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Investigation of fluid surface waves with a new microwave resonance technique Pike, Robert L. 1967

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The Uni v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of R„L. Pike B.Sc, The Uni v e r s i t y of B r i t i s h Columbia IN ROOM 301, HENNINGS BUILDING WEDNESDAY, SEPTEMBER 20, 1967, AT 3:30 P.M. COMMITTEE IN CHARGE Chairman: B. N. Moyls B. Ahlborn R.M. E l l i s J.W. Bichard R. Nodwell R.W. Burling C F . Schwerdtfeger A. F o l k i e r s k i External Examiner: V.R. Malkus Woods Hole Oceanographic I n s t i t u t i o n , Woods Hole, Massachusetts. Research Supervisor: F.L. Curzon INVESTIGATION OF FLUID SURFACE WAVES WITH A NEW MICROWAVE RESONANCE TECHNIQUE ABSTRACT A new microwave technique has been developed for the experimental study of small amplitude surface waves on an e l e c t r i c a l l y conducting f l u i d . The f l u i d forms one' of the walls of a resonating, microwave ca v i t y . Surface waves with amplitudes as small as -3 10 cm. can be measured by observing the r e s u l t i n g change in the resonant frequency of the c a v i t y . This technique has been s u c c e s s f u l l y used to measure the viscous and' magnetic damping c o e f f i -c i e n t of a small amplitude, standing, surface wave i n l i q u i d mercury. The magnetic damping c o e f f i c i e n t for a v e r t i c a l , magnetic f i e l d was found to be i n good agreement with a c a l c u l a t i o n that was made for low magnetic Reynolds numbers. When the viscous damping c o e f f i c i e n t was compared with the standard theory, which allows h o r i z o n t a l motion of the surface, a disagreement of up to a factor of four was found. I t , however, showed excellent agreement with a modified theory which assumes that there i s no horizontal, motion of the surface. GRADUATE STUDIES F i e l d of Study: Plasma Physics' Applied E l e c t r o n i c s Elementary Quantum Mechanics Waves Electromagnetic Theory Plasma Dynamics Plasma Physics Analysis of Linear Systems Theory of Ideal F l u i d s Advanced Plasma Physics M. Kharadly G.M. Volkoff R.M. E l l i s P„ R a s t a l l F.L. Curzon L. . Sobrino E.V. Bdhn G.V. Parkinson R. Nodwell AWARDS 1960-64 B. C Government Scholarships 1965-67 National Research Council Scholarships 1967 National Research Council Postdoctorate Fellowship INVESTIGATION OF FLUID SURFACE WAVES WITH A NEW MICROWAVE RESONANCE TECHNIQUE • -;- \ - by Robert L. Pike .B.Sc. U n i v e r s i t y o f B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTORATE OF PHILOSOPHY i n the department of PHYSICS We accept t h i s t h e s i s as conforming t o the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e 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 , I a g r e e t h a t t h e 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 r e f e r e n c e a nd S t u d y . I f u r t h e r a g r e e 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 t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h.i>s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The 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 V a n c o u v e r 8, C a n a d a (ii) ABSTRACT A new microwave technique has been developed for the experimental study of small amplitude surface waves on an electrically conducting fluid. . The fluid forms one of the walls of a resonating, microwave cavity. Surface waves with amplitudes as small as 10~3 cm. can be measured by observing the resulting change in. the resonant frequency of the cavity. This technique has been successfully used to measure the viscous and magnetic damping coefficient of a small amplitude, standing, surface wave in liquid mercury. The magnetic damping, coefficient (for a vertical, magnetic field) was found to. be in good agreement with a calculation that was made, for low magnetic Reynolds numbers. When the viscous damping coefficient was. compared.with the standard theory, which allows horizontal motion of the. surface, a disagreement of up to a factor of four was found. It, however, showed excellent agreement with a modified theory which assumes that there, is no horizontal motion of the surface. ( i i i ) TABLE OF CONTENTS ABSTRACT LIST OF ILLUSTRATIONS TABLES ACKNOWLEDGEMENTS Chapter 1 INTRODUCTION 1 Chapter 2 MICROWAVE. THEORY . 6 Sec. 1. Slater's.Theorem 6 Sec. 2 Cavity resonators 7 Sec. 3 Calculations of the resonant 10 frequency change for a surface wave. in. a cylindrical cavity Chapter 3 EXPERIMENTAL. PROCEDURE* PARAMETERS, U VARIABLES, AND ASSUMPTIONS Sec- 1 Experimental procedure 14 Sec. 2 Parameters, variables, and assumptions 24-Chapter 4 STUDY" OF THE OSCILLATION FREQUENCY" OF 27 SURFACE VAVES IN MERCURY Chapters STUDY OF THE VISCOUS DAMPING OF 34 SURFACE. WAVES:. IN MERCURY , Chapter. 6. THEORETICAL CALCULATION: OF' MAGNETIC DAMPING 43 OF SURFACE WAVES FOR LOW MAGNETIC REYNOLDS NUMBERS Chapter 7 EXPERIMENTAL:.STUDY! OF THE MAGNETIC DAMPING 49 OF SURFACE WAVES IN MERCURY Chapter 8 FUTURE WORK 54 SUMMARY 56 REFERENCES 58 i i v vi v i i - i v -CALCULATIOM. OF THE RESONANT .FREQUENCY. CHANGE FOR A SURFACE WAVE IN A.RECTANGULAR CAVITY THE TEST OF SLATER'S THEOREM THEORETICAL CALCULATION OF THE MAGNETIC DAMPING OF SURFACE.WAVES. FOR LOW MAGNETIC REYNOLDS NUMBERS MUMMERY SUMMARY -V-LIST OF ILLUSTRATIONS Figure Page 1 Cylindrical microwave cavity 8 2 Microwave system and test cavity 15 3 Details of microwave test cavity 16 A Details of the oscilloscope trace 18 5 Method of monitoring the resonant frequency changes 18 6 Method of producing air pulses to set up surface waves 20 7 A record of the time dependence of a surface wave amplitude 22 8 Method of producing a uniform constant,magnetic field 23 9A,B Oscillation frequency results 28, 29 10 Sinusoidal time dependence of a typical surface wave 30 11 Surface distortion 32 12 The exponential decay of a typical surface wave 35 13 Viscous damping coefficient results 36 14 Description of the magnetic damping problem to be solved AA 15A ,B ,C. Magnetic damping coefficient results 50, 51, 52 16 Rectangular microwave cavity 65 17 Details of the perturbation used to test Slater's theorem 65 18 Graphs showing the change in resonant frequency as a 68 function of the perturbation amplitude 19 Description of the magnetic damping problem to be solved 71 - v i -TABLES Page I . Parameters, variables, and assumptions 26 used during surface wave experiments. II. Comparison between theory and experiment 67 III. Comparison between theory and experiment 67 -vii-ACKNOWLEDGEMENTS I wish to thank Dr.. F. L. Curzon for the excellent supervision I received while carrying out this investigation. The assistance of. Mr. W. Ratzlaff and Mr. J . Dooyeweerd in the field of electronics and the assistance of. Mr. R. Ninnis, Mr. W. Morrison, Mr. V. Stonebridge, and other members of the technical staff in the construction of the apparatus is gratefully, acknowledged. Helpful information and suggestions by. members of the plasma group and members of my Ph. D. committee are also gratefully acknowledged. I wish to thank the National Research Council, of Canada for financial assistance, during the course of this. work. Chapter 1 INTRODUCTION Surface waves are studied i n many s c i e n t i f i c f i e l d s : oceanography, geophysics, plasma physics, etc. They are of interest to the plasma physicist with respect to the confinment of plasmas. A l l plasmas i n one way or another must have surfaces to be separated from our normal, h o s t i l e room temperature environment. These surfaces, produced under the influence of gravitational and magnetic f i e l d s , tend to become unstable or distorted. This dist^rjotion can be described by a series of sine and cosine waves. Hence, the study of a plasma surface involves the study of surface waves. The effect of g r a v i t a t i o n a l and magnetic^ f i e l d s on the surface motion i s of great interest. The study of t h i s effect i s complicated, however, by temperature and mass density gradients which are present i n most laboratory plasmas. The effect can be computed and observed more eas i l y i f an e l e c t r i c a l l y conducting f l u i d , such as mercury, i s used instead of a plasma. For t h i s reason a t h e o r e t i c a l and an experimental study of surface waves on an e l e c t r i c a l l y conducting f l u i d have been made. The time dependence for a l i n e a r (small amplitude) surface wave i s usually expressed i n the form exp ( C T f ) cos (2jt£"£ ). CT is called the damping c o e f f i c i e n t and J i s c a l l e d the o s c i l l a t i o n frequency of the surface wave. I t was the aim of t h i s thesis to study the damping c o e f f i c i e n t of a surface wave under the influence of a magnetic f i e l d , but i t was found necessary, f i r s t , to investigate the damping of a surface wave free of magnetic f i e l d s and to consider the methods used to measure t h i s wave. I t was found that the l i n e a r theory of surface waves has not previously been properly tested and v e r i f i e d i n many cases (ref. 18). This i s at least par t l y due to the problems associated with the wave guages used i n the past. Devices such as capacitance (ref. 41) and resistance (ref. 42 and 43) wave guages are available to study surface waves. They, however, suffer from the -2-difficulties common to e l l immersion devices. The difficulties are attributable to: (1) the erratic, dynamic behaviour.of the. meniscus, (2) the.existence, of a of fluid, on the gauge as the free surface, recedes3 and. (3) large disturbances around .the gauge, when fluid velocities, are. large (e.g. the upward directed jet at the. stagnation.point and. caviation in the wake). Also,, the usefulness of, these and other, devices is. limited, in. many cases.,., by the lack of the. sensitivity required to study linear surface waves.. For example.,, the. linearity restriction., 5 „ K « (HK)^, (where. So is the wave amplitude.,. K is the. wave, number, and H is the fluid depth) requires, that §t»K< 10 ^ for shallow fluid (KH—0.1) experiments. Investigation of shallow fluid, surface waves, as a result, has been restricted, .because reliable methods of measuring $* when KS« ^ 10 ^ have not been available. In view of these problems Curzon end Howard (ref. 39 and.4-0) developed theoretically, a new technique to study...linear, surface, waves on a column, of an electrically conducting fluid.. . The essential idea behind this, technique is the. following. If an electrically, conducting fluid acts as a wall of a microwave cavity, any change, in shape, of that fluid (e.g. surface waves) causes a change .in. the. shape of the cavity which,,, in turn, causes a change in •the resonant frequency of. that cavity. It i s , therefore,, possible to monitor the motion of the fluid. by monitoring the change in the resonant frequency of the microwave cavity. This idea was the author to develop theoretically and experimentally a new technique, to study surface waves on a flat surface. In chapter 2 Slater's theorem (ref. 2) is used to calculate the. change in frequency of. a microwave cavity from CO to OJet. when a surface wave of amplitu'de. So is. set up on the end plate, of the cavity of length. L. Results show that (frJa. -UJ.) L — 1. There are. two important features here. (J S o First , the change, in. the resonant frequency is directly proportional to the surface wave...amplitude.. . This will. make, the observation of. the time dependence of S„ very easy. Secondly,., the system is small amplitude surface, waves. • Since values of 0.5 x 10"^  for CO are easily observed i t appears, that surface, waves such-that S«k-— 0.5 x 10 ^ can be detected. This technique, therefore, can be used to study, linear surface waves, even in the shallow fluid region. Another interesting feature i s also found. Calculations show that a. rectangular., cavity, when study surface waves acts as en. automatic Fourier analyser. -It appears that, a rectangular, cavity constructed so the change, in re sonant frequency, of. the cavity is directly proportional, to only one. Fourier component... of. the surface... wave under investigation.. This feature, should eliminate the.tedious and complex procedure of Fourier analysing arbitrary surface waves. These features,together.with the fact that the system is free of. the: problems... associated with, immersion devices^ make the: microwave cavity a useful diagnostic, tool for" studying surface waves. A cylindrical microwave cavity has. been used".to measure.the oscillation frequency, , and viscous damping coefficient, 0"o , of a standing,; axisymmetric wave; in shallow, and. deep ;,pools of "mercury. This .work, described in chapters 3, A and 5, shows that the. observed oscillation, frequencies are higher, than, predicted, by theory by up to 15$. The results indicate that the •meniscus is. the cause of this discrepancy. The observed ..viscous, damping coefficients are..higher than predicted .by the. standard theory of Case and Parkinson, (ref.. 7) by up to 4.0.0%.. In. view of this discrepancy the assumption that there horizontal, motion of the. surface is introduced. Lamh,and, (ref. 21 and. 22) this is. the condition that must be used, when, an incompressible,., insoluble, surface, film is present. The author calculated that, the additional, viscous..damping, coefficient resulting from the above equal to • -l/y 2TTS' K cosh2(KH) . i) is the 1/2 sinh (2KH) kinematic viscosity of. the fluid... The. experimental, results are found to be in. excellent, agreement with the theory when this effect is included. It i s , therefore, concluded, that there, horizontal, surface motion. It is suggested .that. this. was. due to the. presence of a surface, film. Since, the literature indicates, that surface., films are. present.on fluids., in.ordinary laboratory experiments.,, the above calculation should be useful, to those studying surface, waves. The study of magneiohydrodynamic surface waves is initiated by the author's calculation of the magnetic damping, coefficient, CJB", for low magnetic Reynolds, numbers. It is chapter6 for an. axisymmetric, linear, standing,, surface wave on an incompressible, ideal, fluid with, a finite electrical conductivity, <^  , and mass.densityyO . The effect of a vertical constant, applied, uniform,, magnetic field, , is considered. Detailed 2 calculations are made, for the. case Q Bl << 1- The calculations show that ^2TTf the oscillation frequency is independent of B^ and that ^ is given by <T z. -ff B* ( / + £KH ) This, calculation is 8 A/O ( I exp (2KH) - exp (-2KH)) experimentally verified by a series of experiments with, liquid mercury. Chapter 7 describes this work. The magnetic damping coefficient, OJ , is measuring the total damping coefficient,C , and assuming that <5~o + @e ' E xp e r i i n - e t l 'tal results confirm, this assumption. In agreement 2 with. the. theory (TB is found, to be. proportional to ^ B ? and. to vary from 2/° - 5 --J&Q (for KH 4 0.3) to - B?0 (for KH > 3.0) as. the f lu id .depth,, fl,. is varied.from, small, to large, .values. The rate at which Cfjs varies, with fluid found to increase with K as. predicted. A l l experimental, values are within 10% of. the theoretical, values. This, percentage is. less than the maximum...systematic percent error.. These results, therefore,..are considered to be in excellent agreement with our. theory for the damping.of surface waves, in. a vertical,, magnetic field. Chapter- 2 MICROWAVE .-THEORY The theory for the proposal, to use a microwave cavity, to study, surface waves is. developed in this. chapter.. Slater's theorem.relating, the change, in the. resonant frequency of a microwave cavity to small changes in shape of that, cavity is given. His theorem is used to calculate frequency shifts for a surface wave perturbation of the cavity. Section 1 SLATER'S THEOREM Slater's theorem (ref. 2) describes the relation between the shape of a microwave cavity, the electromagnetic f ie ld distribution in that cavity, and the resulting change, in resonant frequency of. that, cavity. The theorem i s stated, as: 6Ja2 - ^) 2 (1 + jjj^ (Ba2 - Ea2)cVv* )• (hi.I) Q is the re'sonant frequency, of the cavity before its shape is changed Q% is the. resonant frequency of the cavity after, i t s changed. Ba and Ea are given by Ba2 - IBI 2 / l l £ Ea 2 - IEI 2/N 2 — D """" E vc vc JT] IBI 2 dv and N E 2 - J / f l E I 2 dv , V p is the change in volume of the. cavity . V c is the original volume of the cavity . E and.B. are the original, e lectric and magnetic fields in the cavity. It should be noted here that a perturbation, method was used to develop, this theorem.. . It was assumeithat changing the cavity volume by the amount. A A, causes only a small change, in E and B. It is. evident that the theory of resonant cavities must be considered before this theorem, can be used. -6--7-Section 2 GAVITY RESONATORS A microwave cavity is a container made of a material of high, electrical conductivity. . The following, definitions.are made: ytt is the. magnetic permeability in the cavity. € i a the. electric, permittivity in the cavity. £ w i s the. electric permittivity in. the cavity walls. ^e i s the electrical, conductivity in. the cavity, w^ is, the. electrical conductivity in the, cavity walls., Q is the oscillation frequency, of the electromagnetic, (em..) wave in the. cavity. It is assumedthat: (i) The material in the cavity and. in the cavity walls i s . homogeneous and isotropic. (ii) There is no free charge..present. ( i i i ) The depth of. penetration of the fields, the skin depth, in the cavity is very.much, larger, than the.wavelength of the fields, and the skin depth in the cavity walls is very much, smaller, than the wavelength , (i .e. yzCLO^jL 1 and fr/2£„(X)>? 1 ) . The nature of. the em. field in the cavity is found .by solving Maxwell's equations using the above assumptions. The solution'for. a. cylindrical, cavity can be found in most texts on electromagnetics (ref.. 1). It wi l l be stated here.without proof. Figure 1 shows the geometry of the. cavity. There are two situations possible. First , the magnetic field in the cavity can. be. transverse to. the axis. of. the cavity.. This. Is called a transverse, magnetic, or T.M.. mode.. Secondly, the electric field can be transverse to the axis of. the cavity. This is called a transverse electric Figure I. Cylindrical microwave cavity -9-or T. E. mode. For the M. modes i t can be shown that B - ( Bel© + Brl )e AND E = ( Ee le + Er lr + Et l t )e Ju)t where Be = Ko JjL'( K o r ) Cos (J^G ) Cos ( y 7 7 T z/L) B r = i Jk^K.r) Sin (ie) Cos (riTTi/L) Ee - - J / y i l y?7r 5 (K.r)Sin ('© ) S i n ( wrt/i) E r = Suu^JLjt Jx'(Kor) Cos (jf©-) Sin ( W T E / L ) = - j a) Jx C K 0 r ) cos Ue ) cos ( W Z / L ) is a Bessel function of order A (1.2.1)' Also, these standing em. waves are set up in the cavity only when of= K , 2 - + im/ij . a.2.2) is the resonant frequency of the cavity and the three variables 1, m and n are integers which describe the azimuthal, radial and axial dependence, respectively, of the em. fields in the cavity. A specific T. bi. mode is usually denoted by " ^ A l j ^ ^ • Similarly for the T.E. 1, m, n, i t can be shown that B=(Bele *Bt\Y + dnd E = (Eele + E r lr)e where B© = - 7T/??J(Xr) Sin (/6 ) Cos (Y)7TZ/L ) Br = >)77 ^ ' (K rO Cos ( i 0 ) Cos ( fflfe/l) L K 7 Bi = Jickr) cos(ie) sin (Y)TTZ L) Ee = - i ^ L j / ( K r ) c o s ( i e ) S m C y m z / L ) FR2- , • • 1 (1. 2. 3) J Z "\ (K (1, m) A) - 0 and where CO 2 - K 2 •» ( YITT/L ) 2 . ( l . 2. A) In the above equations for B and E a constant amplitude factor has been omitted..-.. This factor i s the. output, power, of the microwave system, unimportant in the following calculations.. . Slater.' s theorem can now be used to compute the. change in resonant, frequency of. a cavity, for a surface wave perturbation of that cavity. Section 3 CALCULATION . OF THE. RESONANT. FREQUENCY! CHANGE FOR A SURFACE: WAVE IN A CYLINDRICAL: CAVITY The surface wave is. assumed to be. on the end plate (at £ z_ 0) of. the cavity. . In reference. 6 i t is shown that the.amplitude of a linear, standing,, surface wave in an. incompressible fluid with small.viscosity in a cylindrical basin .can. be. expressed by £ - £I3"(S, P) X (K (S, P » COS (S0) where J 1 (K (S; P) A) - 0. The calculation for wi l l be made for. a simple wave ( £ - J s (Kr) COS ( S e ) ) . It is assumed that . TTWSO/'L.**-I and (Gtia -O})/^)^ /• Equation (1.2.1.) is used in. (1. 1. .1.) for T. M. modes and. (1. 2. 3) is used,in .(1....1. 1) for T. E. modes.. . The integration variable, dv, is the form,. rd6 drdZ., In. this way i t can be shown that, for T. E. modes ,1* KU,Jn)f\ 2{6J„-0) - K CO 71 TT s in 2 (19) cos (se)^e J J 2 (x)Js(fol)dx o x (1. 3- 1) TVTT L i -J where (x) - dl Ji (x) d * io i I COS^ IQ) C o s ( s e ) d e [j/(x)J J S ( M x dx -11-and - K (S.. P) . K (1,. m) Similarly i t can be shown that for T. M. modes k(j? >w)A 2(0)* -Q) - i o _ j cos2 (16) cos ( s e ) de (x)l 2 J s (ou) xdx. * i 2 So \ s in 2 ( i e )coS (se ) c^e \ J , 2 (X)_JS (O<x) X CL 'X - Cd 2 K , 2 f0 \ cog2 (10) cos (se ) do f j 2 (x) Js (otx)xrfx k*Mt J ° o 1 U- 3. 2 ) where o( - K (S, P) . K G (1, m) It is. only necessary, to. compute, of. the norma.lijza.tion.. constants, p p Ng and N£ ,. because, they are. simply related... This can be shown, by. using Poynting's theorem (ref. 1). Poynting's theorem.states that f ( E x B ) - d s + j [ f B<B.+ E'-Ee]dv - . f - E-J 1/1/ . V S i s the surface enclosing the volume V . J is the current density in V. Let V be the volume of the microwave cavity, V c. As a result,. J _^ 0. ^ ^ JA> Also, E x B • (X s -_ 0 because, the electric field, tangential to the surface. S is zero.. This, is a consequence of the assumption.that the conductivity of the cavity very high.(assumption i i i above). Therefore, f r A A i A A T j \ B-B + E-E6 CCv - constant. J^A 2 J Equations, ( l . 2. 1.) and (1. 2. 3) show that B and.E. are out of phase by 90° . A* s\ s\ y\ y\ s\ That is , , when E - 03B - , B max. and., when B ^  0. , E _^ E max. Therefore, f F JLjl + £ i l £ 1 d v ^ ( B max . B max dv -_\ E max - E max £ (L v • J. 2 / 2 J J 2 yu. J 2 C A A. ^ Y 4 - ^ Ax ^ A. 9 But,. B max > B max.. - IBI2" and E max • E max.. I EI''. As a result, [ IBI 2 d v - yU€ ( UH2dv and so Ng 2 ,A/£ 2 -12-It can easily be shown (by the use of equation 1. 2. 1) that for T. M. modes \ 2> Jo Jo ~7* <5«-0 1 i - j and A// = Nl/{€jJL) "(1. 3. 8) Similarly for T. E. modes i t can be shown (by use of equation 1. 2. 3) that A/£ = NE ~~ and M i = 6 i ^ i ( r/^ /o ) f [ ^ V K n ] r ( l r +J2, ( l^XKOdr K T I JO j r ' (1. 3. 4) Equations ( l . 3. 1) to (1. 3. 4) relate the change in resonant frequency of a cylindrical microwave cavity to the amplitude of a surface wave on a fluid acting as the end plate ef that cavity. The above calculations were made for a cylindrical cavity because this type of cavity was used during the experimental work. This type was used because i t was easy to construct and to set up surface waves on a fluid within i t . A rectangular cavity, however, has also been considered. Calculations similar to those above are given in appendix I. They show that a rectangular cavity acts as an automatic Fourier analyser. It appears that a rectangular cavity can be constructed so the change in resonant frequency of the cavity is directly proportional to only one Fourier component of the surface wave under investigation. The advantage of this feature is as follows. The theory of surface motion usually expresses the time and spatial dependence in terms of Fourier components. It is necessary, therefore, to Fourier analyse experimental data to obtain the time and spatial dependence of each Fourier component in order to make a comparison with theory. This tedious and complex analysing procedure may be eliminated by the automatic Fourier analysing feature of a rectangular cavity. -13-In. the calculations for the rectangular.and.the cylindrical cavities i t was found that the change, in resonant frequency was proportional to the surface wave amplitude... This means...that the .observation, of the surface, wave . s time dependence wil l be very easy. Also, an order of magnitude, calculation wave sensitive to very small, amplitude, waves. . These, featureswith the. fact, that the. system is free of the problems associated with, immersion devices., make, the microwave system, an. extremely useful diagnostic: tool for studying surface waves. In order to use Slater's theorem to study surface waves i t was considered.necessary to test the important features of the theorem and: to examine the. conditions. for which they are valid. This, was done, by introducing, various,, well defined.shapes .of known dimensions into, the cavity at known positions. The work is reported in appendix 2 . The results, were found to be in agreement .with the theory and i t was concluded, that Slater's theorem can be used to predict the change in resonant frequency, of a microwave cavity caused by a small, amplitude perturbation of the shape.of that cavity. The.theoretical results of this, chapter were used to develop a microwave system to study surface waves on. mercury. The equipment and techniques used to study these waves are the topic of the next chapter. are easily means that the micro-Since values 0) Chapter. 3 EXPERIMENTAL .PRO.CEDIIRE, PARAMETERS VARIABLES, AND ASSUMPTIONS Section 1 EXPERIMENTAL PROCEDURE • The microwave system that given in a block diagram in figure 2. Liquid mercury was. placed in the. bottom, of. the test cavity, as shown in. figure 3- The. test cavity and microwave system were mounted.on a large, (lm.xim.xlm) cement block to prevent any vibrations that might set up unwanted surface, waves in the mercury. The test cavity was. made of brass. and. nickel pla.ted to prevent the mercury from, reacting with the brass. Three centimeter (8.6. to 9-6 KMc./s.) microwave equipment was. chosen because, i t was easy to build, the required test cavities and. because..of the availability and. low cost of the components. The klystron (723 A/B) was the. source of microwave power, (a few. milliwatts). Its output frequency was electronically modulated by a.sawtooth voltage, (from, the oscilloscope) which was added to the repeller voltage, of-the klystron.. In this way the output frequency of. this klystron was amounts., up. to 65 Mc../s. at modulation rates, of up to. 10,000. c./s. It was. found, that. the. change in output, frequency of the klystron, was directly proportional to. the sawtooth voltage of the scope for changes, up .to 65 Mc./s,. The isolator was prevent reflections caused by the. wavemeter.,.. magic, tee,, test, cavity, etc. from .changing, the output power of. the klystron. The calibrated.wavemeter was used to determine the output frequency range of.the klystron and. to determine the. resonant frequency of the microwave test, cavities. The-magic tee split the power, from the klystron: one-half, to the power terminator (which dissipated the energy) and-one-half to the test cavity which dissipated, or reflected the energy depending upon whether, or not the output frequency of. the klystron was equal to or not equal to the resonant frequency of the test cavity." The difference .in power reflected from the: power - U --15-detector klystron isolator wavemeter / /power terminator magic tee klystron klystron power repeller supply voltage o input scope waveguide attenuator sawtooth out test cavity Fig. 2 Microwave system and test cavity. -16-waveguide / pulsed air 5 L J L - _ 4 * r i s ^ .63 cm mercury ZZZZZZZZZZ777////7777"< plunger 7> seal T H J L mercury reservoir valve Fig. 3 Details of microwave test cavity. -17-terminator and the test cavity was detected by a crystal detector. The output, voltage of the detector was a monotonic increasing function of the power detected and was applied to the vertical input of the scope. The nature of the resulting, trace on the scope i s shown i n figure U-The Qu of the. system ( Qu = ^/(banwidth of the resonant dipj)was found to. be. about 2000,. as expected. An order of magnitude calculation (ref. l ) shows that Qu ^ Cavity volume ( where i s the ele c t r i c a l conductivity of the cavity wall surface, S; ? andyS i s the coupling coefficient. The coupling between the wave guide and cavity was varied u n t i l Q L was a maximum. This corresponded to having /S « J . It was. calculated from the above equation that the Q L value could be increased by about 50% i f copper or silver plating i s used instead of nickel the cavity. A Q L of 2000, however, was more than adequate for measuring resonant frequency changes of .5Mc./s. or larger. The depth of. the mercury, HjWas determined by the position of the plunger as.shown i n figure 3- The length of the cavity was varied (by varying the amount of mercury in the cavity) u n t i l an appropriate resonant em. mode was set up in the cavity (i.e. u n t i l a resonant "dip" was observed on the scope as in figure U)- This signal was expanded and the scope face masked off as shown in figure, 5- The entire trace was masked off except for two .small pieces which appeared, as dots.. The horizontal motion of these dots corresponded to the change in resonant frequency of the cavity. The horizontal motion of the peak of the resonant dip also corresponded to the change in resonant frequency of the cavity. The dots, however, were precisely defined whereas the peak of the resonant dip was not. Measurements -13-Power refl the test c jcted by i v i t y Bandwidth of the k l y s t r o n Resonant frequency of the t e s t c a v i t y F i g u r e U' D e t a i l s of the o s c i l l o s c o p e t r a c e . scope t r a ce Frequency ( KMcys.) of the k l y s t r o n cope face scope tra c e resonant f r e q . d i p Figure 5 Method of monitoring the resonant frequency changes. . The h o r i z o n t a l a x i s represents the k y l s t r o n frequency and the v e r t i c a l a x i s represents the power r e f l e c t e d by the t e s t c a v i t y . The motion of the v i s i b l e p a r t of the scope t r a c e was f i l m e d . -19-were made to ensure that the horizontal motion of the dots (for resonant frequency changes of less than. 2 Mc/s..) was due only to a change in resonant frequency and not due to a change in the shape of the resonant dip. To set.up axisymmetric, standing, surface, waves on the mercury, air was pulsed into the. cavity through a hcle in the center of the cavity top (figure.3). Other workers such as Keuligan, Case and Parkinson (ref. 10 and 7) mechanically rocked, the basin containing the fluid to set up surface waves. Fultz and Taylor, on the other hand, (ref. 11 and 12) used "flapgenerators." That i s , they moved portions of the basin's walls to create surface waves. It was considered desirable to avoid these methods because the theories for surface, waves assume ...that the basin is. not in motion and that its walls are rigid.. . As. a result, the method of setting up surface waves with pulsed air was developed. The repetition rate, of this, pulsed air was varied (figure 6) until i t was equal to the oscillation frequency of one of the "allowed" surface waves. At this point the "allowed" surface wave was set up. This caused the resonant frequency of the cavity to change. The air pulses were then turned off and the. mercury wave was allowed, to .oscillate freely. It has been shown (chapter.2,. section 3) that the change in resonant frequency,. 6)<*. - 6J , is proportional to. the amplitude of the surface wave, So . (Recall that 5 e(t) - S o(0) cos {27TH ) exp {(Ji ) •) Therefore, the time dependence, of the.horizontal motion of the two dots, seen on the scope, face, is proportional to the time dependence of the surface wave in the cavity. This time dependence was continously recorded by continuously rolling .film through a camera., that was. mounted on the scope face. In this way successive traces were photographed for an interval of about ten seconds. -20-metal disk Air pulse to cavity 15.3 cm. 1.27 cm. tubing 100 volt Sorensen Power Supply f h.p. Bodine D. C. motor air valve to air supply Figure 6. Method of producing air pulses to set up surface waves. -21-The nature 'of the resulting negative is seen in figure 7. From similar.negatives the values of So(0), i and 0~were found for various conditions. The oscillation frequency, was found by counting the number of traces that occurred during ten complete oscillations of the surface wave.. The trace time was. measured by a time mark generator. . The number of traces per ten oscillations. could, be measured to within \%. Therefore, when the trace time was kept constant, oscillation, frequency changes of %% or larger, could be detected... The damping coefficient, <J~, was measured by counting the number of oscillations, of the surface wave that occurred as the amplitude decayed to Q of its original value and then using the oscillation". frequency. It was. found that periodic surface waves were set up when the repetition.rate, of the air pulses was within 5% of i . Furthermore, no change in % (I.e. less than \% change in f ) was found when the repetition rate was varied within 5% of f . This indicates that the oscillation frequency, f , of the. particular surface wave that was created was independent of the method used to create.the..wave. During a number of experiments the effect of a uniform, constant, vertical, applied, magnetic field was studied by placing the test cavity and magnetic field at the' center of a solenoid (figure 8). The solenoid consisted of 5 coils of #11 copper wire in parallel. Each coil consisted of 150 turns. Twenty-four volts across the solenoid caused an input current of about 600 amperes. This resulted in a magnetic field of about 2200 gauss at the center of the solenoid. . This field was independent of position (i.e. varied by less than 3%) in the volume indicated in figure 8.. The mercury was placed in this volume when the effect of a magnetic field on a surface wave was studied. Switching the magnetic field on or off caused unwanted surface waves. These waves were allowed to damp out before -22-Figure 7- A record of the time dependence of a surface wave amplitude. The above i s a 120 f i l m negative taken of the ho r i z o n t a l motion of the scope trace shown i n f i g u r e 5. The h o r i z o n t a l p o s i t i o n of each dot represents the change i n the resontnt frequency of the t e s t c avity which i s proportional to the amplitude of the surface wave i n the c a v i t y . The v e r t i c a l axis represents time. -23-cooledjWire wound resistor Figure 8 Method of producing .a uniform, constant magnetic field. a controlled wave was. set up and investigated. The strength, of the magnetic field was measured,.by a Bell 240 incremental gaussme.ter. Section. 2 PARAMETERS., VARIABLES, AND ASSUMPTIONS The oscillation frequency and. the .damping.coefficient of a surface wave have been measured for a variety of. parameters. During a l l experiments described in this, chapter and in.the following chapters, this, surf ace, wave was. of the. form J - J.o ({) Js (K (S, P) IT ) cos (SO )• The. standard.theory of. fluid surface, waves (ref. 6 or equation (4.. 1- 1)) was used, to. compute K (S, P) from the measurement of the oscillation frequency. The valves of S and P were then found by consulting the tables for K (S, P) A (ref. 5 ) - Recall that Js (K. (S, P) A) ^ 0 where A is the ca.vity radius.. The study was restricted, to waves where S ^ 0 and P ^ 2. By keeping.S and P constant the "shape" or "type" of wave was kept constant. From the. tables of reference 5 i t was found that K (0, 2) A ^ 1.2197 TT . Three different values of K were, used by using three different cavity radii (A. - 2.54, 3.17 and 3-64 cm..). In a l l surface, wave experiments the. em. mode in. the resonating cavity was either the T E ^ or the T E , ( < mode. A computer program was used to calculate ( £)a -tJ ) L from equations (1. 3 - 1) and (1. 3 - 2 ) . This OJ So S calculation was. .only used to measure .the. magnitude of 3o . This .was. done to ensure that the. linear or small,amplitude assumptions were, valid for,the waves under investigation.. The. study of the damping, coefficients,.and... the oscillation frequencies^ however, relied, only upon the. fact, that is proportional to Table.1 summarises, the parameters .and,variables for...which,.the oscillation frequencies and,damping,.coefficients were measured. -25-In order, to isolate the. magnetic damping coefficient, , from the observed, total damping, 6~ , (6^•== (5 o + (5~B ) the viscous damping coefficient, and also the oscillation frequency have tc be measured. The following two chapters describe this work. SUMMARY OF. DEFINITIONS.AND CONSTANTS USED IN. TABLE 1 A K S T /° D 9 9 H cr 5 S. L 6Ja - 0 " CO L = -cavity radius radial wave number of surface waves 0 P = 2 KA = 1,2197 TT surface tension of mercury = 490 dynes/cm. mass density of mercury = 13.6-^./cm. kinematic viscosity of mercury 0,11 cm.^/o. gravitational constant = 980 dynes/9, electrical conductivity of mercury 1.04 x 1 0 6 (ohm meter)-1 depth, of mercury damping coefficient of the surface wave oscillation frequency of the surface, wave surface wave amplitude applied vertical magnetic, field length of the microwave cavity axial mode number of em... mode in the cavity change in resonant frequency of the cavity resonant frequency of the microwave cavity magnetic Reynolds number -26-Table I Parameters.,. Variables and Assumptions used during surface wave, experiments Variables and Parameters Assumptions used in theory Experimental values Cavity number I I I I I I A (cm) 2.54 3.17 3-64 K (cm."' ) 1.51 1.21 1.05 em. mode used TE„, TE w 0 range (cm.) ..04 - -01 .0015 - .0007 .0015-.0007 H range (cm.) 0.4 - 4-0 Q.4 - 4-0 0.4-4-0 B 3 (gauss) 0 - 2200 0 - 2200 0-2200 h (c/s.) 4-6 - 7.0 3-5 - 5.8. 3.5-5.2 TOTT^o / L . « 1.0 <0.1 ^0.004 ^0.004 ( 6 J * - u)/(J « 1.0 ^ .002 < .002 ^ .002 So / H « 1.0 ^ .06 <.0005 < .0005 « 1.0 <C .03. ^ .002 < .002 S „ K / ( K H ) 3 < < 1.0 ^ :5 < .04 < .04 i : o <^ .0006 ^.0006 < .0006 « 1.0 ^ .015 < .015 < .015 <<r 1.0 ^ .001 ^ .001 ^ .001 « 1.0 ^ .02 < .02 < .02 /O = CONSTANT mercury jsed « 1.0 ^ 2 ^.2 ^ .2 //.(jjVl = R^ « 1.0 ^ .01 <.01 ^ .01 Chapter. 4 STUDY OF THE OSCILLATION FREQUENCY OF A SURFACE WAVE IN MERCURY The microwave cavity technique .was used to investigate, the.. relation between the oscillation frequency, ^ , the fluid depth, H, and the radial, wave number, K, of a linear, standing, gravity' surface wave. The oscillation frequencies, were studied for both deep and shallow fluid waves. KH'was varied! from about 0 .4 to about 3-0. Figure 10 is typical of. the time dependence of the waves that were studied. . It shows that the.time dependence, was sinusoidal,, as expected. The oscillation frequencies, were measured and ..summarised in figures 9A and 9 B . The linearized, theory of standing surface, waves on an. ideal,, incom-pressible fluid (ref. 6) shows that (2TTS ) 2 - (TK3 + 0 k0 tanh (KH)• (4.1.1) P D Table. 1 contains the relevant definitions and..a l i s t of the assumptions which were made in order to develop the above equation. Equation ( 4 . 1. I) was plotted, using the valves for T,^ o , and kA given in table 1 for various A's. These plots (fig. 9a) indicate that the oscillation frequency depends, on the basin,radius, A, and the fluid depth, H, as predicted by the theory.. It was noted., however, that the observed, frequencies, were, larger .than. the. theoretical ones, by 10 to 15$. Discrepancies of this nature have been noted by, other workers (ref. 7)-The difference, between experiment ..and theory may be due to: (1) experimental error in measuring 5 J (2) the assumptions of the linear theory not being valid, (3) an error in the value of T/^ o that was used, ( 4 ) surface tension effects associated with the meniscus". Each of. these possibilities will now be considered. (l) The maximum,percent error in measuring £ was less., than. 2%. Therefore, the differences., between the experiment and the theory (approx. 10$) O S C I L L A T I O N F R E Q U E N C Y , f ( C Y C L E S / SEC . ) o > o > > H ro In o 3 CD 33 J> TJ X M -2Z-A * 2.54 cm DEPTH OF FLUID r H (cm) -oc--31-cannot be explained in this way. (2) Table 1 shows that a l l of the assumptions of the theory were satisfied during the experiment. • (3) It is well.known that contamination of a .surface. ..can decrease the,surf ace, tension, T. (ref.. 9) • However, TK3 < 10$ of (j K for our experiments. Therefore, T would have to be increased by 200$ to account for the difference between experiment and theory. (See equation (4..1.1.).) ( 4 ) The equation for J results from evaluating a linearized,form of. the equation of motion of the fluid at the, surface. The value.of K is determined. by using the boundary condition, Vy- 0 at Y" - A ( Vr is the radial velocity of the fluid). , It is conceivable, however, that because of the meniscus this boundary condition is not true at the surface. It is suggested that the condition \Zy- - 0 at r = A -6 , (where £ is defined in figure 11) is more realistic at the surface. This would result in K and, therefore, £ being evaluated, by A - € instead, of A. During, the. experiments corresponding to. graphs, I and, II. .of figure 9 £ was measured and,found to be of the order of 0.2 cm.. Figure 9b shows that the use of. € values, of this, size to calculate 5" for these two experiments, results in agreement between theory and. experiment. During the experiment of graph III, figure 9b,' € was less than .02 cm.. This low value was. presumably, due to surface contamination during,this, experiment. Figure 9b shows that the use. of an € value of this size to calculate 6" for this experiment again results in agreement, between theory and, experiment. Another experiment to test the dependence of S on 6" was performed for H -_ 1.00 cm. and A ^  3 . 6 4 cm. The results showed that € was of the order of 0.2 cm. when, fresh, clean mercury was. used The oscillation frequency agreed with the. theory when A -0.25 cm., was used instead.... of A. -32-container wall Figure 11 . Surface distortion due to surface tension. _33-After a film of oil> dust and mercury oxide was introduced the value of £ was found to be of the order of 0.02 cm. The oscillation.frequency was then found to agree, with the theory using A - 0.02 cm. These, results suggest that the meniscus, is responsible.for. .the 1056 discrepancy between, the theoretical and experimental values for the oscillation frequency. CHAPTER 5 STUDY OF THE VISCOUS DAMPING OF A SURFACE WAVE IN. MERCURY The time dependence of ...a small, amplitude j standing. ..wave. is. given, by exp ) sin (27T§{ ). The microwave cavity technique was. used to investigate the relation between the viscous.damping-coefficient, 0"o , the fluid depth, H, and the radial wave number,. K, of a linear, standing, gravity, surface wave. The viscous damping, of surface waves, was studied for. both deep and shallow fluid waves. KH was. varied from about 0.4. to about 3-0. Figure. 12 shows how the. surface, wave amplitude decayed.. It shows, that, the waves decayed exponentially, as expected. The damping coefficient was. measured and the., results summarised, in figure 13-The standard, linearized theory of axisymmetric, standing waves in an incompressible fluid is given, by Case and Parkinson (ref. 7). They show that (?° - crv + <y* + (5"b + • (5.1.1) &f is the. damping coefficient due. to the energy the body, of the...liquid.... (Svj is due to the. boundary layer, at the. side, walls. . (5h . is due to the boundary layer at the bottom wall. of. the fluid container and (5^ is. due to the dissipation in the boundary layer at the free surface of the fluid.. They find the following: 6~ - ~ \\^2 K ( 1 + (S /KA) 2 _ 2KH ) (5. 1. 2) 11 ^ (AK)2 ( 1 - ( S /KA) 2 sinh (2KH) ) \ &t> - - \I/J2 7r5r ' K ( 1 ) » ^ ~ • (sinh (2KH)) where (2 7r5) - (TK3 +- q K ) tanh ( KH) and j^is the.kinematic viscosity p d of the. fluid. The assumptions that were used to develope. this, theory., are given in table 1. - 34--35-H 1.01 C I L . K = 1.21 cm-' 3 2j) _3j) 4.Q Time units Fig. 12 graph showing...the exponential decay of a typical surface wave. -3b-Q\ I I I I ! i i i i I i i i I I J ' i i 0 0.5 1.0 1.5 D E P T H O F FLUID, H ( cm) Fig. 13 Viscous Damping Coefficient Results The boundary conditions were the following: (i) The fluid velocity is zero at the. rigid boundaries, (ii) The flow is irrotational at the free, surface_ The above equation for-.| (Jl I , (5- 1. 1) and (5- 1- 2), was plotted in figure 13 using the experimental values given in, section.2, chapter 3-Figure, 13 indicates that the experimental values, of CTJ" were larger than predicted hy Case and Parkinson's theory by up to a factor of L. Discrepancies of this nature have been, noted and, in .some cases.,., accounted for by people., working on other, surface, wave experiments (ref., 10, I 4 , . 16, 17, 18). The difference between experiment and.theory may be due to (1) systematic errors (I.e. experimental error, in. measuring (T0 ' )} (2) a damping, effect due to em. fields,in the microwave cavity, (3) violations of the linearity assumptions,used: to develop the j theory; ( 4 ) rough boundaries, or (5) the use of an invalid, boundary condition. Each of,these, possibilities will now be, considered. (1) The systematic percent error, (i .e. the percent-error in the measurement of (5^  ) could be as high, as 15$. The graph of figure 13 shows that there was a difference between experiment and theory of up to about 400%. The systematic error, therefore, could, not (2) An experiment was. performed, .to. investigate. the effect, of. the microwave cavity's em. field on the damping,of the.surface, waves. An attenuator was. used to vary (by a factor of 3) the power, absorbed by the cavity during resonance. In this way was. measured.for various values of the cavity's em. field. Results showed that (Jo varied by less than 3%. It was concluded, therefore, that (Jl was independent of the cavity's em. field. -38-(3) Other workers have, not always • s a t i s f i e d , a l l the. l i n e a r i z i n g assumptions given i n table 1... The assumption £oK<^ (HK)^ is.,, i n some cases, not even included i n the t h e o r e t i c a l discussions ( r e f . 7, 10, 1.7). Hunt and U r s e l l ( r e f . 18 and .13) have pointed out this, omission. In. many cases, ( r e f . 7, 14 and. 17.) workers have v i o l a t e d this., assumption during, t h e i r experiments i n shallow water ( i . e . KH £ 0.1) . : This happened, because they had, d i f f i c u l t y , observing surface,waves such..that ioK.^* (0.1)^. Table 1, however,, shows that a l l of the assumptions (including i o X « . (KH)3) were s a t i s f i e d . i n our experiments. The high, s e n s i t i v i t y of the microwave system made, this p o s s i b l e . ( 4 ) Case and Parkinson . (ref'.. 7) noticed an e f f e c t which was apparently due to wall, roughness. They found that the experimental damping.coefficient was. two to three times l a r g e r than, their, theory predicted u n t i l they highly polished the container, walls. Even a roughness of. depth l e s s than the / boundary la y e r thickness • to increase the rate of damping. A f t e r p o l i s h i n g the container they found that t h e i r experimental r e s u l t s agreed, to-within 15% of t h e i r theory. Other experimenters ( r e f . 19. and 20),. however, found that (J0 was independent of wall, roughness. In view of t h i s and. i n view of l a t e r work on surface wave damping., due. to surface f i l m s , i t i s suggested, that the Case and Parkinson discrepancy was, i n f a c t , due-to a f i l m on the water surface. For our experiments, the. c a v i t i e s were plated, with a hard n i c k e l and then highly polished. (5) The theory of Case and Parkinson (equations (5. !• 1) and (5- 1. 2)) assumes, that the f l u i d motion i s i r r o t a t i o n a l at the f l u i d surface. I f , however, t h i s assumption i s . replaced by the. assumption that there i s no ho r i z o n t a l , f l u i d v e l o c i t y at the surface then the theory w i l l be a l t e r e d . The following i s an. o u t l i n e of how the Case, and Parkinson, theory was -39-modified by the. author, using, .this new boundary, condition in an attempt to explain the. experimental .results of figure 13. Case, and. Parkinson, used standard, boundary, layer, theory, and let V , the A. / A. A A. I fluid, velocity, equal - Vy) + V x A. . Vf is. the fluid velocity, that would exist i f ]/ ^ 0 and V x A is the. correction that.must be added. ,to to satisfy the additional boundary condition that occurs when V7^ 0. When )) ^ 0 the fluid velocity, tangential, to a. wall is zero. Case and Parkinson show that 0 - C, cosh. ( ) cos. ( S 6 ) Js ( K r ) cos ( 2 I T 5 i ) exp ( (5>t ) (5-1.3) where. C, is a constant. The. approximations used in boundary layer theory allow one to calculate 6^ from (X - (5v + + tfb + 6~z (ref. 2 1 ) where (5. Ii, 4) 6S - c ^ f C v x v xt)dv s C^ is an unimportant constant.... Vb , Vw , V$ are volumes in the vicinity of the bottom, wall, the side, walls and the 'surface respectively. A. /V. /N-It is convenient to define.As., Ab and Aw by s\ A J A ^ As in volume Vs A A A A . / A - Ab in volume V b A /» A - Aw in volume. Vv (5. 1. 5) Case and Parkinson solved for Ab and.Aw. by. using the boundary condition resulting from, the fact, that A- , A. A A. ¥ 0 . That i s , they solved for At and Aw from ^  x ( V x A b ) ^ 1^ x ( 2 = 0 ) (5. 1 . 6 ) and l \ x (V x Aw) - i \ x V 0 (T= A) (5. 1 - 7) using equation (5- 1. 3)- They then solved for (Ty and by using equations (5- 1- 5) and (5- 1- 4) --40-/ A A, They, assumed that, the flow, was irrdiational, at the surf ace (i .e. V ^ - V0 )• A This, meant that As ^ 0 and, therefore, - 0. If, however, i t is assumed that there is no horizontal fluid velocity at, -the., .surface then will not be zero because As wil l not be. zero. This A A I is because " \J ^  - yu) at surface.!' i s .replaced by *l x. V ( i - H) - 0 - - I 4 x V 0 d - H) + 1^ x V x As . . (5. 1. 3) A A l l that, to use equation ( 5 • 1• 3) in (5- 1. 8) to solve for A s and then, use ( 5 . 1. 5) in (5.. 1. 4 ) to solve for. 6$ . This can. be easily accomplished by noting in equation (5- 1- 3) that 0(1 - H ) - 0(1 -.0) cosh (|<H). ( 5 . 1 . 9 ) Using...equation ( 5 . 1.-9) in (5- 1. 8) and (5. 1". 6) i t is obvious, that Aa - At. cosh ( K H). Usingthis with equations. (5- 1. 4 ) and ( 5 . 1. 5) 2 gives 6^ " z. 6b c ° 3 h (V<H) that 61 --IM-ZTTS ' K cosh2 ( KH) I 2 sinh (2. KH) From equation. ( 5 . 1. 2) we find, therefore, (5. 1. 10) It is reassuring to. note that 6S - - | ^ 2 7 T $ ' ^- for K H /7 1 and that. this,.agrees., withthe 2 calculations made, by Lamb, and Levich .(.ref.: 21 and. 2 2 ) . The total damping coefficient is found by using (5- 1. 10) in (5. 1. 1) and. (5- 1- 2 ) . It is 60 - - 2 ^ K 2 VzirS K 1 . 2 I 2A + cosh^ ( KH) 1 + ( s / k A ) 2 - 2 KH 1 -(S/Kf\f sinh (2KH) 1 (5- 1. 11) sinh ( 2 X K) sinh (2 k H) J Equation ( 5 . 1. ll) for I 6~0 1 was. plotted in. figure 13...... As figure 13 shows there is. excellent, agreement between this,modified theory, and. the experiment. It was, therefore, concluded that the. horizontal,fluid..velocity at the surf ace...was. zero. -41-The question now i s , why i s there no.horizontal,, f l u i d , velocity, at the surface? According... to. Lamb and. Levi cih ( r e f . 21 and 22) this, may be due .to an.incompressible, i n s o l u b l e , massless, surface f i l m on t h e , f l u i d . A surface film. of. t h i s type acts as a thi n metal plate would. The motion of f l u i d can, e a s i l y bend the. surface f i l m , that i s , Cause the. film, to move v e r t i c a l l y . The f l u i d , motion cannot., however, cause the f i l m to. move h o r i z o n t a l l y because the f i l m i s incompressible. Because of. the. f i n i t e v i s c o s i t y of the.fluid the f l u i d at. the surface must move with the f i l m . This means that, the f l u i d a t the surface can.move v e r t i c a l l y but.,, because, of the fi l m ' s presence, cannot move h o r i z o n t a l l y . I n other words, there can be no h o r i z o n t a l , f l u i d v e l o c i t y at the surface i f an. incompressible, i n s o l u b l e , surface, f i l m i s present. Lamb ( r e f . 21) considered theoretically, the e f f e c t of. an incompressible, insoluble,, surface f i l m for. deep f l u i d waves. Van Dorn ( r e f . 14) too, considered t h i s type of film... He worked t h e o r e t i c a l l y and. experimentally on progressive, g r a v i t y waves,, in. a. rectangular, tank.. Van Dorn claims to have . calculated the. t o t a l , viscous, damping., .coefficient, including...the. surface f i l m e f f e c t f o r a l l values, of KH. This., unfortunately, i s not true because... h i s c a l c u l a t i o n s .use a formula taken, from. Landau and. L i f s h i t z ( r e f . 44) which i s only true f o r deep f l u i d s , ( i . e.-X H ^ 1). His experimental .results do not c l o s e l y agree, with...his theory. Surface f i l m e f f e c t s were, discussed t h e o r e t i c a l l y .as.a function of the. compressibility and s o l u b i l i t y of the film. by. Levich and Dorrestein ( r e f . 22 and 24) f o r deep f l u i d waves ( i . e. KH» 1). Experimental work to, investigate, these.effects..for deep f l u i d , c a p i l l a r y waves was carried, out by Davies and. .Vose ( r e f . 16.). Their experiments demonstrated, i n agreement with theory, that damping,.is increased by factors of two or three, by surface f i l m s . Only when the. experimental, equipment an d . f l u i d -42-were scrupulously cleaned, did the damping, coefficient, agree .with, theoretical calculations which neglected surface film.,effe.cts. It can be concluded from our,work that, the observed,viscous,damping coefficients agree with .the linear, theory when the assumption that, the, flow is irrotational. at the fluid surface is replaced .with the, assumption i that, there is no horizontal., fluid velocity at the surface. . Also.,, the comments and,work of others, in the field .suggest .that, an .incompressible, insoluble,, surface film made this .latter assumption valid, during our experiments. This, concludes the study of surface, waves free of.magnetic fields. In the following two chapters the effect, of a. vertical,, magnetic field, is theoretically and experimentally studied. CHAPTER 6 THEORETICAL. CALCULATION OF MAGNETIC .'DAMPING OF" SURFACE: WAVES TOR LW MAGNETIC 'REYNOLDS^ .'NUMBERS In this chapter the damping.coefficient, is derived for standing, axisymmetric., surface waves, on an ideal, incompressible fluid .of finite conductivity and depth in a vertical,... uniform, constant,, applied magnetic field,. 6 3 . The theory in this chapter was developed to explain the results of the. experiment described in chapter. 7- A more general discussion of damping, by magnetic appendix .3. Figure U+. shows, the nature of the problem to be solved.. We. define, the following: w. 8 = magnetic field. E •= electric field. % ~ electrical, conductivity of. the fluid. J = current density, v = fluid velocity. P — fluid.density § = surface, wave amplitude, H. = fluid depth. A = radius, of fluid container. T = fluid surface tension. P — pressure-B 3 = applied magnetic field. 9 — gravitational... constant. K- =• radial wave number, of the.surface wave, V = fluid, viscosity. We use. the following assumptions and. equations: Maxwell's equations (assuming no free charge) V- B - 0 b B - - v x E V x B - yUJ yu€ <^ E (6. 1. 1) A A. 6t (The above, equations give... V-J 0) Ohm's law^(neglecting Hall, currents) J - q (E + v x B.) (6 .1 . 2) In this, work we make the. assumption. that,..the .magnetic. Reynolds...number.., Rm, is. much, less than one. . (Rm ^ yU(^VJL where, V and. J are the characteristic -43--44-S, applied magnetic field, I rigid walls Fluid with.conductivity, , and mass densityy£> I I H Rigid bottom Figure 14. Description of the magnetic damping problem to be solved. -45-velocity and length,, .respectively)... This means, that . EX-4- V x B in equation...(6... 1. 2) and. that. the. induced.magnetic f ield, , is much smaller, than the applied magnetic f ield, . The fluid equations are be + V- (PV ) - 0 (6. .1. 3) and/O ( V -V) \/l - - V P - vO VG. + J x B +•" ^(other terms) ^ J ( 6 . 1. 4) we assume jj ^  0 ( i . e. ideal f lu id) 3 that^o is a constant, (i.e.. incompressible fluid) } and & — (i .e. in a constant gravitational, field)-The Linearity conditions, required are KS " u 5/H "1, \ ( 6 . 1 . 5) and <-<- (HK) • The boundary conditions are V-~n - 0 - (at rigid walls), (6. 1 . 6) P - P0 T (1) <3_( ) (at the free surface), (6... 1 . 7 ) and (at the free, surface) • (6. 1 . 8) The surface, wave, is assumed to be axisymmetric The problem will be solved using, a perturbation technique. . (i.e.. we .assume. B. - B.s + O B where | 6 B I « } B 31 . ) The oscillation frequency, ^ , and the damping .coefficient, C^e , are found by assuming, that the.time, dependence .is of the form exp. (.-int) where Yi z. 2.TT$ + i (5~B • Fraenkel. (ref. 24) has solved this problem with the restriction k H <r<- 1. We will solve this problem for a l l values of k. H . By using ( 6 . 1. 5) in (6. 1. 4), taking the curl cud. of ( 6 . I . 4), dropping terms of f irs t order, or, higher in. OB/B3 , and. using ( 6 . . .1 . 3) we find that /Oy\\ h _ { - 2Ni ) - 1 f) (r cWt_)l -.. (JVfc . (6. 1. 9) Ve recall, that 1 c) f r 3 j p ( Kr )f - -K2jo ^ r ) ( w h e r e J° (Kr) i s a Bessel function r dr\ dr J of order 0 ) . Therefore, we can use V4 Jo ( K r ) |A,cosh ( . k£ ) + sinh ( ki )) as a solution, to (6. .1. 9) We. note, however, that ^2.— ® because, a boundary condition (equation 6 .1.6) gives, us. - 0 at £ ^ 0. Using.. V 4 in (6- 1. 9) we find that k (1 + i0B3 ) - x2- (6.1.10) The value of |< is found in.the. following, manner. . By taking, the curl of (6. 1. A) we find that <3Vr (1 + 1 0 B) ) - (jVk Therefore, V r °C Jo ( K r ) -d i ( V ' <3r The boundary condition at the walls gives us. Vr ^ 0 a t r - A-Therefore, J 0 ' ( K A ) - 0 . (6. 1.11) The value of K } a s a result, is determined by the roots, of. J c The, dispersion relation (relating n to. K, k , B 3 , H etc.) is found by taking...the time derivative, of (6. I. 4-) and. evaluating .its.. £ component at the. surface. Note that, the J x B- term in the rc component, of. equation (6. 1. U) is of the order of ^ BjSB and,, therefore, can be neglected. This gives / ° % - i i _ ( r c ) i " ) - AO ^ (.6.. 1-12) / _ * dr dr ^ di We recall that 5 at the surface. This means that. % °< Jo (K r ) £ , We define lj)so that <^  IjJ- \jh . Therefore (6. 1. 12) becomes A4/ - ( T K 2 + ) ^  2jJ ( . .£ - H) . (6. 1.13) di di  1 However, /j) <=4 cosh .(. k £ ) because .^.^J- V^. o( sinh ( k H , ) . di hi -47-As,..a result, (6. 1. 13) becomes "H 2 - | l K 2 + ^ | ^ tanh ( k H ) (6. 1. 14) The problem is solved,. ..therefore, by. considering the three equations, (6.... 1. 1 4 ) , (6. 1. 11) and (6. 1. 10) . They are as follows: 7\ P TK •+ J ^ 2 . 1 ^ K tanh ( k H) K (6. 1.15) k ( 1 - . 1 0 8 / ) : J-'o (KA) - o where ri -' 2 T T ^ + i^B and 5 <* Jo ( KIT ) & . For B 3 - O ^ e a ^ o v e results agree, with. Coulson (ref. 6). The problem of finding ~Y\ in terms of K , H, and B 3 is simplified i f we assume « . / • It is clear later that -this corresponds, to assuming that /27rb ^ i* This assumption is consistent with the experimental results described, in chapter 7 tanh ©^*)(e** - e2Jl) ^ for & « / Using the above assumption and. using the fact A (1 + 10)1 - /tanh (A)l (1 + 4A61 t J I /( (1 + A we find that (6. 1. 15) reduces to -YX - 2-ni + ±61 (2.7T5 ) 2 - | T K . 3 + ^ K j tanh ( K H ) I/O j 'o (KA) - . . 0 4 K H e (6. 1.16) -iht So<. jo (Kr ) e The interesting points to note are the following: (i) £ is independent of B 3 for a l l K H. ( i i) (3B is proportional to B^Q for a l l K H 1 ( i i i ) For VCH^ 1 CT - -Bjfl -48-(iv) For K H « 1 - -B2Q This agrees with the theoretical calculation by Fraenkel (ref. 24). (v) The shape... of. the. wave is. independent... of B3 • In the next chapter,the results, of. the experiment to test this theory are given. 5 and (J^ were measured for different values of B3 > K- > and,.H..The., experimental. results agree, well with, the above theoretical calculations. 7 The problem of solving ( 6 . 1. 15) for a l l values of. OQ2 was not considered. CHAPTER 7 EXPERIMENTAL: STUDY- OF "THE" MAGNETIC" DAMPING OF A SURFACE"WAVE IN"MERCURY This,chapter describes the experimental study of the magnetic damping coefficient, (T^ , of linear,, gravity, standing, surface waves in mercury innaeraecLin.. an ..applied,. Vertical, uniform, constant, magnetic field, B$ . The, microwave cavity technique was used -to,investigate, the. relations between (5^ , H, K , and B 3 A discussion of the results anda comparison with.the theory (developed in chapter 6) is also included in, this chapter. The values of OJ were, found from the measurement of the observed, total damping coefficient, 6~, by assuming 0~ - (J~o + (j^ . The values of Ol that were used here were obtained,from the,.experiment, described in Chapter. 5.- The results are summarised in figure 15, A, B and C. This experiment, studied both, deep and; shallow fluid waves. KH was varied from about 0.4 to about .4.0. In figure 15 the magnetic field values are described as, -for example,,, "approx. 2100. gauss." This means that the magnetic field..value was -in the ,2100,.gauss range (i.e., between 2050 and 2150 gauss). . The, measured.value with an error .of less than 5% was used, however, to. calculate OB / g ^  The linearized, theory of standing waves, in a vertical, magnetic field has been developed in chapter 6. for, this experiment... Equation ( 6 . 1..16) . gives 61 - - $ML [1 + ) . . (7-. 1. 1) JyO ( exp (2.KH) - exp (-2KH) ) The assumptions ..used, this., theory, are. summarized-in..-table '1. The above equation.was used to plot OB as a function,of depth,with wavelength as. a. parameter, (figure. 15 A, B, C).' The plots show that the agreement between the theory and.,the experiment is very. good. . . Any. difference can be explained .by. the experimental..error In measuring, OJ and. B2 . ' The EXPERIMENTAL RESULTS © 6 APPROX. 2IOO GAUSS A « • 1600 " O • •• 1200 » A » w 700 » (max. experimental error in ^ B/ 2 Is 25%) /B-» THEORY (EQN. 7.1.1 ) Fig. 15 a Magnetic Damping Coefficient Results ( K = 1.51 c m " 1 ) 0.6 L2 1.8 2.4 3.0 DEPTH OF FLUID, H (cm) -51-E X P E R I M E N T A L R E S U L T S O B APPROX. 2000 G A U S S A « a 1500 « o « « 1200 « A « « 8 0 0 » (max. experimental error in °^B 2 is 2 5 % ) T H E O R Y ( E Q N . 7.1.1 ) O . 15 b Magnetic Damping Coeff icient Results ( K = 1.21 c m - 1 ) J — 1 — L — » — ' — - L — i — i — L — i — i — I i , I • , 1.2 1.8 2.4 3.0 D E P T H OF FLU ID , H ( c m ) -52-4 8 4 2 " — • o UJ (O 1 CVJ UJ o r h- LU CO > • 3 6 3 0 b CO UJ o o o o H K < o o I-Ul e> < 2 4 o Ui cc < ZD O i/> o _ l UJ u. o >-Ul 2 o < 18 12 J L E X P E R I M E N T A L R E S U L T S © B APPROX. 2 2 0 0 G A U S S A N N | 4 0 0 » O « « 1 2 0 0 II . A N II 7 0 0 " (max. experimental error in ^BZ , 2 is 2 5 % ) T H E O R Y ( E Q N . 7.1.1 ) Fig. 15 c Magnetic Damping Coeff ic ient Results ( K = 1.05 c m " 1 ) J L 1 _ J _ 0.6 1.2 1.8 2.4 D E P T H OF F L U I D , H ( cm) I , , 3.0 -53-percent error in the measurement.of <TB could be as high as 15$ and, the percent, error in the measurement of could be .as high as, 10%. The oscillation frequency, $ , was found to be independent of S3 to within 1%. This agrees with the theory developed in chapter 6. CHAPTER 8 F U T U R E - W O R K I t i s the intention, of. this, chapter to l i s t a number of questions which a r i s e natura l ly from the work reported i n t h i s t h e s i s . . The answering of these questions is., regarded as. future work. The. microwave cavi ty technique has proved, very useful for studying the viscous and.magnetic damping of a surface, wave on a conducting f l u i d . Can i t also be used to study surface.motion caused by density gradients, temperature gradients, and. .current discharges i n a., conducting . f l u i d ? ' In appendix I the theory for the use of a rectangular microwave cavi ty as. an automatic. Fourier, analyser of surface waves was. developed. Can this, theory be.put into, pract ise? In chapter 4 i t was suggested.that, the observed o s c i l l a t i o n frequency of e. standing surface wave, was larger than .that predicted by the standard theory because of the meniscus. Can a detailed, theory be developed to account for t h i s effect? In chapter 5 experimental.evidence indicated, that there was no hor izonta l motion of the surface when the viscous damping, of a surface wave on mercury was studied.. . Was t h i s due. to an incompressible, i n s o l u b l e , surface f i lm? Could an o i l or mercury oxide f i l m have caused t h i s effect? Could the em. f i e l d s i n the microwave cavi ty have caused t h i s effect? In chapter 6 and. appendix 3 a theory for the magnetic damping of surface waves was developed by using the r e s t r i c t i o n s . Pan 1 and without these r e s t r i c t i o n s ? What would we expect i f Rm » i or i f 1 ) . Can e. theory be developed (cjEV/Cmip)» i ? -54-- 5 5 -As chapter 7 shows,, the experiment on the damping of a surface wave by a. vertical magnetic, field confirms the theory that was developed for i t . Will a similar, experiment, .using, a horizontal magnetic field, agree with, the theory developed in appendix 3? SUMMARY A new, convenient, method of studying small amplitude,, surface waves has been, developed theoretically and. experimentally,. This method is free of the problems associated, with immersion devices. It is, capable of measuring surface waves where., $ oK % 10-A. In certain, cases i t can,,be used to automatically Fourier analyse arbitrary, surface. waves. This method was used, to study a.simple,, axisymmetric surface,wave in liquid, mercury. The, wave, was studied for values.of. « H between 0.4 and 4.0. v Experiments showed .that.the oscillation frequencies of the wave were., higher than predicted by the linear, theory, by up to 15/6. It was suggested that this.discrepancy can be accounted for by considering the effect .that the meniscus, has upon the boundary condition at the walls. Experiments, also showed, that the viscous damping, coefficient was much higher than predicted by standard, theory. In view of this the author computed the additional damping .coefficient which resulted when the assumption that "the flow is irrotational at the fluid surface" was replaced by,, the assumption that "the,horizontal fluid .velocity is zero at the. surface."' This additional term was found to be - lhj _ 2 J T j ' X. cosh^ ( K H ) . The experimental, results were in excellent ' 2 sinh ( 2 K H ) agreement with the theory when this effect was included. Finally, the damping of surface waves by, a.vertical, magnetic.field.was considered for low magnetic, Reynolds Numbers. The" magnetic damping ...coefficient and the oscillation frequency were the author, for an axisymmetric, linear,, standing, surface wave on an incompressible }electrically conducting fluid. Detailed calculations were made for the case QB£ CC / They showed that the. oscillation.frequency was independent of the. magnetic -56 -- 5 7 -f ie ld and that .the, magnetic damping coefficient was given by L k H ) . An experiment with liqu 0 U/O { exp ( 2 K H ) - exp ( - 2 K H ) ) mercury was performed.. The results are. in excellent agreement, with, the above calculation. REFERENCES 1 . Langmuir, R. V. ( 1 9 6 1 ) , Electromagnetic f i e l d s and. Waves, McGraw-Hill s e r i e s i n Engineering.Sciences .2, S l a t e r , J . C. (1963), Microwave.Electronics,. B e l l L a b o r a t o r i e s S e r i e s 3 Ginzton, E. L. (1957), Microwave Measurements, McGraw-Hill. Book Company, Inc. p. 4 4 5 4 Whitmer,. R. M.. (1962).,. Electromagnetics,..Prentice-Hall, I n c . 5 Morse, P. M. and. Feshback, H. (1953), Methods of T h e o r e t i c a l P h y s i c s , McGraw-Hill Book Company, Inc.. p. 1565 6 . Coulson, C . A. (1961), Waves, U n i v e r s i t y Mathematical Texts. 7 . Case,. K. M . and Parkinson, W. C. (1956)., X of F l u i d . Mech. 2,. 172 8 Tad j bakhsh, I. and. K e l l e r , J . B, (1959), J. o f F l u i d Mech. 8 , U 2 9 Rayleigh, Lord (1390), P h i l . Mag. XXX,. 386 1 0 Keulegan, G. H. ( 1 9 5 8 ) , <X of F l u i d Mech. 6, 33 11 F u l t z , D. (1962), J. F l u i d Mech. 13, 193 12 T a y l o r , S i r G. (1953), Proc. Roy. Soc. A, 218, 44 13 U r s e l l , F. (1953), Proc. Cam. P h i l . Soc. 4 9 , 685 14 Van Dorn, W. G. (1965), J. F l u i d Mech. 24, 769 15 Kranzer, H. C. and Keller,, J . B. (1959), J- A p p l . Phys. 30, 398 16 Davies,. J . T". and V'ose, R". W. (1965), Proc. Roy. Soc. A, 236, 218 17 Grosch, G. C. and Ward, L. W.. and Lukaslk, S. J . (I960), Phys. of Fl u i d s . 3, 477 18 Hunt, J . M. ( 1 9 6 3 ) , Phys. F l u i d s 7, 156 19 S p i e s , R. (1953), A e r o j e t - General Corporation Rept. 1508 20 Eagleson, P. S. (1959), M. I. T. Hydrodynamics Laboratory T e c h n i c a l Rept. No. 32 21 Lamb, H., (1945), Hydrodynamics, Dover P u b l i c a t i o n s , Inc., New York. 22 L e v i c h , V. G. (1941), Acta. Physicochim. U'.S.S.R.' L 4 , ,307, 321; Phys i c o c J i i m i c a l Hydrodynamics., New. Jerse y : P r e n t i c e - H a l i Inc. (1962). 23 Dorrestein,. R. (1951), Proc. Acad. S c i . Amst. B, 54, 260, 350 - 5 8 -- 5 9 -2 4 . Fraenkel, L. E. (1959), J- of Fluid Mech. 7, 31 25 Levich, V. G. and Gurevich, Yu. Ya. (1962) Dokl. Akad.. Nauk. S. 3. S. R. 143, 64 26 Roberta, P. H. and Boardman, A... D. (1962), Astrophys. J. (U.S.A.) ' 135, 552 27 Wentzell, R. A. and Blackwell,. J.. H. (1965), Can. J. of Physics, 43, 645 28 Kukshas, B. and Ilgunas,. V. and Barshauskas,, K.. (1961), Litov. Fiz. Sbornik (U.S.S.R.) 1, 135. 29 Gupta, A. S. (1964), Proc. Roy. Soc. A (G. B.) No. 1374, 214 30 Vandakurov, Yu, V. (1963), Soviet Physics - Technical Physics (U. S. A.) 2, IO4. 31 Peskin, R. L . (1963), Phys. of Fluids (U. S. A.) 6, 643 32 Nayyar, N... K. and Trehan, S. K. (1963) Phys. of Fluids (U.S.A.) 6, 1587 33 Murty, G. S. (1963), Ark., Fys.. (Sweden) 24, 529 34 Bickerton, R. J . and Spalding, I. J . (1962), Plasma Phys. -Accelerators; - Thermonuclear Res. (G. B.) 4, 151. 35 Dattner, A. (1962), Ark. Fys. (Sweden) 21, 71. 3.6 Murty, G. S. (1961), Ark. Fys. (Sweden), 19, 483 37 Tandon, J . N. and Talwar, .S. P. (1961), Plasma Phys.. - Accelerators -Thermonuclear Res. (G. B". ) 3, 2bl 38 Lehnert, B. and Gjogsen,. G. (I960), Rev. Mod. Phys. (U. S. A.) 32, 813-39 Curzon,. F. L. and. Howard, R. (1961), Can. Journal of Physics 39, 1901 40 Curzon, F. L. Howard, R. and Powell, E. R. (1962), J. of Electronics and Control 14, 513 41 Tucker, M. J . and Charnock, H. (1955), Proc. Fifth Conf. Coastal Engineering (University of California,. Berkeley). 42 Morrison, J . R. (1949), Bull. Beach Erosion Bd. 3, 16 43 Wiegel.,. R. L. (1947), University of California. Wave Project. Report No. HE 116 - 269 (Berkeley). 44 . Landau, L. D. and, Lifshitz, E. M. (1959), Fluid Mechanics., Ch. 6 Reading, Mass: Addison-Wesley 45 Winsor, F. and. Parry, M. (196.3), The Space Child's. Mother Goose.,, Simon and Schuster, Inc. APPENDIX. I CALCULATION OF THE RESONANT... FREQUENCY CHANGE FOR A SURFACE WAVE IN A RECTANGULAR CAVITY The method for .calculating, the. change, in resonant frequency of a microwave cavity for a surface.perturbation on the end plate of the cavity is the same as that in chapter 2. Details of the cavity are given in figure 1.6. In a rectangular cavity i t can be shown (ref. 1) that ) where E* = E , cos E , = E a = Z3 Sin B„ = By = ~ J UJ E,nt -{ L B i = ~ j \ 00 { sin ( w y / B ) sin ( Y)HZ/L ) Cos (fax/ft) Sin ( W T T - Y / B ) Cos {yffft/L) B JULEL. B YITT E? = 0 L where E", , 2^ and H 3 are the peak values of E„ , EY and ^ and where In order to specify completely FT and B (except for the arbitrary power factor) i t is necessary to let either H3 = 0 (T. E. modes) or Bi= 0 (T. M. modes) . In reference 6 i t is shown that the amplitude of a linear, standing surface wave on an ideal, incompressible fluid in a rectangular tank can be expressed by S = TISoCs P) COS LHJO C O S ( STIY )(See fig. 16). S,p J ft B S and P are integers. The calculation for 2{(j)orU)) wil l be made assuming Q -60--62-that only one simple standing .wave is present (i .e. $ • — cos (PTTX ) Cos ( S7TJ/ ) ). By using equation (A 1.-1) in A B (1. 1. 1) i t is easy to show - ABL E, 2 (1 - 6n,o )(1 - 8-n)0 ) d * ) + E a 2 4r,o ^ (1 - 6n)0 )(1 + 6M,O ) t E - 3 2 (1 - 8„t0 )(1- 6X)0 )(1 + <5njo)f and A/2 -.€/UA/e- ( A 1-3) As before,, if. the, amplitude of the surface wave. , $ 0 , is small (i .e. UTT « . ] ) and _ « 1 then can be. calculated from 2(CJL-tJ ) - Jo AB "7— TTTT - / 1 - -rri-, cOW; 7Tl[. B ( J E3ITm - Ez.ft.7r a; AB 6t)1 A/*1 F 2 - 1, A 7TJD, U ^ J Z L ^ ( w , s ) f X ) D, (w.SjjTD^ (i,P) ^ 4 / 4 — So AB E ^ W ^ V D , (J?,,P) D. (m,s) (Al. 4 ) where AC and are given by equation ( A l . 3) / E, t TnEi. + ~n.Es and either E 3 — 0 (for T'. E. modes) or / Ea, - WE, x 0 (for T. M . modes). Also.,. \ Sin 2 (19) Cos ( j e ) c f e - TO , ( i , j) - ^ Jo ' . * I Cos2(ie) cos (je-)^ e £ _ 7 £ D a ( i , j)= 7 f A 0 L < ' 0 for i - 0 7T(1-Sip) j - 0 2 i - 0 J T ; (1 + di, 0) j , - 0 2 _ 7 T C $ 2 i , 3 i 4 7^ 0 and j ^ 0 - 6 3 -Therefbre, i f an em., wave is set up in the cavity such that X - 0 and ~yn.^= 0 then (d(x- id — 0 unless. P - 0 and S - 27R. (In other words, is proportional to the P ^ 0, S ^ _ 2m, Fourier component of the CO surface wave in the cavity) . Similarly,, i f - 0 and JL 0 then 0 unless S ^ 0 and P. - 2& . (In other words, iii-cd is CO CO proportional to the S - 0 , P ^ 2J? , Fourier component of the, surface wave in the cavity). This allows the »P - 0, S - 2m" and »S - 0, P - 2i " Fourier components of. an arbitrary surface wave to be studied with ease. If an em. wave is set up in the cavity such that $ ^ 0 and M 9^0 then (A. CO ^ 0 unless one or more of the following is true: 0J (i) 2 i - P and S - 0 (ii) 2Tn - S and P ^ 0 (Iii) 2 i - P and 2m - S . In other words,, em. modes of this type "react" to three different types of surface waves ( i . e. three different Fourier components). In this case COa- CO - c y £ o(s - 0, P - 2'i ) + c 2^ o(s - 2m, P - 0) CO ( -+ C 3 J o(.S - 2>n, P - 2i ) where C.-j_, C2 and C3 are found from equation ( 1 . 4« 2). Therefore, once J o(S ^ 0 , P ^ _ 2l ) and i~o(S ^ _ 2}ny P ^  0) have been measured (by using the 14 0, Tn - 0 and 7 n ^ 0, £ - 0 em, modes ) ^o(S - 27n, P - 2i ) can be studied by the use of the Jl 0, » l ^ ' 0 em. modes. In this way the rectangular microwave cavity can be used to automatic Fourier analyse an arbitrary, standing, surface wave in a rectangular tank. APPENDIX 2 THE TEST OF SLATER'S .THEOREM A pie shaped, perturbation ( f i g . 17) was introduced into the bottom of a microwave, cavity to test Slater's theorem. The change in resonant frequency for such a perturbation, was computed from the theorem i n the same way. as in chapter 2. The assumptions, Ctia. - OJ/_< 1 and Y\T\ £« ^  ^ 6) 1 are used again. It can be shown that for i '- 0 (T. M . modes) e 2(6l-Q) - K* §o 0 feo ( K.rjlrdr - kt 6J Lio0 [ jo (Kr)lrdr • Similarly f or £ ^ 0 2(6^L-6J ) - K ^ i o f 2 i ^ + Sin (2J?H©/ +^J) ~ sin (2 ie j } [ [ V ( K r f c d r + f$« I 2U - s i n ( 2 J ? r e / ^ J ) + s i n ( 2 i e ( ) ) f T 2 ( K 0r)dr - f f i f f i ^ * J-.2J0 * s i n ( 2 i c e , - s i n u i e , )J| Rjj lCK 0r)rir. ^ . (A2.2) A/B and are given by equation (1. 3- 3). It can be shown that for Jl ^_ 0 (T. E. modes) 2 ( 6 i - £J )/o; - (r)lT) Jo 0 [ f r o ' (Kr)rdr. (A2.3) \ L) J C / V ^ J 0 ; Similarly for i. ^ 0 2(lU-£J ) - (WIT) Jo / -sin ( 2 i 0 + 2.M ) t s i n ( 2 i © / )1times ^ 4 i I Z. /K"/VB I J { i y ^ } + fmtjo.(2H0 t sin ( 2 l 0 + 2 / 6 , ) - sin(2J?e, ) jtltties (kfr)]rdr| • (A2.4) )\lg i s given by equation (1. J.. 4). -64-Figure 17 Details, of the perturbation used to test Slater's theorem. -66-A pie shaped, perturbation was placed on the bottom, plate of the ca v i t y . The azimuthal. p o s i t i o n , , (see f i g . 17) was varied. A l l . other parameters were held constant. As the perturbation was rotated 360° about the axis of the cavity the resonant, frequency varied from a minimum tc a maximum a c e r t a i n number of. times. The difference between the maximum and the minimum resonant frequency, OF, and. the. number of v a r i a t i o n s , yVO, were measured.. Equation (A2..4) was used to calculate F and NO from Slater's theorem and. a comparison was,made, between, the t h e o r e t i c a l and. experimental values. This experiment, was repeated f o r various L, 's ( i . e . f o r d i f f e r e n t em., modes, in,the. cavity).. Equations (1. 2. 2) and.(l. 2. 4) were used to i d e n t i f y the modes. The r e s u l t s are shown i n table 2. A s i m i l a r experiment was performed but, i n t h i s case, the p o s i t i o n of the. perturbation was. kept.constant and the perturbation amplitude, So , was varied from 0 to .35 cm. ' A l l other parameters were held constant. The change i n resonant frequency per change i n perturbation amplitude y was.measured f o r 3 0 < .4 cm. Equation (A2-4) was used to calculate 0)0,-0) from Sla t e r ' s theorem. This experiment was repeated f o r Joi7T various L 's ( i . e . f o r d i f f e r e n t em. modes i n the c a v i t y ) . The experimental results, f o r the, TE'jj, and TE Zl, modes are given i n f i g u r e 18. I t shows that, as,.expected, the. change i n resonant frequency is. d i r e c t l y proportional to. the. amplitude of the perturbe..tion... Table. 3 shows, that,.the experiment and,.theory are i n agreement.,for the four modes, that were tested. Varying the si z e or p o s i t i o n of the perturbation also resulted i n small, undesirable changes, i n the length of the cavity. These unavoidable changes sometimes caused resonant frequency changes that were almcst as large as those, caused by. changing the perturbation size, or p o s i t i o n . , This e f f e c t can account f o r any diffe r e n c e between the t h e o r e t i c a l and experimental values of tables 2 and, 3 • Parameters for. Table. I I (see f i g u r e 17) = .310 cm. R, = 1.46 cm ft = 2.54 cm. Experiment Theory L (cm'.) r (Gc./s.) No Mode JL n F (Gc./s) No 1.83 .018 2 • TE 1 1 1 .020 2 2.20 .008 4 TE 2 1 1 .006 4-45 .003 4 TE 2 1 2 .003 4 Table H i Comparison between theory and experiment Parameters f o r Table I I I (see f i g u r e 17) /?, = .97 cm. 0 = 4.30 A =2-54 cm. _£j = 8.9 KAlc./s 27T Experiment Theory 4 (cm.)' (Gc/ Mode X >n n (60a.-6J)/zirSo (Gc/ 1.83 0.14 TE 1 l 1 0.15 2.20 0.017 TE 2 1 1 0.018 2.90 0.16 ' TM 1 l i 0.12 3.66 0.071 TE 1 1 2 0.063 Table I I I . Comparison between theory and experiment. 0 = 43° (jj/ = 8.9 KMc./s. /2TJT -67-8.924 O in £1 8.913 8.902 O 8.891 8.906 fc O 0 8.904 Qi . , ^ 8.902 O 8.900 -63-MODE RESULTS o / / o o o / o / o / / o / / o / / o / / o / / / .127 .254 .381 .508 TE 2.11 PERTURBATION AMPLITUDE, J . MODE RESULTS (CM.) o .635 O o o o o o O/ / o o 0 .127 • 254 .381 .508 • 635 PERTURBATION AMPLITUDE,: So (CM.) Fig. 18 Graphs showing .the change in resonant frequency as a function of the perturbation amplitude. - 6 9 -In some cases when a perturbation was introduced i n t o the ca v i t y , two resonant frequencies were observed d i f f e r i n g by l e s s than 20Mc./s. The r e l a t i v e changes i n these resonant frequencies, as the perturbation was being moved about i n the c a v i t y , suggested that they corresponded to i d e n t i c a l em. modes ( i . e . same i , TYi and YI ) with d i f f e r e n t azimuthal or i e n t a t i o n s . As expected, the electromagnetic f i e l d s near a coupling hole or probe (between the waveguide and the cavity) were found to be d i f f e r e n t than given by equations (1. 2. 1) or (1. 2. 3). Probes were found to d i s t o r t the f i e l d s i n the c a v i t y more than holes. A coupling hole, therefore, was used. I t was made as small as possible and placed as f a r from the perturbation as possible (see f i g . 5). I t was found i n agreement with the theory that the change i n resonant frequency was proportional to the perturbation amplitude f o r amplitudes l e s s than .4 cm. When So was made greater than .4 cm., however, the change i n resonant frequency was no longer proportional to . This i s to be expected because the small amplitude assumption that was made ( i . e . Y)TT So « 1 ) i s not v a l i d i n t h i s case. L The above r e s u l t s , therefore, show that S l a t e r ' s theorem can be used to p r e d i c t the change i n resonant frequency of a microwave cavity caused by a small amplitude perturbation of the shape of that c a v i t y . APPENDIX 3 THEORETICAL CALCULATION OF THE. MAGNETIC DAMPING. OF SURFACE WAVES FOR LOW MAGNETIC" REYNOLDS NUMBERS In t h i s appendix the damping c o e f f i c i e n t i s derived f o r l i n e a r ^ s t a n d i n surface, waves, on. an i d e a l , .incompressible f l u i d of f i n i t e conductivity and depth i n a uniform, constant, applied magnetic f i e l d . Figure 1 9 shows the. nature of the problem to be solved. The d e f i n i t i o n s are the same as those used i n chapter 6 except that k| = 2 T T / < A . i and, kz. = 2 ' J T / < x 2 . -The assumptions and equations ( 6 . 1 . 1 ) , ( 6 . 1 . 2 ) , ( 6 . 1 . 3 ) , ( 6 . 1 . 4 ) , and ( 6 . 1 . 5 ) i n chapter 6 are used to solve the above, problem. The boundary conditions are as follows: Vt~ 0 AT £ = 0 . (A3.6) P = % + T j . ^ | t + } A T T H E F R E E S U R F A C E - ( A 3 - 7 ) Vi = ^ at the free surface . ( A 3 - 3 ) The. problem w i l l , be solved using a perturbation technique. We assume B - B 0 + <5B where 6§ Bo . g 0 i s the applied A - A A A magnetic, f i e l d . Bo = B, lx + ly * B3 J £ . The os.cillati.on. frequency, -f , and. the. damping, c o e f f i c i e n t , CJJ, are found by assuming that, the time dependence, is. of the form e)Cp ) where n = Z77f + }(% • X Freenkel. ( r e f . 2U) has solved t h i s problem f o r the case ( k, + kz < < B, = O } Bz - 0 . Roberts and Boardman ( r e f . 26) have solved i t . f o r the case ( ^  + $H >> 1 ', B, = 0 = Bt . Wentzell and Blackwell ( r e f . 2 7 ) have solved.this problem f o r the case k , H ^ ^ > = 0 = B 3 = k 2_ and Kukshas, Ilgunas. and Barshauskas ( r e f . 2 3 ) have considered various aspects of the case B3 - BL = ^ - i = O . We w i l l solve the problem f o r a l l values of ( + k^H y k, ^  A Z ) /3, } - 7 0 --71-F i g . 19 Description of the magnetic damping problem to be solved. -72-By taking .the curl curl of (6. I. A ) , dropping terms of second order or higher, in \SB\ Bo. and.using the. fact that V-V = 0 = V-B = V - J we find that in/oVP - qil-m- U 3 . 9 ) Let V& be of the form exp( ik,X + ikx^-IYXt ) times J A, cosh ( k 2. ) + A z sinh ( k i n . We note, however?that A,= O because a boundary condition requires that = 0 at H. = 0. Using, this form for \4 in (A3-9) we find that The dispersion equation (relating .71 to , kj. , k 3 6. , H) is found .by taking the time, derivative, of (6. 1. 4) and at. the surface. This gives ^ <• (A 3-11)' It is more convenient to consider two separate.cases,I ( B, = 0 = ) and II ( S3 = 0)3 when solving (A 3-10) and (A3-H) simultaneously. Case I ( 8, = 0 = Bz. ) Using sinh ( ) Q '** we find that (A3JL0) and (A3 J-l) reduce, to (A 3-12) 5 ex EXP (ik,x + iki_y - irii) • For = 0 the above results ...agree with Coulson (ref. 6) The problem of finding .71 in terms of K , kt. , H, and S 3 is simplified i f we assume Q B3 « j • Using this assumption and -73-u s i n g the f a c t that tanh[.A ( 1 + 1 e )^ .1 ~ ( t a n h l A ) U 1 + J A Q 4  we f i n d t h a t (A 3-12) becomes ^ = k!" + kt F O R e «• 1 exp czKH) - e x p ( . - £ k H ) (A 3-13) 5 °< E X P ( ik,X + ik*.y - i n O • The i n t e r e s t i n g p o i n t s .to note, are the. f o l l o w i n g : ( i ) 5 i s independent of B 3 f o r a l l values of kH, ( i i ) 0 7 i s p r o p o r t i o n a l to ff-Bj f o r a l l kH. ( i i i ) — - Q di fop kH » 2 • This r e s u l t agrees w i t h the t h e o r e t i c a l c a l c u l a t i o n o f Roberts and, Boardman. ( r e f . 26)-( i v ) (Ji n -JilJiLL— * " o r kH <^<- 1 . This r e s u l t agrees w i t h the t h e o r e t i c a l , c a l c u l a t i o n s of Fraenkel. ( r e f . 24.). Case I I ( Bi = 0) Vr OA i °< s i n h ( k £ ) e reduce to ??= { 9 + T. ) J k TBNhl(kH) -'m^(B^Bl) ( J 1 c V + k i ) J we f i n d t h a t (A 310) and (A 3-11) (A3-H) n = ins * i c r 8 . Again the problem of fi n d i n g . y\ i n terms of k, , kz. > H, B , and Bz i s s i m p l i f i e d i f " we assume Q ( Bf * BL ) ,/< \ . Using, t h i s assumption -74-and using the f a c t that tanhJA( / + / 9)J ~ 1 ^ <- o+ AlelK expcifl).- expc-zfl) we f i n d that ( A 3 . I 4 ) reduces to ( Z T f#={n 3 + y k ] T f l N H ( k H ) B - ^ 5 J k > 4 xT * C X P ( Z k H ) - e X P C - 2 . K H ) J S * e x p e i K . X H k ^ - i r r t ) • The i n t e r e s t i n g ..points note are the, following: ( i ) 5 i s independent,,of Bi and Bj. f o r a l l ( i i ) (X - --£_ U (BSB.)- Bfk* + BUt I F O R k H » i . \ V + ki J For the case. 8 1 = ^ = 0' t h i s agrees with the t h e o r e t i c a l calculations..of Wentzell and Blackwell ( r e f . 27) and also Kukshas et a l . ( r e f . 23). ~ ( i i i ) a; - -47 ( i c ^ ^ \ ) - m < i ± ^ ) ( z - z c r f i l F O R . k u « i For the case., 6 2 = 0 = t h i s agrees with Kukshas et a l . Kukshas. et a l . claim that CfB z. 0 i f kz. = 0 and B, = 0 or. i f k , = 0 and Bj, = 0. ( i . e . i f the applied, magnetic f i e l d , i s p a r a l l e l to the wave f r o n t ) . This claim does not agree with our c a l c u l a t i o n s . The problem of, solving ( A 3 . 1 2 ) . a n d (A3.I4) f o r a l l values of C B,z + + Bs ) was not considered. / o m i APPENDIX 4 MUMMERY, SUMMARY The following is,based on work by F.. Winso.r ( r e f . 45) This i s the theory Jack b u i l t . This i s " t h e flaw that lay i n the theory Jack b u i l t . This i s . t h e mummery hiding the flaw that lay. i n the theory, Jack, b u i l t . This i s the, summary based on. the mummery hiding the flaw that l a y i n the. theory This i s the constant K that saved the summary based on the mummery hiding the flaw that lay i n the theory Jack b u i l t . This i s the erudite verbal haze cloaking constant K. that saved the summary based on the mummery hiding, the flaw that l a y i n the theory Jack b u i l t , -75--76-This is the turn of a plausible phrase that thickened the erudite verbal haze cloaking constant K that saved, the summary based on the mummery hiding the flaw that lay in the theory Jack built. This is chaotic confusion and bluff that hung on the turn of a plausible phrase that thickened the erudite, verbal haze cloaking constant k. that saved the summary based on the mummery hiding the flaw that lay in the theory Jack built. This is the electromagnetics and stuff that covered chaotic confusion and bluff that hung on the turn of a plausible phrase and thickened the erudite, verbal haze cloaking constant K that saved the summary based on the mummery hiding the flaw that lay in the theory. Jack built. -77-This is the microwave cavity, machine to make with the electromagnetics and stuff to cover chaotic confusion and bluff that hung on the turn of a plausible phrase and thickened the erudite verbal haze cloaking constant K that saved, the summary based on,the mummery hiding the flaw that lay in the. theory Jack built. This is the fool with brow serene who started the microwave cavity machine that made with the electromagnetics and stuff without confusion, exposing the bluff that hung cn the turn of. a plausible phrase and, shredding the.erudite verbal haze, cloaking constant K wrecked the summary based on the mummery hiding the flaw and demolished, the theory Jack built. This is the theory Rob built. This is the flaw that lay in the theory Rob built 


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