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Capacitative Fourier analyzer of hydrodynamic surface waves. Langille, Brian Lowell 1970

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A CAPACITATIVE FOURIER ANALYZER OF HYDRODYNAKIC SURFACE WAVES  by BRIAN LOWELL LAKGILLE  B . S c , U n i v e r s i t y of B r i t i s h Columbia,' 1969 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department  of PHYSICS  We accept t h i s thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA September, 1970  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 requirements  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 f r e e l y a v a i l a b l e f o r  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  r e f e r e n c e and copying o f t h i s  It  i s understood that copying o r  thesis  permission.  Depa rtment The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  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  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 of my Department by h i s r e p r e s e n t a t i v e s .  for  (ii)  ABSTRACT  A t e c h n i q u e h a s b e e n d e v e l o p e d f o r s t u d y i n g s u r f a c e waves on The m e a s u r i n g d e v i c e e m p l o y e d F o u r i e r studied.  This  property  independent t e s t s . of  t h e s u r f a c e wave b e i n g  t e c h n i q u e has been v e r i f i e d by  three  The method d e v e l o p e d h a s b e e n a p p l i e d t o t h e  the R a y l e i g h - T a y l o r  this  of the  analyzes  i n s t a b i l i t y of f l u i d s u r f a c e s .  s t u d y a r e i n good agreement w i t h  theory.  liquids.  The  study  results  of  (iii) TABLE OF CONTENTS Page ABSTRACT  (ii)  TABLE OF CONTENTS  (iii)  LIST OF FIGURES  (iv)  LIST OF PHOTOGRAPHS  (v)  ACKNOWLEDGEMENTS  (vi)  CHAPTER 1  INTRODUCTION  1  CHAPTER 2  THEORY  2(a) Fourier Analysis of Surface Waves  5  2(b) Rayleigh-Taylor Instability  8  CHAPTER 3  THE EXPERIMENTS Experiment 1 - Fourier Analysis of Surface Waves  CHAPTER k REFERENCES APPENDIX  11  Experiment 2 - Rayleigh-Taylor Instability  20  CONCLUSIONS AND FUTURE WORK  30  •  33 3J4.  (iv) LIST OF FIGURES Page 1  Capacitor Plate  k  2  Surface Modes on a Liquid i n a Rectangular Cavity  6  3  Growth of Rayleigh-Taylor Instability  9  h  Pulsed A i r Jet System  5  Amplitude Measuring Device  6  Surface Mode Approximations  19  7  Accelerating System  21  8  Triggering System  22  9  Acceleration Measuring Technique  25  10  Growth of Rayleigh-Taylor Instability  28  11  Fourier Analysis of Surface Waves on Liquid Metals  32  12  Optical Fourier Analysis of Surface Waves on Liquid Metals  32  Al  Excitation of Surface Waves  35  13  >  Hi  00 LIST C F PHOTOGRAPHS  ;  Page Photo  1  CAPACITANCE BRIDGE OUTPUT vs. TIME  2  "  »  3  "  "  1;  »  II  5  "  6  »  11  11  it  16 l  6  "  "  II  It:  II  «  «  "  »  2k  »  "  11  11  2ii  11  2.7  1  7  7  ACCELERATION MEASUREMENT  26  8  CAPACITANCE BRIDGE OUTPUT vs. TIME  26  9 10  11  n  11  11  it  26  "  "  it  it  it  29  :  (vi)  ACKNOWLEDGEMENTS • I would very much like to thank Dr. F.L. Curzon for his stimulating and informative supervision throughout this work. I would, also like to thank Dr. W. Westphal for his guidance and encouragement i n supervising the i n i t i a l stages of this research. I have benefitted greatly from the technical assistance of Mr. D. Sieberg and Mr. R. Dickson.  The machine shop instruction  of Mr. R. Haines has enabled me to carry out portions of this work which otherwise would have been extremely d i f f i c u l t .  -1CHAPTER 1 INTRODUCTION Until recent years the study of surface waves on liquids has been restricted to a few unsuitable methods.  Commonly used are pressure  probes which track the pressure variation under the liquid surface. Relating the pressure variations to surface height i s then both d i f f i c u l t and inaccurate (see ref. 1).  Electromechanical techniques involve  d i f f i c u l t i e s i n wetting electrodes and are usually too insensitive to study surface waves which have amplitudes small enough for linearized theory to apply (i.e. waves for which g^r i s much less than one where 3  a i s the wavelength,? the amplitude of the surface wave and h the depth of the fluid).  Capacitative probes can be made very sensitive  but respond indiscriminately to a l l modes of oscillation of the surface (see ref. 2).  Optical methods of studying surface waves are expensive,  fragile and although easily employed for the study of periodic phenomena are not readily applicable to the study of instabilities on liquid surfaces. A method has been developed by Curzon and Pike using a microwave technique to study surface waves on liquid conductors (e.g. mercury).  This method was highly sensitive and gave a continuous  tracking of non-periodic phenomena(see ref. 7 ) » In this report a method of studying surface waves applicable to non-conducting liquids i s developed. The chief advantage of this technique i s that i n studying an arbitrary surface wave the technique responds to only one of the Fourier components of the wave, i . e . the wave i s Fourier analyzed by this technique.  Since, previously, whenever  -2-  a study of surface waves is made the wave must be Fourier analyzed mathematically i n order to compare results with theory i t i s obviously a great advantage to have this work done automatically by the experimental apparatus.  This technique has a number of other advantages.  The most  signifigant are l i s t e d below. (a) The technique i s sensitive enough to study waves i n the linear region  much less than unity).  /  (b) Exponentially growing wave amplitudes (e.g. instabilities) can be followed continuously. (c) The c r i t i c a l equipment i s inexpensive and very rugged. That i s , the measuring device w i l l endure any impulse to which the l i q u i d container can be safely subjected.  This property i s not shared  by any of the previously mentioned techniques. The basic idea of this method i s to have two sides of the tank containing the liquid form the plates of a capacitor. Each plate i s made up of matching vertical strips, equally spaced, but of unequal widths so that the area of the plates varies sinusoidally down the length of the tank (fig. 1).  If the tank contains a liquid of high dielectric constant  (e.g. for d i s t i l l e d water the dielectric constant, K, is about 80) then the capacitance i s defined by the underwater portion of the plates since capacitance, C, i s  /Aw Ky/ -  J U and  d  £o  I  ~t"~  /AA £Q d  are the areas of the plates separated by d i s t i l l e d water and a i r  respectively.  Y ls the dielectric constant of the water, d the separation M  of the plates and £; the permittivity of free space.  Since A can.easily A  be made less than A the f i r s t term, corresponding to the underwater portion w  of the plates, dominates.  If a surface wave.is initiated on the liquid  then, as i s shown i n section 2, the capacitance only varies i n response to that harmonic which has a wavelength matching the wavelength of the sinusoidal pattern on the capacitor plates.  I f the capacitance i s  monitored with a capacitance bridge then a continuous tracking of this harmonic can be displayed on an oscilloscope by connecting the oscilloscope across the bridge null detector.  In this report the a b i l i t y of this  technique to follow surface waves and the Fourier analysis property of the plates are experimentally verified.  This method i s then applied  to a problem requiring a l l of i t s basic advantages; the study of the Rayleight-Taylor instability of the surface of a liquid accelerated downward at an acceleration greater than g, the acceleration due to gravity.  When a liquid i s accelerated i n such a manner there i s a net  body force on the liquid i n the upward direction, i f viewed from the frame of reference of the container, hence the surface i s unstable. The experiment obviously requires a rugged technique by -which the i n s t a b i l i t y can be observed i n the linear region, where the known theory applies.  The Fourier analysis property of the technique makes  simple the analysis of results.  cross-hatching indicates non-conducting portion of plates Fig. 1  Capacitor Plate  CHAPTER 2 THEORY (a) Fourier Analysis of Surface Waves When standing waves are present on the surface of a liquid i n a rectangular container the boundary condition to be applied is that .antinodes of the waves appear at the walls of the container.  Hence the  f i r s t few harmonics of surface waves i n a rectangular container are as shown i n f i g . 2. The theory indicating how an arbitrary surface wave can be Fourier analyzed into a sum of these harmonics i s now presented. The two walls of the container i n f i g . 2 which run parallel to the direction of propagation of the surface wave (i.e. perpendicular to the z direction) form the plates of a capacitor of the form shown i n f i g . 1. The variation i n area of the plates and depth of the liquid i s i n the x direction only and hence capacitance i s calculated as an integral over the x variable (the co-ordinate system used i s shown i n f i g . 2). For a small increment, dx, the capacitance i s given by  (I) where K i s the dielectric constant of the liquid,  the permittivity of  free space, d the separation of the capacitor plates, N the number of strips per unit length, W(x) the strip width as a function of x and y(x) i s the depth of the liquid as a function of x.  Now the plates are constructed  such that W(x) varies as  where W is a constant, n,, i s an integer and L i s the length of the tank i n the x direction. y(x) varies as  Fig. 2  Surface Modes on a Liquid i n a Rectangular Cavity  by Fourier expansion of the surface wave. Here y i s the equilibrium 0  depth of the liquid,£ i s the amplitude of the  harmonic of the surface  wave and ^ t ) i s the time dependence of this harmonic. We thus have U  )  j> ( I - c o s H f * ) + (, - c o s uz*)f  dC =  f^fl] Ha.-'  To integrate over the plates we must take the limit as N-» , W->o such that w  NW=A, a constant.  Then integrating over x we have  since J^*. J-pc<**o  for integer j . Now the orthogonality of cosines states  that  <  for integer j and k where § is K  the Kronecker delta.  Therefore equation  1  (3) becomes  This expression i s of theform  (f)  C = 3 + Dfn,(t)  We see that the time dependence of the capacitance i s proportional to' the time dependence of the inharmonic of the surface wave, i.e. capacitance follows the development of one harmonic thus Fourier analyzing the surface wave. Since the standard capacitance balance bridge responds linearly to small changes i n C the null detector of the bridge produces a signal with an amplitude which follows the amplitude of the n' harmonic. h  The sensitivity of this technique i n measuring a given harmonic can be found very easily.  If water i s added to the container such that  the depth of water i s increased by an amount A . then the bridge response to this change w i l l be simply twice the response of the bridge to an nf  h  harmonic wave of amplitude A (recall n,-is the harmonic the measuring device  •••••  -8-  responds to). Hence the sensitivity of the technique can be easily found by taking one half the sensitivity of the technique to changes i n water depth. This can be seen by carrying out the integrations to get the response to each perturbation, i . e . we have, by equation 2 and the definition of W(x)  fespome  tc  *Y C W . / i -  hat-men ic  SoO  j  o  =  L  <o  S  !L«j  A  cos ^2L  <ix  ~ cos  cos ±jL*  (l  -  coi  dx  2  (b) Rayleigh Taylor Instability When a liquid i s i n a container at rest the body force exerted on the liquid by gravity i s  r  (s)  -  f  s_  where ^ i s the density of the liquid and g i s the acceleration due to gravity.  If the container i s accelerated downward with an acceleration g  (  then the effective body force exerted on the liquid i n the frame of reference of the accelerated container is clearly  When |g|> |g| then the net body force exerted on the liquid i s i n the upward (  direction.  Since the upper surface of the liquid i s free the surface i s  unstable and any perturbation of the surface w i l l grow (see f i g . 3)« The theory predicting the development of the instability i s well known (see ref. 3}h) and results are just quoted here.  I f the parameter ^ defines  the position of the surface of the liquid then »? i s given by  -9-  Fig. 3  Growth of Rayleigh-Taylor Instability  -10where C and D are constants and k Is the wave number of the perturbation of the surface.  The x dependence of the surface can be written as a simple  cosine since any perturbation of the surface can be Fourier analyzed into a sum of cosines and the growth of each cosine wave treated as a separate problem.  Using other measuring devices a mathematical Fourier analysis  of the i n i t i a l surface perturbation i s required to study the,growth of a particular mode. The method under examination performs this step automatically. found to be  The dispersion relation for u> i n the above equation i s  CHAPTER 3 THE  EXPERIMENTS  Experiment 1 - Fourier Analysis of Surface Waves The capacitor plates were constructed as printed circuits on the standard plastic circuit boards.  The pattern i s illustrated iri f i g . 1.  The wavelength of the sinusoidal pattern of strips was set equal to that of the second harmonic surface wave. The plates were sprayed with Krylon clear acrylic spray to isolate them electrically from the liquid dielectric and hence greatly decrease electrical dissipation effects i n the capacitor. The container used was 9.7 cm long (in the direction of wave propagation) by U.l cm wide by 10 cm high.  The dimensions were chosen to ensure that a  deep f l u i d approximation i s valid for a l l modes of oscillation of the surface i . e . to ensure that the equation  i s obeyed.  Here k i s the wave number of the surface wave and h i s the  depth of the f l u i d .  The tank was f i l l e d with d i s t i l l e d water since this  has a dielectric constant of 80 and hence, effectively, the area of the capacitor i s defined by the cross-section of the f l u i d .  In this arrangement  any wave propagated lengthwise down the tank should be Fourier analyzed by the capacitor, i . e . capacitor variations should be i n response to the second harmonic wave only.  This effect was tested as follows.  The capacitor  plates were connected to a Type 1650 A impedence bridge (General Radio Company) and the bridge was balanced using the 1 kilohertz signal of the internal oscillator of the bridge.  The bridge has an external detector output to  which an oscilloscope was connected allowing direct observation of the 1 khz  -12balancing signal.  The minimum amplitude of this signal indicates balance  of the bridge. The oscilloscope allows a more sensitive balance than the meter on the bridge and also allows a complete monitoring of any fluctuations i n capacitance seen as a change i n amplitude of the signal on the oscilloscope. Waves are excited by pulsing an a i r jet onto the surface of the water. The pulsed effect i s achieved by interrupting a steady a i r jet with a semicircular disc rotated by a 1  h.p. Emerson motor (see f i g . 4 ) . The  frequency of the pulses i s varied using a mechanical reduction gear on the motor. We thus observed the effect of exciting the various resonant frequencies of the container. The amplitude of the excited wave i s not expected to be the same for each mode i f the same a i r pulse amplitude i s used.  For this reason i t i s neccessary to monitor the amplitude of the  excited wave. This was done by the following procedure.  A travelling  microscope was found which had a telescope of 1 ^ in. diameter mounted i n a bracket which travelled up and down a vertical gauged post. The vernier scale allowed measurements of vertical displacements to an accuracy of -,ooicfw  . The telescope was replaced by a plexiglass rod of the same  diameter i n which was imbedded a sharp needle. This needle was electrically connected to a voltage source (14 volts), a load resistor then to an electrode immersed i n the tank. at one end of the tank. resistor.  The needle i s positioned above the antinode  An oscilloscope i s connected across the load  The whole arrangement i s illustrated i n f i g . f>. The wave  amplitude i s measured by lowering the needle into the undisturbed water until a pulse i s seen on the oscilloscope signifying electrical contact with the water.  This defines the equilibrium depth of the water. The  needle i s then raised and the waves excited by the pulsed a i r .  The needle  i s then lowered again until periodic pulses are observed, thus defining the amplitude of the wave. • It was found that this amplitude could be  -13-  Fig. k Pulsed A i r Jet System  -Ik-  oscilloscope voltage supply-  load resistor  needle  travelling microscope stand  Fig. 5  Amplitude Measuring Device  -15measured to within + .002 cm . The oscilloscope connected to the capacitance bridge was then checked to see i f the capacitance was fluctuating. When the second harmonic wave was exoited the output of the bridge fluctuated at the frequency of this harmonic. This frequency was measured by "stopping" the motion of the motor turning the disc which interupted the a i r jet using a Strobotac calibrated strobe light.  The measured  frequency of 270 r.p.m. matched the frequency at which'the bridge output fluctuated.  A polaroid camera mounted on the scope face photographed  this fluctuation.  The output i s displayed i n photograph 1.  The amplitude  of the surface wave was found to be .030 cm. + .002 cm. The third harmonic wave was excited and the a i r jet pulse amplitude increased until the amplitude of the wave exceeded..030 cm. The frequency of the surface wave was found to be 330 r.p.m. The output of the capacitance bridge did not fluctuate at 330 r.p.m., however instead there was a small fluctuation at 270 hz.  This i s shown i n photograph 2.  This result  indicates that when the third harmonic wave i s excited a small second harmonic component i s also s t i l l excited.  The bridge responds to this  component but does not respond to the larger third harmonic component. A similar result was obtained when the experiment was repeated at the fourth harmonic.  I t was desireable to see the effect of off resonance  waves at arbitrary frequencies.  The a i r pulse frequency was set at 150,  200, 250, 300, 350 and li00 revolutions per minute. the form shown i n photographs 3 and U.  Results were a l l of  In each test the output of the  capacitance bridge oscillated at the frequency of the second harmonic. We thus have excellent evidence that the capacitor plates are, i n fact, only responding to the second harmonic component of the surface wave, A further test of this property was carried out as follows.  The cavity  -16-  Photo 1 Unless otherwise stated a l l photographs have the following scales: vertical scale: .1 v./cm. horizontal scale: .0/ sec./cm.  Photo 2  Photo 3  (air pulse at 300 R.P.M.)  •  Photo k  (air pulse at 250 R.P.K.)  -18was devided into three chambers by installing two temporary walls as illustrated i n f i g . 6(a).  The liquid surface was. perturbed by transferring  water from the center chamber to the two outside chambers (equal amounts to' each side). If this perturbation of the surface i s Fourier analyzed then the main Fourier component i s the second harmonic. The change i n the output of the bridge was found to be .0li2 v./mm. of amplitude of the perturbation.  The container was divided into two chambers as shown i n  f i g . 6(b) and water transferred from one to the other.  The main  Fourier component of this perturbation of the surface i s the f i r s t harmonic.  There i s also a zero contribution from the second harmonic  hence we predict that the capacitor w i l l not respond to the perturbation. The response of the capacitance bridge i s found to be less than .002 v./mm. of amplitude of the perturbation, much less than the response to the second harmonic.  The f i r s t harmonic was also approximated by simply -  tipping the container as shown i n f i g . 6(c). Again the second harmonic contribution to this perturbation i s zero.  In this case no response of  the capacitance bridge could be observed for any amplitude of the perturbation (amplitudes over 1 cm. were tested). The results of these three tests indicate that we can say very confidently that the measuring technique i s successfully Fourier analyzing perturbations of the surface of the liquid. As suggested i n section 2 the sensitivity of the system to the surface mode i t analyzes was found by simply taking one half the sensitivity of the system to an increase i n water depth equal to the amplitude of the wave. It was found that the bridge response was of the surface mode.  .05 v./mm. of amplitude  -19•  6(a)  v  y—  %  'N  «*  %  6(b)  6(c)  Fig. 6  Surface Mode Approximations  ~>"  *  ^  '  Experiment 2- Rayleigh-Taylor Instability The Fourier analysis properties of the capacitor technique has been applied to the study of the Rayleigh-Taylor instability discussed i n section 2. A shaft approximately 1 metre long was constructed i n which the tank used i n experiment 1 could slide freely up and down. A spring was connected from the bottom of the tank to the.bottom of the shaft.  Extended the f u l l length of the shaft the spring accelerated the  f u l l container at over 2 g accelerations. Sponge rubber pads half way down the shaft safely stopped the container.  The capacitor plates on  the sides of the tank were connected to an impedence bridge with leads long enough to allow the tank to move freely up and down the shaft. The detector output terminals of the bridge were connected to an oscilloscope. The system i s illustrated i n f i g . 7. The tank i s fastened to the top of the shaft with a wire lead which i s connected i n series with a l ^ volt 1  battery and a 100 ohm resistor.  The trigger input of the oscilloscope  is then connected across the resistor.  This i s illustrated i n f i g . 8.  When the wire fastening the tank i s broken, allowing the spring to pull the container down the shaft the output of the battery appears across the open circuit rather than the 100 ohm resistor.  The voltage pulse triggers  the oscilloscope trace hence starting the trace at the same time as the tank starts i t s f a l l down the shaft.  The oscilloscope i s set to make only  one sweep. The output of the bridge displayed on the oscilloscope should then i l l u s t r a t e the growth of the surface wave as a proportionate increase i n the output of the capacitor bridge.  Since the bridge i s driven by a  1 khz oscillator the output signal i s an amplitude modulated 1 khz signal. Runs were made both with surface waves excited at the second harmonic, as in experiment 1 and by just allowing the surface waves to develop from  -21-  water container  spring  Oscilloscope  sponge stoppers  capacitance bridge  Fig. 7 Acceleration System  -22-  oscilloscope  trigger °" input o  To^ohms to fasten tank  1.5 v.  Fig. 8  Triggering System  ( opening circuit releases tank  small perturbations due to the accelerations technique. . The sweep speed of the scope was kept on a range at which the container would f a l l no more than 10 cm. by the completion of the sweep so that corrections for changes i n acceleration as the spring relaxed could be neglected.  When  an a i r jet excites the second harmonic before the tank i s released then photo 5 typifies the results obtained.  The growth of the instability i s  depicted very cleasrly by the envelope of the 1 khz. signal i n this case. YJhen no a i r jet pulses the surface the instability s t i l l develops but the effect i s much smaller (see photo 6).  This could be due to either a small  i n i t i a l perturbation of the surface or a perturbation representing chiefly the higher modes of oscillation of the surface.  Three different  springs were tested so that different accelerations could be checked. The accelerations were measured by having a metal contact attached to the tank make contact with another metal rod at a given position down the shaft. The metal contacts were connected so as to short out the capacitance bridge for an instant thereby throwing i t off balance (see f i g . 9). This resulted i n a pulse appearing on the oscilloscope trace (see photo 7). Since the trace was triggered when the container started i t s f a l l the position of the pulse gives the time (t) the tank was f a l l i n g .  Measuring  .the distance (d) travelled before contact -is made gave the accelerations, g,, ef the tank by the formula  The three springs tested were found to give accelerations of spring 1  g  (  « 20l|0 cm./sect'  spring 2  g  t  = 2k$0 cm./secf  spring 3  g, = 2630 cm./secT  Photograph 5 shows the growth of the i n s t a b i l i t y when spring 1 i s used.  -2ii-  Photo 6  -25-  Fig. 9 Acceleration Measurement Technique  -26-  Photo  7  Photo 8  Photo  9  Photos 8 and 9 show the development of the i n s t a b i l i t y with springs 2 and 3 respectively.  From these photographs data can be taken to be compared  with the theoretical predictions given i n section 2.  We recall equation  (7). of Chapter 2 defining the surface of the l i q u i d ^  =  (C  coskfu/f) +  D <;inh(uf))  CO£  \n X  Photos 5, 8 and 9 exhibit small slopes for t close to zero and hence the cosh( t) term i s taken to dominate the expression (recall that AiccsUx)\  _  0  <U<.inkx)\ -  ~£  C cosh(out)  )  #  Hence we have  cos  kx  The photographed results are compared with theory by plotting h, the amplitude of the envelope of the 1 khz. signal against coshwt where CJ x •[ [ (cj t  l  i s found from the acceleration measurements,  number of the second harmonic, i s k = ^=c = . £ ^ 7  cm~\  k, the wave  These graphs,  shown i f f i g . 10 demonstrate the good agreement of the experimental results with the predicted theory.  There i s a residual out of balance  signal from the bridge. This signal varies from one experiment to the next.  For this reason the graphs i n f i g . 10 do not go through the origin.  Photo 10 i s a run made with a slower sweep setting on the oscilloscope and shows the complete growth of the instability climaxed by the tank striking the rubber stoppers and the instability dropping into a bounded oscillation.  -29-  Photo  10  -30CHAPTER k  -  C O N C L U S I O N S  Fourier Analysis of Surface Waves A method of studying surface waves on liquids has been developed which Fourier analyzes the waves.  The technique has been experimentally  verified by three distinct tests which check the a b i l i t y of the technique to Fourier analyze the waves under study.  The sensitivity of the  technique has been measured and found good enough easily to be applied to the study of linear surface waves. Rayleigh-Taylor Instability . .. " The method developed for studying surface waves has been applied to the study of the Rayleigh-Taylor instability on the surface of a l i q u i d . Good agreement has been found between experimental results and theoretical predictions. ...FUTURE  WORK  The work presented i n this thesis can easily be extended to other experiments.  Different geometries could be studied for example.  The technique developed here could be applied to the study of other i n s t a b i l i t i e s of l i q u i d surfaces. A suitable example would be the study of electrostatic hydrodynamic i n s t a b i l i t i e s .  These are the instabilites  arising when a strong electric f i e l d i s applied to the surface of liquid (see ref. 5 ) .  The study of surface waves on liquid metals may be a  suitable application of a variation of this technique i n which the liquid forms one of the capacitor plates.  A practical arrangement i s shown i n  -31-  fig. 1 1 .  One plate i s formed by the side of the liquid and the other by  a plate made of strips identical to the plates used i n this experiment (see f i g . l ) . These plates are separated by a thin dielectric barrier. Again we find that the area of the plates, and hence the capacitance, varies only i n response to that harmonic of the surface wave on the liquid metal which matches i n wavelength the strip pattern on the solid plate. Fig. 1 2 shows one more way i n which the principle presented here can Fourier analyze surface waves on a liquid metal.  Black strips are  painted on the side of a glass tank i n the sinusoidal pattern used throughout this thesis.  I f a light source i s shone on the tank as i n f i g .  1 2 the liquid metal showing between the strips acts as a mirror. The reflection from this mirror provides the light source for the photomultiplier. I f there i s a wave on the surface of the liquid then the area of the mirror varies only i n response to the harmonic defined by the pattern of black strips and hence the photomultiplier responds only to this mode of oscillation.  -32-  solid capacitor plate — (metal strips) dielectric ~ l i q u i d metal-  side view of tank  Fig. 11 Fourier Analysis of Surface Waves on Liquid Metals  photomultiplier lens black strips liquid metal  light source vertical view of system  Fig. 12  Optical Fourier Analysis of Surface Waves on Liquid Metals  -33REFERENCES ' 1.  J. Isaacs and C. Iselen, Oceanographic Instrumentation National Academy of Sciences - National Research Council  2.  W.  3.  G. Taylor, Proc. Roy. Soc. A 201, P. 1 9 2 (1950).  h.  D.J. Lewis, Proc. Roy. Soc. A 2 0 2 , P. 8 1 (1950).  5.  Landau and Lifschitz, Electrodynamics of Continuous Media Addison - Wesley ( i 9 6 0 ) .  6.  H. Lamb, Hydrodynamics, Dover (1945).  7.  Chester and J. Bones, Proc. Roy. Soc. 3 0 6 , P.23  (1968).  F.L. Curzon and R.L. Pike, Canadian Journal of Physics 4 6 , P.  2001  (1968).  -3UAPPENDIX Excition of Surface Waves on Conducting Liquids In the outline of future work to which the results given i n this thesis could be extended i t was suggested that studies of surface waves on conducting liquids could be carried out.  In such studies the problem  of exciting the surface waves, i n particular the pure harmonics may be considerable.  This i s especially true i f the liquid i s i n the presence of  a strong magnetic f i e l d .  In this case there is a strong damping of the  waves (see ref. Al) which w i l l frustrate attempts to excite waves by normal mechanical methods. What i s needed i s a method of exciting the pure surface modes of oscillation -which makes use of the applied magnetic field.  Such a method has been developed.  The basic idea of this method  i s to run currents through the conducting liquid i n such directions that the jxB forces support the motion of the surface wave desired. i s current density and B i s magnetic f i e l d .  Here j  This i s done by positioning  electrodes just above the surface of the liquid so that they are wetted i f the surface i s perturbed.  Since the wetting of the electrodes is  not immediate upon contact with the liquid the current i s flowing for a greater time when the liquid i s falling than when i t i s rising.  If  the electrodes are positioned so that antlnodes of the desired wave are between them and the £xB  force directed downward then the net effect of  the applied current i s to support the motion of the wave. Such an arrangement for exciting surface waves down a rectangular tank i s shown i n f i g . A l . The electrodes are vertical; rectangular plates of the same width as the tank spaced lengthwise down the tank. problem of exciting the n  th  We consider the  harmonic surface wave. On a common lead  -35-  voltage source  electrodes  liquid metal  o  B  Fig. Al  Excitation of Surface Waves  •  -.;.  •'-  ;•:  -36-  : .• .  electrodes are positioned at point's a distance  and on another common lead at a distance d  -  ( +lf " * 3  ) L  °> t >  • • •  Here L i s the length of the tank and ^ i s a small displacement from the node to ensure that the electrodes are wetted (a typical value would be & = .2). The two common leads are then connected across a voltage source as shown i n f i g . A l . I f the surface i s perturbed a r b i t r a r i l y so that i n i t i a l wetting of the electrodes i s attained then the desired wave w i l l be excited. -  The method was attempted for the case n = 2,  = .2.  The tank  used was it i n . long by 2\ i n . wide by 5 \ i n . deep. A Trygon model M36-30A power supply delivered currents from j? to 20 amps to the electrodes. The magnetic f i e l d was supplied by an electromagnet powered by a Miller model 2^0 A.C./D.C. welder power supply. S> kilogauss.  The magnetic f i e l d applied was  The i n i t i a l perturbation of the surface could be generated  by jarring the container s t i f f l y enough to cause a single i n i t i a l wetting of the electrodes. The method proposed was found to satisfactorily excite pure harmonics on the surface of mercury i n the presence of a magnetic  field.  R E F E R E N C E S  Al.  F.L.  Curzon and R . L .  Pike, Canadian Journal of Physics 1J7, P. 10£l  (1969)  

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