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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.Sc, University of British Columbia,' 1969 A THESIS S U B M I T T E D IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September, 1970 In p resent ing 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 f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date ( i i ) 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 w a v e s o n l i q u i d s . 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 a n a l y z e s t h e s u r f a c e wave b e i n g s t u d i e d . T h i s p r o p e r t y o f t h e t e c h n i q u e h a s b e e n v e r i f i e d b y t h r e e i n d e p e n d e n t t e s t s . T h e m e t h o d 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 s t u d y o f t h e R a y l e i g h - T a y l o r i n s t a b i l i t y o f f l u i d s u r f a c e s . T h e r e s u l t s o f t h i s s t u d y a r e i n g o o d a g r e e m e n t w i t h t h e o r y . ( i i i ) TABLE OF CONTENTS Page (ii) ( i i i ) (iv) (v) (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 1 1 Experiment 2 - Rayleigh-Taylor Instability 2 0 CHAPTER k CONCLUSIONS AND FUTURE WORK 3 0 REFERENCES • 3 3 APPENDIX 3J4. ABSTRACT TABLE OF CONTENTS LIST OF FIGURES LIST OF PHOTOGRAPHS ACKNOWLEDGEMENTS (iv) LIST OF FIGURES Page 1 Capacitor Plate k 2 Surface Modes on a Liquid in a Rectangular Cavity 6 3 Growth of Rayleigh-Taylor Instability 9 h Pulsed Air Jet System 13 5 Amplitude Measuring Device > Hi 6 Surface Mode Approximations 19 7 Accelerating System 21 8 Triggering System 22 9 Acceleration Measuring Technique 25 1 0 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 00 LIST CF PHOTOGRAPHS ; Page Photo 1 CAPACITANCE BRIDGE OUTPUT vs. TIME 16 2 " » 11 11 it l 6 3 " " 1 1 " " 2.7 1; » II II It: II 1 7 5 " « « " » 2k 6 » » " 11 11 2 i i 7 ACCELERATION MEASUREMENT 26 8 CAPACITANCE BRIDGE OUTPUT vs. TIME 26 9 11 n 11 11 it 26 1 0 " " 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 in 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 difficult. -1-CHAPTER 1 I N T R O D U C T I O N 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 is then both difficult and inaccurate (see ref. 1). Electromechanical techniques involve difficulties in 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^r3 is much less than one where a is 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 is developed. The chief advantage of this technique is that in studying an arbitrary surface wave the technique responds to only one of the Fourier components of the wave, i.e. the wave is Fourier analyzed by this technique. Since, previously, whenever - 2 -a study of surface waves is made the wave must be Fourier analyzed mathematically in order to compare results with theory i t is 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 listed below. (a) The technique is sensitive enough to study waves in the linear region much less than unity). / (b) Exponentially growing wave amplitudes (e.g. instabilities) can be followed continuously. (c) The critical equipment is inexpensive and very rugged. That i s , the measuring device will endure any impulse to which the liquid container can be safely subjected. This property is not shared by any of the previously mentioned techniques. The basic idea of this method is to have two sides of the tank containing the liquid form the plates of a capacitor. Each plate is 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 distilled water the dielectric constant, K, is about 80) then the capacitance is defined by the underwater portion of the plates since capacitance, C, is /Aw Ky/ £o I /AA £Q - d ~ t " ~ d J U and are the areas of the plates separated by distilled water and air respectively. YMls the dielectric constant of the water, d the separation of the plates and £; the permittivity of free space. Since AA can.easily be made less than Awthe f i r s t term, corresponding to the underwater portion of the plates, dominates. If a surface wave.is initiated on the liquid then, as is shown in section 2, the capacitance only varies in response to that harmonic which has a wavelength matching the wavelength of the sinusoidal pattern on the capacitor plates. If the capacitance is 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 ability of this technique to follow surface waves and the Fourier analysis property of the plates are experimentally verified. This method is then applied to a problem requiring a l l of its 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 is accelerated in such a manner there is a net body force on the liquid in the upward direction, i f viewed from the frame of reference of the container, hence the surface is unstable. The experiment obviously requires a rugged technique by -which the instability can be observed in 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 T H E O R Y (a) Fourier Analysis of Surface Waves When standing waves are present on the surface of a liquid in a rectangular container the boundary condition to be applied is that .antinodes of the waves appear at the walls of the container. Hence the fi r s t few harmonics of surface waves in a rectangular container are as shown in fig. 2. The theory indicating how an arbitrary surface wave can be Fourier analyzed into a sum of these harmonics is now presented. The two walls of the container in fig. 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 in fig. 1. The variation in area of the plates and depth of the liquid is in the x direction only and hence capacitance is calculated as an integral over the x variable (the co-ordinate system used is shown in fi g . 2). For a small increment, dx, the capacitance is given by where K is 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) is 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,, is an integer and L is the length of the tank in the x direction. y(x) varies as (I) Fig. 2 Surface Modes on a Liquid in a Rectangular Cavity by Fourier expansion of the surface wave. Here y0 is the equilibrium depth of the liquid,£ is the amplitude of the harmonic of the surface wave and ^ t ) i s the time dependence of this harmonic. We thus have U ) dC = j> ( I - c o s H f * ) + (, - c o s uz*)f f^fl] Ha.- ' To integrate over the plates we must take the limit as N-»w, W->o such that 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 §Kis the Kronecker delta. Therefore equation 1 (3) becomes This expression is of theform (f) C = 3 + Dfn,(t) We see that the time dependence of the capacitance is 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 in C the null detector of the bridge produces a signal with an amplitude which follows the amplitude of the n'h harmonic. The sensitivity of this technique in measuring a given harmonic can be found very easily. If water is added to the container such that the depth of water is increased by an amount A . then the bridge response to this change will be simply twice the response of the bridge to an nfh 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 in 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 hat-men ic SoO ~ cos joL cos ±jL* (l - coi dx = 2 (b) Rayleigh Taylor Instability When a liquid is in a container at rest the body force exerted on the liquid by gravity is (s) r - f s_ where ^  is the density of the liquid and g is the acceleration due to gravity. If the container is accelerated downward with an acceleration g ( then the effective body force exerted on the liquid in the frame of reference of the accelerated container is clearly When |g(|> |g| then the net body force exerted on the liquid is in the upward direction. Since the upper surface of the liquid is free the surface is unstable and any perturbation of the surface will grow (see fig. 3)« The theory predicting the development of the instability is well known (see ref. 3}h) and results are just quoted here. If the parameter ^  defines the position of the surface of the liquid then »? is given by *Y C W . / i - < o S ! L « j A cos ^2L <ix -9-Fig. 3 Growth of Rayleigh-Taylor Instability -10-where 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 is required to study the,growth of a particular mode. The method under examination performs this step automatically. The dispersion relation for u> in the above equation is found to be CHAPTER 3 T H E E X P E R I M E N T S Experiment 1 - Fourier Analysis of Surface Waves The capacitor plates were constructed as printed circuits on the standard plastic circuit boards. The pattern is 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 in 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 fluid approximation is valid for a l l modes of oscillation of the surface i.e. to ensure that the equation is obeyed. Here k is the wave number of the surface wave and h is the depth of the fluid. The tank was f i l l e d with distilled water since this has a dielectric constant of 80 and hence, effectively, the area of the capacitor is defined by the cross-section of the fluid. In this arrangement any wave propagated lengthwise down the tank should be Fourier analyzed by the capacitor, i.e. capacitor variations should be in 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 -12-balancing 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 in capacitance seen as a change in amplitude of the signal on the oscilloscope. Waves are excited by pulsing an air jet onto the surface of the water. The pulsed effect is achieved by interrupting a steady air jet with a semi-circular disc rotated by a 1 h.p. Emerson motor (see fig. 4 ) . The frequency of the pulses is 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 is not expected to be the same for each mode i f the same air pulse amplitude is used. For this reason i t is 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 in a bracket which travelled up and down a vertical gauged post. The vernier scale allowed measurements of vertical displacements to an accuracy of - , o o i c f w . The telescope was replaced by a plexiglass rod of the same diameter in 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 in the tank. The needle is positioned above the antinode at one end of the tank. An oscilloscope is connected across the load resistor. The whole arrangement is illustrated in fig. f>. The wave amplitude is measured by lowering the needle into the undisturbed water until a pulse is seen on the oscilloscope signifying electrical contact with the water. This defines the equilibrium depth of the water. The needle is then raised and the waves excited by the pulsed air. The needle is then lowered again until periodic pulses are observed, thus defining the amplitude of the wave. • It was found that this amplitude could be - 1 3 -Fig. k Pulsed Air Jet System -Ik-oscilloscope travelling microscope stand voltage supply-load resistor needle Fig. 5 Amplitude Measuring Device -15-measured 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 air 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 is displayed in photograph 1. The amplitude of the surface wave was found to be .030 cm. + .002 cm. The third harmonic wave was excited and the air 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 is shown in photograph 2. This result indicates that when the third harmonic wave is excited a small second harmonic component is 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. It was desireable to see the effect of off resonance waves at arbitrary frequencies. The air pulse frequency was set at 150, 200, 250, 300, 350 and li00 revolutions per minute. Results were a l l of the form shown in photographs 3 and U. 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, in 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.) -18-was devided into three chambers by installing two temporary walls as illustrated in 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 is Fourier analyzed then the main Fourier component is the second harmonic. The change in 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 in fi g . 6(b) and water transferred from one to the other. The main Fourier component of this perturbation of the surface is the f i r s t harmonic. There is also a zero contribution from the second harmonic hence we predict that the capacitor will not respond to the perturbation. The response of the capacitance bridge is found to be less than .002 v./mm. of amplitude of the perturbation, much less than the response to the second harmonic. The first harmonic was also approximated by simply -tipping the container as shown in fig. 6(c). Again the second harmonic contribution to this perturbation is 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 is successfully Fourier analyzing perturbations of the surface of the liquid. As suggested in 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 in water depth equal to the amplitude of the wave. It was found that the bridge response was .05 v./mm. of amplitude of the surface mode. -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 in section 2. A shaft approximately 1 metre long was constructed in which the tank used in 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 is illustrated in f i g . 7. The tank is fastened to the top of the shaft with a wire lead which is connected in series with a l 1 ^ volt battery and a 100 ohm resistor. The trigger input of the oscilloscope is then connected across the resistor. This is illustrated in f i g . 8. When the wire fastening the tank is 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 its f a l l down the shaft. The oscilloscope is set to make only one sweep. The output of the bridge displayed on the oscilloscope should then illustrate the growth of the surface wave as a proportionate increase in the output of the capacitor bridge. Since the bridge is driven by a 1 khz oscillator the output signal is 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 Oscillo-scope sponge stoppers capacitance bridge Fig. 7 Acceleration System -22-oscilloscope trigger °" input o To^ohms to fasten tank ( 1.5 v. opening circuit releases tank Fig. 8 Triggering System 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 in acceleration as the spring relaxed could be neglected. When an air jet excites the second harmonic before the tank is released then photo 5 typifies the results obtained. The growth of the instability is depicted very cleasrly by the envelope of the 1 khz. signal in this case. YJhen no air jet pulses the surface the instability s t i l l develops but the effect is 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 fig. 9). This resulted in a pulse appearing on the oscilloscope trace (see photo 7). Since the trace was triggered when the container started its f a l l the position of the pulse gives the time (t) the tank was falling. 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 gt = 2k$0 cm./secf spring 3 g, = 2630 cm./secT Photograph 5 shows the growth of the instability when spring 1 is used. -2ii-Photo 6 -25-Fig. 9 Acceleration Measurement Technique - 2 6 -Photo 7 Photo 8 Photo 9 Photos 8 and 9 show the development of the instability with springs 2 and 3 respectively. From these photographs data can be taken to be compared with the theoretical predictions given in section 2. We recall equation (7). of Chapter 2 defining the surface of the liquid ^ = (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 is taken to dominate the expression (recall that AiccsUx)\ _ 0 <U<.inkx)\ ~£ ) # Hence we have - C cosh(out) 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 •[ [t(cjl - is found from the acceleration measurements, k, the wave number of the second harmonic, is k = =^c = . £ ^ 7 cm~\ These graphs, shown i f f i g . 10 demonstrate the good agreement of the experimental results with the predicted theory. There is a residual out of balance signal from the bridge. This signal varies from one experiment to the next. For this reason the graphs in fig. 10 do not go through the origin. Photo 10 is 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 1 0 - 3 0 -CHAPTER 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 ability 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 liquid. Good agreement has been found between experimental results and theoretical predictions. ...FUTURE WORK The work presented in 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 instabilities of liquid surfaces. A suitable example would be the study of electrostatic hydrodynamic instabilities. These are the instabilites arising when a strong electric field is 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 is shown in - 3 1 -fig. 1 1 . One plate is formed by the side of the liquid and the other by a plate made of strips identical to the plates used in 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 in response to that harmonic of the surface wave on the liquid metal which matches in wavelength the strip pattern on the solid plate. Fig. 12 shows one more way in 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 in the sinusoidal pattern used throughout this thesis. If a light source is shone on the tank as in f i g . 12 the liquid metal showing between the strips acts as a mirror. The reflection from this mirror provides the light source for the photomultiplier. If there is a wave on the surface of the liquid then the area of the mirror varies only in 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 ~ liquid 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 -33-R E F E R E N C E S ' 1 . J. Isaacs and C. Iselen, Oceanographic Instrumentation National Academy of Sciences - National Research Council 2 . W . Chester and J. Bones, Proc. Roy. Soc. 3 0 6 , P.23 ( 1 9 6 8 ) . 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 . F.L. Curzon and R.L. Pike, Canadian Journal of Physics 4 6 , P. 2 0 0 1 ( 1 9 6 8 ) . -3U-A P P E N D I X Excition of Surface Waves on Conducting Liquids In the outline of future work to which the results given in 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, in particular the pure harmonics may be considerable. This is especially true i f the liquid is in the presence of a strong magnetic field. In this case there is a strong damping of the waves (see ref. Al) which will frustrate attempts to excite waves by normal mechanical methods. What is needed is 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 is to run currents through the conducting liquid in such directions that the jxB forces support the motion of the surface wave desired. Here j is current density and B is magnetic field. This is done by positioning electrodes just above the surface of the liquid so that they are wetted i f the surface is perturbed. Since the wetting of the electrodes is not immediate upon contact with the liquid the current is flowing for a greater time when the liquid is falling than when i t is 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 is to support the motion of the wave. Such an arrangement for exciting surface waves down a rectangular tank is shown in f i g . Al. The electrodes are vertical; rectangular plates of the same width as the tank spaced lengthwise down the tank. We consider the problem of exciting the n t h harmonic surface wave. On a common lead -35-voltage source o B electrodes liquid metal Fig. Al Excitation of Surface Waves • -.;. •'- ;•: -36- : .• . electrodes are positioned at point's a distance and on another common lead at a distance d - (3 +lf " * ) L °> t > • • • Here L is the length of the tank and ^ is 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 . Al. If the surface is perturbed arbitrarily so that i n i t i a l wetting of the electrodes is attained then the desired wave will be excited. - The method was attempted for the case n = 2, = .2. The tank used was it in. long by 2\ in. wide by 5 \ in. deep. A Trygon model M36-30A power supply delivered currents from j? to 20 amps to the electrodes. The magnetic field was supplied by an electromagnet powered by a Miller model 2^ 0 A.C./D.C. welder power supply. The magnetic field applied was S> kilogauss. 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 in the presence of a magnetic field. A l . R E F E R E N C E S F . L . Curzon and R . L . Pike, Canadian Journal of Physics 1J7, P. 10£l ( 1 9 6 9 ) 

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