ELECTROMAGNETIC WAVE PROPAGATION THROUGH comm LINE OF PARTICULAR caoss-secnon Thesis m the Mm af M. s. macmw STATE coma Karlis Kmkiitis W54 1'. v fi..‘.n 4 (,3 I7; This is to certify that the thesis entitled “Electra-Magnetic Wave Prepagation Through Particular Coaxial System“ presented by Karlie Kruklitla has been accepted towards fulfillment of the requirements for _!_°_S_'_ degree in ___E;_B_-_ W Major profésorfl Date-Mat 20. 1951+ 0-169 ELECTROAMAGNETIC WAVE PROPAGATION THROUGH COAXIAL LINE CF PARTICULAR CROSS-SECTION by Karlie Kruklitie A THESIS Submitted to The School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfilment of the requirments for the degree of MMSTER OF SCIENCE Department of Electrical Engineering 195A Tame M4- \r’ I 110% ‘ ACKNOWIEDGMENT The writer wishes to express his very sincere thanks to Prof. Dr. Joseph A.Strelzofi’ for his many valuable criti- cism and aid on this problem. ‘A B 8 I RNA C T The prepagation of the electromagnetic waves in a parti- cular coaxial system is investigated,eee figure l.The e-m waves are assumed to be confined in the dielectric bet; ween perfect conductors. For the solution of fields governing differential equati- ons relaxation methods are enployed.By applying them the potential distribution is found for the principal node. Corresponding B field components are computed by employ- ing the relationship‘betveen . : . - - - ' .. t l ._ 36’) Al, 3/40." poten ial gradient and electric 1_ --...';-,- ~ field intensity.H field compo- nents come out from intrinsic ' _ . impedance expression.ror solu- ‘<3 F;:ht .,. .:f L. tion of characteristic impedan- ' . . es the capacity per unit length is computed by the flux mapping. J‘__""z”—“—_fi* Computing eigenfunctions and ei- FCchf‘fl I genvalues for particular size of given kind coaxial system,Ez components are compueted for TM mode and Rs components for TB mode.!he other field comp— onents are solved by employing the relationships given by Maxwell's equations. - The relaxation.methods as employed in this paper may be applied to coaxial systems of any cross-section,contrary to analytical methods which.are ap- plicable only for very simple geometry cross-sections. l. 2. 3. l... l. 2. 3. A. 5. TABIE of CONTENTS INTRODUCTION Statement of the Problem . . . . . . . . 1 Brief Historical.Background . . . . . . 2 Methods Suitable to the Boundaries of this Problem . . . . . . . . . . . h Relaxation.Msthods . . . . . . . . . . . 7 II THE PRINCIPAL MODE Introduction . . e . . . . . . 10 Solution of Laplace's Equation . . . 12 Characteristic Inpedanee . .. . . . 17 Numerical Example . . . . . . . 20 ‘Dieeuseions of Computation of 2° . . 27 III l. 2. 3. I... 5. 6. 7. 8. l. HIGHER ORDER MODES Introduction . . . . . . Eigenfuntions and Eigenvalues . . The Lowest Mode . . . . . Transverse Magnetic Mode . . . Transverse Electric Mode . . . Second Order Mode . . . . . Higher Order Modes . . . . . Evaluation of Particular Example . IV CONCLUSIONS Discussions and Conclusions . . APPENDIX . e BIBLIOGRAPHY e . e 31 33 36 LO #2 1+3 My 1&5 67 81 l. SINIEMENT OF THE PROBLEM AND ASSUMPTIONS The aim of the present work is to investigate and formulate mathematically the physical pro- perties of a particular system for the transfor- mation of guided electromagnetic waves. The sy- stem consists of two coaxially placed, semi-in- finite long conductore: metalic inner cylinder , and thin rectangular metalic outer tube,as shown in figure 1. They are perfect conductors; elec- tric conductivityG'; do. Dielectric with dielec- -.. . .. trio constant ' 6/3/6711; .. 3‘ é’and magnetic r . Q permeability fl, :- which is equal 6 . to that of emp- « 1 :. i: f ty space/11°, .‘5 : is located in the region be- tween both conductors. It is assumed that die- lectric is lossless or nonabsorbing, its electr. conductivity 0": 0.1‘he electromagnetic waves are assumed to be confined to the interior of the co- axial line and are prepagated along its axis. The discussion is limited to the problem. of determining the waves that may exist in this par— ticular system. It is also asked to express the characteristic impedance and out-off conditions. The question of reflection and radiation that may occur at discontinuity in the interior die- lectric or in the conducting part, will not be discussed here. 3 . BRIE! -roarcu. BACKGROUND The mathematical theory of electromagnetic wa- ve propagation along a conductor with an external return part is very old, and is going back to Ray. leigh and Heaviside. The first discussion of such systems for guiding electromagnetic waves appears in a paper by lord Rayleigh,published in the Phi- -2... losOphical Magazine, in 1897. Much important work has been done in develo- ping and extending this theory. In 1931+ Dr.S.A. Schelkunoff published the paper ”The Electromag- netic Theory of Coaxial Transmission Lines and Cylindrical Shields". In this article he discus- sed the prepagation of electromagnetic waves through cylindrical coaxial lines.A few years la- ter the same problem‘was solved by a Frenchman, S.Pollaczek, paying more attention to the condi- tion of prepagation from the physical viewpoint. This example,investigated by Schelkunoff and Pollaozek, is ideally adopted to mathematical so- lution because the conductorrs shape fits perfec- tly into the cylindrical system of coordinates; thereby making it entirely feasible to carry out a discussion on the basis of the electrometio field theory instead of using ordinary circuit concept. The circle and the rectangle are the only domains for which the exact solution of the boundary value problem,for the Laplace's equaticn -3- in the plane,can be given in terms of elementary functions. The cross-section of coaxial structureof this problem is confined by two different lines sir- cular inner,and the rectangular outer partADif- ferontial equations for such kind boundaries can not be solved in classical form. In solving the second order partial dofferenta ial equation by product trial (variable separati- on).method in rectangular coordinates, the bounp dary conditions on the inner surface oftne outer rectangular tube can not be introduced into Bes- sers functions,which results from partial diffe- rential equation solved in cylindrical coordina- tes, applying separation of variables. For the above.mentioned or rather practical reasons the coaxial guide with cylindrical inner and rectangular outer conductive surface was not solved yet, at least was not advertised. 3. METHODS SUITABIE TO BOUNDARIES OF THIS PROBIEM much work has been done by number of mathema— - g - ticians to derive the methods for solution of di- ferential equations, which can not be solved by classical methods in finite terms, and therefore approximate methods of solution must be employed. Sp called numerical and graphical methods are ap- plied tn much more problems than other methods. Especially numerical methods have become particu- larly popular in resent years. The main difference between numerical methods and analytical methods of solution of differenti- al equations is that numerical methodeare concer- ned only with obtaining answers to particular pro blem, but analytical methods gave the general so- lution. With the introduction of the idea of relaxati- on, originated and developed by Sir R. Southwell, the numerical methods have become as practical me- thods for solution of problems in physics and me- chanics. Sir Richard Southwell and small group of his followers, since 1938, have published many scientific papers to investigate how the relaxa- -5- tion methods may be employed to solvethe problems of different branches of physics and mechanics. The principal mode of given coaxial structure - for our problem.- may be solved numerically by relaxation.methods - by replacing the Laplace equ- ation by Laplace's finite difference equation", and graphically by flux plotting or conformal map- pins. Eugher order modes.may be investigated by variational methods, or by relaxation methods, which.can be applied to the theory of eigenfuncti- ons ( approximate solution of Helmholz equation) after appropriate preliminary steps. If cut-off 'oonditions are the only aim.Ray1eigh.methods can be employed to find the solution. Relaxation.methods will be used to discuss this problem. These methods, for solving the se- cond order partial differential equations are ge- nerally not concerned directly with electromagne- tic wave propagation, but they are used to solve the problems in physics and engineering,which are governed by the same differential expressions. - 6 - Therefore it is reasonable to apply these methods directly to the electromagnetic wave phenomenon because of analogy. The phisical preperties of the heat flow, and of vibration of clamped membrane are formulated mathematically by Poisson's (at limiting case La- place's), and Helmholz's equations respectively. The same kind partial differential equations go- vern two-dimensional field distribution and trans- verse waves in electromagnetics. Relaxation me- thods are introduced to investigate the heat flow problems as well as membrane vibration. So, also electromagnetic field problems ,characterized by the same differential equations, may be solved by similar means. There is no solved heat flow problems for sy- stems which would be direct analogues to the coa- xial electrical devices. Further, there are no solved problems of vibration of clamped membrane, which would be direct analogue of our coaxial sy- stem. -7- Thus we cannot solve this problem.by direct analogy, but we will apply relaxation.methods di- rect to this problem.and employ the significant properties found by solutions of heat flow and vibration of membrane. h. REIAXAIIONWMETHODS Essentially the relaxation methods are the grou up of numerical methods for the solutions of a simple system of linear algebraic solutions,with finite number of unknowns, by succesive approxi- mations. The most convenient way to explain the procedure of relaxation is to solve an example, as shown in appendix 1 and 2. Other mathematical expressions,whioh.may be put in a form.similar to the mentioned algebraic expressions, also can.be solved by relaxation me- thods. In the solution by numerical methods of ordi- nary and partial differential equations, the de- rivatives are replaced by their finite-differen- cs equations,generally infinite series of diffe- rences.For the employment of relaxation. methods it has been customary to ignore all but the very dominant first terms.In recent years the methods are develOped to use the full difference equa- tions instead of reduced only.It certainly helps to increase the accuracy of solution.‘We ‘will not discuss this in detail now, but will observe separate parts when it will be neccessary. ‘Different techniques,theoretically based on the preperties of MacLaurin or Taylor series,are develOped to derive the "finite-difference equa- tion”,which represents the "differential equa- tion" by some approximation. In solving this pro- blem the square type not were employed. Solving the plane potential problem.the dimensionless va- lues at the nodal points of square not will be found instead of continuous function as solution of differential equation. The particular values for all nodal points are found by relaxation.pro- cedure.The relationship between differential and -9- finite-difference equations is expressed as fol- 4:"; (“j— a, =£(V'“/j 17] where N are meshes lows which adjoin to each.nodal point and h is the di- stance between two adjacent nodal points.For squa- re net Nah. Two-dimensional Laplace's differenti- al equation,which characterizes the field distri- bution in this problem, is related with correspon- ding finite-difference equation in such way V144, =(Vfccj, :39. (a, #0. Hz, Mount.) [.2] 4 Figure 2a, The solution of particular eiample is a conve- nient way to explain the relaxation methods in a practical application to partial differential equ- ation as it is shown in appendix 2. II THE PRINCIPAL MODE 1. INTRODUCTION Rectangular coordinate system is employed in discussion of this problem. We are interested in the propagation of waves which vary with distance and time ,in the posi- tive z direction (the axial direction of the gui- de) . Therefore we introduce the requirement that electric and magnetic coanonents involve time on- ly in the roxjapée’wtand distance in the form The factor a .1 z-represents a wave traveling in positive 2 direction, with frequencyfxé—of , and a prepagation constant I: “ff/3} a( being the attenuation constant, and [5w the phase constant. If propagation factor @160 M, is introduced into the wave equation, we obtain the wave aqua - tion in the form with the time variation excluded as .11- Vi? E =- -—([2- w/Zé] 12': £an V13}: 1??- -—//’-+ £92911"th For the Transverse electromagnetic 'waves 1. [4 60/546 vanishes, therefore for the principal .mode of the problem in discussion the field equa- tions 83° Vy?§:0£4‘4] Vifffso [:46] Electric and magnetic fields both satisfy La- place's equation under static conditions.But the field in the transverse plane is exactly a sta- tic distribution, if it can be shown, that the boundary conditions,to be applied to the diffe- rential equations Wylie-om.) Vfif‘7=0 £96] are the same as those for a static field distributionflhe boundary conditions for the transverse electro- magnetic wave (TEM) on a conductor of coaxial li- ne may be obtain a normal component only, 'which is the same as the condition at the conducting boundary in statics. The line integral of the - 12 - electric field between both conductors is the sa- me for all paths lying in a given transverse play ne,and may be thought of as corresponding to a p0- tential difference between the conductors for that value of z. Thus,Laplace's two-dimensional e- quation may be used to investigate a TEM mode. 2.801U'ITON OF LAPIACE'S EQUATION Electric field distribution is expressed by La- placian and is denoted by Vz‘ézo. The aim of this section is to solve the Isplace's equation forthe two dimensional dielectric region with given boun- dary values. Boundary conditions are given, gene- rally,as follows it -.=.— C, on the outer surface of the inner conductor and gm = (:1 on the inner enl- faoe of the outer conductor. To apply the relaxation.methods to our transver se plane let's replace the Iaplacian by finitedif- ferenoe Iaplacian (vs We 412 9‘1?— // ‘6 .// ‘79191 2375:227 [:2] - 13 - " .. (Viéjoz 9*93— +33 222:" 425.455] where the nodal points of the square not (it is called al- so "relaxation pattern”) under investigation,and its adjacent points are expressed as shown by fi- sure 3. S? L Before we commen- ce the procedure of relaxation, it it °“ To 5’ is necoessary to recognise the dif- ficulties, which are raised by com- élr . plexity of the par- Figure 3-” tioulsr geometry of the transverse plane in consideration. we are working with.rectangular coordinate system.There- fore high accuracy is asured only for rectangulr regions.To obtain the numerical values of nodal ochfs P I -11..- where the boundaries are curved and therefore do not fit in the square type relaxation pattern, the irregular star Operators are employed. In general the multiple-connected (hole inthe transverse plane) boundaries ask for special tre- atment,but it is not necoessary for finite diffe- rence Leplaoian. For the process of relaxation,the point relax- ation method. is applied and the following Opera- tors are used,see figure a and 5. Both Operators are derived from.the finite-di- ference equations.As customary,the numbers at the left of nodal points denote the displacement and at the right - residuals or appr0priate multip- liers of residuals. The electric field vector at the pivotal point for each point separately is computed by intro- duoing the numerical values (computed by the pro- cess of relaxation) into the known relationship between a gradient of potential (scalar) field , and field intensity:_ E=—Vé=—- amaze [6] Thus we have é as a function of x and y (x,y as finite values for the pivotal points); field intensity as negative gradient of scalar values may be computed. Taking in consideration also propagation fac- tor we obtain the field intensity of electric field as a function of space and time,for finite number known points §=_v§€J'wf-/a [7] Propagation constant I may be sinnplified,sin- ce dielectric is lossless (no dissipation)1he at- tenuation constant 4.: a or / = 0 4/73 Finally, electric field intensity is g =__ Vé cjwt JAE: _V§eJ/wt-/33) £8] The magnetic field intensity at eachruflal point in x,y plane may be expressed by introducing the electric field intensity at that nodal point into intrinsic impedance expression fi‘fié"? E5" @vée”“'p“ 1721 Eat_____§_'g—____ __ ‘__ BM 7;" #7 74/? [75—7 3. CHARACTERISTIG WE To obtain the characteristic impedance zo,for L. the ideal case it is it)“ E , consider the ex / I pression 7) -722.)v may be computed from U 2 7, fiend éare given. I The capacitance per unit length is = 0/ , CL er’n (calm; C 45,-- gr, ‘7‘fi‘z M945. we 5"] Field plot,see figure 6, gives the chargesni the potential difference.Thus,capacitance per unit length may be readily coinputed. Nf is the number of flux tubes.Multiplied by flux per tubeA (P it gives the total flux ending on the conductor. According to Gauss lav total -18- flux,ending on the surface of oonductor,is equal to the charge induced on the conductor.Nb is the number of potential divisionsdtmnltiplied by po- tential difference per division 43’ gives the to- tal potential difference between both conductors. Assdistance between two adjacent field lines. Algdistance between two adjacent equipotentials. (— dielectric; constant of dielectric. .6 _ . _ 46 i l c _ I F63“?! 66 Fc’gurelfa - _ 0.. A” A C“ fizz/f (6 2-5 [”3 -19.. For plot pattern with small squares A‘le/ is equal to unity. ‘ 1” After v and C are computed, I. may be evaluated / ‘ I from 7/‘=-—- and it is = —--- r22; .VCZ ZfVET' I. and 0, both per unit .leng‘th,arexfound,there- fore 20 (ohms per unit length) may be evaluated. To save the labour we solve numerically a spe- cial case of coaxial line (outer conductor is squ- are: a: b, but not rectangular in general). Prin- cipially the relaxation's procedure is the same. See figure 7 e 1». NUMERICAL HAMPLE The technique of computing the characteristic inpedance for coaxial line with irregular surfa- ces of conductors by method discussed in this pa- per may be demonstrated by numerical example.Also characteristics of principal mode will be sedved. In doing this the relaxation methods are employed, - 20 .. therefore the solution can not be in general form,but related with structure of particular sl- ze. To reduce the amount of numerical work let's solve the problem,which is a special casect our general problem,see figure 7. an, -. - ,. Given: ash-10 on; - 2 .6}. - J. .. j - r=l chasm that ...-.' 1;: .1- 6-: {9/ 3/4qu 01'- ” " I " ©r -:' fect of spacers, . {511.52 " which were used be- -' i . cause ofair-dielec- J___ , tric,is negligible. 9" #3 IRS units are em- nguvc 7, played. Find: 3 and H field couponents, characteristic impedance, and characteristics of m wave for this particular problem. R1,)“, Z In general 2° = G- 1,ij For a line with low losses or completely 1&- al case, like this one, - a - If we know capacitance 0 per unit length,then inductance I. per unit length may be computed and characteristic impedance found. To compute o the flux mapping method is employ- ed. Equation 18 states that C’ = e- Wyy/lfi . Because of symmetry the treatment will be applbd to one fourth of the coaxial region, see figure Band9. Aw,- Wm ”#24; '-22- '- : .r-i-is.‘ .. 1.1/15: 6 £5 l C“ e ”/0 g ' ' Li— C=z3.dzvo"’7,,,; I I I ___ flea, ________ _} W: CI , Fc‘ c . go - V C C ’ / “J .. - o .. a 23. 3‘6 #0"; F—__————____I =/4/.3.0_; C =23,5g;// 5%,; 231° =- Mull; For the solution of field components the fini- te difference Laplacian relaxation is employed,as it was developed in section 2 of chapter II. Figu- re ll shows the transverse region of ‘EEM mods. Assume that d :1 cmggiven a=lo cm and r=l cm. -23.. Because of symmetry we may apply the treatment of relaxation to one eigth of the region or interest, a. r— "" "" ”-IL 3"};7{ by such step reducing: I I I I I I I I great amount or numeri- cal workJ'igure of lla shows the sector of tran averse region to which the relaxation is ap- plied. Employing theeq. 5,applying it to region in AnB-CAD in figureiua, by technique as indicated and demonstrated at ap- pendix 2 and k,the numerical values of potential field for nodal points are computed and given in figure 12 on the separate sheet. According to equations 10.11.11. and 15 tb tiild components are 't4fif“ fluff - )3; £:%}(7h2@9 hmfiéh] ==-—-%?§§ -éi ') } - 24 - 7 C an. 2:. Q. ‘r— 5% go} 91.7 9—519. 2233,, g éw Sif"! ’er a," a” 2Pro} ' if). 14%?” 4 ’ F2; 7a. y-e/ // 42,, ? wav‘éi- z~ HM ”‘40 ”74%: “Egg—#5 fl / Hy/My,md’;vj z“ Zigeflwz~fi£j -25.. NdI and may indicates the particular nodal point, for which we express the field component. For example, the H: comPonent at the point P is, as shown by figure 12b, w 0 Phase constant may be derived from ge- ral case , where ,_ .51... a £1. = lL-rw'yré- If \& 4p ‘ 9’ axial component In 4'” m is equal zero . . then there It: vani— "‘i ”(V 3’" ehes, therefore Fc?‘cre fl 5. [Ls-w? g— ; H7Zldnjfij=fb5§§1fi€gxdfiai OY /=/x/w7<-; m m 0; jfi =/w W,- W F’ WV??? CW 2/4.: 71“.: % mama/Av. fl”: ”pg; -26.. 24.33-“? / =0. 5. DISCUSSIONS OF GCMPUTATION OF 20 Several methods are develOped for the compu- tation of characteristic impedance of coaxial syb stem. By combination of conformal mapping and theory of images,the.method for evaluation of characte- ristic inpedance was develOped by Sidney Frankel, who did this by investigating the parallel wires in rectangular troughs. Other methods are similar to each other. Kho- wing v from.given.dielectric prepertiee expres- / / sions 2r:- 7:: 'and :— __.._....— are taken in /" VAC the consideration. If one of both.l.or C .may be computed by other means, then the other can be found from above relationships. Mbssinger and Higgins tables, consisted from empirically obtained measurements, supplys I.va- lues for different size coaxial systems. Anderson has derived a method for computing the capacitance per unit length for the rectangu- lar coaxial structure. - 27 - Gandy,who published some articles to pOpulari- ze relaxation methods, has develOped an interes- ting technical procedure for computing the capaci- tance per unit length..Actually in Gandy's case the relaxation methods are employed for conformal mapping purposes. uandy solved capacitance for three coaxial structures,as shown.in figure 10 a, b and 00 Flux.mapping.method is employed in.this paper, because of irregular surfaces of both conducting cylinders the mentioned previous methods were as- sociated with considerable difficulties. .e .s e a. -‘ ' ’ " ' 4 .‘ o... c .. ' 5" o... FH?.A04. u.Thé F6; , /0{. Ffiy/Zk: Naturally, it may be asked which method is bet- ter,.more advantageous? Which.method brings to ex- pected aim quicker,which upholds higher accuracy? To deside which.method would be more desirable for particular Problem.the wanted accuracy is im- portant factor.0nly Anderson's method lrings exact solution, but this method is associated with very great amount of work. The mathematicall errors 'which are very possible in the long procedure do not appear in the process of computation. Gandy's suggested method relaxation involved also.msy'as- sure high accuracy if very fine net of relaxation is employed. But as finer net is used as greater becomes the numerical work. The advantage of this .method is that that in process of computation al- ways may be seen the range in which.lies the cor- rect answer. After solution of the problem of fi- gure lu a,b and c, Gandy indicates that accuracy range is between .65 and .98 %. The.method employed in this paper doesn't pro- .nise very high accuracy, but this method is appli- cable tc surfaces with irregular shape. The camp puter, who has gain considerable skill,for apply- ing flux mapping, may reach satisfying Intresults very quick for cases where high accuracy is not - 29 - of first degree necessity. It seems t9 this writer that there is no par- ticular the best universal method for all kinds of problems and for all catnputers. The wanted ac- curacy and specific features of the problem indi- cates the more suitable method, and each unputer develOps his own practical ways to solve the pro- blems by some particular method. Gear and Miller have done reasonable Job to supply the tools for direct approximate solution of characteristic impedance. By taking practical measurements fromzo different kind and different size coaxial structures, they have supplied spe- cial curves. Those may be used for determination of characteristic impedance for coaxial structup res with known ratio between outer and inner con- duotor e -30- III E I G HIE R 0 END I R MIO?D E 8 1. INTRODUCTION In addition to the ordinary transmission line wave in a coaxial line, there may exist, under certain conditions, higher order waves with elec- tric or magnetic field in the direction of the line axis, in z direction. ' If the prepagation factor is assumed to be in ‘Juvé‘jyir the form 6 then the finite axial compo- nent must satisfy the wave equation,as shown be- 1 . 0' VLv-4 (/1+w/‘aéjV'=o [/9] V‘v-I-éI—Vso CZQ] where v represents or suply the numerical value of Ez for m wave, and Hz. for the TB wave. The problem is to express the £2 and Hz as functions of space coordinates, and to find the -31.. rest of the field components 31,1353; and By in terms of L'z and Hz. Taking a special case, when a==b (outer con- ductor as square),we would try to solve the par- tial differential equation which governs the we- ve phenomena by the separation of variables. As- suming that outer conductor is approximately a cylinder with radius a/2 we would have identi- fied our constructure with normal cylindrical co- axial line. Observing the difference between the geometry of both conducting surfaces, it is obvio~ us, that by such arbitrary approximations the so- lution will be too far from the actual, and will not be satisfactory in no occasion. In general ca- se the results would be much more far away fromn tisfying ones. let's turn to the numerical methods. The free, transverse vibration of metalic mem- brane with uniform density is governed mathemati- cally by the partial differential equation of the “m Viv-#AW-co/ [‘20 .. 32 .. z. 291; where the Operator stands for VLF. 3?, +3—1- The physical behavior of vibrating membrane is analised.mathematically by applying the numerical methods,to solve the governing differential equa- tion. It is found, that the work of computation is reduced reasonably by employing the relaxation methods. Equation V17} + {:9 = 0 which governs the oscillations in an electromagnetic system.is ana- lagous to that of membrane vibration Vim-Idle; =0 The solution of expressions 1:: V‘M {She—01s identieal for TM and TE waves as relaxation me- thods are concerned. Therefore we will investiga- te a general case (v will represent the variable), and the particular features of TM and TE we ‘will disduss separately. 2. EIGENFUNCTIONB AND EIGENVBIUES In the case of principal (TEM) mode the Lapla- ce's equation had only unique solution for each -33.. particular guide. Expression Vzb" 4' {c.LV‘0 has many solutions. V1 values as the functions of nodal points are called the eigenfunctions or natural modes.Corres- ponding :3, values are denoted as the eigenvalues or characteristic values. From the theory of vibration it is known ,that an equation in the form V11} +4270 and at our particular boundary conditions, is not satisfied for any arbitrary value of k: . There exists an infinite sequence kfil 4 kggé 1:334 ... of values for what the equations can be solved. For each ei- genvalue xii corresponds one particular eigenfunc- tion. v1. Likewise of solution to Laplace's equation the finite difference equation is introduceth's dif ferent now. Again, finite difference equations are solved by relaxation methods. The latter can be applied to eigenvalue problems after simple extension of relaxation technique used before.The changes will -31,- be observed at the residual and at the unit re- laxation operators as it will be shown later. The boundary conditions: vau cm the outersur- face of the inner conductor and on the inner sur- face of the outer conductor. Corresponding to equation 912- 0-3: 37p!» 7}: +£ 0 [22.] the finite dif- ference equation, in general form, is 2. enigma-+717, —-zr.(4 -{Lfij ...— o 1:23] The appropriate operators of relaxation methods suitable for eigenvalue problems are shown in fi- sure 13 and 12.. m = 4-1421,. cos/4,. - 35 - 57a?! /4 3e THE LOWEST MODE In general, the higher order mode which pos- seses the greatest cut-off wave length or the lo- west cut-off frequency of all modes in particular transmission guide is denoted the "lowest" or the "dominant" mode. The eigenvalue, which characte- rizes the cut-off wave length, comes out from the process of relaxation. As for all treatments of differential equati- ons by relaxation methods the preliminary steps are those of replacing the governing differential equation by its finite difference representations. Form.this the general residual and the unit point relaxation Operators are derived. For a case like thiswhere the value of the eigenfunction on the boundary is everywhere zero we have to deal with as many eigenfunctions as many internal nodal points lie in.the relaxation pattern. To start the computation of eigenfunctions and eigenvalues, at first we assume the values ofall - 36 - set of eigenfunctions. An example is shown in fi- gure 15. V1 are inter - L nal nodal values, where q J iv-l,2,3 and 1.. Accor- I ‘L dingly we have v1, v2, Flynn: ,5— V3 and 744.138.1118 the nu- merical integration,in- troduce the assumed v values into Rayleigh quo- 1. tient, l: ,-,—__.. //+ VVIVM¢$5_)£M] //. W- W e 2 so the initial,first approximation, value of k0 is obtained. We will denote this by kg”). Inser- ting kiu) into given residual and unit relax- ation Operators the first,initial, set of residu- als are obtained. The process of relaxation relax- ation is continued until there is only little clan- ges in residuals .Using the eigenfunctions obtai- ned after the first series of relaxation the nu- merical integration is employed again. The new ei- genvalue, which certainly is closer to correct va- -37- lue is denoted by k§(2). The process of adjust- ment of E0 value is continued until the eigenva- lue becomes stable and residuals are zero or ve- ry close to it. The final v1 values and the corresponding ei- genvalues (g) are the seeked solutions. This procedure was employed by the origina- tors of relaxation methods and is employed en.ma- ny occasions yet. ‘Without this general method, which is related by a great amount of numerical work, there are 0- me improved methods.By applying them.quick results are available for some particular kind problems. One of such kind method, suggested by group of instructors of Harvard University, may be applied with success to the problem discussed in this Iris By this method the finite difference equation is expressed in little different way ‘§]£:v_ zit Application of this indicates what is ‘3}: 1:- M61 The different k§1,called as local eigenvalues, ‘wrong with our arbitrary initial assumptions. - 33 - are not equal by the first assumption. The .main work is to change the eigenfunctions v1 so that all the kgi become equal. Before the final kg; are found the correct kg value is somewhere between local kfils. In such ‘way is indicated the range in which the correct kfi value is confined. we can see the range at e- very stage of relaxatione‘Depending of the accu- racy needed, we can step the process of relaxa- tion at any stage if the desirable range is ob- tained. When the values of wanted function are pres- cribed on a curved boundary, as in this problem on the inner cylindrical conducting surface,then the nodal points of relaxation pattern do not co- incide with the boundary..A special treatment is necessary for such cases. An example is shown in appendix A. By introducing the numerically expressed ei- genvalues kg and eigenfunctions of transverse electric and magnetic fields ( 32 or H; represen- - 39 - ted by v) into the relationships, which define the electromagnetic phenomenon,the prOperties of TM and TE modes can be determined. s. TRANSVERSE MAGNETIC mans TM mode or Ez wave is defined by 323-0 and 32ft). With assumed prOpagation factor ejwé—[d- the finite axial component of electric field for the m waves must satisfy the wave equation in a the form UV} Ea; +£L5e = 0' [-26] [1- 1 t 2 In.” 57%:— or s =/ 4 a, 57 k0 should be constant for a par- ticular mode and is determined by the boundary conditions to be applied. ror TM mode the boundary conditions arezonihe surface of conductors Ezzu. Knowing Ez, the rest of the field components, spews; and fly, at harmonic time variations,may be determined with the help of Maxwell's equa- tions. All these components are expressed in -l+0- TM mode \\@ [ c __.__ E {fair/(l / ~— H {1164/ ‘7 Fifi/6. Co 72.149 and £0 n 01/ fie/0L disivc'éuéio‘w terms of E2 ~1- DE” ' E J/ 35537] ——. I ”’- 5y=“4§' by f? 11‘ Jive 352., =-~ZE;§E_€§ [15’ H7=T5§’ ? Ac “934' J Let's determine the cutoff conditions. The pro- pagation constant is A prOpagation occurs if radical brings out a real value. Itb is a prOpagation occurs if k2 kg; if g k2 than there is no prOpagation. k The limit, boundary, between a prOpagation and non-prOpagation is then, when k2 kg . From this the critical wave length may be determined. 1.:- co/‘trc- , 1:: (LifijiaaCZ/gj xfi = 211/34 W -.- 2.1734 3’: £50.] 25' __Z”. ' 5 [gr-xv} 4C’I’ Ll] -41- 5. TRALBVERSE EIECTRIC MODE TE mode or Hz wave is defined by Ez =0 and Hsz. If everything is assumed to be invariant in axial,z,direction, then the wave equation may be written in the form Vif/‘J/z- {-éLIL/ =0,[‘52] ...... {it/"+1? The boundary conditions are different thanfor Ez wave and are: on the surface of conductor 'DIL/s 7n. =0 ['55] The field components are expressed in terms of axial components H2 and they are ‘ . J“ 311/; (DH 57‘: Ti??? E7 =fey [34‘] / “326’ I 244, ”1 ”a???“ ”#119? Gut-off conditions are evaluated by the follo- ,_ 9 wing considerations: I‘VL-{By a real radical VB -l+2- [.5 5'] ode, TE m Fog. ’7 Mac ‘ a raft Con 0746 CO night 6 u‘éz‘ok/ value a prOpagation may take place. It will hap- pen when k§>k2 . Condition 1:3sz determine the boundary case, or cut-Off conditions. The criti- L , £7" cal wave length is at. = I [3‘] b. SECOND ORDER MGUE The determination of the second mode may be reduced to the solution Of the first mode as fol» lows.?Draw’an arbitrary nodal line over the tran- sverse plane (cross-section) of the coaxial line. Imagine that along this line the perfect conduc- ting surface is located into coaxial system. New we have two guides,see figure 18 a and b. ‘Determine the lowest mode for each of them. The next step is to adjust the nodal line by succes- sive modifications of relaxation that each of the separate guides has the same characteristic value; they can be considered as parts Of one and the same coaxial line.0ertainly, if the ad- - 43 - Justment is completed the nodal line will devide the cross-section into two symmetric regions. For example, the second mode of TM wave would be like shown in figure 18 a and b. 7. HIGHER ORDER MODES The procedure for determining the third and still higher order modes essentially is the same as for second one. To obtain a sufficient accurep cy the finer nets are needed,.AlsO, an amount of labour depends much.more of the initial assump- tions of the numerical nodal values. It is characteristic that the third mode must be "orthogonal" not only to the second but also to the first one. In general, each higher order mode has orthogonality prOperty to all lower.mo- des. fi’llll. 4 u/IIE— a. EVAIIIATION of PARTICUIAR mm To demonstrate the practical application of methods develOped in this paper, let's compute the characteristics of the lowest TM modefor the definite (given size) coaxial transmission sy- stem, the cross-section of which is shown in fi- 6m “ ik——- a, F437“. re /9, at- gure 19.9}lggm: 20 cm, r :1 cm and 6- 4744. 153951: E and H field com- ponents; fo - cut-off frequency ; Ac - out-off wave length; vp -phase velocity; v - group ve- 8 loo ity; o(,- attenuation constant andfl - phase constant. In discussing the higher order modes theoreti- cally (section 3 and A of chapt. III), the deriva- tion of eigenfunctions and eigenvalues was star- ted from given wave equations. In order to follow the detailized features of the derivation of fish .15.. distribution by our.method for particular prob- lem, this time we will start with.Maxwell's equ- ationS. For a dielectric containing no charges and 'with no conduction current in it Maxwell's equ- ations are W -__)§__ 3H. ‘Eu ~- /V’ )5 ...-E .. E,- \ Q/ me = I xl Mathematically it is a group of simultaneous partial differential equations. Beginning the so- lution we separate the variables Vx{§7¥§} = (74-31—5- Vuvii =—/4<7»:-§—g- - he - For the Operator 46f a curl (being space de- rivative) may be considered as constant,therefo- re DH =B(V“H) V)‘ 5? and Vyvgg 2.7an 2—(Vwé’j Replacing by its equivalent in iaxwell's equations (eq. AU) D ()5 -... 25.... D" VXVYEs‘fha-z-(Dé /"§€{65_- $22.52.} From vector analysis mm: = v(v.§) -- (7‘5 but V7.13: V.€—§= O Then, expanding V 5"va 251+ng 53,4" 0; V Eé According to the definition 'IM mode is defined by Hz: U and E2 720 . Ez should be computed, the rest of the field components ( ExaEyaHx and By ) may be expressed in terms of Ez by employing the relationships between axial and transverse compo- nents as it is defined by Maxwell's equations. 2. D a". We are interested in E ’~ g ...... z V Eg- /16' 362 Z ‘l 2 2. 2 9 £2 3 El. 2 £1- 07‘ V 5i; '97:. +SE1+ 3:2. J'w'é ~18: Introducing the prOpagation factor 8 wa-J’a— (1115 E23. 3 Ema. e ) where Emz is a function of x and y only, but not of z and t,see figure 20. jwt_/2_. Now, insert E4. = E77”; 6 into the wa- ve equation, which already is found -hg- V E’- "‘ v7; 5% + 3)sz 7695-;— 5" V1; Ez hilartjeh/f 9‘5 2:62 5 z 97“ ”L"- 93" I /Z and 13 ngta’fze f ’1 {20”54/(25'e [apgaq'fllz f, F43?“ re 2 0. .151 .. €JWt— -J2~ ape—mi 6Jwt‘vlé flame '2’: DE 2; ind-Ia 3-771}: D7‘['D)£ *1 3%[33‘1—6 ] =9 2'54; {Jeff "(l- Dxt ,_ 66a; ”VIM )2 £4 .. Jw£~lz2§31fi Dr- ” z 97” -49- Similarly it is obtained 2- . z >___£z, = €Jw£~lé £271!- . 93‘ DZ ’ ’ As stated before Emz is not function of z,the- refore, applying% it must be considered as con- stant )E j Jame-X} Jw‘c "/3” 32L=3v€[£mi'€ J=7£mee 215 3 DE __?_ __ wa—Iz 3;i=oz[afl VJ 155"“ J le"—-J2- ‘w-é- =(-J)(-d’)£m%e =12Em%ed [27 'sz lot-[Z- 7552 =- / £76; Ema- 7‘}: a [6251‘ mem bar of “23.02.1441, WW Won wfamd 6Z4 £77m dew/@614 3%! zap—“(Ema a)“: "43] J t“ 2" ge-‘EZI'JWEmtE/ w J ; a saw-3 4m ’7 jwf ~J’: 2. M1572- = (J“)(J‘V)Eme 6 -w 6»; e ‘/ 2 ° f, fies-...- 002'..er [agave— flew 925%: ——w/2‘hé- e 6»;— Jnse'zzf ”so dzraZo/mcx 604. m6.“ me Mr Wm wick a}; entree/4r». 9‘ El. 2) 5.. V77 EE 9a?— "" .96" 6M"; 2 £2» '3 lg. f— 4% 3x13743734/ Lghij‘g‘é/w «H.651»;- -51- 2. DE... 2‘5... 2 . var' 3;;E; 1F ‘Eyg;;_-f4(9f -+£O>A?f/(E)n1ys=cp ,4g, lpz¢vacizfaL222nk (Mace/V5} {or am Osage. /=J/3 - 55 - fl-J .,, f 51:) or c. -56- 3f {) { lcompfe‘é, affentcdian occur: I: defl -£= go fix '{j d {We M or 247=fl 3““ Q 7/ {KW 24:... /:: (//z I dw __ _ g I 71% “g; " 7" // /%/ jht’M-ducan; numerical Mgu er 21' Zr 117' do I =- c ‘ W 3-7-8 =0J3/6m1 hm" 1r" 7" 18/6 ......- Mfl} ”mic/’5- Q 74- .= /6'03 #65, In order to obtain corresponding E and H field components at particular point of the cross-secti- on or our coaxial structure,Just computed Emz va- lue (on figure 25 represented by v) at that parti- cular point is introduced into eq.36. I resp.k§ - 55 .. value was determined in this section before, the- refore Ez value is known.Equation 27 qnd 28 are expressed in terms of Ez, thus the rest of field components are determined and they are M. 2a; 1%{;1€nk{§9 77hflgb) '-' ;‘ fE;" / Z: Dy' w6¢r¢ 7211 49:4 7774/ (bodicai 5&4. ”add! Pod’hé thtr‘ Co‘hgfde‘yqjc'a’t. - 59 - IV C 0 N'C L'U S I 0 N'S i. DISCUSSIONS AND CONCLUSIONS The writer of this work believes that the me- thods of relaxation will be employed in the fu- ture much more than now.In reasoning this it is desirable to discuss some characteristics of re- laxation methods in general and observances of this problem as well. About a decade and a half has past since Sir Southwell started to pOpularize actively the ap- plication of general computational process ,known as relaxation methods.Series of articles written by Southwell himself or together with some 11' his collaborators chiefly have aimed to extend the use of relaxation methods to more and more pro- blems of dngdneering interest. The discussions , covered by those series,mostly indicate the ge- neral way to apply the relaxation methods to so- me of the engineering fields. The general methocb -bU- don't discover authomatically the advantages gi- venlby special consideration of particular pro- blem or particular field of science.Taking into consideration the particulars of each problem the amount of computational work.may be conside- rably reduced. As it can be observed at conside- ration the lowest mode of our coaxial system the quick solution by relaxation methods very much.dr pends of good initial guessing. Good pro-estima- tion,as base for the initial guessing, may be ma- de only by knowing particular features of the pro- blems For example: in considering the eigenvalue, for computation the lowest cut-off frequency or wave length of our coaxial line ,we notice that not £;-r‘ but longer distance,somewhere between the following length a 7’ > a r, 2 ff would tell us the distance between both conductors if our mixed boundary system would be replaced by cylindrical,circular,coaxial line with the same cut-off conditions. See figure below. Assuming the characteristic length right in the middle between A.and B and assuming the circular coaxial line of such distance between I conductors we can easily ‘ as ,which also will be ap- evaluate the nodal valu- vol proximate but close r____ dL’———~——9; the correct ones. Apply- F'Z'g‘wé 11/. ing numerical integration we will find out the ap- proximate eigenvalues and check it by comparison 'with value for circular coaxial lins.Such preesti- ‘mation leads computer to very quick solution. Only in few sections of physics and engines - ring principially significant problems are entire- ly investigated by employing the relaxation me- thods.These solutions ShOW’that for a particular kind of problems the relaxation.methods are much more preferable than other.methods,because relaxa- tion methods allow to employ particular characteri- - 62 - stics. The writer of this work could not find written evidence that it would be done very much.to deve- lOp satisfying ways for applying the relaxation methods to the problems of electrical engineering. Several outhors have indicated that relaxation me- thods may be applied to circuit and fieldpoblems. Solving by general forms of relaxation methods the se problems require very great amount of numerical work,therefore other methods may be better amploy- ed4More investigation should be done to develOp ad- vanced methods for solution xexlxlxtllxllt problems in electrical engineering by relaxation.methods in quicker way than by other methods. Anbther advantage of relaxation methods is that that computer always can imagine the physical pic- ture of the problem,which helps to see timwly the big computational errors and helps to find the ban way toward the aim.from.every stage of computation process. Since the analytical methods supplys the soluti- on for the simplest geometrical shapes and boun- dary conditions only, and general numerical me- thods do request a great amount of computatio- nal work,the relaxation methods are more promi - sing. The real strength of these methods lies in the fact that numerical results can be obtained for guided waves ‘with.more complicated and ir- regular boundaries, where analytical methodsfail to bring the solution or is very complex. Irregularity of boundaries (geometry of outer conductor entirely different from.geometry of Ln ner conductor) is the reason why this problemmy not be solved satisfactory by analytical methods. It would be misleading to believe that the re- laxation.metheds are the best suited for every circuit or field problem. Undoubtly there are many problems which can be solved easier and qui- sker by other methods. For example, the separa- tion of variables is very suitable for differen- tial equations where boundaries are expressed by circular or rectangular cylinders. - 54 - Also for this problem the relaxation methods were not used in parts where other methods were more suitable. For example, to evaluate the cha- racteristic impedance it was desirable to find the capacity of the coaxial line per unit lengtl For solution of this the relaxation methods may be employed as it has been done by Gandy. In opi- nion of this computer the flux ploting.method is more suitable for particular problem.and that was applied. ‘As for the method which gives approximate re- sults it is important to see what is the accura- cy of solution when it is found by relaxationxe- thods. For finite systems,applying the relaxation methods thods,the problem.is completely solved;for differ rential equations it is also necessary to investi- gate the accuracy of finite difference approxima- tions and of the solution of finite difference eqk- ations. Since this question is not in scope of our paw - b5 - blem.we will not go into details to discuss this. 'We note only this: the finite difference equati- on is more and more identical to the correSpon - ding differential equation by taking finer and finer nets in the relaxation pattern. To lessen the error made by approximate solution of finite difference equation some succesive methods are derived, very suggestive one by.Andrew' Vazsony, instructor of Harvard University. -bb- APPENDIX 1 A system of linear algebraic equation 51: - 2y =- 21 2X - 3y 1' h is solved by relaxa- tion methods. First, rewrite a given system of equations in the resi- dual form to have the residual operator 5x -2y - 21 = R1 21! -3Y - h = R2 To form the unit relaxation operator indicate how the residual R1 and R2 Will be changed, when x is changed by unit and y kept constant the same time, or otherwise TABLE I Unit change in residuals operation number Displacements R1 R2 r== S Operat. l x: l 5 2 Operat. 2 y = l -2 -3 U‘nit relaxation operator -57- The computation of x and y values is shown in the table of relaxation: TABLE II Operation Operation R 9 3 i d ‘1 ‘1 1 3 Step number multiplier R1 R2 1. All variables as zero -21* -h 2. I h 20 8 -1 14* -3 l h. I 1 S 2 2 3* 5. II 1 -2 -3 O O Solution for x: operat.2 plus operat h: h+l= 5 x: 5 Solution for y: operat. 3 plus operat. al-kl: 2 Y:2 -65... APPENDIX II The problem, shown on figure A, is solved by point re- laxation. The following notation is used: Displacement in- crements are written to the left of the vertical lines, and progressive totals of the residuals to the right. The boundary values of u are given and they are equal to zero. Residuals for all nodal points are given too, and each of them R1 is equal to 100. O O O 9 O O 100 0 100 100 o 0 1! b 4 U 1 O 100 O 100 0 100 o - I O O 100 0 100 0 100 O O r—J————£Jh—2Jh——OJ In g.n.r.1 hzmzu/da + We?) a gem) u1+ u2 +u3 +uh = huo For partial differential d are used instead of) because there is no Greek letters on the typewriter. -69.. T o o o o 125 100 125 30 130 o 25 o o 25 30 125 o 125* 125* so 5 o o o o 125 100 125 30 130 STEP 1 STEP 2 For step 1: 30:111.; 112+ 113 +uh,.— hu°+ 100 - 11110:. 100/ 14': 25 o o o o 30 o 130 125’ 1,30 160 5 160 30 25 3D 30 25 30 5 60 5‘ S 90 S ‘ o o o o o o 130 125 130 130 12? 130 scrap 3 STEP h -70.. o 30 o 160 o 30 s 160 ‘ 30 O as 160 30 5. 16 5. 120 5 30 25 30 0 o 16 30 o 5- 120 145 0 o 5 160*4} 30 no 160* 1.5 0 STEP 5 STEP 6 o f 30 o 1607 ‘ 30 5 160 ho 30 0 1‘5 25 30 0 us 160* ‘ 30 ho 120 us 25 30 0 30 no 85 120 1.5 he 85 o 30 no 0 85 0 STEP 7 STEP 8 -71.. he 30 ho o h 30 85 0 o 30 ho 25 30 0 n5 0 85 1 * 30 25 no 20 85 30 30 30 ho 115 o 115* o 85 0 ho 30 ho o 115 0 STEP 9 STEP 10 he 30 he 30 0 i he . 115* 2 25 25 25 15 50 115 30 30 ho 25 15 30 25 3o ho ’ 115'“ 25 15 0 ho 1.15 25 30 1‘0 o 115 25 STEP 11 STEP 12 -72.. ho 30 5 25 ho 30 O h 30 25 15 50 0 25 ho 25 30 30 5° 15 25 30 25 15 50 25 3 ho 50 15 30 22 30 15 so 7 v 25 115* no 30 5* 15 25 25 ho 50 15 50 STEP 13 STEP 1h ho 30 5 25 he 30 o as he 30 30 50 25 25 i2 3° no 25 50 25 30 :5 30 30 50* o m 30 15 30 1; 32 25 no 30 30 15 ho so ho he 30 50* 25 10 50 ho 1 f ; ‘ 0 STEP 15 f STEP 16 -73.. 30 ho ho 25 10 55$ 25 ho 10 30 30 30 25 15 25 30 15 50 30, ho ho 25 10 50 ho 10 STEP 1? no 30 ho 10 25 10 10 25 50 10 30 30 30 25 15 2 5 so. 15 sar“ ho 30 ho 10 25 10 10 50 10 STEP 19 -74.. ho 30 ho 10 25 10 10 2 s 50 ho 30 3o 2 5 25 15 30 he 15 50 30 ho ho 25 10 50* ho _ 10 STEP 18 ho 30 ho 10 25 10 10 25 503‘ 30 2o 30 30 25 25 15 A 15 50 25 410 ho 30 ho 10 25 10 10 50 20 STEP 20 30 30 ho 25 ho 10 104 10 20 25 10 30 30 30 30 25 25 15 10 50* 35 10 ho 30 ho 10 25 10 10 50 20 STEP 21 30 ho 25 ho 10 10 10 30 30 as 10 30 30 25 30 25 10 15 10 A 10 30 55’ 10 ho 25 ho 10 10 10 30 10 30 STEP 23 ho 25 ho 10 10 10 30 30 25 10 30 30 25 30 25 10 15 10 10 h5 10 ho 30 ho 10 25 10 20 50* 20 STEP 22 30 ho 25 30 10 10 20 30 30 55 20 30 30 25 15 25 10 1a 10 20 30 15 20 ho 25 ho :10 10 10 30 20 30* STEP 2h 65 50 50 65 55 65 3O 65 30* 65 30* 25 10 65 65 15 55 25 50 20 65 15 30 55 30 25 10 30 25 10 STEP 25 55 65 55 STEP 26 65 80 30 65 55 65 55 50 25 65 L15 55 30 65 10 80 30 65 55 30 65 15 30* 55 30* 25 10 STEP 2? 10 30 10 -70.. .0 STEP 28 55 65 55 55 70 55 65 10 80 30* 7O 15 65 80 10 7O 20 55 30 65 2O 55 10 55 30* 65 25 55 10 10 30 15 STEP 29 10 30 15 55 70 55 STEP 30 70 80 10 70 55 7O 55 7O 2O 80 10 7O 55 10 65 3O 55 10 55 10 7O 35* 55 10 15 15 STEP 3]. 20 10 20 -77.. STEP 32 The given steps demonstrate the relaxation technique,employing the pints: point relaxation method; to solve this problem comple- tely there is 51 more steps or total 83 steps of relaxation. Undoubtly this is very long ww but it helps much more than different improved operators to understand the practical appli- cation of relaxation. Without complete understanding of point relaxation it would in not be easy to employ the multiline and block operators. Because the rest of the relaxation steps ( 51) don't bring anything new we will demonstrate last two steps only. 67 8h 67 67 8h 67 0 1 an 102 8h 8h 103 8h 8- A 3 -1 67 8h 67 67 8h 67 0 1 ..‘lo— APPENDIXh Eigenvalues and eigenfunctions for curved boundaries may be ccnzputed by the same way as for the straight line boun- daries if the pivotal points of the relaxation pattern en- tirely conceide with the boundary contourline.It can be done by some approximation when pivotal points are pret- ty close to the boundary contour. The latter can be al- lowed if very high accuracy is not requested. when the values of the wanted function are prescribed on a curved boundary,an extra problem is presented. it A the function is already known, given or previous]: com- puted, but the distance a-A is some proper fraction of the tabular interval h.See figure B. , d ___._I I vn :‘ ‘. Figu‘vc {5. le‘ A—I Otherwise the additional treatment for conducting boun- dary is necessary. It is required to express a derivative, for example the second one, at the point a in toms of the fractional values at internal points. This can be done by producing the line to the plint o, where c-q is equal to the tabular interval h, expressing the value at o in terms - '/9 - of internal points and the known value at A, and substitu- ting for in the expression 2‘¢_ v v A Dy" "éflé‘25‘5‘14fl +534" Neglecting differences higher then the fourth in equati- on, the required expression for 2’ becomes :fif {—)+3(§j- 66-4)}1" +¢[‘[§]+9(:/] . /"(T)-+/-}:-}(Z gj-ztg) 21“} 7-(3‘j-+(>2)—71;)-+% FEM! mappina {30v com putaiton I an awn, \..° (P _ ...... ...... ”7...... ...... .__.,...T__ ......— ...y. . .\ l R l r I l l | I . L._. __._L.. FLgQYé ll (5.2 pagedi} Alamzu'cal méus if.» {Phi/'4' 74200 f 61 I e In. '5 75;? made IVameu'dae mace: often/1n? Ea Com/07227915 of 777 fan/11f m rate £33101. :5— [See pay 5553‘ s 57y Name YCCaZ [Nae/ass: 0266! 11/226 5% 0072/ mm of (cu/eff 7/7 r5612 figure, 225:: 1. 2. 3. 7. 8. 9. BIBLIOGRAPHY Barlov, 3., M., “Microwaves and Waveguides". 19147, Gen.- stable and Company LTD, London. Blckley, We. 3., “Formulae for Numerical Integraticn',Math. Gazette. l9ul, vol. 23. Cellatz, I... 'Eigenwertprobleme und ihre numerische Be- handlung', 1945, Akad. Verlag, Leipzig. entrant, B. , Advanced Methods in Applied Mathematics“, 1911'1, New Yorke Den Hartog, Jo. Po. “Mechanical Vibrations" Emmcns, 3., Wu “The Numerical Solution of Heat Conduction Preblems", Trans. American Society of Me- chanical Engineers. 1943, vol. 65. IrOCht, "a. He. and. ‘0va N" "a. “A Batienal Approach to the Numerical Se- lution of leplace's Equation". .7. of Ap- plied Physics, 1938, vol. 168. Green. Jo. Re. and Scuthvell, Re. '0. 'Preblems Relating te large Transverse Dis- placements of Thin llaetic Plates“. Phil. Truae Boy. 800e, “nan. 1943. "lo 23%0 Guillemin, leg ‘e. 'Oemmunicstien Networks“, vel.2,0hapt.TI. 19148. John Wiley do Sons Inc.. Rev York. lOe tam. van, He. “Lech und Sclitzkepplmngen swischen kcaxi- alen Leitungssystemen". Zeitschrift fuer angeWendte Physik, Qitober, 1950. 110 Ian-@1111!“ 10, Adams, 1'... Q... and. MBiklO. Ge, Se. “Flow of Heat Through Furnace Halls', Trans. American Electrochemical So- ciety, 1913, vol. 211'. 12. IneV‘ln, 1... “Advanced Theory of waveguides“, 1951. Lon- don. Iliffe and Sons LID. ' 130 ?011aczek. '0’ "Theory of the Coaxial Cables“, 3'. Phya.Ba- dium. 1947, vol. 8. 11‘". mm. TOD. ‘0. “Symetrisierung und Eranaformfiionen nit [unalleitungenfl Ielefunken Zeitung,Eeft 93, D00. 1951. 15. 39.1110, 8., and. “binary. :0. Be. ”Fields and Waves in Modern Radian", 1953, 2nd.ed.. John Wiley and Sons, New York. 16. 831.736.0131. no. Go. and. 331.011. M" In, “Numerical Methods in Engineering“. 1952, Prentice Ha11.Inc.. New York. 17. SChOJkUnOff, SO. ‘0. *"!he Electromagnetic Theory of Baxial Tras- mission Lines and. Cylindrical Shields', Bell.8ya.!echn.J.. vol. 13. 19311,. 18. Smerfeld, ‘0. “Partial Differential Equationsi in Physics“. 19249. NW York. 190 Bauthvell. Re. '0. - "Relaxation Methods in Eheoretieal Physics". 19%, Olarendon Press,0fiord, kndon. 20. Temple, GO. 'fhe General Theory of Relaxation Methods Applied to Linear Systems“. Proc. Roy. Soc. 1939, vol. 1691., London. ~82- 21. 23. 2h. 25. 26. 27. 28. 29. 30. m, 10". M Miller, 10,00, haracteristio nnpedanoe of Rectangular anal Transmission urea", Trans. of 5.3351952, 701071 Inont, He, Re, In, ‘I Have Guideal, 1950, New York, John Riley and 30118 Inc. 301101, In, Go, He, '1 Survey of the principles and Practice of mm Guidea", 19M, New York, The Macmllan Company. Anderson, 0., 3., " The Calculation of the Capacitance of Go- uial Cylindera or Reoatangular Gran-Sec- tion", Trans. 0: 1.1.8.3.,1950, 70]» 690 Meaainger, H., P., and Higgins, 1., J., IFomulae for the Inductance of coaxial Bua- eea Composed of Square Tubular Conductors", True Of LeIaEeEe, 19,46, 7010 65e King, Re, We, 15,1611», Be, Be, m, ‘0, He, fil'ransmiaaion Lineagntennae and Wave Guides", 1915, New Iork,McGraI- Hill. Brilloudn, "Theoretical study or Dielecytflc Cable", Electrical Communication, April, 1938. Hankel, Sidney, ‘ “Characteriatic Impedance of Parallel Wi- -rea in Rootangular Troughs", Proc. of IOROEO,19h-2, V01 30. Poritsky, 'Graphical Field Kiwi-a". Trans. of 1.1.3.3., .1938e, ”la 57e GW’ Re, “0360, and Southwell, Re, '0, "Conformal Transformation of a Region in Plano Space',Phil. Trans. of Royal Soc. of London, 1910, sect. A 238. -83- r} A O .’ 31. For, LO, "Mind Boundary Conditiona in the Relaxa- tione Treatment of Bihamonic problemsu , Proc. of Royal Soc. of London, A. 189. 32. Woods, H. ,W. , and Warlow-Daviea, "On the Application to Tubular Framework: of the Method of Syatanatic Relaxation of Constraintel,Aero Res. ctee. ~81.— No l 5" “TH ifixiflmjlflig’uju {111M @113 “[11 M W