The U n i 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. P i k e B.Sc, The U n i 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. A h l b o r n J.W. B i c h a r d R.W. B u r l i n g A. F o l k i e r s k i B. N. Moyls R.M. E l l i s R. Nodwell C F . Schwerdtfeger E x t e r n a l Examiner: V.R. Malkus Woods Hole Oceanographic I n s t i t u t i o n , Woods Hole, M a s s a c h u s e t t s . Research S u p e r v i s o r : F.L. Curzon INVESTIGATION OF FLUID SURFACE WAVES WITH A NEW MICROWAVE RESONANCE TECHNIQUE ABSTRACT A new for microwave technique has been developed the e x p e r i m e n t a l study of s m a l l amplitude waves on an e l e c t r i c a l l y conducting f l u i d . surface The fluid forms one' of the w a l l s of a r e s o n a t i n g , microwave cavity. S u r f a c e waves w i t h amplitudes as s m a l l as -3 10 cm. can be measured by o b s e r v i n g the r e s u l t i n g change i n the resonant frequency of the cavity. T h i s technique has been s u c c e s s f u l l y to measure the v i s c o u s and' magnetic c i e n t of a s m a l l amplitude, l i q u i d mercury. a vertical, magnetic Reynolds c o e f f i c i e n t was damping c o e f f i c i e n t f o r f i e l d was agreement w i t h a c a l c u l a t i o n numbers. found to be t h a t was i n good made f o r low When the v i s c o u s damping compared w i t h the s t a n d a r d which a l l o w s h o r i z o n t a l motion disagreement coeffi- s t a n d i n g , s u r f a c e wave i n The magnetic magnetic damping used theory, of the s u r f a c e , a of up to a f a c t o r of f o u r was found. I t , however, modified showed e x c e l l e n t agreement w i t h a t h e o r y which assumes t h a t there i s no h o r i z o n t a l , motion of the s u r f a c e . GRADUATE STUDIES Field of Study: Plasma P h y s i c s ' Electronics M. Kharadly Elementary Quantum Mechanics G.M. Volkoff Applied Waves R.M. E l e c t r o m a g n e t i c Theory P„ Plasma Dynamics L. . Sobrino E.V. Bdhn A n a l y s i s of L i n e a r Systems Advanced Plasma P h y s i c s Rastall F.L. Curzon Plasma P h y s i c s Theory of I d e a l F l u i d s Ellis G.V. Parkinson R. Nodwell AWARDS 1960-64 B. C Government S c h o l a r s h i p s 1965-67 N a t i o n a l Research C o u n c i l Scholarships 1967 N a t i o n a l Research C o u n c i l Fellowship Postdoctorate INVESTIGATION OF FLUID SURFACE WAVES WITH A NEW MICROWAVE RESONANCE TECHNIQUE • -;- \ - by R o b e r t L. P i k e .B.Sc. 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 , 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTORATE OF PHILOSOPHY i n t h e department of PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1967 In presenting for an that advanced thesis shall I further agree for scholarly Department my in partial make that permission may be written thesis Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada granted by Columbia requirements Columbia, I agree for reference f o r extensive t h e Head shall and copying It i s understood f o r f i n a n c i a l gain permission. of the of British i t freely available purposes of this fulfilment at the University o r b y h.i>s r e p r e s e n t a t i v e s . publication without thesis degree the Library Study. or this of of this my that n o t be copying allowed (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. waves with amplitudes as small as 10~ 3 Surface 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 i n 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. (iii) TABLE OF CONTENTS ii ABSTRACT LIST OF ILLUSTRATIONS vi TABLES ACKNOWLEDGEMENTS Chapter 1 INTRODUCTION Chapter 2 MICROWAVE. THEORY Chapter 3 v vii 1 . 6 Sec. 1. Slater's.Theorem 6 Sec. 2 Cavity resonators 7 Sec. 3 Calculations of the resonant frequency change for a surface wave. in. a cylindrical cavity EXPERIMENTAL. PROCEDURE* PARAMETERS, VARIABLES, AND ASSUMPTIONS 10 U Sec- 1 Experimental procedure 14 Sec. 2 Parameters, variables, and assumptions 24- Chapter 4 STUDY" OF THE OSCILLATION FREQUENCY" OF SURFACE VAVES IN MERCURY 27 Chapters STUDY OF THE VISCOUS DAMPING OF SURFACE. WAVES:. IN MERCURY , 34 Chapter. 6. THEORETICAL CALCULATION: OF' MAGNETIC DAMPING OF SURFACE WAVES FOR LOW MAGNETIC REYNOLDS NUMBERS 43 Chapter 7 EXPERIMENTAL:.STUDY! OF THE MAGNETIC DAMPING OF SURFACE WAVES IN MERCURY 49 Chapter 8 FUTURE WORK 54 SUMMARY 56 REFERENCES 58 -iv- 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 a i r 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 9A,B Oscillation frequency results 23 28, 29 10 Sinusoidal time dependence of a typical surface wave 11 Surface distortion 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 30 32 50, 51, 5 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 function of the perturbation amplitude 68 19 Description of the magnetic damping problem to be solved 71 -vi- TABLES Page I. Parameters, variables, and assumptions used during surface wave experiments. 26 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 f i e l d 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 i n t e r e s t t o the plasma p h y s i c i s t with respect t o the confinment of plasmas. A l l plasmas i n one way or another must have surfaces t o be separated from our normal, h o s t i l e room temperature environment. These surfaces, produced under the influence of g r a v i t a t i o n a l and magnetic f i e l d s , tend to become unstable or d i s t o r t e d . This dist^rjotion can be described by a s e r i e s of sine and cosine waves. Hence, the study o f a plasma surface involves the study of surface waves. The e f f e c t 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 i n t e r e s t . The study of t h i s e f f e c t i s complicated, however, by temperature and mass density gradients which are present i n most laboratory plasmas. The e f f e c t can be computed and observed more e a s 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 f o r a l i n e a r ( s m a l l amplitude) surface wave i s u s u a l l y expressed i n the form exp ( C T f ) cos (2jt£"£ ). damping c o e f f i c i e n t and wave. CT is c a l l e d the 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 I t was the aim of t h i s t h e s i s t o 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 , t o i n v e s t i g a t e the damping of a surface wave free of magnetic f i e l d s and t o consider the methods used t o 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 ( r e f . 1 8 ) . This i s at l e a s t p a r t l y due t o the problems associated with the wave guages used i n the past. Devices such as capacitance ( r e f . 41) and r e s i s t a n c e ( r e f . 42 and 43) wave guages are a v a i l a b l e to study surface waves. They, however, s u f f e r from the -2difficulties 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 viscous.film of fluid, on the gauge as the free surface, recedes 3 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.. 5„K« example.,, the. linearity restriction., For (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 f l u i d (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 adopted.by 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) i s 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 (J S o — 1. There are. two important features here. F i r s t , 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 sensitive-..to 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. feature i s also found. Another interesting Calculations show that a. rectangular., cavity, when used..to study surface waves acts as en. automatic Fourier analyser. -It appears that, a rectangular, cavity can.be. 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 i s 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 i n chapters 3, A and 5, shows that the. observed oscillation, frequencies are higher, than, predicted, by theory by up to 15$. •meniscus is. the cause of this discrepancy. The results indicate that the 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 is.no horizontal, motion of the. surface is introduced. Accor.ding....to. Lamh,and, Le.vi.ch (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 assumption.is equal to • -l/y 2TTS' K cosh (KH) . 1/2 sinh (2KH) i) 2 kinematic viscosity of. the fluid... i s the The. experimental, results are found to be in. excellent, agreement with the theory when this effect is included. therefore, concluded, that there, was.no horizontal, surface motion. suggested .that. this. was. due to the. presence of a surface, film. It i s , It is 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 i s initiated by the author's calculation of the magnetic damping, coefficient, CJB", for low magnetic Reynolds, numbers. It is calculated.in chapter6 for an. axisymmetric, linear, standing,, surface wave on an incompressible, ideal, fluid with, a finite e l e c t r i c a l conductivity, <^ , and mass.densityyO . constant, applied, uniform,, magnetic f i e l d , 2 calculations are made, for the. case Q Bl ^2TTf << 1- The effect of a vertical , is considered. The calculations show that the oscillation frequency is independent of B^ and that ^ <T 8 z. -ff B* A/O ( / ( I £KH ) exp (2KH) - exp (-2KH)) + Detailed is given by This, calculation is experimentally verified by a series of experiments with, liquid mercury. Chapter 7 describes this work. The magnetic damping coefficient, OJ , is measured.by measuring the total damping coefficient,C , and assuming that <5~o + @e ' x p i - ' t a l results confirm, this assumption. In agreement 2 with. the. theory (T i s found, to be. proportional to ^2 B ? and. to vary from E B eri n etl /° -5- -J&Q (for KH 4 0.3) to - B?0 (for KH > varied.from, small, to large, .values. 3.0) as. the f l u i d .depth,, fl,. is The rate at which Cfjs varies, with fluid depth.is found to increase with K as. predicted. are within 10% of. the theoretical, values. maximum...systematic percent error.. A l l experimental, values This, percentage is. less than the These results, therefore,..are considered to be i n excellent agreement with our. theory for the damping.of surface waves, in. a vertical,, magnetic f i e l d . 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 change.in shape of a microwave cavity, the electromagnetic f i e l d distribution i n that cavity, and the resulting change, in resonant frequency of. that, cavity. theorem i s stated, as: 6J 2 a + jjj^ - ^ ) (1 2 (Ba 2 Ea)cVv* 2 )• The (hi.I) Q is the re'sonant frequency, of the cavity before i t s shape i s changed Q% is the. resonant frequency of the cavity after, i t s shape.is changed. Ba and Ea are given by Ba - I B I / l l £ 2 — JT] IBI dv vc 2 E a - IEI /N 2 and N 2 D 2 E - J/flEI vc 2 """" 2 2 E dv , V p is the change i n volume of the. cavity . V c is the original volume of the cavity . E and.B. are the original, e l e c t r i c and magnetic fields i n 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- -7Section 2 GAVITY RESONATORS A microwave cavity i s a container made of a material of high, electrical conductivity. . The following, definitions.are made: ytt i s the. magnetic permeability in the cavity. € i a the. electric, permittivity i n 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 i s assumedthat: (i) The material in the cavity and. in the cavity walls is. homogeneous and isotropic. (ii) (iii) There is no free charge..present. 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 i s found .by solving Maxwell's equations using the above assumptions. The solution'for. a. cylindrical, cavity can be found in most texts on electromagnetics here.without proof. (ref.. 1). It w i l l be stated Figure 1 shows the geometry of the. cavity. There are two situations possible. F i r s t , the magnetic f i e l d i n the cavity can. be. transverse to. the axis. of. the cavity.. transverse, magnetic, or T.M.. mode.. This. Is called a Secondly, the electric field can be transverse to the axis of. the cavity. This is called a transverse electric Figure I. Cylindrical microwave cavity -9or T. E. mode. For the B- M. modes i t can be shown that ( Bel© Brl )e + Be = Ko JjL'( Ko r ) Cos (J^G ) C o s ( y 7 7 T z / L ) Br = i Jk^K.r) Sin (ie) Cos ( r i T T i / L ) Ee - -J/yil E = Suu^JLjt where r = - j a) Ju)t AND E = ( Ee le + Er lr Et lt)e + y?7r 5 (K.r)Sin ('© )S i n ( Cos (jf©-) Sin Jx'(Kor) J x CK r) 0 cos is a Bessel function of order Ue ) cos (1.2.1)' wrt/i) ( W T E / L ) ( W Z / L ) A Also, these standing em. waves are set up i n 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 a x i a l dependence, A specific T. bi. mode i s respectively, of the em. fields i n the cavity. usually denoted by " ^ A l j ^ ^ • Similarly for the T.E. 1, m, n, i t can be shown that B=(Bel *B \ e t where B© dnd + Y E = (Eele + Er lr)e = - 7T/??J(Xr) Sin (/6 ) Cos (Y)7TZ/L Br = Bi = Ee = ) >)77 ^ ' ( K r O Cos ( i 0 ) Cos ( fflfe/l) L K Jickr) cos(ie) sin (Y)TTZ L) 7 - i ^ L j / ( K r ) c o s ( i e ) FR - , 2 S m C y m z / L ) • • 1 (1. 2. 3) Z "\ (K (1, m) A) - 0 J and where CO 2 - K •» ( YITT/L ) . 2 ( l . 2. A) 2 In the above equations for B and E a constant amplitude factor has been omitted..-.. This factor i s determined...by the. output, power, of the microwave system, and.is unimportant in the following calculations.. . Slater.' s theorem can now be used to compute the. change i n 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) - The calculation for (£ - J s w i l l be made for. a simple wave (Kr) COS (Gtia -O})/^)^ /• 0. ( S e ) ) . It is assumed that . TTWSO/'L.**-I and 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 written.in the form,. rd6 drdZ., T. E. modes 2{6J„-0) In. this way i t can be shown that, for KU,Jn)f\ ,1* - K 71 TT sin 2 (19) cos (se)^e J J x o CO (1. 3- 1) TVTT io i L i -J where (x)Js(fol)dx 2 (x) - dl Ji (x) d * I COS^IQ) C o s ( s e ) d e [j/(x)J J ( M S x dx -11and - K (S.. P) . K (1,. m) Similarly i t can be shown that for T. M. modes 2(0)* -Q) * i - i o _ So \ 2 - Cd K , 2 2 sin cos j ( i e )co 2 f \ cog S 2 ( 1 6 ) cos ( s e ) de (se ) c^e (10) cos (se 2 0 k(j? >w)A \ J, ) do k*Mt ° x j o J (X)_JS (O< ) 2 f (x)l () 2 x 2 J ( o u ) xdx. s X CL'X Js (otx)xrfx U- 3. ) 2 1 where o( - K (S, P) . K (1, m) G It is. only necessary, to. compute,-.one. 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 £ +j f(ExB)-ds [ 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 walls.is very high.(assumption i i i above). fr \ A Ai B-B + J^A Therefore, A A T j E-E6 CCv - constant. 2 J Equations, ( l . 2. 1.) and (1. 2. 3) show that B and.E. are out of phase by 9 0 ° . A* s\ s\ y\ y\ s\ That i s , , when E - 0 B - , 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 2 yu. J 2 3 J C A ^ Y A. 4 - ^ Ax ^ A. 9 But,. B max > B max.. - IBI" and E max • E max.. I EI''. As a result, [ IBI d v - yU€ ( UH2dv and so Ng 2 2 2 2 ,A/ £ -12- It can easily be shown (by the use of equation 1. 2. 1) that for T. M. modes \ 2> Jo ~7* Jo and Nl/{€jJL) A// = <5«- 0 i - j 1 "(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 = 6 i ^ i ( r / ^ o ) f [ ^ V K n ] r ( l r +J KTI JO i 2, / j (l^XKOdr 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 it. A rectangular cavity, however, has also been considered. similar to those above are given in appendix I. cavity acts as an automatic Fourier analyser. Calculations They show that a rectangular 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 w i l l be very easy. Also, an order of magnitude, calculation Since values are easily observed..it means that the micro- 0) wave system..is 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 i n appendix 2 . The results, were found to be i n 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. Chapter. 3 EXPERIMENTAL .PRO.CEDIIRE, PARAMETERS Section 1 VARIABLES, AND ASSUMPTIONS EXPERIMENTAL PROCEDURE • The microwave system that was..used.is. given in a block diagram in figure 2. Liquid mercury was. placed i n 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 changed.by 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 used.to 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. tee s p l i t the power, from the klystron: The-magic 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 waveguide klystron repeller voltage klystron power supply o attenuator sawtooth out test cavity input scope Fig. 2 Microwave system and test cavity. -16- waveguide pulsed air / 5 T rL J L - _ 4 * i s ^ .63 cm H mercury ZZZZZZZZZZ777////7777"< plunger JL mercury reservoir 7> seal valve Fig. 3 Details of microwave test cavity. -17terminator and the test cavity was detected by a c r y s t a l detector. The output, voltage of the detector was a monotonic increasing function of the power detected and was applied to the v e r t i c a l input of the scope. The nature of the resulting, trace on the scope i s shown i n figure UThe Qu of the. system ( Qu = ^/(banwidth found to. be. about 2000,. as expected. of the resonant dipj)was An order of magnitude calculation (ref. l ) shows that Q u ^ Cavity volume ( i s the e l e c t r i c a l conductivity of the cavity wall surface, S; where andyS i s the coupling c o e f f i c i e n t . cavity was varied u n t i l Q /S « J . L ? The coupling between the wave guide and was a maximum. This corresponded to having I t was. calculated from the above equation that the Q L value could be increased by about 50% i f copper or s i l v e r plating i s used instead of n i c k e l p l a t i n g . i n the cavity. A Q L of 2000, however, was more than adequate f o r measuring resonant frequency changes of .5Mc./s. or l a r g e r . 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 i n the cavity) u n t i l an appropriate resonant em. mode was set up i n the cavity ( i . e . u n t i l a resonant "dip" was observed on the scope as i n figure U)masked o f f as shown i n figure, 5- This signal was expanded and the scope face The entire trace was masked o f f except f o r two .small pieces which appeared, as dots.. The horizontal motion of these dots corresponded to the change i n resonant frequency of the cavity. The horizontal motion of the peak of the resonant dip also corresponded to the change i n resonant frequency of the cavity. The dots, however, were precisely defined whereas the peak of the resonant dip was not. Measurements -13- Bandwidth o f t h e k l y s t r o n Power r e f l j c t e d by the test c i v i t y scope t r a ce Frequency ( KMcys.) of the k l y s t r o n Resonant f r e q u e n c y o f 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 o f the o s c i l l o s c o p e trace. cope f a c e scope trace resonant f r e q . d i p Figure 5 Method o f m o n i t o r i n g the r e s o n a n t f r e q u e n c y changes. . The h o r i z o n t a l a x i s r e p r e s e n t s t h e k y l s t r o n f r e q u e n c y and t h e v e r t i c a l a x i s r e p r e s e n t s t h e power r e f l e c t e d by t h e t e s t c a v i t y . The m o t i o n o f t h e v i s i b l e p a r t o f t h e scope t r a c e was f i l m e d . -19were 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, a i r 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 i t s 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 a i r 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 (t) e S (0) cos o {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- 100 volt Sorensen 15.3 cm. Power Supply metal disk f h.p. Bodine D. C. motor air valve Air pulse to cavity Figure 6. to a i r supply 1.27 cm. tubing Method of producing air pulses to set up surface waves. -21The nature 'of the resulting negative is seen in figure 7. From similar.negatives the values of various conditions. S (0), i and 0~were found for o 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 i t s original value and then using the oscillation". frequency. It was. found that periodic surface waves were set up when the repetition.rate, of the a i r 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 fluid.in.the magnetic field at the' center of a solenoid (figure 8). solenoid consisted of 5 coils of #11 copper wire in parallel. consisted of 150 turns. The Each coil 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. caused unwanted surface waves. Switching the magnetic field on or off These waves were allowed to damp out before -22- F i g u r e 7A r e c o r d o f the time dependence o f a s u r f a c e wave amplitude. The above i s a 120 f i l m n e g a t i v e taken o f the h o r i z o n t a l motion o f t h e scope t r a c e 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 o f each d o t r e p r e s e n t s the change i n the r e s o n t n t f r e q u e n c y o f the t e s t c a v i t y which i s p r o p o r t i o n a l t o the amplitude o f t h e s u r f a c e wave i n the c a v i t y . The v e r t i c a l a x i s 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.o ( { ) J - Js (K (S, P) IT ) cos ( S O )• 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 ) that Js (K. (S, P) A) ^ 0 where A is the ca.vity radius.. restricted, to waves where S ^ 0 and P ^ 2. Recall The study was 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 - 6 4 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 ) . OJ So This 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, observed, total damping, 6~ , (6^•== (5 o + (5~B coefficient, ) the viscous damping 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 cavity radius radial wave number of surface waves A K S T 0 /° D 9 9 H 5 S. L 6Ja - 0 " CO P = L=- KA = 2 1,2197 TT surface tension of mercury = 490 dynes/cm. mass density of mercury = 13.6-^./cm. kinematic viscosity of mercury 0,11 centipois.es. cm.^/o. gravitational constant = 980 dynes/9, electrical conductivity of mercury 1.04 x 1 0 (ohm meter)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 6 cr , from the 1 -26- Table I Parameters.,. Variables and Assumptions used during surface wave, experiments Variables and Parameters Experimental values Assumptions used in theory Cavity number I II III A (cm) 2.54 3.17 3-64 K (cm."' ) 1.51 1.21 1.05 em. mode used TE„, range (cm.) TE w ..04 - -01 .0015 - .0007 .0015-.0007 H range (cm.) 0.4 - 4-0 Q.4 - 4-0 0.4-4-0 B 0 - 2200 0 (gauss) 3 h (c/s.) TOTT^o / L . (6J*So / 3-5 ^0.004 5.8. 3.5-5.2 1.0 <0.1 u)/(J « 1.0 ^ .002 < .002 ^ .002 H « 1.0 ^ .06 <.0005 < .0005 « 1.0 <C .03. ^ .002 < .002 << 1.0 ^ < .04 < .04 i o <^ .0006 ^.0006 < .0006 1.0 ^ .015 < .015 < .015 <<r 1.0 ^ .001 ^ .001 ^ .001 « ^ .02 < .02 < .02 ^.2 ^ .2 <.01 ^ .01 3 : « = 4-6 - 7.0 0-2200 « S„K/(KH) /O 0 - 2200 1.0 5 mercury jsed CONSTANT //.(jjVl = R^ : ^0.004 « 1.0 ^ 2 « 1.0 ^ .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. varied! from about 0 . 4 to about 3-0. KH'was 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,, incompressible fluid (ref. 6) shows that (2TTS ) 2 - (TK + 0 k0 tanh (KH)• (4.1.1) 3 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$. Discrepancies of this nature have been noted by, other workers (ref. 7)The difference, between experiment ..and theory may be due to: 5 (1) experimental error in measuring (2) the assumptions of the linear theory not being valid, (3) an error in the value of T/^o that was used, (4) surface tension effects associated with the meniscus". J Each of. these possibilities will now be considered. (l) The maximum,percent error in measuring £ was less., than. 2%. Therefore, the differences., between the experiment and the theory (approx. 10$) OSCILLATION FREQUENCY, f ( C Y C L E S / SEC.) > H ro In o 3 o o > > -2Z- CD 33 J> TJ X M A * 2.54 cm DEPTH OF FLUID r H (cm) -oc- -31cannot 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,surface, tension, T. (ref.. 9) • experiments. However, TK < 3 Therefore, T would have to be increased by 200$ to account for the difference between experiment and theory. (4) 10$ of(jK for our (See equation (4..1.1.).) The equation for J results from evaluating a linearized,form of. the equation of motion of the fluid at the, surface. determined. by using the boundary condition, ( Vr Vy- The value.of K i s 0 at Y"- A 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. (where £ It is suggested that the condition \Zy- - 0 at r = A -6 , 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. experiment of graph III, figure 9b,' € was less than .02 cm.. During the 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. order of 0.2 cm. The results showed that € was of the 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. _33After 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. shows how the. surface, wave amplitude decayed.. decayed exponentially, as expected. Figure. 12 It shows, that, the waves The damping coefficient was. measured and the., results summarised, in figure 13The standard, linearized theory of axisymmetric, standing waves in an incompressible fluid is given, by Case and Parkinson (ref. 7). They show that (?° - (5.1.1) &f cr + <y* v + (5" • + b is the. damping coefficient due. to the energy dissipation..in 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.. 6~ &t> - They find the following: ~ \\^ 11 ^ K 2 ( 1 + (S/KA) (AK)2 ( 1 - ( S / K A ) - \I/J2 7r5r ' K ( 1 » ^ ~ • (sinh (2KH)) where (2 7r5) of the. fluid. _ 2 - (TK +- q K p d 3 2 2KH ) sinh (2KH) ) (5. 1. 2) \ ) ) tanh ( KH) and j^is the.kinematic viscosity The assumptions that were used to develope. this, theory., are given in table 1. - 34- -35- H K 3 2j) _3j) 1.01 C I L . = 1.21 cm ' - 4.Q Fig. 12 graph showing...the exponential decay of a typical surface wave. Time units -3b- Q\ I I 0 I I ! i i i I i 0.5 DEPTH Fig. 13 Viscous i i 1.0 OF FLUID, Damping H i I I 1.5 (cm) Coefficient Results J' i i The boundary conditions were the following: (i) (ii) The fluid velocity i s zero at the. r i g i d boundaries, 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 3Figure, 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 (T ' ) (2) a damping, effect due to em. fields,in the microwave cavity, (3) violations of the linearity assumptions,used: to develop 0 theory (4) or } the j ; rough boundaries, (5) the use of an invalid, boundary condition. Each of,these, possibilities w i l l 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 accountfor.it. (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. of the cavity's em. f i e l d . It was field. In this way was. measured.for various values Results showed that (Jo varied by less than 3%. concluded, therefore, that (Jl was independent of the cavity's em. -38- (3) Other workers have, n o t 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 g i v e n i n t a b l e 1... £oK<^ The assumption (HK)^ is.,, i n some cases, not even i n c l u d e d i n the t h e o r e t i c a l d i s c u s s i o n s Hunt and U r s e l l ( r e f . 18 and .13) have p o i n t e d out this, o m i s s i o n . cases, ( r e f . 7, 14 and. 17.) workers experiments i n s h a l l o w water have v i o l a t e d this., assumption ( i . e . KH £ 0.1) . had, d i f f i c u l t y , o b s e r v i n g surface,waves made, t h i s In. many during, t h e i r T h i s happened, because (0.1)^. T a b l e 1, ioX«. (including they (KH)3) were The high, s e n s i t i v i t y o f the microwave system Case and P a r k i n s o n . (ref'.. 7) n o t i c e d an e f f e c t which was a p p a r e n t l y to wall, roughness. was. two 1.7). possible. (4) due our experiments. : such..that i o K . ^ * however,, shows t h a t a l l o f the assumptions satisfied.in ( r e f . 7, 10, They found t h a t the e x p e r i m e n t a l d a m p i n g . c o e f f i c i e n t to t h r e e times l a r g e r than, t h e i r , t h e o r y p r e d i c t e d u n t i l p o l i s h e d the container, w a l l s . Even a roughness they h i g h l y of. depth l e s s than the boundary l a y e r t h i c k n e s s • app.ea.red to i n c r e a s e the r a t e o f damping. polishing / After the c o n t a i n e r they found t h a t t h e i r e x p e r i m e n t a l r e s u l t s agreed, to- w i t h i n 15% o f t h e i r t h e o r y . found t h a t (J 0 was Other e x p e r i m e n t e r s independent o f wall, roughness. ( r e f . 19. and 20),. however, In view o f t h i s and. i n view o f l a t e r work on s u r f a c e wave damping., due. to s u r f a c e f i l m s , i t i s suggested, t h a t the Case and P a r k i n s o n d i s c r e p a n c y was, f i l m on the water a hard surface. The however, t h i s assumption horizontal, The polished. theory o f Case and P a r k i n s o n ( e q u a t i o n s (5. assumes, t h a t the f l u i d motion due-to a F o r our experiments, the. c a v i t i e s were p l a t e d , w i t h n i c k e l and then h i g h l y (5) i n fact, i s i r r o t a t i o n a l a t the f l u i d i s . r e p l a c e d by the. assumption !• 1) and (5- surface. If, t h a t t h e r e i s no f l u i d v e l o c i t y a t the s u r f a c e then the t h e o r y w i l l f o l l o w i n g i s an. o u t l i n e o f how 1. be the Case, and Parkinson, t h e o r y altered. was 2)) -39modified by the. author, using, .this new boundary, condition i n an attempt to explain the. experimental .results of figure 13. Case, and. Parkinson, used standard, boundary, layer, theory, and let V , the A. / fluid, velocity, equal - Vy) A. + A A. I 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^ When )) ^ 0 the fluid velocity, tangential, to a. wall is zero. 0. Case and Parkinson show that 0 ) cos. ( S 6 ) Js ( K r ) cos - C, cosh. ( (2 IT5i ) exp ( (5>t ) (5-1.3) where. C, is a constant. The. approximations used i n boundary layer theory allow one to calculate 6^ from (X - (5v + (ref. 2 1 ) where tfb + 6~ + z (5. Ii, 4) 6 - c^fCvxv S xt)dv s C^ is an unimportant constant.... V b , Vw , V$ are volumes in the vicinity of the bottom, wall, the side, walls and the 'surface respectively. /V. A. /N- It is convenient to define.As., Ab and Aw by s\ A A ^ As in volume Vs A A A - A J AA . / /» A (5. 1. 5) Ab in volume V b - Aw in volume. Vv Case and Parkinson solved for Ab and.Aw. by. using the boundary condition ¥ resulting from, the fact, that A- , A. A Aw from ^ x (V x A ) ^ and x ( V x Aw) - l\ b using equation (5- 1. 3 ) - 0. A. 1^ x i \ That i s , they solved for A t and ( 2 = 0) x V 0 (T= A) They then solved for (Ty and equations (5- 1- 5) and (5- 1- 4 ) - (5. 1 . 6 ) (5. 1- by using 7) -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 A w i l l not be zero because As w i l l not be. zero. A i s because " \J ^ - yu) at surface.!' i s .replaced by *l x. V ( i - H) - 0 - - I x V 0 d - H) + 1 ^ 4 All This I x V x As . . ( 5 . 1. 3 ) A that, remains.is to use equation ( 5 • 1• 3) in (5- 1. 8) to solve for A and then, use ( 5 . 1. 5) in (5.. 1. 4 ) to solve for. 6$ . s 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) From equation. ( 5 . 1. 2) we find, therefore, gives 6^" z. 6 ° 3 h 2 (V<H) c b that (5. --IM-ZTTS ' K cosh ( KH) 61 2 I 2 sinh (2. KH) 1. 1 0 ) It is reassuring to. note that 6 S --|^27T$' ^- for K H /7 1 and that. this,.agrees., withthe 2 calculations made, by Lamb, and Levich .(.ref.: 2 1 and. 2 2 ) . The total damping coefficient is found by using (5- 1. 10) in ( 5 . 1. 1) and. (5- 1- 2 ) . 60 --2^K VzirS 2 K 2 + sinh ( 2 X K) Equation ( 5 . 1. ll) 1 .1 + ( s / k A ) I 2A 1 cosh^ ( KH) sinh (2 k H) J 2 1 -(S/Kf\f It is - 2 KH sinh (2KH) (5- 1. 11) for I 6~ 1 was. plotted in. figure 13...... As figure 13 0 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. -41The surface? q u e s t i o n now i s , why i s t h e r e n o . h o r i z o n t a l , , f l u i d , v e l o c i t y , a t t h e According... to. Lamb and. L e v i cih ( r e f . 21 and 22) t h i s , may be due .to an.incompressible, i n s o l u b l e , m a s s l e s s , s u r f a c e f i l m on t h e , f l u i d . f i l m . of. t h i s type a c t s as a t h i n m e t a l p l a t e would. can, e a s i l y bend the. s u r f a c e f i l m , The A surface The motion o f f l u i d t h a t i s , Cause the. f i l m , t o move v e r t i c a l l y . f l u i d , motion cannot., however, cause t h e 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 i n c o m p r e s s i b l e . the f l u i d Because of. the. f i n i t e at. the s u r f a c e must move w i t h the f i l m . v i s c o s i t y o f the.fluid T h i s means that, the f l u i d a t the s u r f a c e can.move v e r t i c a l l y but.,, because, o f the f i l m ' s presence, cannot move h o r i z o n t a l l y . I n o t h e r words, t h e r e can be no h o r i z o n t a l , fluid is v e l o c i t y a t the s u r f a c e i f an. i n c o m p r e s s i b l e , i n s o l u b l e , surface, f i l m present. Lamb ( r e f . 21) c o n s i d e r e d t h e o r e t i c a l l y , the e f f e c t of. an i n c o m p r e s s i b l e , i n s o l u b l e , , s u r f a c e f i l m f o r . deep f l u i d waves. considered t h i s type o f f i l m . . . Van Dorn ( r e f . 14) t o o , He worked t h e o r e t i c a l l y and. e x p e r i m e n t a l l y on p r o g r e s s i v e , g r a v i t y waves,, i n . a. r e c t a n g u l a r , tank.. Van Dorn c l a i m s t o have . c a l c u l a t e d the. t o t a l , viscous, damping., . c o e f f i c i e n t , including...the. s u r f a c e f i l m e f f e c t f o r a l l v a l u e s , o f KH. This., u n f o r t u n a t e l y , i s n o t t r u e because... h i s c a l c u l a t i o n s .use a f o r m u l a (ref. 44) which i s o n l y t r u e f o r deep f l u i d s , ( i . e.-X H ^ experimental Surface .results do n o t c l o s e l y agree, with...his 1). His theory. f i l m e f f e c t s were, d i s c u s s e d t h e o r e t i c a l l y .as.a f u n c t i o n o f the. c o m p r e s s i b i l i t y and s o l u b i l i t y (ref. taken, from. Landau and. L i f s h i t z o f the f i l m . by. L e v i c h and D o r r e s t e i n 22 and 24) f o r deep f l u i d waves ( i . e. K H » work to, i n v e s t i g a t e , t h e s e . e f f e c t s . . f o r deep f l u i d , c a r r i e d , out by Davies and. .Vose ( r e f . 16.). 1). Experimental c a p i l l a r y waves was T h e i r experiments demonstrated, i n agreement w i t h t h e o r y , t h a t damping,.is i n c r e a s e d by f a c t o r s o f two o r three, by s u r f a c e f i l m s . Only when the. experimental, equipment a n d . f l u i d -42were 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 i s 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 f i e l d .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 L W MAGNETIC 'REYNOLDS^.'NUMBERS In this chapter the damping.coefficient, is derived for standing, axisymmetric., surface waves, on an ideal, incompressible f l u i d .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 fields...is..made..in appendix .3. Figure U+. shows, the nature of the problem to be solved.. We. define, the following: w. 8 E % J v P § H. A T P B 9 KV 3 = •= ~ = = — = = = = — = — =• = magnetic field. electric field. electrical, conductivity of. the fluid. current density, fluid velocity. fluid.density surface, wave amplitude, fluid depth. radius, of fluid container. fluid surface tension. pressureapplied magnetic f i e l d . gravitational... constant. radial wave number, of the.surface wave, fluid, viscosity. We use. the following assumptions and. equations: Maxwell's equations V- B - 0 b B - - vxE (assuming no free charge) V x B - yUJ yu€ <^E A A. 6t (The above, equations give... V - J (6. 1. 1) 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 -43- are the characteristic -44- applied magnetic field, I rigid walls S, Fluid with.conductivity, and mass densityy£> , H I I Rigid bottom Figure 14. Description of the magnetic damping problem to be solved. -45- This means, that . EX- - velocity and length,, .respectively)... 4 equation...(6... 1 . 2 ) and. that. the. induced.magnetic f i e l d , smaller, than the applied magnetic f i e l d , V x B in , is much . The fluid equations are V- (PV + be ) ( V-V)\/l - and/O ^ - 0 (6. .1. 3) -VP - vO VG. + J x B +•" ^(other terms) (6. J we assume jj ^ 0 ( i . e. ideal f l u i d ) 3 that^o is a constant, (i.e.. incompressible fluid) and & — 1. 4) } ( i . e . in a constant gravitational, field)- The Linearity conditions, required are and KS " u 5/H "1, <-<- (HK) • \ ( 6 . 1 . 5) The boundary conditions are V-~n P - P 0 - (at rigid walls), T (1) <3_( 0 and ) (6. 1 . 6) (at the free surface), (6. 1 . 8 ) (at the free, surface) • The surface, wave, is assumed to be axisymmetric using, a perturbation technique. . | 6 B I « } B 31. ) (6... 1 . 7 ) The problem w i l l be solved (i.e.. we .assume. B. - B. + s O B where 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 • this problem with the restriction k H <r<- 1 . Fraenkel. (ref. 24) has solved We w i l l 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 i r s t order, or, higher i n . find that OB/B 3 , and. using ( 6 . . . 1 . 3) we /Oy\\ h _ { - 2Ni ) - 1 f) (r -.. cWt_)l (JVfc . ( 6 . 1. 9 ) Ve recall, that f r 3 j ( Kr )f r dr\ dr J 1 c) of order 0 ) . V 2j ^ ) r ( w h e r e (K) r J° i s a Bessel function Therefore, we can use Jo ( K r ) 4 -K o - p |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. k (1 - 0 at £ ^ + i0B3 ) Using.. V 4 0. in ( 6 - 1. 9) we find that - x2 (6.1.10) The value of |< i s found in.the. following, manner. . By taking, the curl of ( 6 . 1. A) we find that <3V di B) ) - V ' r (1 + 10 ( (jVk <3r Therefore, The boundary condition at the walls gives us. ( K A ) - Therefore, J ' 0 Vr V r °C Jo ^ ( K r ) - 0 a t r - 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 , H etc.) is found 3 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 / ° % / ( r c ) i " ) - AO ^ - i i _ _ * dr We recall that 5 dr at the surface. We define lj)so that <^ IjJ- \j . (TK 2 + ) ^ jJ However, /j) <=4 cosh .(. di k£ ) This means that. % °< Jo (K r ) £ , (evaluated.at..£ 2 di di Therefore (6. 1. 12) becomes h A4/ - ^ hi (.6.. 1-12) - H) . ( 6 . 1.13) 1 because .^.^J- di V^. o( sinh ( kH, ) . -47- As,..a result, (6. 1. 13) becomes "H - 2 | l K +^ | 2 ^ (6. 1. 14) tanh ( k H ) The problem is solved,. ..therefore, by. considering the three equations, (6.... 1. 1 4 ) , (6. 1. 1 1 ) and (6. 1. 1 0 ) . P ^ 2 . k (1 - .10 8 / J-'o (KA) where For B 2 T T - O 3 ^ ) (6. 1.15) K : o - ri -' 1 K tanh ( k H) •+ J TK 7\ They are as follows: ^ i^B + ^ e a ^ o v e and 5 <* Jo ( KIT ) & . results agree, with. Coulson (ref. 6). of finding ~Y\ in terms of K , H, and B «. / • /27rb ^ i * 3 The problem is simplified i f we assume It is clear later that -this corresponds, to assuming that This assumption is consistent with the experimental results described, i n chapter 7 Using the above assumption and. using the fact tanh A ( 1 + 10)1 - /tanh (A)l (1 + t J I 4A61 (1 + A /( ^©*)(e** - e ) ^ for & « / 2Jl we find that (6. 1. 15) reduces to 2-ni + ±61 -YX - (2.7T5 ) 2 - |TK.3+^K j tanh ( K H ) (6. 4 KH I/O j'o (KA) So<. jo e -.. 0 (Kr -iht )e The interesting points to note are the following: (i) (ii) £ is independent of (3B B 3 for a l l K H. i s proportional to B^Q 1 (iii) For VCH^ 1 CT - -Bjfl for a l l K H 1.16) -48- (iv) For K H « 1 - -B Q 2 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 B 3 > 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. OQ considered. 2 was not CHAPTER 7 EXPERIMENTAL: STUDY- OF "THE" MAGNETIC" DAMPING OF A SURFACE"WAVE IN"MERCURY This,chapter describes the experimental study of the magnetic damping (T^ , of linear,, gravity, standing, surface waves in mercury coefficient, B$ innaeraecLin.. an ..applied,. Vertical, uniform, constant, magnetic f i e l d , . The, microwave cavity technique was used -to,investigate, the. relations (5^ , H, K , and B 3 between 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~ Ol (J~ + (j^ . o The values of that were used here were obtained,from the,.experiment, described in Chapter. 5.- The results are summarised in figure 15, A, B and C. experiment, studied both, deep and; shallow fluid waves. about 0.4 to about .4.0. This KH was varied from 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, i n a vertical, magnetic field has been developed in chapter 6. for, this experiment... 61 - - $ML [ JyO ( 1 Equation ( 6 . 1..16) . gives ).. + exp (2.KH) - exp (-2KH) (7-. 1. 1 ) ) The assumptions ..used ..to..develop., this., theory, are. summarized-in..-table '1. The above equation.was used to plot OB as a function,of depth,with wavelength as. a. parameter, (figure. 15 A, B, C).' The plots show that the agreement between the theory and.,the experiment is very. good. . . Any. difference can be explained .by. the experimental..error In measuring, OJ and. B 2 . ' The EXPERIMENTAL RESULTS © 6 APPROX. 2IOO GAUSS A « • 1600 " O • •• 1200 » 700 » A » w (max. experimental error in ^B/ 2 Is 2 5 % ) B-» THEORY (EQN. 7.1.1 ) / Fig. 15 a Magnetic Damping Coefficient Results ( K = 1.51 c m " ) 1 0.6 L2 DEPTH OF 1.8 FLUID, 2.4 H (cm) 3.0 -51- EXPERIMENTAL O B A « a 1500 « o « « 1200 « A « « 800 » (max. APPROX. RESULTS 2000 experimental error GAUSS in THEORY °^B is 2 25%) ( E Q N . 7.1.1 ) O . 15 b Magnetic Damping ( K = 1.21 c m J - 1 Coefficient ) — —L—»—'—-L—i—i—L—i—i—I 1 1.2 DEPTH OF 1.8 FLUID , 2.4 H Results (cm) i , I 3.0 • , -52- EXPERIMENTAL 48 © A 42 B N RESULTS APPROX. N 2200 GAUSS |400 » O « « 1200 II A N II 700 " (max. experimental . "—• o UJ (O CVJ UJ 1 or error in THEORY 36 ^BZ, 2 (EQN. is 7.1.1 2 5 % ) ) h- LU CO > • 30 b CO 24 UJ o o Ui cc < o ZD o O o H i/> o _l K UJ < u. o 18 o 12 o I- >Ul Ul e> < Fig. 2 o < 15 c Magnetic Damping ( K = 1.05 J L 0.6 J L DEPTH 1_J_ 1.2 OF cm" ) I 1.8 FLUID, 1 Coefficient 2.4 H (cm) , , 3.0 Results -53- percent error in the measurement.of <T could be as high as 15$ and, the percent, error in the measurement of could be .as high as, 10%. B The oscillation frequency, $ , was found to be independent of to within 1%. This agrees with the theory developed in chapter 6. S3 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 n a t u r a l l y 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 c a v i t y technique has proved, very u s e f u l f o r 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 f o r the use of a rectangular microwave c a v i t y as. an automatic. Fourier, analyser of surface waves was. developed. Can this, theory be.put into, p r a c t i s e ? 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 l a r g e r t h a n .that predicted by the standard theory because of the meniscus. Can a detailed, theory be developed to account f o r t h i s effect? In chapter 5 experimental.evidence indicated, that there was no h o r i z o n t a 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 l m ? Could an o i l or mercury oxide f i l m have caused t h i s e f f e c t ? Could the em. f i e l d s i n the microwave c a v i t y have caused t h i s effect? In chapter 6 and. appendix 3 a theory f o r the magnetic damping of 1 and surface waves was developed by using the r e s t r i c t i o n s . Pan 1). without these r e s t r i c t i o n s ? (cjEV/Cmip)» Can e. theory be developed What would we expect i f Rm » i? -54- i or i f -55- 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,. problems associated, with immersion devices. surface waves where., $ oK % 10 A. - This method is free of the It is, capable of measuring 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 Experiments showed .that.the oscillation frequencies of the wave 4.0. v 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 - l h j _ 2 J T j ' X. ' 2 cosh^ ( K H ) . sinh ( 2 K H ) The experimental, results were i n excellent 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 calculated...by 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- -57- f i e l d and that .the, magnetic damping coefficient was given by ) L kH 0 U/O { exp mercury was performed.. above calculation. ( 2 K H ) - exp . An experiment with liqu ( - 2 K H ) ) The results are. in excellent agreement, with, the REFERENCES 1. Langmuir, R. V. ( 1 9 6 1 ) , E l e c t r o m a g n e t i c f i e l d s and. Waves, M c G r a w - H i l l series i n Engineering.Sciences .2, S l a t e r , J . C. ( 1 9 6 3 ) , M i c r o w a v e . E l e c t r o n i c s , . B e l l L a b o r a t o r i e s S e r i e s 3 G i n z t o n , E. L. (1957), Microwave Measurements, McGraw-Hill. Book Company, I n c . p. 4 4 5 4 Whitmer,. R. M.. (1962).,. E l e c t r o m a g n e t i c s , . . P r e n t i c e - H a l l , I n c . 5 Morse, P. M. and. Feshback, H. (1953), Methods o f T h e o r e t i c a l P h y s i c s , M c G r a w - H i l l Book Company, Inc.. p. 1565 6. 7 C o u l s o n , C . A. (1961), Waves, U n i v e r s i t y M a t h e m a t i c a l T e x t s . . Case,. K. M . and P a r k i n s o n , W. C. (1956)., X o f F l u i d . Mech. 2,. 172 8 Tad j bakhsh, I. and. K e l l e r , J . B, 9 R a y l e i g h , L o r d (1390), P h i l . Mag. XXX,. 386 10 K e u l e g a n , G. H. 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), P r o c . Roy. Soc. A, 218, 44 13 U r s e l l , F. ( 1 9 5 3 ) , P r o c . Cam. P h i l . S o c . 4 9 , 685 14 Van Dorn, W. G. (1965), J. F l u i d Mech. 24, 769 15 K r a n z e r , H. C. and K e l l e r , , J . B. (1959), J- A p p l . Phys. 30, 398 16 17 Davies,. J . T". and V'ose, R". W. ( 1 9 6 5 ) , P r o c . Roy. Soc. A, 236, 218 G r o s c h , G. C. and Ward, L. W.. and L u k a s l k , S. J . ( I 9 6 0 ) , (1958), <X (1959), J. o f F l u i d Mech. 8 , U 2 o f F l u i d Mech. 6, 33 Phys. o f F l u i d s . 3, 477 18 Hunt, J . M. 19 S p i e s , R. ( 1 9 5 3 ) , A e r o j e t - G e n e r a l C o r p o r a t i o n Rept. 1508 20 E a g l e s o n , P. S. (1959), M. I. T. Hydrodynamics L a b o r a t o r y T e c h n i c a l (1963), Phys. F l u i d s 7, 156 Rept. No. 32 21 Lamb, H., (1945), Hydrodynamics, Dover P u b l i c a t i o n s , I n c . , New York. 22 L e v i c h , V. G. ( 1 9 4 1 ) , Acta. P h y s i c o c h i m . U'.S.S.R.' L 4 , ,307, 321; P h y s i c o c J i i m i c a l Hydrodynamics., New. J e r s e y : P r e n t i c e - H a l i I n c . (1962). D o r r e s t e i n , . R. (1951), P r o c . Acad. S c i . Amst. B, 54, 260, 350 23 - 5 8 - -59- 24. Fraenkel, L . E . (1959), J- of Fluid Mech. 7, 31 25 Levich, V. G. and Gurevich, Yu. Ya. S. 3. S. R. 1 4 3 , 64 26 Roberta, P. H. and Boardman, A... D. (1962), Astrophys. ' 1 3 5 , 552 27 Wentzell, R. A. and Blackwell,. J.. H. (1965), Can. J. of Physics, (1962) Dokl. Akad.. Nauk. J. (U.S.A.) 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. 30 Vandakurov, Yu, V. (1963), Soviet Physics - Technical Physics (1964), Proc. Roy. Soc. A (G. B.) No. 1374, 214 (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. ( 1 9 6 2 ) , J. of Electronics and Control 14, 513 Tucker, M. J . and Charnock, H. (1955), Proc. Fifth Conf. Coastal 41 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). Landau, L . D. and, Lifshitz, E. M. (1959), Fluid Mechanics., Ch. 6 Reading, Mass: Addison-Wesley Winsor, F. and. Parry, M. (196.3), The Space Child's. Mother Goose.,, Simon and Schuster, Inc. 44 . 45 CALCULATION OF THE RESONANT... FREQUENCY CHANGE FOR A SURFACE WAVE IN A RECTANGULAR CAVITY APPENDIX. I 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 cos E, E* = E, = Ea = B„ = By = ~ J E,nt UJ { L Bi = Z ~ j sin (wy/B ) sin ( Y)HZ/L ) Sin 3 Cos (fax/ft) - {yffft/L) Sin ( W T T - Y / B ) Cos \ B 00 { JULEL. YITT E? B where E", , 2^ = 0 L and H 3 are the peak values of E„ , E Y 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 B= i 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 )( ee f i g . 16). S,p ft B The calculation for 2{(j)orU)) w i l l be made assuming S J S and P are integers. Q -60- -62that only one simple standing .wave is present (i.e. $ • — cos (PTTX ) Cos ( S7TJ/ ) ). A B By using equation (A 1.-1) in (1. 1. 1) i t is easy to show - ABL E, (1 - 6n )(1 - 8-n ) d * )0 ) t )(1 + 6M,O )0 and A/ 6n,o (1 - 2 ) (1 - 8„ E-32 UTT «. ] a 4r,o 2 6 X)0 ^ )(1 + <5o)f n j ( -3) A1 As before,, if. the, amplitude of the surface wave. (i.e. E )(1- t0 -.€/UA/e- 2 + _« 1 ) and , $ 0 then , is small can be. calculated a; (J from 2(CJL-tJ 7TJD, U ^ J Z L ^ E3ITm - Ez.ft.7r ) - Jo AB l TTTT / -rri, c O W ; 7T [. B "7— - 1 AB 6t) A/* 1 — So AB 1 F 1, A E^W^VD, ^ (J?,,P) 4 / 4 (m,s) D. w D, (w.SjjTD^ (i,P) f X ) 2 - ( ,s) (Al. 4 ) where AC and are and either E 3 / Ea, - WE, Also.,. \ / E, t + ~n.Es TnEi. — 0 (for T'. E. modes) or x Sin 2 Jo Cos (ie) cos 2 given by equation ( A l . 3) 0 (for T. M . modes). ( 1 9 ) Cos ( j e ) c f e ' (je-)^e - TO , ( i , j) - ^ 0 for i . * I 7T(1-Sip) j 2 £_7£D (i, a L j)= 7fA < - 0 - 0 - 0 j, - 0 i 0 ' JT; 2 (1 + di, ) _7TC$2i,3 4 0 i 7^ 0 and j ^ 0 -63- 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 unless S ^ 0 and P. - 2& . - 0 and JL (In other words, CO 0 then iii-cd is CO proportional to the S - 0 , P ^ 2J? , Fourier component of the, surface wave in This allows the »P - 0 , S - 2m" and »S - 0 , P - 2i " the cavity). 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 ^ 0 unless one or more of the following is true: then (A. CO 0J (i) (ii) (Iii) 2i P and S - 0 - 2Tn - S and P ^ 0 2i 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 , CO 2'i ) + c 2 ^ o(s - 2m, P - 0) ( -+ C where P - C.-j_, C2 3 J o(.S - 2>n, P - 2i ) and C3 are found from equation (1. 4« 2). Therefore, once J o(S ^ 0 , P ^_ 2l ) and i~o(S ^_ 2}n P ^ 0) have been measured (by using y 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 p i e shaped, perturbation ( f i g . 17) was introduced into the bottom of a microwave, cavity to test Slater's theorem. The change i n resonant frequency f o r such a perturbation, was computed from the theorem i n the same way. as i n chapter 2. The assumptions, Ctia. - OJ/_< 1 6) and Y\T\ £« ^ ^ 1 are used again. It can be shown that for i '- 0 (T. M . modes) 2(6l-Q) - K* §o 0 feo ( K.rjlrdr - kt e 6J io0[jo (Kr)lrdr • L Similarly f or £ ^ 0 K ^ i o f 2i^ 2(6^L-6J ) - + f$« and B (2J?H©/ ^J) + / J-.2J0 ^ A/ Sin -sin(2J?re ^J I 2U -ffiffi^* + sin(2iej}[ [ V ( K r f c d r ~ )+ sin(2ie * s i n ( 2 i c e , - ( ) )f sinuie, K r)dr T ( 2 0 )J| jj CK r)rir. . R l 0 (A2.2) are given by equation (1. 3- 3). It can be shown that f o r Jl ^_ 0 (T. E. modes) 2(6i- £J )/o; - Similarly f o r i. ^ 2(lU-£J ) ^ (r)lT) \ L) Jo 0 [ f r o ' (Kr)rdr. JC/V^ J 0 0 Jo / (WIT) -sin(2i0 + i y ^ / )1times } t sin(2l0 + (kfr)]rdr| • i s given by equation )tsin(2i© J + fmtjo.(2H0 g 2.M 4 i I Z. /K"/VB I { )\l (A2.3) ; (1. J.. 4 ) . -64- 2 / 6 , ) - sin(2J?e, (A2. ) 4 ) jtltties Figure 17 Details, of the perturbation used to test Slater's theorem. -66- A p i e shaped, p e r t u r b a t i o n was The azimuthal. p o s i t i o n , were h e l d c o n s t a n t . o f the c a v i t y p l a c e d on the bottom, p l a t e o f the , (see f i g . 17) was As the p e r t u r b a t i o n was the resonant, f r e q u e n c y v a r i e d c e r t a i n number of. t i m e s . The varied. A l l . other r o t a t e d 3 6 0 ° about from a minimum t c a maximum a minimum r e s o n a n t f r e q u e n c y , OF, and. the. number o f v a r i a t i o n s , measured.. u s e d to c a l c u l a t e F and theorem and. a comparison values. em., modes, in,the. cavity).. repeated f o r various .35 cm. from Slater's L, 's ( i . e . f o r d i f f e r e n t The r e s u l t s a r e shown i n t a b l e was performed the. p e r t u r b a t i o n was. k e p t . c o n s t a n t and from 0 to yVO, were E q u a t i o n s (1. 2. 2) a n d . ( l . 2. 4 ) were used to A s i m i l a r experiment varied NO 2. but, i n t h i s case, the p o s i t i o n o f the p e r t u r b a t i o n a m p l i t u d e , ' A l l o t h e r parameters was.measured f o r 0)0,-0) 3 < 0 .4 cm. from S l a t e r ' s theorem. So were h e l d c o n s t a n t . change i n r e s o n a n t f r e q u e n c y p e r change i n p e r t u r b a t i o n amplitude calculate the was,made, between, the t h e o r e t i c a l and. e x p e r i m e n t a l T h i s experiment, was i d e n t i f y the modes. parameters the a x i s d i f f e r e n c e between the maximum and E q u a t i o n (A2..4) was cavity. The y E q u a t i o n (A2-4) was T h i s experiment , was u s e d to was repeated f o r The experimental Joi7T various L 's ( i . e . f o r d i f f e r e n t em. r e s u l t s , f o r the, TE'jj, and TE , Zl modes i n the c a v i t y ) . modes are g i v e n i n f i g u r e 18. t h a t , as,.expected, the. change i n r e s o n a n t frequency i s . d i r e c t l y to. the. amplitude o f the perturbe..tion... and,.theory a r e i n agreement.,for I t shows proportional Table. 3 shows, that,.the experiment the f o u r modes, t h a t were t e s t e d . V a r y i n g the s i z e o r p o s i t i o n o f the p e r t u r b a t i o n a l s o r e s u l t e d i n s m a l l , undesirable changes, i n the l e n g t h o f the c a v i t y . These u n a v o i d a b l e sometimes caused r e s o n a n t f r e q u e n c y changes t h a t were almcst as l a r g e those, caused by. changing account as the p e r t u r b a t i o n size, o r p o s i t i o n . , T h i s e f f e c t f o r any d i f f e r e n c e between the t h e o r e t i c a l and t a b l e s 2 and, 3 • changes can experimental values of Parameters for. Table. I I = 0 .310 cm. cm ft cm. 2.54 Experiment (cm'.) 43° 8.9 KMc./s. Theory r L = (jj/ = /2TJT R, = 1.46 = ( s e e f i g u r e 17) No Mode (Gc./s.) JL n F No (Gc./s) 1.83 .018 2 • TE 1 1 1 .020 2.20 .008 4 TE 2 1 1 .006 4-45 .003 4 TE 2 1 2 .003 Table H i Comparison between t h e o r y and experiment 2 4 Parameters f o r T a b l e I I I (see f i g u r e 17) 0 /?, = .97 cm. A =2-54 cm. = 4.3 0 _£j = 8.9 KAlc./s 27T Theory Experiment 4 (cm.)' (Gc/sec.cm.) Mode X (60a.-6J)/zirSo (Gc/sec.cm.) n >n 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 t h e o r y and experiment. -67- -63- MODE RESULTS o / 8.924 o o / o / 8.913 o O in £1 o / / 8.902 / o / o / / O 8.891 / o / / o / / / .127 fc .381 .254 .508 8.906 TE 2.11 MODE RESULTS o O o O 0 8.904 o Qi . ^ .635 PERTURBATION AMPLITUDE, J . (CM.) , o o 8.902 o O/ / O o o 8.900 0 .127 • 254 .381 PERTURBATION AMPLITUDE,: So Fig. 18 .508 • 635 (CM.) Graphs showing .the change in resonant frequency as a function of the perturbation amplitude. -69- In some cases when a p e r t u r b a t i o n was i n t r o d u c e d i n t o the c a v i t y , r e s o n a n t f r e q u e n c i e s were observed d i f f e r i n g by l e s s than 20Mc./s. two The r e l a t i v e changes i n t h e s e r e s o n a n t f r e q u e n c i e s , as the p e r t u r b a t i o n was b e i n g moved about i n the c a v i t y , suggested t h a t they corresponded t o i d e n t i c a l em. modes ( i . e . same i , TYi and YI ) w i t h d i f f e r e n t azimuthal orientations. As expected, the e l e c t r o m a g n e t i c f i e l d s near a c o u p l i n g h o l e o r probe (between the waveguide and the c a v i t y ) were found t o be d i f f e r e n t g i v e n by e q u a t i o n s ( 1 . 2. 1) o r ( 1 . 2. 3 ) . f i e l d s i n the c a v i t y more t h a n h o l e s . used. I t was f r e q u e n c y was Probes were found t o d i s t o r t A coupling hole, therefore, 5). p r o p o r t i o n a l t o the p e r t u r b a t i o n amplitude f o r amplitudes When So was change i n r e s o n a n t f r e q u e n c y was made g r e a t e r than « .4 cm., however, the no l o n g e r p r o p o r t i o n a l t o i s t o be expected because the s m a l l amplitude Y)TT So was found i n agreement w i t h the t h e o r y t h a t the change i n resonant l e s s t h a n .4 cm. (i.e. the made as s m a l l as p o s s i b l e and p l a c e d as f a r from the p e r t u r b a t i o n as p o s s i b l e ( s e e f i g . I t was than 1 ) i s not v a l i d in this assumption . This t h a t was made case. L The above r e s u l t s , t h e r e f o r e , show t h a t S l a t e r ' s theorem can be used t o p r e d i c t the change i n r e s o n a n t f r e q u e n c y o f a microwave c a v i t y caused a s m a l l amplitude p e r t u r b a t i o n o f the shape o f t h a t cavity. by APPENDIX 3 In THEORETICAL CALCULATION OF THE. MAGNETIC DAMPING. OF SURFACE WAVES FOR LOW MAGNETIC" REYNOLDS NUMBERS t h i s appendix the damping c o e f f i c i e n t i s derived f o r linear^standin surface, waves, on. an i d e a l , .incompressible f l u i d of f i n i t e those used i n c h a p t e r The d e f i n i t i o n s are the same as k| except t h a t 6 2 T T / = The assumptions and e q u a t i o n s ( 6 . 1 . 1 ) , and A . < and, k z . i 2'JT/ x .- = < V~ AT P = % Vi = The c o n d i t i o n s a r e as f o l l o w s : 0 t + £ = 0 . j.^| T + t (A3.6) } A ^ T T H F E R E E S U R F A C E - ( A a t the f r e e s u r f a c e . B- B + 0 <5B where A magnetic, f i e l d . Bo 6§ - A Bo = . g A B, l + x 3 ( The. problem w i l l , be s o l v e d u s i n g a p e r t u r b a t i o n t e c h n i q u e . assume 2 ( 6 .1 . 2 ) , ( 6 .1 . 3 ) , ( 6 .1 . 4 ) , ( 6 . 1 . 5 ) i n c h a p t e r 6 a r e used to s o l v e the above, problem. boundary and F i g u r e 1 9 shows depth i n a u n i f o r m , c o n s t a n t , a p p l i e d magnetic f i e l d . the. n a t u r e o f the problem t o be s o l v e d . conductivity - A 7 3 ) - 3 ) We i s the a p p l i e d 0 A l * y B J 3 . £ The os.cillati.on. f r e q u e n c y , -f , and. the. damping, c o e f f i c i e n t , CJJ, a r e found by assuming n = that, the time dependence, i s . o f the form Z77f e)Cp ) where + }(% • X Freenkel. ( r e f . 2U) has s o l v e d t h i s problem f o r t h e case ( k, B, = O the B } z - case ( ^ Blackwell = 0 . + 0 = B 3 = k 2_ < z Roberts and Boardman ( r e f . 26) have s o l v e d i t . f o r $H >> (ref.2 7 ) + k 1 B, = 0 ', = B . t W e n t z e l l and k, H ^ have s o l v e d . t h i s problem f o r the case and Kukshas, Ilgunas. and Barshauskas have c o n s i d e r e d v a r i o u s a s p e c t s o f the case B We w i l l ( s o l v e the problem f o r a l l v a l u e s o f - 7 0 - 3 - B + L k^H = ^-i = y ^> (ref. 2 3 ) O . k, ^ A Z) /3, } < -71- F i g . 19 D e s c r i p t i o n o f the magnetic damping problem to be solved. -72By taking .the curl curl of (6. I. A ) , dropping terms of second order or higher, in in/oVP J A, - qil-m- be of the form & cosh ( k 2. ) + A z U exp( ik,X + ikx^-IYXt sinh ( k i n . a boundary condition requires that for \4 V-V = 0 = V - B = V - J Bo. we find that Let V and.using the. fact that \SB\ ) 3 . 9 ) times We note, however that A,= O because ? = 0 at H. = 0. Using, this form 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 evaluating.it at. the surface. This gives ^ <• (A 3-11)' It is more convenient to consider two separate.cases,I ( B , = and II ( S 3 0 = ) = 0) when solving (A 3-10) and (A3-H) simultaneously. 3 Case I ( 8, = Using 0 = Bz. sinh ( ) ) Q '** we find that (A3JL0) and (A3 J-l) reduce, to (A 3-12) 5 For ex EXP (ik,x = + iki_y - irii) • 0 the above results ...agree with Coulson (ref. 6) The problem of finding .71 in terms of simplified i f we assume Q B3 « j • K , kt. , H, and S 3 Using this assumption and is -73- u s i n g t h e f a c t t h a t tanh[.A ( 1 (tanhlA) U 1 + )^ 1e .1 ~ JAQ4 + we f i n d t h a t (A 3-12) F O R e «• 1 becomes (A 3-13) exp exp(.-£kH) czKH) - ^ = k!" + kt 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, a r e the. f o l l o w i n g : (i) (ii) i s independent o f B 5 0 7 f o r a l l v a l u e s o f kH, 3 i s proportional to — - Q di (iii) ff-Bj kH fop » fora l l 2 • kH. T h i s 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 R o b e r t s and, Boardman. ( r e f . 26)(iv) (Ji n *" -JilJiLL— kH <^<- 1 o r Vi r This r e s u l t agrees w i t h the o f F r a e n k e l . ( r e f . 24.). theoretical, calculations Case I I ( Bi = . 0) OA we f i n d t h a t (A 3 1 0 ) and (A 3-11) °< s i n h ( k £ ) e reduce t o ) J k TBNhl(kH) { 9 + T. ??= ( n = ins J cV 1 * icr 8 + -'m^(B^Bl) k ) i J (A3-H) . A g a i n t h e problem o f f i n d i n g . y\ i n terms o f k, s i m p l i f i e d i f " we assume Q ( Bf * B L ) , ,/< \ kz. > H, B , . and B i s z Using, t h i s assumption -74and u s i n g the f a c t t h a t tanhJA( ^ <- 1 we o+ A e K expcifl).- expc-zfl) l f i n d that ( A 3 . I 4 ) (ZTf#={n 3 l reduces to + y k ] T f l N H ( k H ) k > 4 xT -^5J B ~ / + / 9)J CXP(ZkH) - * eXPC-2.KH) J S * e x p e i K . X H k ^ -irrt) • The i n t e r e s t i n g ..points ..to. note a r e the, f o l l o w i n g : 5 (i) (ii) i s independent,,of Bi and Bj. fora l l (X - --£_ U ( B S B . ) - Bfk* + BUt \ V F o r t h e case. 8 1 = ^ = + ki I kH»i. FOR J 0' t h i s agrees w i t h the c a l c u l a t i o n s . . o f W e n t z e l l and B l a c k w e l l ( r e f . 27) and theoretical also Kukshas e t a l . ( r e f . 2 3 ) . ~ - -47 a; (iii) (ic^^\)-m<i±^)(z-zcrfilFOR. For the case., 6 2 = Kukshas. e t a l . c l a i m t h a t k, = 0 and Bj, = the wave f r o n t ) . /o z + m i + Cf t h i s agrees w i t h Kukshas e t a l . B z. 0 i f kz. = 0 and Bs of, s o l v i n g B, = 0 or. i f i f the a p p l i e d , magnetic f i e l d , i s p a r a l l e l to T h i s c l a i m does n o t agree w i t h our The problem C B, 0. ( i . e . 0 = k u « i (A3.12).and ) was calculations. (A3.I4) f o r a l l values of not considered. APPENDIX 4 MUMMERY, SUMMARY The f o l l o w i n g i s , b a s e d on work by F.. Winso.r ( r e f . 45) T h i s i s the t h e o r y Jack This is"the built. flaw t h a t l a y i n t h e theory Jack built. T h i s i s . t h e mummery h i d i n g the flaw t h a t lay. i n t h e theory, Jack, b u i l t . T h i s i s the, summary based on. t h e mummery h i d i n g the flaw t h a t l a y i n the. theory Jack.bui.lt. This i s the constant K t h a t saved the summary based on the mummery hiding the flaw t h a t l a y i n t h e theory J a c k b u i l t . T h i s i s the e r u d i t e v e r b a l haze cloaking constant K. t h a t saved t h e summary based on t h e mummery hiding, the f l a w t h a t l a y i n t h e theory J a c k built, -75- -76This 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. -77This 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 b u i l t . This is the flaw theory Rob built that lay in the
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Investigation of fluid surface waves with a new microwave...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Investigation of fluid surface waves with a new microwave resonance technique Pike, Robert L. 1967
pdf
Page Metadata
Item Metadata
Title | Investigation of fluid surface waves with a new microwave resonance technique |
Creator |
Pike, Robert L. |
Publisher | University of British Columbia |
Date Issued | 1967 |
Description | 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. |
Subject |
Waves Damping (Mechanics) Microwave measurements |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0105202 |
URI | http://hdl.handle.net/2429/38366 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1967_A1 P54.pdf [ 3.97MB ]
- Metadata
- JSON: 831-1.0105202.json
- JSON-LD: 831-1.0105202-ld.json
- RDF/XML (Pretty): 831-1.0105202-rdf.xml
- RDF/JSON: 831-1.0105202-rdf.json
- Turtle: 831-1.0105202-turtle.txt
- N-Triples: 831-1.0105202-rdf-ntriples.txt
- Original Record: 831-1.0105202-source.json
- Full Text
- 831-1.0105202-fulltext.txt
- Citation
- 831-1.0105202.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0105202/manifest