) .  The  instability  and i t s g r o w t h r a t e r e q u i r e d e t a i l e d i n v e s t i g a t i o n t o s u p p l y additional  i n f o r m a t i o n of the dominant dynamical  In c o n n e c t i o n w i t h t h i s , u s i n g a d r i v i n g f i e l d  processes. of constant  p e r i o d , t h e f r e q u e n c y and a m p l i t u d e r e s p o n s e o f t h e nonl i n e a r l y d r i v e n s u r f a c e s h o u l d be i n v e s t i g a t e d . I n t h i s way, i t s h o u l d be p o s s i b l e t o o b s e r v e  "jump p h e n o m e n a , " a  common n o n - l i n e a r p h e n o m e n o n i n a n a l o g o u s  mechanical  problems. In f u t u r e w o r k on t h e n o n - l i n e a r i n t e r a c t i o n o f s u r f a c e modes c a r e s h o u l d be t a k e n t o e l i m i n a t e t h e n o n r e s o n a n t l y d r i v e n s e c o n d a r y modes by m i x i n g o f a f i r s t o r d e r mode w i t h t h e d r i v i n g e l e c t r i c s t r e s s . accomplished in Appendix  T h i s c a n be  w i t h an e l e c t r o d e s u c h as t h e one d e s c r i b e d F.  By g r a d u a l l y i n c r e a s i n g t h e r a t i o o f t h e p r i m a r y mode a m p l i t u d e t o t h e d e p t h , s u r f a c e w a v e s - i n t h e r e g i m e w h e r e a m p l i t u d e d i s p e r s i o n i s s i g n i f i c a n t c a n be s t u d i e d . A t t h e same t i m e , by i n c r e a s i n g D, a d d i t i o n a l e f f e c t s caused  by t h e a p p l i e d e l e c t r i c f i e l d  n e g l i gi b l y smal1.  non-linear c a n be made  146  Again constant  by e m p l o y i n g a d r i v e r w h i c h c a n m a i n t a i n  frequency  over  long periods  of the phases of the primary respect  and  of time,  frequency  response of both  are important i t is probable s e c o n d and  together  d r i v e n modes.  in understanding  measurements  t h e s e c o n d a r y mode  t o t h e d r i v e r c a n be o b t a i n e d ,  a  with  with  the  These measurements  driven systems.  t h a t by d i r e c t l y d r i v i n g two  Moreover,  modes,  both  t h i r d o r d e r wave-wave i n t e r a c t i o n s c o u l d  be  studied. L a s t l y , another  problem that c e r t a i n l y merits  f u r t h e r i n v e s t i g a t i o n i s the damping of s u r f a c e  waves.  In a l l o u r i n v e s t i g a t i o n s a g r e e m e n t o f e x p e r i m e n t a l with  a theory  f i l m was  t h a t assumes a r i g i d , l a t e r a l l y i m m o b i l e  excellent.  processes  In o r d e r  to understand  the  a perfectly clean fluid surface.  l i k e t h e one  u s e d by K e m b a l l  f a c e c a n be c r e a t e d very u s e f u l .  at w i l l  (1950),  physical  An  sur-  i n a s e a l e d c o n t a i n e r would  I t c a n be e n c l o s e d  i n s i d e a microwave  These  l e a v e the p h y s i c a l p r o p e r t i e s of the s u r f a c e An a l t e r n a t i v e m e t h o d i s t o u s e  as a d i a g n o s t i c .  be  reson-  s p a t i a l l y non-uniform, time p e r i o d i c s t r e s s e s  unaffected.  to  apparatus  in which a clean  be e m p l o y e d f o r s u r f a c e wave e x c i t a t i o n .  technique  surface  t h a t c a u s e d a m p i n g i t w o u l d be v e r y d e s i r a b l e  s t a r t with  a t o r and  results  can  stresses  completely the o p t i c a l  Another advantage of  using  147  Kemball's  arrangement  introduced of the  strong  gradually electric  surface  gations  may  mentioned  in  is in  that  surface  a controlled  fields  in (see  lead  useful  some  Appendix  B.  manner,  removing  investigated to  contamination  Appendix  and  the  contaminants B).  applications  These such  can  be  effect  from investias  those  F i q 1. E l e c t r o n i c m i c r o w a v e s y s t e m ; a r r o w s show t h e d i r e c t i o n o f m i c r o w a v e power f l o w . A=Lambda power s u p p l i e s ; K = k l y s t r o n ; I = i s o l a t o r ; WM=wavemeter; PT=power t e r m i n a t o r ; C D = c r v s t a l d e t e c t o r ; MT=magic t e e ; T P = t u n i n g p o s t ; MCR=microwave c a v i t y r e s o n a t o r ; O = o s c i l l o s c o p e ; SO=sawtooth o u t p u t ; D = r e c t i f v i n q d i o d e .  149  F i g 2. M i c r o w a v e c a v i t y p o w e r o u t p u t P v s . f r e q u e n c y F at resonance. B a n d w i d t h i n (a) i s 8703 t o 8733 MHz, w i t h t h e r e s o n a n t o e a k a t 8718 MHz. (b) shows t h e same r e s o n a n ce w i t h t h e f r e a u e n c y a x i s m a g n i f i e d f i v e t i m e s , o u r u s u a l operating condition. c  P i g 3. M i c r o w a v e c a v i t y a n d b a s i c a c c e s s o r i e s . W G = w a v e g u i d e ; e l e c t r o d e ; BC=brass c y l i n d e r ; L P = l e v e l l i n g p l a t f o r m ; FT=fine HV=high v o l t a g e s o u r c e A H = a i r hole. f  BR=bakelite r o d ; BE=brass t a p ; R=mercury r e s e r v o i r ;  151  F i g 4. M o d i f i e d , g a s t i g h t m i c r o w a v e c a v i t y , now f i l l e d w i t h s u l f u r h e x a f l u o r i d e (SFg)' t o i n c r e a s e t h e d i e l e c t r i c b r e a k d o w n s t r e n g t h i n t h e c a v i t y t o a b o u t 90 kV/cm.  BAKELITE  F i g 6. R o t o r and m o t o r c o n t r o l c i r c u i t s q u a r e wave g e n e r a t o r .  of high  voltage  VOLTAGE  *• TIMS  F i a 7. V o l t a g e o u t n u t o f t h e s q u a r e wave g e n e r a t o r ( a ) , and s i n u s o i d a l t y p e o u t n u t f r o m m o d i f i e d g e n e r a t o r ( b ) . I n ( a ) , 50 ms n e r cm on t i m e a x i s , a n d 1 kV n e r cm on voltage axis. I n ( b ) , 0.2 s p e r cm on t i m e a x i s , and 0.5 kV/cm on v o l t a g e a x i s .  F i g 8. S t a t o r m o d i f i e d f o r p r o d u c t i o n o f " s i n u s o i d a l " t y p e o u t put. I n s e r t shows i n t e g r a t i n g c i r c u i t f o r s m o o t h i n g t h e o u t p u t .  T V/, J  -rl  ) and e l e c t r o d e (E) i n r e c t a n g u l a r c o n t a i n e r o f l e n g t h L=97.8 cm. V ( t ) = s i n u s o i d a l t y p e h i g h v o l t a g e ; L P = l e v e l l i n g p l a t f o r m ; V ' a n d J ^' a r e t h e p r i m a r y a n d s e c o n d a r y s u r f a c e modes r e s n e c t i v e l y ( s e e chapter 7). C  1 56  F i g 10. S u r f a c e wave r e c o r d i n g d i r e c t l y on m o v i e together w i t h time markers.  film,  157  F i g 11. Time marker p r o d u c t i o n , s c o p e . TMG=time mark g e n e r a t o r ; DBO=dual beam o s c i l l o s c o p e .  and d i s p l a y on o s c i l l o PO=pulse g e n e r a t o r s ;  (c) F i g 12. O p t i c a l wave m o n i t o r i n g m e t h o d f o r w a t e r w a v e s . L = r e a d i n g lamp; M=microsoDe; WT=water t a n k , ( a ) : a r r a n g e ment f o r wave d e t e c t i o n ; ( b ) : Meniscus a p p e a r a n c e i n e v e piece; ( c ) P a t h s o f some l i g h t r a y s i n t o t h e m i c r o s c o p e .  159  10 V  F i g 13. P h o t o r e s i s t o r c i r c u i t f o r m o n i t o r i n a s u r f a c e wav e s on w a t e r i n a r e c t a n c m l a r t a n k , ( a ) , and t y p i c a l o u t p u t o f a d a m p i n g wave, (b). P R = n h o t o r e s i s t o r ; OS=oseilJos c o n e ; 8 2 0 i l r e s i s t o r f o r p r o t e c t i o n , and 10 kj2 p o t e n t i o meter f o r g a i n adjustment. I n (b), I s p e r cm on h o r i z o n t a l , and 2 0 mV p e r cm on v e r t i c a l ; 10 mV»2xl9" cm wave amplitude. 3  160  in  f F i g 14. C h a n g e o f v o l t a g e a c r o s s t h e p h o t o r e s i s t o r (Av ) w i t h volume o f water added t o the c o n t a i n e r (A v ) . T h i s g r a p h c a n be u s e d f o r t h e c a l i b r a t i o n o f t h e p h o t o r e s i s t o r o u t p u t t o g i v e a b s o l u t e measurements o f s u r f a c e wave a m n l i t u d e . .1 c c o f w a t e r e q u i v a l e n t t o 2 x l 0 " cm c h a n g e i n w a t e r 3  •depth.  1 61  " kV '  4 -  3-5  2 AF  [cm]  3  ->  F i g 15. P l o t o f V„ /I/SF" V S . A F f o r m i c r o w a v e r e s o n a n c e s h i f t A F c a u s e d by a s t a t i o n a r y f l u i d s u r f a c e d e f o r m a t i o n . The d e f o r m a t i o n r e s u l t s f r o m t h e a p p l i c a t i o n o f a s p a t i a l l y non u n i form, e l e c t r o s t a t i c f i e l d normal to the f r e e f l u i d s u r f a c e .  1 62  F i g 16. A l i n e a r i z e d s u r f a c e wave on a c o n d u c t i n g f l u i d s u r f a c e s t r e s s e d by a u n i f o r m e l e c t r o s t a t i c f i e l d n o r m a l t o t h e s u r f a c e . H = f l u i d depth; D=electrode d i s t a n c e from u n d i s t u r b e d surface; V = v o l t a g e a p p l i e d t o e l e c t r o d e by h i g h v o l t a g e s o u r c e HV, w i t h t h e f l u i d g r o u n d e d . a  163  Ri  I  R,  %s//s  / y y //l  //ss/  D  M E R C U R Y  g(t)  CO  0  -Tf  TT  t 0  R  x  R,  F i g 17- G e o m e t r y o f e l e c t r o d e o f r a d i u s (R?) a n d s u r f a c e wave i n a c y l i n d r i c a l c o n t a i n e r o f r a d i u s (R]_) , a n d s p a t i a l ( f ( r ) ) and t e m p o r a l ( g ( t ) ) components o f t h e a p p l i e d e l e c t r i c f i e l d . R i = 3 . 4 2 cm ( c o r r e c t e d f o r s u r f a c e t e n s i o n - s e e t a b l e 2.1); R =2.0 cm. , 2  65 •  60-  F i q 18. P l o t s o f s q u a r e o f s u r f a c e mode f r e q u e n c i e s vs. the s q u a r e o f t h e v o l t a g e a o p l i e d t o t h e e l e c t r o d e (V?) d i s t a n t D a b o v e t h e f l u i d s u r f a c e , ( a ) : (2,2) mode, D=0.170 cm; ( b ) : (0,3) mode, D=0.166 cm; ( c ) : (0,3) mode, D=0.220 cm; (4,2) mode; D=0.285 cm.  165  F i g 19. M o d i f i e d h i g h v o l t a g e wave g e n e r a t o r u s e d i n a p p l y i n g s u p e r c r i t i c a l e l e c t r i c f i e l d s t o t h e m e r c u r y s u r f a c e . I-IVE: s u r f a c e wave e x c i t a t i o n v o l t a g e ; HVC: s u p e r c r i t i c a l v o l t a g e ; C l and C2 a r e t h e two s p r i n g l o a d e d r o t o r c o n t a c t s , and FRS i s t h e fast relay switch.  (a) SURFACE INSTABILITY  F i g 20. F i l m r e c o r d o f u n s t a b l e s u r f a c e ( a ) , w i t h t h e d o t s o c c u r r i n g a t t h e o s c i l l o s c o p e t r i g g e r i n g r a t e s e r v i n g as time m a r k e r s ; (b) i s a l o g ( a m p l i t u d e ) v s . t i m e p l o t f o r t h e u n s t a b l wave d e m o n s t r a t i n g the e x p o n e n t i a l growth.  167  F i g 2 1 . A m p l i t u d e s q u a r e d ( V(03 ) v s . d r i v i n g f r e q u e n c y f , power r e s o n a n c e c u r v e f o r t h e (0,3) mode a t c o n s t a n t d r i ving amplitude of the e l e c t r i c f i e l d ; f i s the resonant frequency. Q  F i g 22. M e t h o d o f r e c o r d i n g s o u a r e wave, d r i v i n g v o l t a g e V ( t ) s i m u l t a n e o u s l y w i t h s i n u s o i d a l , s u r f a c e wave m o t i o n o n f i l m F . D B O S = d u a l beam o s c i l l o s c o p e s c r e e n ; UT=upper t r a c e (microwave r e s o n a n c e p e a k ) ; LT]_ a n d LT2 a r e t h e t w o l o v ; e r t r a c e p o s i t i o n s , c o r r e s p o n d i n g t o t h e t o p and t h e b a s e l i n e o f t h e a p p l i e d voltage.  169  0  2  4  6  (kV)  x  F i g 2 3 . D e p e n d e n c e o f d r i v e n s u r f a c e mode a m p l i t u d e ( a r b i t r a r i l y r e c o r d e d on t h e o s c i l l o g r a m ) on t h e s q u a r e o f t h e a m p l i t u d e o f t h e v o l t a g e a p p l i e d t o t h e d r i v i n g e l e c t r o d e , (a) f o r t h e (0,2) a n d (b) f o r (0,3) mode r e s p e c t i v e l y .  1 70  F i g 24. D i r e c t m e a s u r e m e n t o f s u r f a c e wave a m p l i t u d e . N = n e e d l e ; T M O t r a v e l l i n g m i c r o s c o p e c a r r i a g e ; L=lamp t o i n d i c a t e e x a c t moment o f c o n t a c t o f t h e n e e d l e t o t h e m e r c u r y s u r f a c e .  (b)  (a)  F i g 25. V a r i o u s e l e c t r o d e s h a p e s u s e d f o r p r e f e r e n t i a l s u r f a c e mode e x c i t a t i o n o n a c y l i n d r i c a l m e r c u r y s u r f a c e , (a) e x c i t e s (0,U) modes, (b) e x c i t e s (1,U) modes, (c) i s u s e d f o r t h e e x c i t a t i o n o f (2,U) modes, and (d) was u s e d i n t h e e x c i t a t i o n o f t h e (4,1) mode.  0  10T  ,  20T  30T  40T  50T  60T  F i g 26. L o g ( a m p l i t u d e ) v s . t i m e f o r (1,2) a n d (4,1) modes o b t a i n e d by u s i n g e l e c t r o d e s 25b a n d 25d r e s p e c t i v e l y . P l o t s g i v e d a m p i n g f r e q u e n c i e s f o r t h e two modes; d a m p i n g f r e q u e n c y u s e d t o d i f f e r e n t i a t e b e t w e e n t h e s e d e g e n e r a t e modes. Wave p e r i o d T=155 ms.  172  F i g 27. Damping f r e q u e n c y , c r , v s . s q u a r e r o o t o f s u r f a c e mode f r e q u e n c y , f. . The d e c r e a s e i n f i s c a u s e d by a s t r o n a e l e c t r o s t a t i c f i e l d a p p l i e d normally t o the o s c i l l a t i n g f l u i d surface, ( a ) : (0,2) mode', w i t h kR-^3/83; (b) : (0,3) mode,' w i t h kR =7.02. R l = c o n t a i n e r r a d i u s = 3 . 4 2 cm ( c o r r e c t e d f o r s u r f a c e t e n s i o n - s e e t a b l e 2 . 1 ) . I n t h e deep f l u i d a p p r o x i m a t i o n , c r q i v e n by t/x  1  y  where  V =kinematic  viscosity  o f mercury=0.11  cP.cm .g"'. 3  F i g 29. C o m p a r i s o n o f e x p e r i m e n t a l r e s u l t s and t h e o r e t i c a l p r e d i c t i o n s f o r t h e dependence o f 7 = 'Iffi/D the voltage V applied to the d r i v i n g electrode, (a) f o r a " f r e s h ' and (b) f o r an " o l d " m e r c u r y surface. o  n  :  Q  r  -PS.  17 5  F i g 30. V o l t a g e a p p l i e d t o t h e wave e x c i t a t i o n e l e c t r o de ( u p p e r t r a c e ) , and t y p i c a l s e c o n d a r y mode w a v e f o r m a t x=L/4. 0.5 kV/cm v e r t i c a l l y f o r u p p e r t r a c e , a n d 0.05 V/cm f o r l o w e r t r a c e ( 0 . 1 V e q u i v a l e n t t o 7 . 6 x l 0 * * cm d i s p l a c e m e n t a t t h e f l u i d s u r f a c e ) ; 0.5 s/cm o n horizontal axis. ?  F i g 3 1 ( a ) . D e p e n d e n c e o f p r i m a r y mode ( w a v e l e n g t h = 9 7 . 8 cm) a m p l i t u d e , A ] , on a m p l i t u d e o f v o l t a g e a p p l i e d t o e x c i t a t i o n e l e c t r o d e , V . ( o ) : m e a s u r e m e n t s a t x=L/8; (+): m e a s u r e m e n t s a t x=0. Q  BIBLIOGRAPHY  A b r a m o w i t z , M. a n d I . A . S t e g u n , 1 9 6 5 . H a n d b o o k o f m a t h e matical f u n c t i o n s (Dover P u b l i c a t i o n s , I n c . , N.Y. ) . A b r a m s o n , H.N. ( E d i t o r ) , 1 9 6 6 . T h e d y n a m i c b e h a v i o r o f l i q u i d s i n m o v i n g c o n t a i n e r s . NASA S P - 1 0 6 ( S c i e n t i f i c a n d T e c h n i c a l D i v i s i o n , NASA, Washi n g t o n , D.C.). Ackerberg,  R.C., 1 9 6 9 .  P r o c . Roy. S o c . A , 3 J _ 2 , 1 2 9 .  B e n j a m i n , T . B . a n d F. U r s e l l , 1 9 5 4 . 225, 505.  P r o c . Roy. S o c . A ,  B l o e m b e r g e n , N. , 1 9 6 5 . N o n l i n e a r o p t i c s (W.A. I n c . , New Y o r k , A m s t e r d a m ) .  Benjamin,  B u r d o n , R.S., 1 9 4 9 . S u r f a c e T e n s i o n a n d t h e S p r e a d i n g o f L i q u i d s (Cambridge U n i v e r s i t y P r e s s ) . J . 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A, Z]_8,  McEwan, 1965.  8,  441.  44.  J . F l u i d M e c h . 22,  1.  T h o m s o n , W. ( L o r d K e l v i n ) , 1872. R e p r i n t o f p a p e r s on e l e c t r o s t a t i c s and m a g n e t i s m ( M c M i l l a n and C o . , London). T s y t o v i c h , N., 1970. P r e s s , New  N o n l i n e a r e f f e c t s i n plasma York, London).  (Plenum  W a l d r o n , R.A., 1969. T h e t h e o r y o f w a v e g u i d e s and c a v i t i e s ( G o r d o n and B r e a c h S c i e n c e P u b l i s h e r s , New Y o r k ) . Wolf,  G.H.,  Y i h , C-S.,  1969. 1968.  Z. P h y s i k . , 227_, Phys.  F l u i d s , 1J_,  291. 1447.  Z i m o n , A.D., 1969. A d h e s i o n o f d u s t and p o w e r P r e s s , New Y o r k , L o n d o n ) . Z r n i c , D.S.  and C D . 618.  H e n d r i c k s , 1970.  Phys.  (Plenum  Fluids,  1_3,  APPENDIX  A  RADIAL DISTRIBUTION OF NON-UNIFORM ELECTRIC FIELD ON THE MERCURY SURFACE  The m o s t c o n v e n i e n t field  way o f m e a s u r i n g  d i s t r i b u t i o n o n t h e m e r c u r y s u r f a c e was  the  radial  t o adapt  L o r d K e l v i n ' s (1872) a t t r a c t e d d i s c e l e c t r o m e t e r i n an analog  system.  T h i s e l e c t r o m e t e r uses the f o r c e o f a t t r a c -  t i o n b e t w e e n two  p l a t e s held at d i f f e r e n t p o t e n t i a l s t o  o b t a i n an e s t i m a t e and  o f the p o t e n t i a l d i f f e r e n c e between them,  hence i t i s i d e a l  f o r our p u r p o s e s ,  since i t gives  d i r e c t measurements o f the e l e c t r o s t a t i c s t r e s s a t each charged  s u r f a c e , -in c o n t r a s t t o t h e more c o n v e n t i o n a l  (but  in this  case i n d i r e c t ) method o f u s i n g an e l e c t r o l y t i c  tank. The  a r r a n g e m e n t i s shown i n F i g . A l .  An a n a l o g  o f t h e e l e c t r o d e and m i c r o w a v e c a v i t y a r e made o u t o f b r a s s , and p l a c e d " u p s i d e down" u n d e r a s m o o t h b r a s s p l a t e w h i c h simulates weight  the mercury s u r f a c e .  o fdiameter  larger diameter  0.6 cm p a s s e s  in this plate. 182  A s m a l l , d i s c shaped through  a hole of  The w e i g h t  brass slightly  i s suspended  LEVELLING SCREW  GRADUATED ROD  ©P  CAVITY ELECTRODE ANALOG  c  F i g A l . Analog experiment f o ro b t a i n i n g surface.  FLUID SURFACE "ANALOG IDLZ  the electrostatic  field  normal t o t h e mercury J  CO CO  184  f r o m one reading  arm  of a conventional  t o an a c c u r a c y  l e v e l l i n g screws (see p l a t e and  o f 0.1  microbalance, A simple  mg.  F i g . A l ) enables  move t h e b r a s s  capable  of  arrangement  of  us t o r a i s e t h e  top  r i n g s i m u l a t i n g the c a v i t y w a l l s ,  so t h a t t h e s m a l l w e i g h t c o u l d e s s e n t i a l l y be made t o traverse r a d i a l l y across  the c a v i t y .  In t h i s way,  by  t r a v e r s i n g a d i a m e t e r o f t h e c a v i t y i n s t e p t s o f 1 mm, measuring the excess  w e i g h t A w of the brass  weight  and  caused  by t h e e l e c t r o s t a t i c a t t r a c t i o n o f t h e w e i g h t by t h e  elec-  t r o d e , a d i r e c t measurement of the r a d i a l p r o f i l e of  the  s q u a r e o f t h e e l e c t r o s t a t i c f i e l d was normalized two Fig.  to the u n i f o r m  smooth e l e c t r o d e s  field  made.  T h i s i s shown  t h a t would e x i s t between D apart  a distance  f o r D — 2 . 5 mm  in  A2.  The  f e a s i b i l i t y of the technique  s t r a t e d by m e a s u r i n g A w the v o l t a g e  V  f i r s t demon- ,  as a f u n c t i o n o f t h e s q u a r e  of  a p p l i e d to the e l e c t r o d e at a f i x e d r a d i a l  0  p o s i t i o n of the brass is proportional The  was  to V  F i g . A 3 shows t h a t  weight. , as  0  d i f f i c u l t y with  expected. the technique  described  l i e s i n the f a c t t h a t the weight i s i n u n s t a b l e w h i c h makes f i n d i n g Kelvin eliminated  Aw  a rather d i f f i c u l t  this difficulty  to suspend the weight from.  Aw  We  task.  above  equilibrium, Lord  by u s i n g s o f t l i g h t  springs  used a s l i g h t l y d i f f e r e n t  E  .  ANALOG ®  EXPERIMENT  COMPUTATION  F i g A2. R a d i a l p r o f i l e o f t h e s q u a r e o f t h e e l e c t r o s t a t i c f i e l d at t h e m e r c u r y surface? f i e l d , a p p l i e d b y a n e l e c t r o d e o f r a d i u s 2.0 cm, d i s t a n t <~ 0.2 cm f r o m s u r f a c e ! T h e d i s t r i b u t i o n i s normalized to the uniform f i e l d b e t w e e n -two s m o o t h p l a t e s - - ' 0.2 cm a p a r t .  187  a p p r o a c h , w h i c h a l s o s p e e d s up t h e m e a s u r e m e n t s ably.  A small  peg  P ( F i g . A l ) was  t h e b a l a n c e so as t o p r e v e n t  the  consider-  f i x e d to the frame l e f t h a n d arm  of  from moving  beyond a p o i n t j u s t below the p o s i t i o n of u n s t a b l e  equil-  ibrium.  gradually  The  increased,  e x c e s s w e i g h t on t h e o p p o s i t e the very  point  s w i n g i n g to the o p p o s i t e  side.  was  till  then recorded  electric field.  the c r u c i a l r e g i o n  of high  Aw  to the square of  the  increase  negligible, especially in  electric field.  e x a c t l y t h e same v e r t i c a l p o s i t i o n w i t h  The  the mercury s u r f a c e .  respect This  s i n c e t h e e l e c t r o s t a t i c f o r c e on t h e b r a s s with  i t s v e r t i c a l p o s i t i o n with  m a n n e r shown i n F i g . A 4 ; vertical  respect  i n t h i s we  method  is  has  at  to  the  necessary,  weight  varies  to the p l a t e i n  have p l o t t e d  Aw  t o p p l a t e was  the brass  w e i g h t was  2 mm,  fixed voltage. and  The  t o an a c c u r a c y  of 0 . 0 0 1  cm.  of  arbitrary  p o s i t i o n on t h e s c a l e o f a t r a v e l l i n g m i c r o s c o p e of reading  at  thickness  the v e r t i c a l p o s i t i o n  m e a s u r e d r e l a t i v e t o an  the  vs.  p o s i t i o n of the weight r e l a t i v e to the p l a t e ,  a f i x e d r a d i a l p o s i t i o n and of the  Aw  in  the advantage t h a t the weight i s always suspended  plate simulating  started  excess weight  The  s l i g h t systematic  r e s u l t i n g f r o m t h i s m e t h o d was  was  at which the b a l a n c e  as p r o p o r t i o n a l The  arm  capable  188  mm  F i g A4. P l o t with respect Regions ( a ) , b e i n g above,  o f Aw v s . v e r t i c a l p o s i t i o n o f b r a s s w e i g h t t o the p l a t e s i m u l a t i n g the mercury surface. (b) and (c) c o r r e s p o n d t o t h e b o t t o m o f t h e w e i g h t i n s i d e and u n d e r t h i s p l a t e r e s p e c t i v e l y .  189  The measurements were a l s o c o n f i r m e d t a t i o n a l method. potential  Laplace's  equation  i n t h e c a v i t y was s o l v e d  method ( F o r s y t h e differential  equations  f o r the e l e c t r o s t a t i c by u s i n g  a n d Wasow ( I 9 6 0 ) ) ,  by a compu-  a relaxation  for solving  partial  by t h e m e t h o d o f d i f f e r e n c e s .  V i s the e l e c t r o s t a t i c p o t e n t i a l i n the c y l i n d r i c a l in the d i a m e t r i c a l  plane  o f F i g . 17, t h e n a t u r a l  If cavity,  approxi-  mations at a point ( r , z )  (A.l)  where h i s a small  f i n i t e length,  lead to the d i f f e r e n c e  equation  (A.2)  when s u b s t i t u t e d  in  L a p l a c e ' s e q u a t i on  r  0  (A.3)  190  By m a k i n g h t h e u n i t s i d e o f a s q u a r e mesh i n t h e r - z p l a n e , equation  ( A . 2 ) c a n be p u t i n t h e i t e r a t i o n f o r m  V[(^h)h]  -  ~  f \ / [ ( ( ^ - h }  + V(((^<•  K)h]  Oj  ^  (A.4)  ( w h e r e t h e i n t e g e r s m and n r e f e r t o t h e (m,n) c o r n e r square mesh). standard  This  c a n be s o l v e d  V = V  e  on t h e c o m p u t e r w i t h a  p r o g r a m by i n c o r p o r a t i n g  V = .0 on t h e m e r c u r y s u r f a c e  the boundary  and r e s o n a t o r  on t h e c h a r g e d e l e c t r o d e .  field  strength.  conditions  w a l l s , and  Then t h e v e r t i c a l  o f V c a n be t a k e n a t t h e m e r c u r y s u r f a c e electric  i n the  gradient  to give the applied  T h i s was d o n e i n o u r c a s e by  using  a s q u a r e mesh o f u n i t s i d e h = 0.1 cm i n t h e c a v i t y o f d i a m e t e r ^ 7 .3 cm. strength in  applied e l e c t r i c field  was t h e n s q u a r e d a n d p l o t t e d as t h e d o t t e d  F i g . A2 ( n o h o l e  computed  The computed  i n the electrode).  line  I t i s seen that the  and measured r a d i a l p r o f i l e s o f t h e e l e c t r o s t a t i c  s t r e s s are i n very  good  agreement.  APPENDIX B  E X P E R I M E N T A L D I F F I C U L T I E S AND P R E C A U T I O N S  IN  MEASURING S U R F A C E WAVE DAMPING  In S e c t i o n 4.2.1 mercury surfaces  we p o i n t e d  o u t t h e two t y p e s  ( i . e . a " f r e s h " and a n " o l d " s u r f a c e )  i n measurements i n v o l v i n g the damping f r e q u e n c y modes.  of used  of surface  In t h i s a p p e n d i x we d i s c u s s i n a q u a l i t a t i v e m a n n e r  the p h y s i c a l c h a r a c t e r i s t i c s o f f l u i d s u r f a c e s t h a t r i s e t o t h e v e r y i n c o n s i s t e n t and u n p r e d i c t a b l e  behaviour  of t h e i r v i s c o e l a s t i c properties i n experiments. experience  give  Our  own  h a s b e e n w i t h m e r c u r y , w h i c h i s an e s p e c i a l l y  difficult  liquid  t o work w i t h  (besides  i t s emission  poisonous  fumes) i n t h i s r e s p e c t , b u t most o f o u r  of conclusions  a r e a p p l i c a b l e t o low v i s c o s i t y f l u i d s i n g e n e r a l . I t i s a w e l l known f a c t t h a t m e a s u r e m e n t s o f l i q u i d v i s c o s i t y by o b s e r v i n g waves a r e b o t h  the damping c o e f f i c i e n t o f s u r f a c e  v e r y e r r a t i c and u n r e l i a b l e , a n d c o n s i s t e n t l y  h i g h e r t h a n v i s c o s i t y m e a s u r e m e n t s made i n t h e b u l k o f t h e  191  192  fluid  by o t h e r m e t h o d s .  mechanical  The reason  f o r this i s that the  properties of a free fluid  v i s c o s i t y and s u r f a c e t e n s i o n ) minute traces of contaminants  surface (basically  a r e very s u s c e p t i b l e to even which i n e v i t a b l y f i n d t h e i r  way t o t h e s u r f a c e  (see Burdon (1949),  Langmuir (1961)).  V i s c o s i t y and s u r f a c e t e n s i o n o f a f r e e  fluid due  Zimon (1969) and  s u r f a c e may c h a n g e by as much as 2 0 % , o r e v e n m o r e ,  to the presence  o f very t h i n adsorbed  films ofo i l ,  detergent  or other contaminants;  thickness  ( a m o n o l a y e r ) may h a v e v e r y a d v e r s e  presence  o f d u s t and s m a l l  the l i q u i d processes  even a f i l m o f monomolecu!ar effects.  The  amounts o f m a t e r i a l s o l u b l e i n  f u r t h e r worsens the s i t u a t i o n . involved are s t i l l  not f u l l y  The e x a c t  physical  understood.  In o u r c a s e , when m e r c u r y was l e f t s t a n d i n g i n the microwave c a v i t y f o r s e v e r a l days d u r i n g t h e course o f experiments,  t h e s i m p l e s t method f o r o b t a i n i n g a " f r e s h "  s u r f a c e was t h e o n e d e s c r i b e d i n S e c t i o n 4 . 2 . 1 . taken  f o r the t r a n s i t i o n from  face (around  The  time  the "fresh" to the "old" sur-  3 minutes) v a r i e d from measurement t o measure-  ment, and t h e damping f r e q u e n c i e s  f o r both  t h e " f r e s h " and  the " o l d " m e r c u r y s u r f a c e s were s c a t t e r e d a b o u t t h e c o r r e sponding  values  given i n Table  6.1.  T h e s c a t t e r was a b o u t  5% o f t h e v a l u e o f t h e a v e r a g e d a m p i n g f r e q u e n c y the  table.  quoted i n  193  An o b s e r v e d e f f e c t o f some i m p o r t a n c e was  that  the  a p p l i c a t i o n o f e l e c t r o s t a t i c f i e l d s o f a b o u t 20 kV/cm,  for  a p e r i o d o f a few s e c o n d s , t o an " o l d " m e r c u r y s u r f a c e  r e s u l t e d i n a r e d u c t i o n o f the damping removal the  of the f i e l d ;  the damping  value for a "fresh" surface.  t a t i o n s of the removal of  frequency just  after  frequency tended  towards  The m e c h a n i s m and  limi-  of s u r f a c e contaminants i n the form  s m a l l p a r t i c l e s o f d i a m e t e r a few m i c r o n s h a v e  been  i n v e s t i g a t e d b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y by  Zimon  ( 1 9 6 9 ) , and a r e r e p o r t e d i n a f a s c i n a t i n g C h a p t e r X i n this reference. T h e b a s i c model  f o r removal  of p a r t i c l e s from a  s u r f a c e by a s t r o n g e l e c t r o s t a t i c f i e l d  i s one i n w h i c h  p o l a r i z a t i o n of the p a r t i c l e s i n the f i e l d ,  together with  s o r p t i o n o f i o n s on t h e p a r t i c l e s u r f a c e b e c o m e s t r o n g enough field  t o d e t a c h some p a r t i c l e s f r o m t h e s u r f a c e . n o n - u n i f o r m i t y t e n d s t o make c l e a n i n g m o r e  by a d d i n g t a n g e n t i a l Zimon  detaching forces.  A spatial effective  I t i s s h o w n by  t h a t most e f f i c i e n t c l e a n i n g o c c u r s f o r p a r t i c l e s  a r o u n d 60 m i c r o n s i n d i a m e t e r . s u r f a c e by a l t e r n a t i n g f i e l d s  E x p e r i m e n t s on c l e a n i n g are also r e p o r t e d .  a method used f o r c l e a n i n g r a i l  the  This is  l i n e s ahead o f a t r a i n ( a t  l e a s t i n t h e S o v i e t U n i o n i t i s ) . We f e e l  that  quantitative  194  investigations actual  physical  i n t h i s area w i l l  throw more l i g h t i n t o t h e  processes involved  ( s e e f o r example  t i o n s i n C h a p t e r 8 ) , a n d may l e a d t o some v e r y industrial  applications  d i r t from the surfaces  (electrophotography,  sugges-  useful  removal o f  of molten p l a s t i c s , e t c . ) .  APPENDIX  C  A D D I T I O N A L C A L C U L A T I O N S ON THE N O N - L I N E A R D R I V I N G OF S U R F A C E WAVES BY S P A T I A L L Y PERIODIC  ELECTRIC  FIELDS  In t h i s a p p e n d i x we e x t e n d Chapter  NON-UNIFORM  the c a l c u l a t i o n s of  6 o f t h e c h a r a c t e r i s t i c s o f s u r f a c e waves n o n - l i n e a r l y  d r i v e n by e l e c t r i c  fields.  We s t a r t by p r e s e n t i n g  calcu-  l a t i o n s on t h e n o n - l i n e a r d r i v i n g o f w a v e s i n a r e c t a n g u l a r container. angle  T h e n we d e r i v e an e x p r e s s i o n  <j> b e t w e e n t h e e l e c t r i c f i e l d  f o r t h e ( 0 , 2 ) mode i n a c y l i n d r i c a l with  c a l c u l a t i o n s to assess  f o r the phase  a n d t h e s u r f a c e wave container.  We c o n c l u d e  the v a l i d i t y of our assumptions  of i g n o r i n g the c o n t r i b u t i o n s to the d r i v i n g e l e c t r i c r e s u l t i n g from  C.I.  the curvature  stress  and s l o p e o f t h e s u r f a c e .  N o n - L i n e a r D r i v i n g o f Waves i n a R e c t a n g u l a r  Container  T h i s p a r t i c u l a r g e o m e t r y o f s u r f a c e wave e x c i t a tion i s of s p e c i a l i n t e r e s t since the relevant i n t e g r a l s 195  196  c a n be p e r f o r m e d  a n a l y t i c a l l y , and s i n c e t h e r e s u l t s a r e  a p p l i c a b l e t o t h e same p r o b l e m on t h e s u r f a c e o f an i n c o m pressible fluid j e t . We a s s u m e t h a t t h e w a v e f r o n t s yz plane  of a cartesian co-ordinate  s y s t e m , and  a l o n g t h e x a x i s , as shown i n F i g . C I . are m h a l f wavelengths along the channel " i n f i n i t e " depth full  width  and width  of the channel,  so as t o a v o i d o v e r l a p for  the v e r t i c a l  b.  are parallel  propagate  We s u p p o s e  there  o f l e n g t h L,  Each e l e c t r o d e o c c u p i e s t h e  a n d h a s l e n g t h 2a ( a <  of the e l e c t r o d e s ) .  displacement  to the  of the f l u i d  We t h e n surface  L/2m, have from  e q u i 1 i b r i urn  (CI)  w h e r e k = (m TT) /L . F o l l o w i n g e x a c t l y t h e same p r o c e d u r e substitute this expression i n c l u d e a time a square  as i n S e c t i o n 6.1, we  into the equations  of motion,  p e r i o d i c e l e c t r i c f i e l d p r o d u c e d by a p p l y i n g  wave v o l t a g e on t h e e l e c t r o d e s , a n d i n t e g r a t e t o  obtain J  him)  (C2)  1  Z  F i g C l . G e o m e t r y o f s u r f a c e wave e x c i t a t i o n i n r e c t a n g u l a r c o n t a i n e r o f l e n g t h L . E n d e l e c t r o d e s s u c h a s e ( l e n g t h a) a r e w e i g h t e d b y .a f a c t o r o f 1/2 r e l a t i v e t o e l e c t r o d e s o f l e n g t h 2a.  97  198  where  1(1)  -  Cos(Kx)ol(Hx)  (  T h i s i n t e g r a l c a n be e x p r e s s e d  KT)  o  =  (  JL  5  -—•-  $m  mvroi  (C4)  S u b s t i t u t i n g t h i s i n t h e r i g h t hand s i d e o f e q u a t i o n we p l o t  F/X(Tj  i n F i g . C2.  f o r various  values  F i g . C3, w h i c h i s d e r i v e d  the r e l a t i o n between  T  ( i . e . when  o f ma/L  as shown  t h e s u r f a c e t o become  is  c  unstable  =" 0. 6 ) . of  g e o m e t r y h a s e s s e n t i a l l y t h e same f e a t u r e s  that of c y l i n d r i c a l  C.II.  gives  where V  T h e a b o v e c a l c u l a t i o n s show t h a t t h e c a s e rectangular  (C.2)  f r o m F i g . C2,  and ma/L,  the v o l t a g e which induces  as  geometry.  N o n - L i n e a r Phase S h i f t In S e c t i o n 6.1, i n o r d e r  t i o n s , we a s s u m e d  >  a n a l y t i c a l l y as  lotft f«H  TO-P)"*  c  to s i m p l i f y the c a l c u l a -  t h a t t h e p h a s e d i f f e r e n c e (j> b e t w e e n  the  3  )  199  F i g C2. "J/Kf) d e p e n d e n c e on J f o r r e c t a n g u l a r g e o m e t r y , a: ma/L= 0.25; b : ma/L=0.167 o r 0.333; c: ma/L=0.12 5 o r 0.375; d : ma/L= 0.083 o r 0.417.  1/1(7)  F i g C3. 7/1(7) dependence on ma/L a t J =0.6 (onset o f i n s t a b i l i t y ) . T h i s graph c a n be used t o g i v e t h e v o l t a g e n e c e s s a r y t o induce i n s t a b i l i t y . 7/1(7) (and hence t h e c r i t i c a l v o l t a g e ) tends t o i n f i n i t y f o r a=0 and ma/L=0.5; i n t h e l a t t e r case t h e e l e c t r i c f i e l d i s u n i f o r m , and o u r t h e o r y breaks down.  201  driving  electric field  a n d t h e s u r f a c e wave was 7T/2 a t  r e s o n a n c e , even i n t h e n o n - l i n e a r regime. as a f i r s t  a p p r o x i m a t i o n by e x p e r i m e n t .  In t h i s  appendix  c a l c u l a t e a v a l u e f o r cp .  we s h a l l  We w r i t e and e x p a n d  <f — JT +• c f  equation  6 Vo k X  , assume t h a t  S  i s small,  (6.8a) o f S e c t i o n 6.1, i . e .  7 Msi*>9zcixot9  ft  C  to  T h i s was j u s t i f i e d  ( C  0  '  5 )  obtain  f_  C Vo  f  (  J (zUi»9zJ*JB  (C.6)  0  We e x p a n d t h e d e n o m i n a t o r o f t h e i n t e g r a l ,  to obtain  202  i n t e g r a t e term by term  a n d sum t h e f i n a l  series (using  f o r m u l a 5 1 5 , p. A 1 7 1 , o f "The Handbook o f C h e m i s t r y and Physics"  £ „  The r e s u l t i s  (1965)J.  TrC  Vo f C~* A  J  1^  [c  J p V W z  %  [i-T*J>)] * v  E l i m i n a t i n g t h e c o e f f i c i e n t o f t h e i n t e g r a n d by u s i n g  JL  - -^Jl.  we c a n e x p r e s s /  (equation  as a f u n c t i o n o f J  (6.9))  and t h e e l e c t r o d e  geometry  K D  *  L  J  T h i s f u n c t i o n i s p l o t t e d i n F i g . C4 f o r o u r e x p e r i m e n t a l conditions.  I t i s a very important  and s u g g e s t i o n s  experimental  parameter,  f o r i t s measurement a r e given i n Chapter  8.  8  )  203  F i g C 4 . Dependence o f ^ on f f o r t h e ( 0 , 2 ) mode i n a c y l i n d r i c a l c o n t a i n e r . Phase between d r i v i n g e l e c t r i c s t r e s s and s u r f a c e mode =<f = cf .  204  C.III.  D i s t o r t i o n of E l e c t r i c The  F i e l d by S u r f a c e  Waves  purpose of t h i s c a l c u l a t i o n i s to  demonstrate  t h e v a l i d i t y o f o u r a s s u m p t i o n s o f S e c t i o n 6.1 form of the e l e c t r i c f i e l d Although  at the f l u i d  the e x p r e s s i o n  for  the  surface.  f o r the e l e c t r i c  potential  used i n the c a l c u l a t i o n s does s a t i s f y the boundary tions  (V = V  Q  condi-  on t h e e l e c t r o d e , z = D; V = 0 on t h e  fluid  z = 3* , as shown i n F i g . 1 7 ) , i t d o e s n o t s a t i s f y  surface Laplace's  equation.  s o l u t i o n of Laplace's conditions  where O  For axisymmetric equation  surface waves, a  which s a t i s f i e s the  boundary  is  i s t h e o p e r a t o r JL  i L tr  *  and  £  is a  f u n c t i o n o f r w h i c h i s i n c l u d e d so t h a t V = 0 a t z = T t h e wave  surface. ' The  s o l u t i o n (C.9)  mation to V i n the r e g i o n electrode ments  J  ,  (R^+  D >  r >  will  not p r o v i d e  a good  under the edge o f the - D).  driver  However, i n our  i s p r a c t i c a l l y zero i n t h i s region.  approxi-  Hence  experiquite  205  large errors i nV will  n o t a f f e c t t h e computed s t r e s s , s i n c e  very  e n e r g y i s f e d i n t o t h e s u r f a c e wave  little  electrical  near the displacement  nodes.  To d e t e r m i n e in equation z = T  £  we r e t a i n t h e f i r s t  three  (C.9), l i n e a r i z e with respect t o £  s o t h a t V = 0. T h i s p r o c e d u r e  and s e t  gives  v [_dL (£^2 o(P-rr'] J  0=  The  o  a t i o n o f (C.9) The  with  a t the surface i sobtained  +  respect to £  <7r»-x-f  From t h i s , b y u s i n g e q u a t i o n  Now,  -  ,0)  by d i f f e r e n t i -  r e s p e c t t o z, and again s e t t i n g z = T  r e s u l t , l i n e a r i z e d with  £ = V o [ —  (c  +  electric field  terms  , is  o-n oo>-r;-'] a  (C.10) t o e l i m i n a t e  - - i ^ ^  .  O i D - T ) " ]  (cl1)  £ we g e t  (c.i2)  206  0(D-rrWo-7r\L and  The  2L  f o r r o t a t i o n - a l l y symmetric  y\  7 =  With  i .r  oXJ  the l a r g e s t  =  -  D  (c  2)  (C.14)  0  ( IT"!  .,  modes  (C.12) reduces t o  value o f the correction  o c c u r s when  2  J (Krj  these r e s u l t s , equation  largest  i K o - n ^ g )  cannot  value o f the correction  factors  containing k  exceed  D).  Hence  factor i s  (C.16)  3 (for r = 0).  Therefore, the correction  £  Vo  t>-r  i s  207  For our e x p e r i m e n t a l r e s u l t s (see T a b l e 6.1), s o t h a t t h e maximum p e r c e n t a g e i n S e c t i o n 6.1  i s less than  (kD)  e r r o r i n the s t r e s s 5.8%.  =  3.6%,  computed  APPENDIX D  A D D I T I O N A L  E F F E C T S  I N  SECOND  ORDER  WAVE-WAVE  I N T E R A C T I O N  The  purpose  of Chapter  7 was t o i n v e s t i g a t e t h e o -  r e t i c a l l y and e x p e r i m e n t a l l y second shallow water.  In t h i s a p p e n d i x  order i n t e r a c t i o n s on  we l i s t  other second  order  e f f e c t s t h a t so f a r have been n e g l e c t e d f o r t h e sake o f s i m p l i f i c a t i o n , i n order not to bury the dominant in a p r o l i f e r a t i o n of a l g e r b r a i c equations. we s h a l l  effects  In g e n e r a l ,  show by o r d e r o f m a g n i t u d e c a l c u l a t i o n s t h a t  effects are n e g l i g i b l e .  D.I.  Effect of Surface The  Tension  surface tension T will  to t h e s t r e s s c o n s e r v a t i o n e q u a t i o n  208  contribute  (7.5)  terms  these  209  z  at the f l u i d s u r f a c e .  These  s m a l l e r than the dominant second  a r e a f a c t o r Tk  s t r e s s terms  order equations of motion  (7.10)).  (see equations  )  The c o r r e s p o n d i n g  i n t h e f r e q u e n c y OJ^  E f f e c t o f Water The w a t e r  t i o n o f 2CJ, a n d  shift  employed.  Depth  depth used i n t h e experiments ( s e e  S e c t i o n 7.2) was ^- 3 cm.  which  (7.9) and  o f t h e s u r f a c e modes i s  a l s o o f t h e same o r d e r f o r t h e modes  D.II.  o f t h e f i r s t and  H e n c e t h e i r c o n t r i b u t i o n ( s e e T a b l e 7.1) i s o n l y  about 0.1%, i . e . n e g l i g i b l e . ( T K * /f  terms  This produces  a fractional  separa-  g i v e n by  i s well within the frequency tolerance f o r e f f e c t i n g  n o n - l i n e a r r e s o n a n t d r i v i n g f o r t h e two modes  employed.  210  D.III.  Shift  o f t h e Nodal  Point o f the Primary  Mode  O u r m e t h o d f o r m o n i t o r i n g t h e wave a m p l i t u d e o f the  s e c o n d a r y mode i n v o l v e s o b s e r v a t i o n s t a k e n a t a f i x e d  point  (x = L/4, s e e F i g . 12) o f t h e c o n t a i n e r .  This  intro-  d u c e s a s m a l l v e r t i c a l p e r i o d i c s h i f t i n t h e o b s e r v e d wave due t o t h e c o r r e s p o n d i n g h o r i z o n t a l The v e r t i c a l  s h i f t o f the nodal p o i n t .  d i s p l a c e m e n t i s g i v e n by t h e s e c o n d o r d e r t e r m  ^cos(2k,)c)^-cos(au,b)j ( s e e e q u a t i o n  I g n o r i n g t h e s t a t i c t e r m , we h a v e a n a d d i t i o n a l  7-2{)  contribution  o f a m p l i t u d e A, " /4-H i n t h e s e c o n d a r y wave d i s p l a c e m e n t . 2  T h i s t e r m i s 77/2 o f p h a s e w i t h t h e r e s o n a n t l y d r i v e n s e c o n d a r y component g i v e n by e q u a t i o n ( 7 . 2 1 ) , i . e .  2  .  H  T h e a m p l i t u d e o f t h i s t e r m i s o n l y 2 / ( 3 Q ) , i . e . a b o u t Q% < x  o f t h e s e c o n d a r y mode a m p l i t u d e A<^ , s o i t c a n b e n e g l e c t e d .  D.IV.  Non-Linear  Mixing o f Disturbances o f Wavelength L  In t h i s s e c t i o n we s h a l l S(p,q,r) t o denote  use the n o t a t i o n  the s u r f a c e d i s p l a c e m e n t and s t r e s s  r e s p e c t i v e l y , w h e r e we d e f i n e  J(p,q,r)  211  or  J  and  p,q  and  r  To the  S =  are  (qo/,t)  cos  (rk,  x)  £^ ,  the  solution  for  J" w i l l  contain  terms  The  first  the  driving  Second the  1(1,1,1)  7(1  7(1,0,2)  J(l,l,2)  1(1,2,2)  = 0j  7(2,0,1) 7(2,0,2)  7(2,1,1) T(2,l,2)  7(2,2,1) 7(2,2,2)  7  terms has  no  terms  are  generated  terms,  from  which  is  the  affect  calculate since  these  second because ;  the  order both  L/4).  time  of  Hence  with  in  they  have only  vanish  part  of  the  o  r  d  e  r  . o r d e r  t  t  because  2 OJ, .  products  S(1,1,0).  .  J  frequency  pairwise  terms  (first  However,  7(1,0,1) the  7(2,1,1)  these  of  = 07  The  of latter  surface  space.  of  involved  terms  by  observations.  amplitude  are  in  ,2,1)  J(l,2,2)  dependent  independent  the  and  components  together  constant  Time not  J^(l , 2 , 1 )  stress  order  arises  stress  1(1,0,1)  order  order  first  term  (x  cos  integers.  order  5(1,1,0)  do  £  (and  non-linear  and  nodes cause  7(2,2,1) at  the  column of it  is  necessary  perhaps  be  The" neglected,  observation  vertical  to  J*(l>0,2)),  terms. can  matrix)  point  oscillations  which  21 2  are  a factor  £  s m a l l e r than the c o n t r i b u t i o n s of the  J ( 1 , 1 , 1 ) mode.  As i s s h o w n a b o v e , J ( 1 , 1 , 1 ) c o n t r i b u t e s  l e s s t h a n 4% t o t h e o b s e r v e d d i s p l a c e m e n t a t x = L / 4 . If the d r i v i n g e l e c t r o d e i s very c a r e f u l l y des i g n e d , then  JT(1 ,0,2) a n d .7(1,1,2) s h o u l d be v e r y s m a l l  i n d e e d , b e c a u s e t h e geometry o f t h e e l e c t r i c f i e l d and s u r f a c e mode a r e m i s m a t c h e d . initially  that  7(1,0,2) =  We s h a l l  t h e r e f o r e assume  J ( l , l , 2 ) = 0.  J ( 2 , 1 , 2 ) as t h e r e m a i n i n g s e c o n d  order c o r r e c t i o n ; i t  i s g e n e r a t e d by t h e m i x i n g o f 7 ( 1 , 0 , 1 ) a n d  D.V.  This leaves only  J(1,1,1).  M i x i n g o f Modes o f Z e r o F r e q u e n c y a n d F r e q u e n c y oj  t  The  f i r s t stage i n the c a l c u l a t i o n i s to determine  the f i r s t order s t a t i c deformation.  This i s readily  seen  (from e q u a t i o n (4.11) g i v i n g t h e s t a t i c d e f o r m a t i o n ) t o be  (D.l) r,(1,0,1) w h e r e A,  ~A  COU(K,H)  i s t h e a m p l i t u d e o f t h e r e s o n a n t l y d r i v e n mode  X ( l , 1 , 1 ) and Q  i s t h e q u a l i t y f a c t o r o f t h e same mode  213  To c o m p u t e  J ( 2 , l , 2 ) , the second order  of m o t i o n and k i n e m a t i c and  boundary c o n d i t i o n (equations  ( 7 . 1 2 ) ) h a v e t o be m o d i f i e d  containing  equation  t h e s t a t i c mode  to include non-linear  1(1,0,1).  (7.10) terms  F o r c o m p l e t e n e s s we  a l s o a s s u m e t h a t t h e s u r f a c e s t r e s s S i s a f f e c t e d by t h e s u r f a c e waves, and w r i t e  (3.2)  where iff  and  f ( x ) = 1, j x - || <  = 0,  The  factor containing  trical  t>  %  J^/D  \  * - k \ > k  i s introduced  because the e l e c -  s t r e s s v a r i e s i n v e r s e l y as t h e s q u a r e o f t h e d i s t a n c e  between t h e d r i v i n g e l e c t r o d e and t h e f l u i d s u r f a c e ( s e e S e c t i o n 6 . 1 ) . The f a c t o r f assumes t h a t t h e e l e c t r i c a l vanishes  except  under the e x c i t a t i o n e l e c t r o d e .  stress  214  The tial  0  (f^K  e q u a t i o n s f o r the second  and v e r t i c a l  order velocity  displacement  T-z  poten^  become  <f>  (D.3) +  5'  n  H  I  M  J  (D.4)  dz*  dz  wh.ere C ~ ( t ) i s a g a i n an i n t e g r a t i o n f u n c t i o n . nate  BT^/dt  from e q u a t i o n  (p  lV  (D.3)  and  We  now  write  = Cf ( 2 , 2 , 2 ) + cp (2,1 ,2)  T h i s l e a d s t o t h e f o l l o w i n g e q u a t i o n f o r Co (2,1 ,2) ,  elimi  215  (D.5)  S u b s t i t u t i n g f o r 7(1,0,1), i . e .  from e q u a t i o n  9 ' =  (D.l)  and  ^  k,f/U(K,H)  from e q u a t i o n  ( 7 . 1 3 ) shows  <*r (w>fJL  =  z  The  . Hi—I.  term  T( 2  that  -LJ—.  1 ,0 ,1) jL dz^  cos (iv, t;  /ilt-i ) & ~k  '  becomes  216  (D.6)  A g a i n t h e time d e p e n d e n t terms righthand side o f equation tion  and  collection  of  p a r t i a l l y cancel C^(t).  (D.5)  terms  of  becomes, a f t e r frequency  OJ  The  simplifica-  f  (D.7)  +  S  0  (terms  of  order  c a n be e x p r e s s e d  1/Q,  smaller)  i n terms  of A  order equation o f motion i.e. equations the second  (7.9)  (  by s o l v i n g t h e f i r s t  and kinematic boundary c o n d i t i o n ,  a n d ( 7 . 1 1 ) i n t h e same way we s o l v e d  order equations, to get  A  (cf. also equation  (4.11)).  (D.8)  217  S u b s t i t u t i n g these  expressions  into equation  neglecting corrections o f order final  equation  (D.5) and  1/Q, a n d ( k , H )  gives the  f o r <jp(2,l,2)  (D.9)  1Q, (K,H)  1  w h e r e we a s s u m e t h a t  y(2,1,2)  (jp(2,l,2) i s o f t h e form  = (constant)  A term c o n t a i n i n g  e ^ ^ c o s h ( 2 k , z)cos(2k, x)  cos(26u»,t) i s n o t i n c l u d e d i n t h i s  sion but i s discussed  b e l o w , w h e r e we c o n s i d e r  t o t h e ( ^ ( 2 , 2 , 2 ) mode c a u s e d by t h e e l e c t r i c f i e l d . and  by n o n - l i n e a r  M u l t i p l y i n g both  expres-  modifications  d r i v i n g produced  s i d e s by c o s ( 2 k , x )  i n t e g r a t i n g from 0 t o L gives  CP{1,U2)  =:  c o i h K X k o d 1 2 ? ]  (D.10)  218  S u b s t i t u t i n g back  i n t o e q u a t i o n (D.4) g i v e s  (D.n)  9 QJ  H  L  znaj  w h e r e we h a v e o m i t t e d a v e r y s m a l l s p a c e i n d e p e n d e n t which  i s 1/Qj  times s m a l l e r than the c o r r e s p o n d i n g  in equation (7.21), which  term term  gives the amplitude of the  s e c o n d a r y mode as  /),  The  -  3  r a t i o of the amplitudes  ( e q u a t i o n (D.12))  Q*  A"  1  (  d  -  1  2  )  o f t h e r e s o n a n t l y d r i v e n mode  a n d t h e n o n - r e s o n a n t l y d r i v e n mode o f  e q u a t i on ( D . 1 1 ) i s  12  Q,Qxl  it T h e r e f o r e , second  ~f<>  (D.13)  H o r d e r modes w h i c h  are non-resonantly  d r i v e n h a v e n e g l i g i b l e a m p l i t u d e s , as we a s s u m e d i n t h e t h e o r y o f C h a p t e r 7.  219  D.VI.  Resonant Driving Field  equation  Cjp ( 2 , 2 , 2 ) i s o b t a i n e d t h a t we  Electric  Effects  The  way  o f S e c o n d O r d e r Modes by  obtained  which describes f r o m (D.3) the  t h e mode i s r e s o n a n t l y  and  expression  driven,  the  the  correction  (D.4)  i n the  f o r <^>(2,1,2).  to  same Since  result is  C o t ( l u s , i ) cot  (iK,X)  (D.14) -  j ^ *  A where  (^(2,2,2) denotes the  by n o n - l i n e a r the  equation  of the  driving stresses. by c o s ( 2 k , x ) and  t a n k ( L ) , and  T(ljljl)  using  5l  1  This  correction  Q,  (OS  to t^(2,2,2) produced  Multiplying  integrating  equation  [lus.e)  (D.4)  toi(lK,X)  both sides  over the  of  length  gives  ~  O.Z  J  (D.15)  p  displacement is ideally (i)  TT/2  out  of phase w i t h the  domi  7  nant term  J  discussion  in Section  , and  i t s e f f e c t i s f u l l y dealt with in 7.3.  the  220  N o n - R e s o n a n t D r i v i n g o f t h e F i r s t O r d e r Mode 7 ( 1 , 1 , 2 )  D.VII.  For i d e a l  e l e c t r o d e g e o m e t r y t h e o n l y mode t o b e  excited i n the linear approximation  is  J  (i.e. 1(1,1,1)).  However, t h e e l e c t r o d e geometry i s n o t i d e a l , and t o the appendix  complete  we now e s t i m a t e t h e l i n e a r e x c i t a t i o n o f t h e  7 ( 1 , 1 , 2 ) mode.  The e q u a t i o n f o r & ( ! , ! , 2 ) i s (from  above)  (D.16)  l.e  M u l t i p l y i n g both s i d e s b y c o s ( 2 k j x) and i n t e g r a t i n g o v e r the l e n g t h o f t h e tank enables of (^(1,1,2).  us t o d e t e r m i n e  the amplitude  Using the f i r s t order boundary c o n d i t i o n  r e s u l t s i n t h e r e q u i r e d e x p r e s s i o n f o r 7*0,1,2)  coi(oj t)cos(%k,x) 3 0 J t  3  Q,  (Q.17)  221  (the i n e q u a l i t y follows because t h e e l e c t r o d e  length  i s less  than L/2 and f ( x ) £ 1 ) .  Non-Linear  Mixing  of A r b i t r a r y  In A p p e n d i x  First  Order  D.V we s h o w e d t h a t t h e n o n - r e s o n a n t  d r i v i n g o f t h e 7 ( 2 , 2 , 2 ) mode b y m i x i n g "7(1,0,1) i s n e g l i g i b l y s m a l l . stress i sproportional included  Modes  J ( l , l , l ) and  Int h i s case t h e d r i v i n g  to Aj^/Qj . I f 7(1,1,1) i s not  i n t h e mixed modes, t h e d r i v i n g s t r e s s i s r e d u c e d  to (A,/Q ) ;  . T h e r e i s t h e r e f o r e no n e e d t o d o d e t a i l e d  c a l c u l a t i o n s f o r o t h e r m o d e s , b e c a u s e t h e y w i l l make r e c t i o n s which a r e 1/Q and  7(1»1»1)  (  cor-  time s m a l l e r than t h e 7(1,0,1)  corrections.  APPENDIX  DAMPING ON END WALLS OF OF  E  A RECTANGULAR CONTAINER  WIDTH S M A L L COMPARED TO  ITS  In S e c t i o n ( 4 . 1 2 ) we n e g l e c t e d  LENGTH  the damping at  the end w a l l s o f a r e c t a n g u l a r c o n t a i n e r whose w i d t h much s m a l l e r t h a n  i t s length.  t h i s f o r the case  used  was  I n t h i s a p p e n d i x we j u s t i f y  i n our experiments  on s h a l l o w  water  waves. The v i s c o u s d i s s i p a t i o n a t a w a l l i s p r o p o r t i o n a l to i t s a r e a , and t h e square  o f the. t a n g e n t i a l f l u i d  Hence, i n the n o t a t i o n o f S e c t i o n  E_  E  E  S  T h u s , when t h e w i d t h its  &  Hb  ^  T* HL  (4.12)  K H b a  x  L  b o f t h e c o n t a i n e r i s much s m a l l e r  l e n g t h L, and e s p e c i a l l y i n t h e case  waves where (kH) «  velocity.  of shallow  water  1, t h e d i s s i p a t i o n a t t h e e n d w a l l s  222  than  223  is negligible.  In o u r c a s e o f w a v e s o f w a v e l e n g t h  100 em on w a t e r o f d e p t h 3 cm, t h e r a t i o i s  about  2 x 10"  •a J  .  APPENDIX F  EXCITATION  OF SURFACE MODES BY S P A C E AND  T I M E PERIODIC  The  FIELDS  purpose of t h i s appendix i s to demonstrate a  f u r t h e r i m p r o v e m e n t o f t h e wave e x c i t a t i o n t e c h n i q u e s p a t i a l l y non-uniform e l e c t r i c f i e l d s . in Sections simplest  In o u r e x p e r i m e n t s  4.2 a n d 7 . 2 , t h e e l e c t r o d e s h a p e s w e r e t h e  possible f o r resonant  e x c i t a t i o n o f s u r f a c e modes.  T h i s r e s u l t e d i n some m i n o r d i f f i c u l t i e s at second order  e f f e c t s on s h a l l o w  water;  d r i v i n g o f s e c o n d a r y modes by m i x i n g primary This  modes ( s e e S e c t i o n  c a n be a v o i d e d  (1 + c o s ( k x ) )' /x  by d e s i g n i n g  specifically,  of spuriously  excited  t h e e l e c t r o d e s o as t o proportional to  where k i s t h e wavenumber o f t h e d e s i r e d  Thus t h e s u r f a c e s t r e s s w i l l  1 + cos(kx),  i n t r y i n g to look  7.3 a n d A p p e n d i x D f o r d e t a i l s ) .  carry a s p a t i a l l y dependent voltage  mode.  by  and t h e o r t h o g o n a l i t y  be p r o p o r t i o n a l t o o f t h e modes i n t h i s  s u r f a c e wave s p e c t r u m g u a r a n t e e s t h e e x c i t a t i o n o f t h e mode w i t h wavenumber k o n l y .  The f e a s i b i l i t y o f t h i s i n p r a c t i c e  224  225  was  d e m o n s t r a t e d as f o l l o w s , u s i n g t h e r e c t a n g u l a r  container of F i g .  water  9.  The wave e x c i t a t i o n e l e c t r o d e o f F i g . 9 was p l a c e d by an e l e c t r o d e made up o f f i f t y c o p p e r 1.6  width  cm,  separated  by a b o u t 0.3  l e n g t h o f t h e t a n k ) l o n g , 5 cm  p l e x i g l a s s b a r as s h o w n i n F i g . F l .  values  chosen to give a v o l t a g e d i s t r i b u t i o n + cos(kx))'^  When t h e f r e q u e n c y  (the  s t r i p s , and  the  con-  resistor  proportional  along the length of the  electrode.  o f t h e e l e c t r i c f i e l d was  set equal  the n a t u r a l frequency was  cm  R e s i s t o r s were  between a d j a c e n t  (1  of  of the tank) wide  nected  to  copper  strips  crn on a 97.8  (the width  re-  to  o f t h e mode o f w a v e n u m b e r k, t h e mode  e x c i t e d on t h e f l u i d  surface.  226  F i g F l . S e c t i o n o f a s p a t i a l l y F o u r i e r a n a l y z i n g wave e x c i t a t i o n ' e l e c t r o d e . C S = C o p p e r s t r i p s . P,^=nth r e s i s t a n c e i n t h e r e s i s t o r chain. Resistances chosen so t h a t t h e v o l t a g e dependence along the e l e c t r o d e i s p r o p o r t i o n a l t o (1+cos(kx) j ' ^ , where k i s t h e wavenumber o f t h e d e s i r e d s u r f a c e mode.  •3  APPENDIX  G  DAMPING OF A PRIMARY MODE BY THE PRESENCE OF A SECONDARY MODE - A P O S S I B L E  STABILIZATION  TECHNIQUE  S t u d i e s o f n o n - l i n e a r w a v e - w a v e i n t e r a c t i o n on shallow water  ( S e c t i o n s 7.1 a n d 7.2) h a v e d e m o n s t r a t e d  that  e x c i t a t i o n o f a s e c o n d a r y mode by s e l f i n t e r a c t i o n o f t h e p r i m a r y one i s p o s s i b l e .  I n t h i s way, e n e r g y i s f e d f r o m  t h e p r i m a r y t o t h e s e c o n d a r y mode. d u c e d by t h i s e f f e c t i s s u f f i c i e n t l y  I f t h e damping  intro-  l a r g e , then t h i s  would  be a g o o d way o f s t a b i l i z i n g t h e p r i m a r y mode, s h o u l d i t b e c o m e u n s t a b l e due t o a m e c h a n i s m t h a t f e e d s e n e r g y t h e p r i m a r y mode a l o n e . damping  into  I n t h i s a p p e n d i x we c a l c u l a t e t h e  o f t h e p r i m a r y mode by t h e s e c o n d a r y . From e q u a t i o n ( 4 . 2 2 ) , i n t h e s h a l l o w f l u i d a p p r o x i -  m a t i o n , t h e e n e r g y i n t h e s t a n d i n g waves i s  E=JLbHL(KcO* = - £ - b H L ( ^ f 227  (3.1)  228  S i n c e f o r s h a l l o w modes OJ/K = c o n s t a n t , t h e p r i m a r y a n d secondary  mode e n e r g i e s a r e  -  E  f  CA?  ;  E  a  =: Cfi£  (G.2)  where C i s a c o n s t a n t o f p r o p o r t i o n a l i t y . electrical  Assuming a l l t h e  e n e r g y i s s u p p l i e d t o t h e waves t h r o u g h  the p r i -  m a r y m o d e , we c a n w r i t e t h e e q u a t i o n o f c o n s e r v a t i o n o f energy as  B V  0  A  l  =  ^ C A . + ^ - C A i  where B i s a l s o a p r o p o r t i o n a l i t y c o n s t a n t .  ( G  -  3 )  T h i s c a n be  simplified to  (G.4) A,  1  where m i s another  V  /q  <  J  c o n s t a n t o f p r o p o r t i o n a l i t y (m h a s a v e r y  weak d e p e n d e n c e on t h e e l e c t r i c f i e l d slight  strength through a  c h a n g e i n t h e s u r f a c e mode f r e q u e n c y  average s t a t i c f i e l d  - s e e S e c t i o n 3.2.3).  caused  by t h e  In our case,  w i t h A^/A, .—- 1 0 % , t h e d a m p i n g i s t o o s m a l l t o a f f e c t  229  appreciably analogous tion worth  by  an  instability.  situations  extracting  looking  into.  This  need  not  be  so  in  and  this  method  of  stabiliza-  however,  energy  from  the  primary  mode  is  other  well  

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