L I B R.AR.Y OF THE UNIVERSITY Of ILLINOIS 621365 U655-te no. 3 1 ~3G cop. < The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN APR 1 8 11)76 MAR 2 8 1976 L161 — O-1096 Digitized by the Internet Archive in 2013 http://archive.org/details/dielectriccoated34week ANTENNA LABORATORY Technical Report No. J)h DIELECTRIC COATED SPHEROIDAL RADIATORS Walter L. Weeks* 12 September 1958 Contract AF33(6l6)-5220 Project No. 6 ( 7-4600 ) Task *K)572 Sponsored "by: WRIGHT AIR DEVELOPMENT CENTER Electrical Engineering; Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ^Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering at the University of Illinois, 1958° > ACKNOWLEDGMENT The author wishes to thank all of the members of the Antenna Laboratory staff for their help and encouragement, especially his advisor, Professor E.C. Jordan, P.E. Mayes and R.L. Carrel. Mr. T. Lahey also deserves special mention for his help in the preparation of the codes for the ILLIAC. Financial support for the investigations was provided by the Wright Air Development Center, under Contract AF 5220, for which the author is very grateful. ABSTRACT Analytical methods are presented for the determination of the changes in the radiation patterns brought about by dielectric coatings on spheroidal radiators. A description is given of a machine computation system which makes the analytical results useful. This system includes a program for the semiautomatic calculation of the radiation patterns of spherical radiators, a practical method for the determination of the eigenvalues of all of the orders of the spheroidal functions which are needed to obtain convergent representations, and a new method for the cal- culation of the spheroidal "radial" functions. Numerical results are included for spheres and prolate spheroids, having sizes of the order of several wavelengths, which are excited by axially symmetric slots. The effects of the dielectric coating are most marked when the excitation is latitudinally asymmetric. One typical effect is an increase in radiation in the general direction of the poles. For spherical rad- iators, the important effects show a complicated dependence on the size of the radiator and the value of the permittivity. Comparison of the patterns of the coated prolate spheroidal radiators herein calculated with the patterns of spherical radiators indicates that the effects are similar in both cases. ii CONTENTS Page 1. Introduction 1 2. Spheroids with Axially Symmetric Excitation 7 2.1 Symmetrically Excited Spheres 11 2.1.1 Boundary Value Problem of the Conducting Sphere (no coating) ik 2.1.2 Boundary Value Problem of the Coated Conducting Sphere 16 2.1.3 Numerical Computations Associated with the Spherical Radiator 22 2.2 Symmetrically Excited Prolate Spheroids 31 2.2.1 Conducting Prolate Spheroid without Dielectric Coating 3^ 2.2.2 Conducting Prolate Spheroid with Dielectric Coating 37 2.2.3 Numerical Calculations Associated with Prolate Spheroidal Radiators k2 3. Sphere with Arbitrary Source Configuration 59 3.1 The Slot Excited Conducting Sphere in Free Space 62 3.2 The Slot Excited- Conducting Sphere Coated with Dielectric 67 k. Summary and Recommendations for Further Work 73 References I36 Appendix A 138 Appendix B 1^6 iii ILLUSTRATIONS Figure Number Page 1 Basic Antenna 1 2 Coordinate System 7 3 Spherical Coordinate System 12 h Spherical Radiator 17 Dielectric Coated Spherical Radiator 17 6 Simplified Flow Diagram for Radiation Pattern Calculations (Spherical Radiator) 27 7 Prolate Spheroidal Coordinate System 32 8 Spheroidal Radiator 36 Spheroidal Radiator with Dielectric Coating 36 10 Simplified Flow Chart for Machine Calculation of Spheroidal V-Functions ^8 11 Simplified Flow Chart for Calculation of the Spheroidal U-Functions 52 12 Slot Excited Spherical Radiator with Dielectric Coating 66 13 Radiation Patterns, H^vs^t ©, of Spherical Antennas with Different Conducting Sphere Radii r. Excited by Equatorial Slot. Patterns on Left Are with No Coating. Patterns on Right Are with Coating, Thickness. 05X, £ r =3 Qi| l 1 *- Continuation of Figure 15 85 15 Radiation Patterns, H^ vs, 6 of Spherical Antennas with Different Conducting Sphere Radii, r. , Excited by Slot at l60° Latitude. Patterns on Left Are with No Coating, Patterns on Right Are with Coating. Thickness " .1X.£=2. 25 86 16 Radiation Patterns, "S.A vs &, of Spherical Antennas with Different Conducting Sphere Radii, r. , Excited by Slot at l60° Latitude. Patterns on Left Are 1 with no Coating. Patterns on Right AretWith COatihg. ' ThYckne~ss"\l?y., 87 Z = 2.25 ILLUSTRATIONS (CONTINUED) iv 17 Radiation Patterns, H^vs ©, of Coated Spherical Antenna, r/\ = 1.51, Coating Thickness ,1\, for Different Values of Dielectric Constant Slot at l6o D . 88 18 Radiation Patterns, H^> vs 0-, of Coated Spherical Antenna, r/\ = 1.51, Coating Thickness ,1k, for Different Values of Dielectric Constant Slot at l60°. 89 19 Radiation Patterns, Ed vs. 6, of Coated Spherical Antennas, r/X = 1.51, Dielectric Constant^ = 2.56, for Different Coating Thicknesses. Slot at l60°. 90 20 Radiation Patterns, K vs. 6, of Coated Spherical Antennas, r/X = 1.51, Dielectric Constant £= 2.56, for Different Coating Thicknesses. Slot at l60°. 91 21 Prolate Spheroid Radiation Patterns (One Half Symmetric Patterns) ^X = 3, £.="&/ 3, H^ vs. 6 92 Prolate Spheroid Radiation Patterns B H = 5, £ = \T8/3 for Different Slot Patterns, H^ vs. 6° r 95 23 Prolate Spheroid Radiation Patterns B I = 5, £ = \T8/3 for Different Slot Patterns, fflvs. 6 ° l 9k 24 Prolate Spheroidal Radiation Patterns, B £ = 8, R^vs. 95 Prolate Spheroidal Radiation Patterns, B X = 8, H^ vs. € 96 26 Prolate Spheroidal Radiation Patterns, B .£ = 5, 12, H^vs. 6 ° 97 27 Prolate Spheroidal Radiation Patterns, B j£ = 5, 12, H^vs. 0- ° 98 28 A Graph of the Quantity —- — — — = — ■ , which H^(Br) 2 n v ' Indicates Quasi-Resonances and Sets the Rate of Convergence in the Field Expansions in Spherical Coordinates. 99 29 [H ""(2jtr/\) J" 1 as a Function of n. 100 30 [H (2*r/X) ]" as a Function of n. 101 31 [$ '(2jc-r/\) ] as a Function of n. 102 32 [H ^nr/X.)]" 1 as a Function of n. 103 ILLUSTRATIONS (CONTINUED) 33 [H'(2jtr/X)] as a Function of n. 104 3^ [H'(2jtr/\)]~ as a Function of n. 105 35 [E^(2xr/\) V 1 as a Function of n. 10 6 36 [H^(2jrr/\) ]" as a Function of n. 107 37 [H'(23tr/\)]" 1 as a Function of n. 108 38 [H^jtr/x)]" 1 as a Function of n. 109 39 [E (2iCr/x)] as a Function of n. 110 kO Ratio of Coefficients, c. in the Field Expansion for the Coated Spherical Radiator to the Coefficients a in the Expansion for the Uncoated Spherical Radiator with the Same Excitation (g = r/x) Coating Thickness .1X6= 2.25 111 hi Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients a in the Expansion for the Uncoated Spherical Radiator with the Same Excitation (g = r/x) Coating Thickness .IX £ = 2.25 112 ^2 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients a in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/x) Coating Thickness .1X£.= 2.25 113 ^■3 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients a in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/x). Coating Thickness .IX £. = 2.25 11^ hk Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients a in the Expansion for the' Uncoated Spherical Radiator with the Same Excitation, (g = r/x). Coating Thickness .IX £..= 2,5 H5 ^5 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients a in the Expansion for the Uncoated Spherical Radiator with the Same Excitation (g = r/x) . Coating Thickness .IX £.= 2.25 116 vi ILLUSTRATIONS (CONTINUED) k6 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/\). Coating Thickness ,1\£ = 2.25 117 Vf Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator 11 with the Same Excitation- (g = r/\). Coating Thickness .IX £. = 2.25 118 48 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation. (g = r/\) . Coating Thickness .IX €. = 2.25 J -U9 ^9 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation* (g = r/\). Coating Thickness .IX £..= k 120 50 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/\) . Coating Thickness .1\ E.= k ' 121 51 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation* ( g = r/\) , Coating Thickness ,1\ £. = k 122 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/\) . Coating Thickness ,l\£,= k 123 53 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/\). Coating Thickness .l\£-=4 12k vii ILLUSTRATIONS (CONTINUED^ 54 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator 11 with the Same Excitation 5 (g - r/X) . Coating Thickness .L\ £ = 6.25 125 55 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation. (g = r/x) . Coating Thickness .IX £,= 6.25 126 56 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/x) . Coating Thickness .IX £.= 6,25 137 37 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation } (g = r/x) . Coating Thickness .IX £ = 6.25 128 58 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation, (g = r/x). Coating Thickness ,1\ £,= 6.25 129 59 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator 11 with the Same Excitation. £. = 2.25, r = 1.6. Thickness t/x Varied in Successive Graphs X 130 60 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation. £. = 2.25, r = 1.6. Thickness t/X Varied in Successive Graphs X 131 61 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation. L- = 2.25, r = 1.6. Thickness t/X Varied in Successive Graphs X 132 viii ILLUSTRATIONS (CONTINUED) 62 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation. £ = 2.25, £ = 1.6. Thickness t/\ Varied in Successive Graphs \ 135 63 Ratio of Coefficients, c , in the Field Expansion for the Coated Spherical Radiator to the Coefficients, a , in the Expansion for the Uncoated Spherical Radiator with the Same Excitation. £ = 2.25, £ = 1.6. Thickness t/\ Varied in Successive Graphs \ 13^- ix LIST OF TABLES Table Page Number 1. Scale Factors for Spheroidal Coordinate Systems 8 2. Facsimile of a Typical TLLIAC Print Out of the Spheroidal Functions, U 55 3. Comparison of the Coefficients Calculated from Different Numbers of Equations in the Coated Prolate Spheroid Problem 58 k. Excitation Coefficients for an Axially Symmetric Slot in a Spherical Radiator 77 5° Resonance and Convergence Factors for Spherical Radiators 'jQ 6. Numbers Which Indicate the Relative Importance of Dielectric Coatings on Spherical Radiators 79 7. Eigenvalues of the Spheroidal Wave Function 80 LIST OF SYMBOLS /\ In section 2, introduction, auxiliary wave function ^ r\ — JL r\ Section 2.1.3* expansion coefficient in field expansions for f\ spherical radiators Bn. Symbol introduced to save writing, see definition pg. ho Bj^ Symbol introduced to save writing, see definition pg. ^0 El Electric field intensity Applied field, presumed to be known on the slot, zero elsewhere — Electric field vector transverse to the radial coordinate in E T spherical coordinates E Applied field (known in the slot, zero elsewhere) E^ 1L — component of electric field intensity E-y- oj — component of electric field intensity C vx E^ in dielectric region E-irfr *-v- i- n f" ree space region H Magnetic field intensity /r* Quantity which shows the distant field variation with angle H

in dielectric region Haj Ha in free space region — : Magnetic field vector transverse to the radial coordinate in H T spherical coordinates K fJ S^H4> = i«-^« 4*- Symbol in coated sphere calculation, defined pgc 26 Xtvt Imaginary part of J n+ ^ Bessel function of the first kind, half integral order A, A *• J^ & *X. (p-r) except in formulas, pg. 26 £. Symbol for matrix in spheroidal function eigenvalue calculation LIST OF SYMBOLS (continued) • Elements of the matrix <£ . The determinant of the matrix '-n-r.. of these elements is equal to zero for the consistency of the set of equations for the expansion coefficients for the spheroidal functions N^ , Bessel function of the second kind, half integral order N' AM/P 1 ^ except in formulas, pg. 26 p Associated Legendre polynomial of order, \ , degree one r^ Associated Legendre function of the first kind Q v . Associated Legendre function of the second kind K Solution to radial differential equation in spherical coordinates Real part of Symbol in coated sphere calculation, defined pg. 26 Re SL b aL~> is element of area on the surface of a sphere y/v j* cos mcb "^ It ""n. * \ sm wC ^J where upper subscript means even and goes ° with the cos -m# j_ n brackets, and lower subscript means odd and goes with the sin m<£> in brackets. When both are present, the symbol implies that both even and odd solutions are being considered U Solutions to differential equation involving the coordinate -u- oj One independent solution to the differential equation in .... &■ prolate spheroidal coordinate, 'u. (analogous to J^ ) "(JL Second independent solution to the differential equation in 11 the prolate spheroidal coordinate ; -u. (analogous to N^ ) 'h Solution to the differential equation in the prolate spheroidal ^ 3 coordinate, u , which satisfies the radiation condition (analogous to {\ ) \J Solution to the differential equation which involves the coordinate v \ / Characteristic solution to the differential equation in the «■ coordinate, v , which satisfies the boundary conditions LIST OF SYMBOLS (continued) "If Excitation coefficients in the vector mode function expansions fc ° in spherical coordinates vv Wronskian determinant of radial functions 2 Bessel function of first, second or third kind, of half A *=- integral order £„ i^z^ a. A constant in the field expansions, to be determined a Constant in the field expansion in the problem of the uncoated sphere with axially symmetric excitation 0.. Constant in the field expansion in coated spheroid problem q. Section 2.2.3, expansion coefficient for the spheroidal radial function j Imaginary unit . Spherical Bessel function of first kind as used by Stratton, 3 U and Morse and Feshbach A Eigenvalue symbol JL Eigenvalue symbol in spherical coordinates n. Spherical Bessel function of second kind as used by Stratton, and Morse and Feshbach X. Semi-focal distance in prolate spheroidal coordinates ■f t Radius of conducting sphere tj_ Outside radius of spherical shell of dielectric A T Unit vector in r direction in spherical coordinates tx Coordinate symbol. In prolate spheroidal coordinates, it is the reciprocal of the eccentricity of the generating ellipse. IX, 11 -surface which coincides with the conducting spheroid 11. IX -surface which coincides with the outside of the spheroidal shell of dielectric y Coordinate surface symbol „ In prolate spheroidal coordinates, it is the reciprocal of the eccentricity of the generating hyperboloid 2 Any of the spherical Bessel functions as used by Stratton, and ' L Morse and Feshbach cX. Excitation coefficient in spherical problems cCl Excitation coefficient in spheroid problems £ Phase constant, ^ or Uj(i<£) 3 ' B, P for dielectric region 6. (3 for free space region LIST OF SYMBOLS (continued) \j, Kronecker delta; equals one if L = A , zero otherwise £. Permittivity or dielectric constant B^ E for dielectric region £„ £. for free space "X Wavelength aj Permeability XJI Constant in the field expansion for uncoated prolate spheroid UJ Frequency times att Note: In section 3, the quantities associated with TM fields are denoted by single primes; the quantities associated with TE fields are denoted by double primes, mimun i i See note page 100 1. INTRODUCTION The object of this work is to provide at least a partial answer to the following question; What changes in the radiation pattern of a moderately large antenna are brought about in the event that for some reason, either within or beyond the control of the designer, the structure is given a coating of dielectric material? The specific antennas considered are conducting bodies of the spheroid family, excited by slots or gaps. In the pages which follow, an analytical method is presented for the determination of the changes in the radiation patterns brought about by spheroidal dielectric casings on spheroidal radiators. The basic geometry is indicated in Fig. 1. Slots for Excitation Conducting Spheroid Dielectric Coating FIGURE 1. BASIC ANTENNA It is shown herein that numerical results can be obtained, in some cases rather easily, with the aid of a digital computer, and some numerical and graphical results so obtained are included. The impetus for the study was provided by the fact that the shapes included in the spheroid family provide reasonable approximations to certain space vehicles or missiles, some of which may be given coatings of (nonconducting) refractory material in order to withstand the temperatures developed by tremendous speeds through the atmosphere. But this is by no means the only interesting aspect to the problem. The effects of dielectric coatings on cylinders and planes have been the subject of much study in recent years in connection with the propagation and launching of surface waves, and there is the interesting question of a possible connection or relationship between the effects in spheroidal geometries and those in plane and cylindrical geometries. Moreover, the general problem is of interest to the antenna engineer, since the effects of dielectric coatings on the one hand introduce new possibilities for the control of radiation and on the other hand may present difficulties to him when the coatings (such as ice or radomes) are beyond his control. The spheroidal radiators are particularly attractive for a study of this kind since they are the only antennas of finite dimensions for which exact analytical formulations of the electromagnetic boundary value problems are possible (at least at the present state of knowledge). In spite of this, there has been suprisingly little published work concerning dielectric coated spheroids . The problem of the scattering \ of a plane wave by a dielectric coated sphere having a size near that of the first resonance has been studied by Scharfman and Aden and 2 3 Kerker. Elliot has discussed the possibility of surface waves on a dielectric coated hemisphere set on a perfectly absorbing pad. As far as is known to the author, there are no previous publications on the subject of the dielectric coated prolate (or oblate) spheroids. The method employed in the present work is the classical method for the solution of electromagnetic boundary value problems . The fundamental idea is that an electromagnetic field is uniquely determined in a source free interior region by the values of the tangential components of electric (or magnetic) fields on the bounding surfaces of that region* In the antenna problem, the space exterior to the antenna is effectively converted into an interior region by the device of regarding space as bounded at one extreme by the antenna surface and at the other extreme by a large enclosing surface located at a distance from the antenna which is so great that the fields can safely be assumed to be those of outward traveling waves ° In the present problem, this exterior (rendered interior) space is not homogeneous (viz., it is partly filled with dielectric material). As a result, since solutions are easily obtained only in homogeneous regions, the problem is handled by splitting the complete region into two homogeneous regions and obtaining solutions for each of these regions by requiring that the tangential components of the electric and magnetic, fields be continuous at the boundary between the homogeneous regions* In outline the solution proceeds as follows: Maxwell's equations for the electro- magnetic field are first postulated., These vector partial differential equations are transformed to ordinary scalar differential equations by introducing suitable potential functions and/or by taking advantage of known symmetries. The ordinary differential equations so obtained are of the Sturm-Li ouvllle type, and as such, with the appropriate boundary conditions, define characteristic functions and characteristic numbers (eigenvalues) in terms of which the solutions can be expressed. The fields are expressed in terms of these functions multiplied by (at first) unknown constants. Fortunately, in the boundary value problems, the number of independent boundary conditions which can be applied is exactly equal to the number of linear equations which is required for the determination of the unknown constants. Unfortunately, however, the number of equations which must be solved is seldom small. In the sections which follow, the foregoing outline is developed in detail. The organization is such that the problems are separated first according to whether or not the fields have rotational symmetry, and then into the specific coordinate systems. In Chapter 2, the Maxwell field equations in a general coordinate system having rota- tional symmetry are considered, and it is shown that the partial differential equations can be separated in spheroidal coordinate systems. The conditions for separability are exhibited. Following this, general expressions for the field components in such systems are developed and, in Section 2.1, these expressions are applied to spheres with rotation- ally symmetric excitation (latitudinal slots). Section 2.1.1 includes a review of the solution for the radiated field of a spherical antenna in order to standardize the notation. The solutions for a dielectric coated spherical radiator are presented in Section 2.1.2. Also in that section, an expression is developed which makes it possible to evaluate the effect of a dielectric coating, independent of the type of (rotationally symmetric) excitation. It is interesting to note that because of the ease with which the different orders of the characteristic functions of spherical coordinates (spherical Bessel functions and associated Legendre functions) can be generated, the sphere calculation is almost ideally suited for a digital computer* Almost all of the results included herein were obtained with the aid of the University of Illinois Digital Computer (ILLIAC). The system of calculation is discussed in Section 2.1.3. In Section 2.2, the general equations which are developed in the first part of Chapter 2 are applied to the prolate spheroid. The problem of the uncoated prolate spheroid is reviewed for the purpose of standard- izing the notation. In Section 2.2.2, the solutions for the dielectric coated prolate spheroid are presented. A description of a way in which the spheroid calculations can be programmed for a digital computer is included in Section 2.2, 3* together with the radiation patterns for a few specific spheroids. Most of the results are new in that they are based on function values which heretofore had not been tabulated . The system of calculation for the spheroidal functions is different from that which has been reported before and it appears to have some advantages. Chapter 3 is devoted to the development of the equations for spherical ' antennas with arbitrary (known) excitation — that is, the restriction of rotational symmetry is removed. The equations for both coated and un- coated spheres are developed, and it is shown that the ratio of the coefficients in the series expansions for the TM fields is the same as that developed in Section 2 for the case of rotationally symmetric excitation, The results included in the following pages show that the effect of a dielectric coating on a spherical radiator follows a rather typical pattern, the net result usually "being an enhancement of the relative magnitudes of the higher order Legendre polynomials in the representations. One effect then is the introduction into the patterns of coated spherical antenna of a given size some of the characteristics of the patterns of an uncoated antenna of somewhat larger size, but there are other more interesting effects. The solutions for the prolate spheroid are somewhat less satisfactory in that the calculations require a relatively large amount of computer time, but more than this, they seem to be too involved to provide insight into the physical processes involved. The results ind: that of spherical antennas. We now turn to the task of developing the equations which apply to general orthogonal coordinate systems in the special case that the fields have rotational symmetry. 2. SPHEROIDS WITH AXIALLY SYMMETRIC EXCITATION In the special case that the sources of the field (and therefore the field itself) have no variation with the coordinate (fi (see Fig. 2), k Maxwell's equations expressed in rotationally symmetric coordinate systems (V'>' ir )4 > ) reduce to (assuming e time variation) the equations ■a-u. — a uJA\ Jw Xv MA (1) (2) (3) together with a similar independent set which can he written down "by inter- changing U and r I while exchanging the permittivity and permeahility FIGURE 2 COORDINATE SYSTEM TABLE 1. SCALE FACTORS FOR SPHEROIDAL COORDINATE SYSTEMS (u,v,(j>) Scale factors and related quantities Coordinate System spherical prolate oblate (r,(-cos 9), (j)) spheroidal spheroidal 2 2 1/2 SL (— ■* ) i (- u - 1 2 2 1/2 u 2 + l d-v 2 ) 1 ^ 2 2 1 - V 2 2 1/2 1 - V '(1 - Y 2 ) l/2 I ( (U 2 -1)(1-V 2 ) ) X / 2 | ( (U 2 + 1)(1-V 2 ) )^ h* h h u 9 V h h h. u v f x (u) f 2 (v) 1-v i(l-v 2 ) i(l-v 2 ) 1 1 i (u 2 -D i(u 2 + l) 3/ 2 2 A i (u -v ) iW) 3 2 i u A 2 3 2 -rv 3 2 i v 9 symbols according to the rule & —— M . The lk>L 's are scale factors of the coordinate system and E A . and H^ represent the components of electric and magnetic fields in the coordinate system (."U-.'ir., (fr) . If a new variable, r\=^%v\ t is introduced, the Eqs. 1, 2, and 5 can be combined into the equation 1-u. ^^ "d**- 3 V ^^v "Bit X<|> Inspection of the table of scale factors (Table I) discloses that in -n'Tr all of the spheroidal coordinate systems, the quantity ( ~n — ^ — ) is independent of 1L , while the quantity ( . -r ) is independent of if , so that Eq. h can be written ^ 'S-tA. 3 - ^v Sir* Assuming that A has the separable form A = U^OVCir) ? it is possible to write the equation in the form V U inr V (5) As before, a glance at Table I shows that the scale factors depend on the coordinates in such a way that Jk^X^Jkfy — -J-, Cu") ■+- ^Cv) (and as before ( JtA* ) is a function of ^l only while ( j^j^ ) is a function of V only) so that the partial differential equation separates into the two ordinary differential equations : 10 ^U U + (j/^uj^^a -k)\J = o V = o (6) (7) where -& is the separation constant. Because of the symmetry of the source distributions which have been postulated, the v~ — and cj) - com- ponents of the fields must be zero on the z-axis. For spheroids, these field components are therefore zero at if — ± 1 , independent of tjl It follows from the identity with Eq. 2, that these field components involve the V -functions directly, and therefore the following conditions on the functions are implied : K = , v = ± 1 . Equation 7 with these boundary conditions constitutes a Sturm-Liouville system, for which it is known that the solutions can be represented by characteristic functions having eigenvalues M.- h which give non-zero solutions of Eq. 7- The general solution to Eq. 6 can be written Lit , and (Ai where Uo^, and (A^ are the two linearly independent solutions. It follows therefore that any solutions to the field equations which meet the specified conditions of symmetry and continuity can be written as follows: 11 The electric field components can "be found by substituting Eq. 8 in Eqs. 1 and 2. The result is (9) E v = - — l ILcq^VIU, -t-^^&oVfe . (io- Having thus derived the expressions for the field components in the general spheroidal system, we can now proceed to the formulation of the boundary value problems for particular classes of spheroids. 2.1 Symmetrically Excited Spheres Among the spheroids, the sphere is usually the easiest to study be- cause of the familiarity of the functions which are involved. This study will therefore begin with the sphere and, in fact, will present more nu- merical details for it than f or v the other spheroids. The coordinate system selected is that one in which U. — ~t~ , v-=— cos n . i.e. ( nT, — c/>s©>4 This selection, in place of the usual ( -f, G>

i>R' c-^u/Ve -1 S )R = (ii) and i-v*)V" +^V =o . (12) Although it seems a pity to do so, Eq. 12 will he transformed into the more complicated hut more familiar associated Legendre equation by means of the substitution V s — Cl-v x ) \ y as follows: (1 -^P" - av p' + (4- ) P =o (l—v*) The functions which satisfy the differential equation, plus the condition V 3 '~ D — u , are therefore the associated Legendre function of degree one, with the eigenvalues Jfl<, equal to "YiCtu + D. With these eigenvalues, Eq. 11 is recognized to be a Bessel-type differ- 7 ential equation which has solutions of the form R = o,^^^ -+■ A^N^pf) ( he - tU^+I) identified as Mjg; - 1 , so -f - * ^ + i , where p is a. "+ the function order ) (3 = U) >m,£ . These functions can now he inserted into the general Eqs. 8 and 10 for the field components which are tangent to the spherical surfaces In doing so, it is convenient to use the notation of Schelkunoff and introduce a radial function *—v^ X /L —n.+.L 3-> where ZL^^^ is any one of the three kinds of Bessel functions. In terms of these functions, the equations for the tangential field components become and from Eq. 2 where A r- az . Ik 2.1,1 Boundary Value Problem of the Conducting Sphere (no coating) Unique solutions of boundary value problems in regions exterior to a spherical surface are obtained from the values of tangential L (or (— | ) over the spherical surface, together with the radiation condition at a larger spherical surface whose radius tends to infinity. Since the radiation condition implies that the fields are propagating in the di- rection of increasing -f , we note from Eq. 13 that the coefficients O-n. and Jcr r ^ must be related so that Jr l = -a Oj^ , in order to give the proper function to represent the outward traveling waves , A £_ A H„P> . (16) The coefficients in these expansion are determined by requiring that the series expansion Eq. l6 represent the electric field at the surface of the conducting sphere ( ■+■ — -fc ) * This latter electric field is of course zero except over the axially symmetric slots. The field in the slot is presumed "to be known, and if it is represented in the form 15 ) ,*_ P« . (17) the orthogonality properties of the Legendre functions make it possible to obtain the following equation for the coefficients : = jL l R. 1 a SIOt3 ' CJL f^ where TcPjAir (18) X Ay > H^^V) (2D In the calculation of the distant field, the asymptotic form for H.^f 3 can be used: H w x— *-©° e 16 Thus, the expression for the calculation of the radiated field is To display the variation of the field with angle 9 (the pattern), only the quantities to the right of the summation sign are of interest This much of equation (22) was programmed for calculation by the ILLIAC. Some of the details of the solution by the digital computer are presented in Section 2.1.3- In a desk calculation, the "book Tables of Spherical Bessel Functions can be used to advantage. Therein, the function Z^Cx) = C-^-T" zi^^is tabulated. With these tables, the following relationships are helpful: Z^P-t) = (5+- zjtprt A <£+■ 2.1.2 Boundary Value Problem of the Coated Conducting Sphere The equation for the distant field intensity is basically the same whether or not the radiator is dielectric coated. However, the coefficients in the field expansions are of course different and these are obtained from the solution of the boundary value problem. One measure of the effect of a 17 E v - FIGURE k SPHERICAL RADIATOR Region I Region II Conducting Sphere Dielectric Spherical Shell FIGURE 5 DIELECTRIC COATED SPHERICAL RADIATOR 18 dielectric coating on the surface of a sphere can be obtained by comparing the coefficients, as given by Eq. 20 with the corresponding coefficients in the coated problem. This is done in the present section. When the sphere is dielectric coated, we must recognize two regions: a region (i) consisting of the spherical shell of dielectric surrounding the conducting sphere, and a region (il) consisting of the remainder of the exterior space. In region (i), the Eqs. 13 and 1^ apply directly. In region (il), the equations have the same form as Eqs. 15 and l6, which can be rewritten as follows: l-L =AVc r H'.rvoP>) (23) Again, the series expansion for t_ , evaluated at -T" = "tj , must be equal to the series expansion for the applied field at that surface, which latter is taken to be that of Eq. 17. Equating the coefficients in these two expansions, term by term, gives the following relationship among the coefficients : a a f a^J.,'/.fU> 4- J^N_,./fv^ =-^£i+,«*o o Thus, we find an equation for the expansion coefficients in region (il) as follows ; A A /\ /\ 20 where A^= o A A A A A A The quantity in traces in Eq. 28 is the Wronskian determinant of /\ A ( J IN^ ) • 1^ is quantity can he determined from the asymptotic solutions of Eq. 11. The result is With this result, the equation for the distant field coefficients takes a simpler form as follows t (29) ^ = -4^^^ A. (30) Now a measure of the effect on the radiation pattern of the dielectric layer on a conducting sphere may be found from studying the ratio of the coefficients, (X^_ (Eq. 20), which apply to the uncoated sphere to the coefficients, C^, (Eq. 30), in the situation that the conducting sphere has the same size and the same excitation hut has a dielectric layer. The ratio is as follows ; cu. '""' i ftl-Ccp.*,) (3D ^ ' ' ** A^ (32) 21 It is interesting to study these ratios in limiting cases. For example, if "both spherical surfaces are large enough so that the Bessel functions can be replaced by their asymptotic forms for large arguments, i.e., A J^ > cos Cp-r-{c*t+xN A e re— » r the result, after a fair amount of rearrangement, can be put into the follow- ing form; c^ ~a^ e . ^,-**\ a. ° sivu p.c-r,.-^,) — ^ ^ cos ftto-r.) It is noteworthy that this ratio is independent of n. This indicates that for such large spheres, the dielectric coating will not influence the pattern variation with angle, but the approximations may not be valid for the highest orders needed in this representation. On the other hand, if ( (3>N~ ) is very small, we can take as approxi- mations the first terms in the power series representations : -r\. C "XT** -*" 1^ I n™ = - C3.Tb)j V -■n. 22 Then, after a fair amount of algebra, the ratio of coefficients becomes For such small spheres, only the term having tl equal to unity is important : a, -L a. C, 3£ r ^F + i^ 1 "^ 1 ] It should be mentioned here that there is a danger of making an invalid observation based on the latter equation. This equation applies in the event that the electric field is the same in both the coated and uncoated state. Normally in practice the constant would be the voltage across the gap, in which case the electric field would be less with the dielectric present. For the intermediate (and more interesting) range of values of (S-r- the ratio must be studied numerically. This was done and the results are discussed in Section 2.1.3. 2.1.3 Numerical Computations Associated with the Spherical Radiator Because of the relative ease with which different orders of the spherical functions can be generated, the sphere calculation is almost ideally suited to a digital computer. The functions which are involved (spherical Bessel functions and associated Legendre polynomials) satisfy very convenient recurrence relationships as follows : 23 A and ? CjO = 3.^+1 7 (x) " 7 <*-> z,>> ■= ^ z>> - z^ x =^^z^-^z r ^l P 1 (%) - ±- \ ca^H-^^P 1 ^) -(ti + i^PV)! Thus, given the two lowest orders of the functions (which involve only sines and cosines), the higher orders are easily generated in a digital computer "loop". The calculations described herein were "based on these recurrence relationships - Qeneration of the functions was done entirely by the machine. With Legendre polynomials and the spherical Bessel functions of the A second kind ( j\J ) this upward recurrence process can apparently he carried out to almost indefinitely high orders of the functions without serious loss of accuracy. However, in the calculation of the spherical Bessel functions of the first kind u^ , there is a serious loss of accuracy if carried out in a straightforward upward recurrence process. In fact, the figures begin to lose significance for those orders ( ru ) which exceed approximately twice the argument -- and this is approximately the number of terms re- quired for the series expansions for the fields to converge satisfactorily. Hence, an alternative system is employed for these functions. The relation- ship, T _ aTl-h i . 7 _ 7 2k is transformed into a continued fraction form for use in a downward scheme as follows : ^j-A-t-i — a.n-t-1 L-li. _ ^ i* X, J, — 7v T .Jtl . ^+1 I 1 a.-n. + i 1 QjfL+3 1 ^ \ \ \ \ 3.(TL+m)-l JC J, n+m-i /A Now if -w\. is taken sufficiently large, the ratio J^+^w/ J^ +Yrv . negligihle in comparison to the term a, (n+m -!)/;& ("because of the nature of the functions of high order and relatively small argument), and hence a good approximation for this ratio is zero. With this approximation, the ratio I is first calculated and then employed to cal- culate the lower order ratio. This process is continued so as to calculate 25 and store the ratios of successive orders of functions from some high order to some low order (say n - 1 ) for which the value of the function in the A denominator is readily obtained ( JT(v.) — sin.% ). The functions them- selves are obtained from the ratios by successive multiplications. When generated in this way, all except the highest several orders calculated can have full machine accuracy. However, the method breaks down if the argu- ment coincides exactly with a zero in one of the (lower) orders. Thus, it is safest to use the upward recurrence form in the oscillating region of the functions and the continued fractions scheme in the monotone region. An alternative is the use of the recurrence form in the downward sense. This latter method is satisfactory except for the arguments which correspond to the zeros of the lowest order function computed (integral mutliples of 2jt A if the process is carried to Jo to )• Except for this difficulty with the J-a, functions, a completely auto- matic digital computer program could easily be written, in that the decision as to the number of terms to be computed could be made by the machine (by a convergence test). The actual program, described below, which was written for the ILLIAC is semi-automatic in the sense that the programmer makes a guess concerning the number of terms required for convergence, and on this A basis, a sufficient number of J^ functions is made available. If the guess which is made is too low, the program stops on a special stop order. (The number of terms required appears to be of the order of twice the argu- ment of the Bessel functions ( fi"f ) or perhaps ten times the radius in wavelengths. Actually, orders of the functions as high as sixteen times the radius in wavelengths were computed in the program, and this number never 26 failed to provide a safe margin). In the calculation of the radiation patterns, the constants and the ■^ variation in Eqs. 22 and 23 are suppressed and an expression having the form y- = yl r kp: is evaluated. The quantity A^ is of course a complex number which is evaluated as follows, in the uncoated and coated problems respectively: A A JT + re " where and in which Calculate n = l6r- orders of functions A n Set pattern angle to be calculated equal to , 2n+l (a n = 2n(n + l); E a P n J (v)dv) zero V lno t Form sum of Calculate Increment pattern angle to be calcu- lated Is four terms in orders 1 + ki S i -s n+4 n S n < e ? series of a) Re H b) Im H of Legendre polynomials 1 (both Re H Tno and Im H) yes Print angle, print Re -H, Im H and number of terms required for convergence Form ((R H?+ (Im nf) and store Are all pattern angles calculated? 27 yes STOP or change parameters for new calcula- tion Display J H J vs . angle on CRT Pick out the largest in set of (Re if + Im H 2 )i Divide by Largest. FIGURE 6 SIMPLIFIED FLOW DIAGRAM FOR RADIATION PATTERN CALCULATIONS (SPHERICAL RADIATOR) 28 A simplified flow diagram for the machine calculation of the radiation patterns is presented in Fig. 6. The computer program is written in such a way that for a given angle in the field, additional terms in the summation (on tl ) are obtained in groups of four. This summation process is carried on to higher and higher orders until a comparison with the cumulative sum shows that the last four terms computed is satisfactorily small in com- parison to the cumulative sum. The excitation coefficients ( oC^ ) are obtained by a five point Simpson's rule integration process on the assumption that the electric field is constant over a slot whose width subtends two degrees of latitude. The machine time required (using a floating point interpretive routine) for a single pattern calculation involving thirty- five points in the field is about five minutes (for greater accuracy than is required for a cathode ray tube display). If several patterns are computed by varying the parameters while the program is still in the machine, the average time per pattern is less. Some of the radiation patterns which were so calculated are displayed in Figs. 13 to 20. The radiation patterns of a given spherical radiator with and without a dielectric coating are presented side by side for con- trast. The number of radiation patterns included herein is not large, since it appears to be ultimately more useful and more economical to present certain key data pertaining to spherical radiators, from which additional c&lculations can be made with a minimum of effort. Thus, Table k gives the excitation coefficients (calculated as mentioned above in the hope that they will be useful in future impedance calculations) for slots whose posi- tion varies from ten to ninety degrees latitude. Another quantity of 29 interest is | rl^fvnl } for the uncoated radiator. The square of this quantity is plotted as a function of ~^/\ for a given Tl in Fig. 28:. The maxima in these curves show the sizes for which the sphere becomes qua si -re sonant for the various orders of the functions. The same quantity as a function of "Tx. , for different values of P ^ is given in Figs. 29 A / I A i i ^. A ' I ^Vl a to 39, and in Table 5 with the quantities so that the surfaces u = constant become spherical, the angle becomes the usual polar angle of spherical coordinates The quantity JL is the semi-focal distance. 52 . 259 -f .8G<2> FIGURE 7 PROLATE SPHEROIDAL COORDINATE SYSTEM 33 With this identification of coordinate variables, the results of the analysis presented at the beginning of Section 2 are immediately appli- cable. It is clear from the results in Table I, and Eqs . 6 and 7, that the differential equations which apply to the prolate spheroidal system are as follows : Ctjl* -dU" ■+ < K K^ ^ ^ncuf-^ci-^)) 1 35 It follows from the boundary conditions and Eq« 37 that the statement of orthogonality among the V -functions is as follows : J_ x (l-v l ) O if A- *: TL so that E, c-^-v 1 ^ V. A- Js/o ^ — _ 1 ::' LA-y±± ^-1 ci-v*) (42) The actual evaluation of the coefficients is done in terms of the series (38) and (kO) so that cr ft *. - _ ^^ YL Kr.Lr E^C^-V*^ f^ 1 A- .v In the numerical calculation of the \J -functions, there is an arbitrary constant to be chosen. It appears from the latter equation that a very convenient choice would be one which leads to the following equation: y k, f < p;^ ^ = y <& ™™* = 1 The arbitrary constant is so chosen in the calculation described in Section 2.2.3. Also, in the calculation of the distant field, it is convenient to have a familiar asymptotic form for the U -functions as (31 u — * (3l~-><=> The arbitrary constant in the Bessel function series for these functions /KN** 56 FIGURE 8 SPHEROIDAL RADIATOR Dielectric Region (I) (€ 1 ) Region (II) (€ Q ) FIGURE 9 SPHEROIDAL RADIATOR WITH DIELECTRIC COATING 37 is chosen so that A that is, the (A -function is made asymptotically equal to the corre- sponding order of spherical Bessel function. With these normalizations, the equation for calculating the dis- tant field becomes 2.2.2 Conducting Prolate Spheroid vith Dielectric Coating Let the conducting spheroid coincide with the surface "U — U a , and let the dielectric coating extend from the surface Ik — u t to the surface "U. ^^ . The dielectric region will he designated (i) and the remainder of exterior space will he designated (il). Differential equations of the type (36) and (37) apply to both regions; however, the propagation constants and the eigenvalues are different for the two regions. This complicates the boundary value problem since the eigen- f unctions on one side of the boundary are different from those on the other side of the boundary, which means that the coefficients cannot be equated term by term. In region (i), the equations for the tangential components are the same as (8) and (10), with £ = £j. , (*3) 38 E„ =- [c^-^)(i-V)] i 7[< ^ ^ + J^ AT^^Vt . (lop) In region (II), the equations are of the type of (\o) and ( I4-1) ; however, in writing the expressions, the eigenvalue symbol, Jk , will be re- placed by the letter -j , to underscore the fact that the functions are different in the two regions : o ^ u;£ o r ' ' ^ 1^'" ' (laa) The field is presumed to be known over the inner (conducting) Spheroidal surface, xl± , (the tangential electric field is zero except over the slots) and is expanded in a series of V -functions : E = f^ = - (c^-^)u-^)] "" V ^\4 The coefficients, otj^ , are determined in the usual way by a multiplication by V- (l-^r 1 , with the result: oC A - d ^-1 Ci-v 1 ) 39 The actual computation of these coefficients can be done with Legendre polynomials and (X -coefficients as indicated in Section 2.2.1, under Eq. k2. Thus, the application of the "boundary condition at the surface U! leads to a set of -& equations in the unknown coefficients cl^ and -&I?. , as follows i TZ Av. (w ° Two more sets of equations are obtained from the boundary conditions at the surface lx =^^ • The condition on the magnetic field gives the result ? while the condition on the electric field gives the equation Equation 1+4 can be employed to remove the variable, Xfj^ , from Eqs. k-5 and k6 as follows : XL** 1;' c/MiO ci-v»>* j ci-v*)» UL'i 40 The quantities V^ and \A' can be replaced as follows: The reason for the latter step is that now the corresponding orders of the Legendre polynomials can he equated term by term, whereas this is not possible for the V -functions. When this is done, and the substitutions are introduced to save writing, the result is in These latter equations, (kj) and (kQ) , must be solved for the unknown coefficients C^ (and CX.^ ). The equations are valid for each value of yv. , so that each of the equations can be written down the number of times which is required to solve for the Jk -4- A unknown coefficients. Of course, the solutions to the equations are strictly correct only for an infinite set. However, the series expansion of the fields, and those for the V -functions, are presumed to be convergent representations. Hence, it will be permissible in an actual computation to replace the infinite sums by sums having a finite number of terms. Then, the number of equa- tions which must be handled is, of course, at least twice the number of terms, N, which are required for a sufficiently accurate distant field representation. The fact that the ^^ -coefficients peak rather sharply about the value, o-jkJk, , (i.e. the \J -function of a given order closely resembles the Legendre polynomial of the corresponding order) means that a finite set of equations can be essentially decoupled from the in- finite set, and this finite set may not be appreciably greater than the minimum number required (2N). Hence, in practice, one can start with some small number of equations (say a number of equal to three times (31 ), obtain the solutions, then increase the number successively in steps of four and repeat until the following two conditions are satisfied: l) The co-, efficients determined do not change appreicably as the number of equations is increased and 2) the magnitude of the highest order coefficients deter- mined are some small percentage of the largest (the latter condition will guarantee the former, but the largest coefficient may sometimes be hard to find). k2 It is worthy of special mention that the odd orders and the even orders are completely decoupled, and this good fortune cuts the number of simultaneous equations in a set in half. However, the O-a. } s and C, ' s (and perhaps the °Cfc '« ) are complex numbers, and the necessity for equating the real parts and the imaginary parts separately doubles the number again. Once the C^ ' s have been determined, Eq. 4la can be used to deter- mine the distant field, wherein the U, 3 -function is replaced by its asymptotic form, as was done in Eq. k^. 2.2.3 Numerical Calculations Associated with Prolate Spheroidal Radiators There are two very serious obstacles to the carrying out of the numerical calculations for spheroidal radiators The first is the sparsity of tabulated values of the functions (especially for large values of (3 A , as well as the lack of a thorough study of the properties of the functions), and the second is the difficulty of handling the large systems of equations which arise in the coated spheroid problem. Both of these obstacles can be surmounted with the aid of a large scale digital computer. The next few pages describe briefly the system by which the spheroidal functions, V^ and U^ were calculated in connection with the present work. The similarity of the differential equation, (37), for the V -functions to the corresponding equation in spherical coordinates, (12), suggests that the V -functions can be determined from a series of the spherical functions, Thus, since the V5 functions which are the solutions to (12) k3 form a complete orthogonal set, it is possible to represent the spheroidal functions in a series as follows : CO \=Y, **~^* • w-1 If this series is substituted into Eq. 37, and the necessary conditions of convergence assumed, the result is TV. "W. Equation 12, with Je s — yiCyi+1), can be multiplied by constants c\fey^ and the results added together to give the equation From these latter two equations, the following result can be obtained: V 2_, K-, 1 K - *^ + « -\ +5) = This is the result which was stated earlier in Eq. 39 These relation- ships can be employed to determine the coefficients, once the eigen- values, J^k } have been determined. In the eigenvalue calculation, for practical reasons which will become clear below, it is helpful to employ normalized Legendre poly- nomials. The integrals are evaluated in the same way as that indicated above for the unnormalized functions. The equations which are so ob- tained can be symbolized as follows : YL a ^l Y\- ' rvmrv (51) or by the matrix equation in which jyz st-n+3 ■m' — "n'-i Lv,^ - ° > otherwise 1+6 The conditions for the consistency of the set of Eqs. 51 is the vanishing of the determinant of the coefficients as follows : = o (52) The eigenvalues, Jkj^ , are those numbers for which Eq. 52 is satisfied. The problem of eigenvalue determination is thus in this case equivalent to the problem of finding the eigenvalues of a symmetric (infinite) matrix. The main object in developing the spheroidal functions in the fashion described is to take advantage of the service routines in the ILLIAC library which are capable of finding the eigenvalues of symmetric matrices as large as 128 x 128, with accuracy of 10 or 11 decimal places. The fact (as is shown in Appendix A) that the foregoing relationships have stationary properties, together with the potential sizes of matrices which can be handled, makes it possible to determine the eigen- values of nearly all interesting orders of the functions in a re- latively short time. In the actual program, the equations (and therefore the eigenvalues) were scaled by ( (6J0 ), and a routine was written which generates the matrix elements in the form required by the matrix eigenvalue service routine. It is worthy of special mention that the equations (coefficients) having TO. odd can be handled independently from those having 7\. even, which means, for example, that to obtain a set of X eigenvalues, one needs to handle only /^ order matrices. The eigenvalues for (B>! — 5,8, and 12 are presented in Table 7- The effect of matrix size was also in- vestigated, with the surprising result that for £1= 8 , for example, 1+7 except for the last five or six values, a 23 x 23 matrix gives results which agree with those given by a to x to matrix. With the eigenvalues so determined , Eq 39 can be employed to find the coefficients. However, as with the Bessel functions of the first kind, (see Section 2.1.3), a straight upward use of the recursion form is not generally practicable. In the programs for this work, the coefficients for the lower order functions were evaluated by a downward continued fractions technique, the form of which can be obtained by rewriting Eq. 39 in a fashion similar to that described in Section 2.1.3 in connection A. with the u-^ functions. For the higher order functions, this scheme was employed for those coefficients <5-j^ in the range "A. = JfL . Then a switch was made and the remaining coefficients A^^ «> 1 — Vi — Jk. , were calculated from the recurrence formulas in a straight upward fashion (this double-barreled approach proved to be absolutely essential in the calculation of the. higher order functions). The coefficients were all generated on the assumption that &k!k~ ^" > an Q (Publications by Schmid imply that such a representation may be mathe - matically proper.) Assuming the necessary conditions of convergence, this series is substituted into Eq. 36. The properties of the function as dictated by Eq. 11 are employed to reduce the result to the statement 50 With a repeated use of the expression for the derivative function, the term involving the second derivative can he replaced, and Eq. 5 4 put into the form OO A A <=ao A, A If now the coefficients of the terms, J^ , are individually put equal to zero, the resulting expressions can he put into a form as follows: Ca.-n.-3X sen- 1) H- (55) Note the similarity of this relationship to that of Eq. 59- The simi- larity is such that if the condition CL^ — <&fejt were imposed, the magni- tudes of the remaining coefficients would he equal. (The signs would he such that and so on > alternating.) As with the coefficients, if Jk is odd (even), only the odd (even) co- efficients are non-zero. 51 A glance at the first term in (55) shows that the positive side of the recurrence chain breaks off at Y\.= l and >x= a, , while from the last term it is clear that the negative side of the recurrence chain "breaks off at Y\. = -a. and ~a=-3 . As a result, the value of CL^ corresponding to one of these particular four values can be assigned, and on the basis of this assignment, a positive (negative) side, odd (even) set of coefficients can be determined. Furthermore, if it happens that Q-j^i. = ^M.-^ * °^, a,— Q-js -3 > "then the coefficients generated on the positive side from Cl^ ^_ ( o~^^ ) are the same as those generated on the negative side by CLj^ ( + ^ > with (Xj^ _.— o .) On the other hand, if into a sum over positive values of TV , ^-ip (58) in which the coefficients are identical to those in Eq. 56 Unfortunately, the series (58) converges rather slowly when 7a. is in the neighborhood of unity. Some difficulties were encountered in the programs which were written for the ILLIAC in the calculations of the higher order functions, A since the magnitudes of the P"v -functions approach machine capacity before the functions converge to a desirable accuracy. A simplified flow chart of the program to calculate the -functions is given in Fig. 11. An interesting feature of this pro- gram is the print out which indicates that there is an insufficient number of terms available and also provides a measure of the over -all accuracy of the calculation. This latter is an internal check which is based upon the value of the Wronskian determinant of Uj^i. and Uj^a. • This quantity, \A/ t has a constant value as can be shown as follows : °° f To accomplish this normalization, it is only necessary to enforce the following condition: or CO E ^ = 1 • Since the sum is over only odd (even) tl , this is accomplished in the machine by making O^fe^ positive, and changing the signs of alternate co- efficients in the summation process. The radiation patterns of the uncoated spheroid are then easily calculated, with the spheroidal functions determined in the way described above. The pattern calculation is based on Eqs. k2 and k^> and is done almost exactly as described in Section 2.1-3, with the spheroidal functions A , ( A , replacing the spherical functions as follows: ilj^t *~~* L -'n, ■> ^Aa.* "' ^vo and y^tl-V*} < ^-^ \\^ . Some of the results are presented in Figs. 21 to 27, with some corresponding coated patterns alongside for com- parison. The patterns in Fig. 21, for the smaller values of ^ , were calculated on a desk calculator using functions modified from those computed q 10 by Myers and Wells , and coefficients tabulated by Stratton, et al. These latter calculations were done early in the investigation to study the feasibility of the method and are subject to greater error than the others; 57 in these calculations, the digital computer was employed only to solve the systems of equations which arise in the coated spheroid problem (12 equations in 12 unknowns was the maximum size in this case). The biggest job in the computation program is in connection with the coated spheroidal radiator. The calculation is based on Eqs. k'J and h8 and the operations indicated therein are carried out in exactly the manner indicated. Fortunately, the odd and even orders can be handled separately. The pair of equations, (V7) and (hS) , are replaced by the four equations obtained by equating separately the real and imaginary parts of (hf) and (k&) . A guess is made as to the number of equations required for convergence, and the augmented matrix for this number of equations is generated by the machine in the form required by the ILLIAC library service routine which obtains the solutions of a set of linear equations (a max- imum of 1^3). The variation of the coefficients, C; , as a function of the number of equations was investigated. The results of one such investigation are pre- sented in Table 3. It is clear from these data that the process, as dis- cussed near the end of section 2.2.2, is actually a convergent one, and in principle could be carried out to any desired accuracy. The coefficients, Re. C, , and "X*a C, , so calculated are fed back into the machine and a 3 used in the basic routine for radiation pattern calculation. We now leave the study of prolate spheroidal radiators, and take up once more the study of the sphere, in order to remove the restriction of axial symmetry on the sources . 58 TABLE 3. COMPARISON OF THE COEFFICIENTS CALCULATED FROM DIFFERENT NUMBERS OF EQUATIONS IN THE COATED PROLATE SPHEROID PROBLEM. The similarity in the numbers shows that the infinite system can he re- placed by a finite system of equations. The example is for the parameters as follows: $,1=5", (3.JL = 8 , u x = l.OTT , 7^= l.i ^ slot oJT V = 0. 20 equations 24 equations R-S iQ equations 32 equations 1 5 5 7 9 .035669646 -.061748146 .009879699 -.000092924 .000000405 • 035669590 - . 061748740 .009881601 -.000092948 .oooooo4o6 .035669589 -.061748754 .OO9881658 -.000092949 .000000406 .035669589 -.06174^750 .009881652 - . 000092947 .000000405 11 -.000000001 -.000000001 - . 000000001 13 . 000000000 .000000000 15 ImCj .000000000 1 3 5 7 9 11 13 15 .048023926 -.009061661 .033134300 -.002072803 .OOOO76310 .048023784 - . 009061355 . 033143465 -.002083607 .000083883 -.000002187 . 048023780 -.0Q906l3^5< .033143727 - . 002083989 . 000084252 - . 000002383 .000000045 . 048023780 - . 009061346 .033143723 - . 002083994 .000084262 - . 000002392 .000000048 - . 000000001 59 3. SPHERE WITH ARBITRARY SOURCE CONFIGURATIONS The analysis for the fields which result from the excitation of a spheroidal conductor by an arbitrary slot is very complicated., In fact, it appears that only the sphere can be treated numerically with existing techniques. However, in spherical coordinates, it is possible to represent the fields in terms of a pair of scalars, as indicated in the following developments. In this process, one or the other of the field variables is eliminated from Maxwell's equations to obtain the vector Helmholtz equation VxV* B - u/Ve B =° (59) where Q is either E or H « The problem is then simplified by the division of all possible fields into parts in which either E or H is transverse to the radial direction, with consequent representation in terms of a vector (or Hertz) potential which has only an -f -component. Thus, for TM fields H = y*f+- (6o) and for TE fields E=V*3^ (6i) The subtitution of (6o) into (59) gives for example yxVxfr-^t = vV ( 62 ) where V is arbitrary. If ^ is chosen so that W = JiL the following equation is obtained for r : 6o (63) and a similar result can be obtained for Q . The scalar Helmholtz equation, (62), is separable in spherical coordinates, and the solutions have the form 4^H =-^v.R"vT" +^'vT" (6k) with the constants selected so that the boundary conditions are satisfied. For the moment, for brevity, let the solutions for be represented as f = (&rtT'ce,+) (6 5 ) where \\Q*) and I (9>4>} are defined appropriately. With this notation, expressions for the fields can be written as follows : a) TM fields ^&E = -^-R'vT +^'vT' (6?) b) TE fields C3 = R.V> T "(. 9^ £ = v * R"T"r ■= -t^'vT" (68) (69) 6i Thus for example, the transverse component of any electric field can be written E T = YL R!i, vT," * £ + aft yt (to) where the constants in R. , \ , f^" , and r y 1 ' , are determined by application of the boundary conditions, and the summation over the index i- is to be interpreted as a summation over terms involving all permissible combinations of V and ~rr^ in Eq. 64. In any complete region (including both B — O and — Tf ) ; the constant cXy^ in Eq. 64 must be zero, in order to avoid infinities on the z-axis, and the index / must be an integer. To simplify the analysis, it is convenient to treat separately the parts of the solution involving snrbTm,^ (odd solutions) from those involving cos -m.4> (even solutions). It can also be seen that the quantity \\_ , as defined from Eq. 64 will contain S~ as a multiplier. It is convenient therefore to represent the radial vari- A A ations in terms of the functions J^) '— ** V &> and Nl^ = ^"V^ which were employed in the analysis in the previous sections (let the constants absorb the quantity p which is also needed as a multiplier in order to change the functions). With these changes, and the definitions, T Lt = PC cos -mc{> and T 10 — f^ > Sm yti$ , Eqs . 70 take the following form: + c <_ c [L + K^, N^ V X * >r (71) + jnk L <™ a J- + A —* ^ N ^ vTU 3 &T" "XT 4 fc cJL^r cJL-T 62 It is convenient in the theoretical discussions to define vector functions as follows : el? = S?X % ^ r ^ T/ s = vXc along with appropriate scalar functions of -f- so that Eq. 71 can be written in the simpler form E T = XL OP^*- +%:^eZ 4-^Cf^ +t(^S , (72) It is shown in Appendix B that the vector functions e^e and e«e have orthogonality properties which permit field expansions of the vector E r rather than one component of it as was done in the case of axially symmetric excitation. In the next section, the use of these expansions in the solutions for a conducting sphere in free space is illustrated. The development serves the dual purpose of providing a review for the reader and standardizing the notation so that comparisons to the dielectric coated sphere problem can be more readily made $.1 The Slot Excited Conducting Sphere in Free Space Let a conducting sphere of radius -^ be excited by applying a (known) electric field in an opening which is cut somewhere on its surface. Then, the electric field is known everywhere on the surface, -t* = -f x , viz., it is zero over most of the surface and E T 00 . This implies that the constants in (71) must be related as follows: 0-™. = ~i Kir. , C n^ = ~ 4 ^ - (78) Thus, the expression for the transverse electric field can be written 6k E T = YL <*4 H^p.^ e.'s -h c^ dL H A CM e/ s . (79) *" ^£» At An evaluation of this expression at -f- = -f t , and subsequent comparison of it to Eq. 73 indicates the manner in which the constants can be evaluated numerically, and the result is XUt.) = Lr e^ e 4 <*-$ (80) U = rf - *gj=l£flig • (81) — — — ^— u -fz.T JL-r A* The best form for the computation of the fields is probably «. — r A f i; ^ r cos w4>"i £ r-^t^r-sivt^^T jjtu&o ^-T ^ -^q- -|- sm 9 since and c ^ * "Xe" -tsm-t A ^-m. C— Siw.™.* -) A n ^ f COS Tn(j) ") r; = 7T« ^ = 1_™_ RTi c*s m * J -4. &?» 1 ^m* 5 65 Also, in view of these latter two relations, the following forms for the evaluation of the integrals in (7^) - (77) can be written down: siu9 £9 JL E T ; K* <^ S = Xol^eo. ART" { snt m* ) £0 ' — (J) 0. \ COS -yn-4> J +, Sm9&0cSL f coS^rvi 1 H A0 ; + TTL C sm a 9 c \i ) i cos- y,^ j SITL -ir 1(4?) jkv * -rf-R* 1 >»e*eit -f- JQ1 16 9m a e J = TT ^ ^+m^l •ntn + 1') (Stratton, p. klf) 3CYI+1 C^ _m ^! J 66 Conducting Sphere Dielectric Coating Slot For Excitation FIGURE 12 SLOT EXCITED SPHERICAL RADIATOR WITH DIELECTRIC COATING 67 3.2 The Slot Excited Conducting Sphere Coated with Dielectric When the conducting spherical antenna is coated with dielectric material, the exterior space is split into two homogeneous regions, the region (i) of the spherical shell of dielectric ( +, — ""f" — ^\ ) and the region (il) exterior to the shell ( -f > -T^ ). As in the problem of the conducting sphere in free space, the tangential electric field is presumed to "be zero on the surface -f — -tj , except over the slot, where it has the prescribed value E T(v .In the region (i), the transverse electric field has the form precisely as given in (7l)« The field over the conducting surface, -T — ^ , can be represented in the form of Eq. 73* in which the If 's are regarded as the excitation coefficients, determined from relations (7*0 - (77) « The expression given in (71)/ evaluated at -f ■=• "Tj 1 , must be equal term by term to an expression of the form of (73)/ to satisfy the boundary condition at the conducting sphere. This implies the following set of four equations: /v a a-L. X C &> ■+ -C S N. A ^ = Xl% (82) (83) Next, the boundary conditions at the outer surface of the dielectric shell; -T = -^ are applied. In region (il), "t* > *t^ , the transverse electric field has the same form as that given in (79)/ since it must consist of outgoing spherical waves as ^ >co , When the condition 68 E Tr — L Tir is enforced (where again the form of Eq. 71 for the field in region (i) is employed), the result is the equation LUu.-i^ +C„N,(p. - // 1 4 — "A+ (85) This equation is satisfied if the following set of four equations is satisfied: <^.Jjfito + *lA&* = i"*Rj|^ (86) " ~£^~ ° A+- £ ° ° A-t- The boundary condition on the magnetic field H _1 — H. (87) must now be enforced. A careful examination of the equations (66) - (69) shows that an equation for H T z can te bitten as follows : ,. 1 u;: e &jcw + 41. an, (88) A A 69 In region (il), the functions must represent outgoing traveling waves. Thus, the boundary condition on the magnetic field gives the following equation: » i»>*>- TT JL+ (89) = 7 -J, {IsiH^^vT. + ^H^WT^* . (The terms have orthogonality as mentioned under Eq. 72, where the notation was used for the vector functions.) This equation is satisfied if the following set of four equations is satisfied : <„ s AjX66 -03 -81800 -03 +94320 -03 -10363 -02 +10955 -02 -11199 -02 +11099 -02 -IO667 -02 +99254 -03 -89065 -03 +76489 -03 -61976 -03 +46023 -03 -29156 -03 +11912 -03 +51733 -04 -21587 -03 +36850 -03 -50534 -03 +62269 -03 78 TABLE 5. RESONANCE AND CONVERGENCE FACTORS FOR SPHERICAL RADIATORS. The values tabulated are different orders of the combinations of the spherical Bessel functions which appear in the series expansions of the fields of a spherical radiator whose radius is 2.^ wavelengths. Data and output tapes for these quantities for different sphere sizes as follows: ~t~/?\ — .l( .05)1*05, tA = l.l(.l)2.^V?i = 2.4(.8)4.8, are on file at the U. of I. Antenna The machine was instructed to calculate successively higher until the exponent of the quantity N >% (Note +123 -02 means .00123) ^ + ^ A. A j: n LaDora orders cory. xue maunxm of the quantitiei -5 became less than 10 1 jAr 1 +10044 +01 2 +10134 +01 3 +10273 +01 1+ +10467 +01 5 +10725 +01 6 +11062 +01 7 +11496 +01 8 +12056 +01 9 +12790 +01 10 +13769 +01 11 +1511^ +01 12 +17019 +01 13 +19758 +01 11+ +23231 +01 15 +23902 +01 16 +13905 +01 17 +40526 +00 18 +88558 -01 19 +17242 -01 20 +30354 -02 21 +48o4o -03 22 +68263 -04 23 +87336 -05 24 +10106 -05 25 +10627 -06 26 +10202 -07 27 +89809 -09 28 +72778 -10 29 +54488 -11 30 +37816 -12 T J'" n! 7 +52857 +00 +91823 +00 -22277 +00 -10219 +01 -39022 +00 +73515 +00 +10276 +01 +29563 +00 -68837 +00 -11699 +01 -96790 +00 -35622 +00 +29995 +00 +75329 +00 +79978 +00 +36431 +00 +69952 -01 +88559 -02 +90006 -03 +75697 -04 +52945 -05 +31008 -06 +15346 -07 +64788 -09 +23545 -10 +74262 +20477 +49690 +10675 +20412 -12 -13 -15 -16 -18 A. A. , . J> ■N'* +85147 +00 -41257 +00 -98875 +00 -48556 -01 +95930 +00 +75214 +00 -30595 +00 -10575 +01 -89730 +00 -90763 -01 +75798 +00 +12550 +01 +13732 +01 +13250 +01 +13231 +01 +11215 +01 +63274 +00 +29746 +00 +13131 +00 +55095 -01 +21918 -01 +82621 -02 +29553 -02 +10053 -02 +32599 -03 +10101 -03 +29968 -04 +85310 -05 +23343 -05 +61495 -06 79 TABLE 6. NUMBERS WHICH INDICATE THE RELATIVE IMPORTANCE OF DIELECTRIC COATINGS ON SPHERICAL RADIATORS. The quantities tabulated are the ratios of the expansion coefficients for the field of a coated spherical radiator to the corresponding coefficients for the field of an uncoated spherical radiator which has the same excitation.. The ratios of the real parts and of the imaginary parts are listed separately. The parameters are as follows: sphere radius l.k wavelengths, coating thickness .1 wavelength, dielectric constant 2.25. Data and output tapes are on file at the U. of I. Antenna Laboratory for other parameters as follows: ^ = 2.25, t/A - .1, -rA : .5(.l)l-6, r/b : 1.6(. 8)4.8; £ T - +.o , */* = ,1, -T/h : .5(.l)i.6 ; £ v = 6.25, V* = 1.6, */* : .l(.05)-3 . order FUc r IvnC w n R*o~ I^a^ 1 +15028 +01 +11 5^1 +01 2 +11765 +01 +178+3 +01 3 -1^931 +01 +12138 +01 k +1301+ +01 +8ll40 +00 5 +10103 +01 +17359 +01 6 +17206 +00 +11925 +01 7 +27870 +01 +86526 +00 8 +18860 +01 ++1378 +00 9 +30099 +01 +2l8l4 -01 10 +1788+ +02 +12306 +01 11 +1896^ +02 +330+6 +01 12 +11221 +02 +25209 +01 13 +91727 +01 +219^6 +01 14 +85991 +01 +2035^ +01 15 +85879 +01 +19 i K)3 +01 8o TABLE 7 . EIGENVALUES OF THE SPHEROIDAL WAVE Differential equation (1-v 2 ) dV 2 /dv 2 + ((piv) 2 - k ) Pi =5 kjW odd 1 .21401687 3 .93590450 5 1 .70632729 7 2 .74589040 .9 4 .10404858 11 5 .78287310 13 7 .78212780 15 10 .10163380 17 12 .74129183 19 15 .70104605 21 18 .98086382 23 22 .58072513 25 26 .50061724 27 30 .74053164 29 35 .30046262 31 40 .18040621 33 35 37 39 41 43 45 47 49 51 53 Pi =» 8 HjAPi) 2 odd .12968845 .58438654 .98196932 1.39873600 1.92299702 2.57428143 3.35241653 4.25662492 5.28647244 6.44172000 7.72222957 9.12791755 10.65873113 12.31463555 14.09560733 16.00163007 18.03269202 20.18878471 22.46990174 24.87603848 27.40719101 30.06335668 32.84453316 35.75071871 38.78191194 41.93811159 45.21931685 FUNCTION V = V(-l) = Pi = 12 k/pi) 2 odd .08528034 .39771381 .67784891 .92487780 .1.16537067 1.44669764 1.78603032 2.18356789 2.63830632 3.14963127 3.71718528 4.34075224 5.02019593 5.75542732 6.54638607 7.39303017 8.29532979 9.25326322 10.26681421 11.33597068 12.46072310 13.64106440 14.87698883 16.16849184 17.51556974 18.91821961 20.37643922 81 (3i = 5 kjXpi)' odd 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 even 2 .58571819 4 1 .29687773 6 2 .18687186 8 3 .38487933 10 4 .90339175 TABLE 7. (CONTINUED) Pi =8 kj/(0i) 2 odd 48.62552680 52.15674084 55.81295838 even .36532339 .78777844 1.18065149 1.64517729 2.23271934 Pi =12 kj/OiV odd 21.89022638 23.45957953 25.08449723 26.76497837 28.50102197 30.29262727 32.13979344 34.04251998 36.00080629 38.01465187 40.08405647 42.20901963 44.38954113 46.62562063 48.91725799. 51.26445283 53.66720516 56.12551471 58.63938135 61.20880496 63.83378541 66.51432263 69.25041649 even .24526082 .54209080 .80489795 1.04264567 1.29907596 82 Pi =5 j < k./([3i) sven 12 6 .74246023 14 8 .90185691 16 11 .38144804 18 14 .18115935 20 17 .30094848 22 20 .74078997 24 24 .50066795 26 28 .58057205 28 32 .98049534 30 37 .70043302 32 42 74038172 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 TABLE 7. (CONTINUED) PI = 8 kj/'((Bi) 2 even 2.94755097 3.78879486 4.75586192 5.84843150 7.06632312 8.40943005 9.87768612 11.47104867 13.18948916. 15.03298816 17.00153171 19.09511000 21.31371552 231.65734288 26.12598793 28.71964731 31.43831867 34.28199994 37.25068942 40 . 34438605 43.56308857 46.90679620 50.37550829 53.96922417 57.68794341 PI =12 ^/(Pi) 2 even 1.60901445 1.97759588 2.40383322 2.88692212 3.42639552 4.02197700 4.67349590 5 . 38084249 6.14394374 6.96274954 7.83722457 8.76734344 9.75308732 10.79444239 11.89139790 13.04394555 14.25207898 15.51579320 16.83508408 18.20994839 19.64038338 21.12638702 22.66795726 24.26509285 25.91779241 27.62605496 29 . 38987949 31.20926533 33.08421174 35.01471814 83 Si ■ 5 kjASi)' TABLE 7. (CONTINUED) Si = 8 kj/(pi)' Si = 12 kj/CPi) 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 37.00078420 39.04240935 41.13959327 43.29233563 45.50063613 47.76449461 50.08391077 52.45888438 54.88941528 57.37550344 59.91714850 62.51435054 65.16710946 67.87542497 70.63929714 FIGURE 13 RADIATION PATTERNS, Ha VS. 0>OF SPHERICAL ANTENNAS WITH DIFFERENT CONDUCTING SPHERE RADII X L EXCITED BY EQUATORIAL SLOT. PATTERNS ON LEFT ARE WT^h no COATING. PATTERNS ON RIGHT ARE WITH COATING, THICKNESS .05 * £ r = 5 f\ ±L z 1.61 85 = 2.41 45° 90" |35° 180° © FIGUEE Ik CONTINUATION OF FIGURE 13 86 RADIATION PATTERNS, HW> VS, 6 OF SPHERICAL ANTENNAS WITH DIFFERENT CON- DUCTING SPHERE RADII , r l3 EXCITED BY SLOT AT l60° LATITUDE. PATTERNS ON LEFT ARE WITH NO COATING, THICKNESS .1\, e = 2.25 PATTERNS ON RIGHT ARE WITH COATING, ii- -.2.01 87 Sl = 2.81 0° 45° FIGURE 16 90° 135° 180° G 0* 45° 90 135° 180° 6 RADIATION PATTERNS, Ha VSe> OF SPHERICAL ANTENNAS WITH DIFFERENT CON- DUCTING SPHERE RADIi; T ± 3 EXCITED BY SLOT AT l60° LATITUDE. PATTERNS ON LEFT ARE WITH NO COATING, PATTERNS ON RIGHT ARE WITH COATING, THICKNESS .IX, €-2.25 FIGURE 17 RADIATION PATTERNS, H* VS e^OF COATED SPHERICAL ANTENNA,- = 1-51, COATING THICKNESS .1\, FOR DIFFERENT VALUES OF DIELECTRIC CONSTANT. SLOT AT 160°. 89 RADIATION PATTERNS, H^ VS 9, OF COATED SPHERICAL ANTENNA, - = 1.51> COATING THICKNESS .IX, FOR DIFFERENT VALUES OF DIELECTRIC CONSTANT. SLOT AT 160°. 90 85* W K FIGURE 19 RADIATION PATTERNS, H^ VS . 6, OF COATED SPHERICAL ANTENNAS, v/X = 1.51, DIELECTRIC CONSTANT 6= 2.56, FOR DIFFERENT COATING THICKNESSES. SLOT AT 160°. FIGURE 20 RADIATION PATTERNS, % VS. e ^ OF COATED SPHERICAL ANTENNAS. r/\ = 1.51, DIELECTRIC CONSTANT € = 2.56, FOR DIFFERENT COATING THICKNESSES. SLOT AT l60°. 92 e--o e--o Slot ' a* cen4< Uncoaled a= I. 00OO Coa+ed a,-- l.ooooi FIGURE 21 PROLATE SPHEROID RADIATION PATTERNS (ONE HALF OF SYMMETRIC PATTERN) 6 * = 3, £*.= J5 i H<> «. © nocoohng U^/07 7 coded u,H077 93 slot of v =-cos£o< PROLATE SPHEROID RADIATION PATTERNS p I DIFFERENT SLOT PATTERNS, IUVS. e no coating U. ,= 1.077 coated u ,- \ 07 7 Sk u,= 1. 1 FIGURE 23 PROLATE SPHEROID RADIATION PATTERNS p i = 5, € DIFFERENT SLOT PATTERNS, H. VS. 6 &, FOR no coating 95 ! o FIGURE 2k PROIATE SPHEROIDAL RADIATION PATTERNS, P Q I H, VS no coating 96 Vo-'COSdO 135° 180° e FIGURE 25 PBOIATE SPHEROIDAL RADIATION PATTERNS, p Q I = Q, Hx VS. 6 no coating TA^s- 1.077 Mote: concJuctoY si*e. tkat of Fiq. aa V.- " COS 30 *5* 90° (35° 160' ^ FIGURE 26 PROLATE SPHEROIDAL RADIATION PATTERNS 3A=5"j 12, H^ VS. e no coating \ Vo = -C0S45° j/ot at Vo^-coseo° 98 coa"ted Note: condacxot siie thai ©?- Fit. jl3L 11,.= 1.077 ^* S0 6 A35 - * /5b FIGURE 27 PROLATE SPHEROIDAL RADIATION PATTERNS g£- 5 5 12, Ha VS.© 2.2 FIGU RE 2£ A l 1RAPH OF 1 HE Q, ELNTI ffilfl i ?*" . Will <"!N TT\m.T f'M'j'E, . n=lo / <<> .a 2-1 QU uSI-B 1GBMC ESONfl E IN NCES THE I AND 'IELD 1™ 3ETS EX PA ^THE F NSIODS IN \TE SPHE )F CO HICAL N- r v 2.0 i C0< 5RDIE ^tes ^=7 r v \ 1 h \ /■y Jl n-6 f \ k / \ \ 18 / rt=5 /^N { f k J \ I ■ lti'-< r / n "i X // \ \ 1 i \ 1-6 \ \ { \ \J \ 15 n= 2 ( N \ I 14 \ k 1.3 1 k \J 1.2 1 si v, - J J ~_\ i I I I 1 i '5 1 / ■4 \ / / / 1* / / / / / i ■2 / / / / / / ■' 1 r / / j 1 y / / / f J y -^ / / — ^ ^ ^ • r X — ' \l .2 .3 • 7 .£ ,9 hO hi / . ^ 7 i // n FIGURE 30.[H / (2rtr/\)]" 1 AS A FUNCTION OF n 103 REAL PART ■IMAGINARY PART (r/\ = 1.0) 13 5 7 3 1/ FIGURE 31-[H / (2nr/\)]" 1 AS A FUNCTION OF n . 104 REAL PART IMAGINARY PART (r/\ = 1.2) FIGURE 32.[H'(2jtr/\)] AS A FUNCTION OF n 105 REAL PART IMAGINARY PART (r/\ - 1A) FIGURE 33. [E'{2xr/\)] AS A FUNCTION OF n 106 REAL PART IMAGINARY PART (r/\ = 1.6) 1 t ~, 1 H n (^ ,■< J t / c^iS > '" " / • / tO~ \ — ' \ / r^\/ r L-- - <* ^A ] /\ 7 \ \ \ / \ \ / y "7" \ \ \ \ I \ 3 \ I \ \ \ / V \ ) i \ V \ \ \ \ \ \ \ \ \ \ \ \ \ \ o \ \ \ \ \ 5 FIG JRE t>K [H^(2: 7 5 trA)]" 1 AS J 5 I I FUNCTION 0) 1 1 P n . 3 '5 107 REAL PART IMAGINARY PART (r/X = 1,8) n 5 A 9 13 i"? FIGURE 35, [H , (2jtr/\)] _ AS A FUNCTION OF n. 21 108 REAL PART IMAGINARY PART (r/\ = 2.0) FIGURE 36. [H / (2itr/\)] AS A FUNCTION OF n 109 1 REAL PART IMAGINARY PART (r/\ = 2.2) 1 o 1 \ I >\ / ( i JF ' 1 • A i \ / 1 J \ l\ / • V ', ft 1 \ 1 • i / V ft / ^ /(' » 1 / V 1/ / I / 1 J 1 ' T T 1 J y / , 1 1 1 / / I 1 I' \ /I /■ 1 1 1 \ Ji li r 1 1 * 1 / 1 \ \ i \ \ 1 ■!■ ( / ( ; : i / \ < \ ,1 i 1 it u 3 ' i I l i I i 1 i 1 1 ' 1 1 _ 1 1 \ > \ \ \ 01 \ \ \ \ \ \ — r\ 3 5 7 9 M 13 15 \7 19 FIGURE 37. [H / (2jtr/\)] _1 AS A FUNCTION OF n . 21 110 REAL PART IMAGINARY PART (r/\ = 2.1+) 5 7 9 II 13 15 II 19 21 FIGURE 38. [H / (2itr/\)]" 1 AS A FUNCTION OF n . Ill REAL PART ■IMAGINARY PART (r/\ = 3.2) 3 5 7 9 II 12 15 n f9 21 It) 2.5 2( FIGURE 39. [H / (2nr/\)]" 1 AS A FUNCTION OF n. 112 6 7 8 9 10 II n- FIGURE kO RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERICAL, RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION (g = r/\) COATING THICKNESS .1\. € = 2,25 113 FIGURE 41 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SFHEROECAL RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UNCCATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. (g = r/\). COATING THICKNESS .IX- € = 2.25 114 JO / / I 1 1 \/ A / s \,.o 1 / V. \ -_ - r Re. 1 1 1 j l i i I 1 / / / 1 I Oi 1 o\ 97 / \ \ / / / 1 1 / ^y i >v / \ / N >X / D Im j M / \ / / ' \ i \ r / A.o Sf 1 1 1 1 1 TL ' (b 8 10 iz 13 FIGURE k-2 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERICAL RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. (g = r/x). COATING THICKNESS .IX. 6= 2.25 FIGURE kj> RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERdCAL RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. (g = r/\). COATING THICKNESS .1\. €= 2.25 i , ^ — * V 1 \ / \ \ / N k 10. % §_ / \ / 9-?/ / - - C / A / / / 1 \ / / / \ / y--// JTm / \ / - k ■/ / S / V \ S <.. L6 \ \ / Nj f 1 \ / 1 ]k 1 — I N| 1 i \ / 1 — \ / \ ~r \ / ' \ 1 \ I ' \ r \ / j ! i [JO 1 1 j \ i I 2 3 + 56 78 <— 40 10 tl / ? t ? s ? /< 9 /; f A g /: 5*~ s 2,| .si i 119 FIGUR E ^7 3ATI0 TOR TH )F COE 3 COAT ITTCIE CD SPH ATS, c l IN ?HE FI] ITOR T( 3LD EX] ) THE ( 'ANSIOI !0- ■ /I CRICAE RADL SFFICI 3PHERI CNTS a :al ra IN T 3lATOR IE EXP WITH INSION rHE SA] FOR T] ffi EXC IE UNC( [TATIOI )ATED I. \ / \ (g = r k). :oatin S THIC (NESS ,1\. £ = 2.S 15 / \ / \ \ \ \ \ 1 V I \ 10. 1 / 1 J -3.r 1 fi | A 1 1 ; \- f- -k- / ': / 4 1 A / 1 / \ / t Jm J \ I \ > *" \ J 1 °i =3£ / I 1 1 \ \ j r" "1 3 \ \ 1 h \ 1 \ \ / f / N iO \ /y j \ j\ ,1 1 1 A \ A._jL \ r L 1 I \ ' \ r/ \ / \ \ / -f \ / \ L_ V \ \ » \ 1 \ \/ ux-rl i / \ \ 5^ \ \ v \ * V K ! il :j I 3 5 7 i // i n- 120 121 to A 10 IP 122 L I \ 1 \ 1 \ 1 \ 1 \ 1 \ 1 i \ V 1 1 \ \ 9= ^Re 1 I I / \ / / / i \ / \ | \ \ / i \ ) \ / I \ / i \ \ / / ^~~*^s ggft > i ! ___ 1 \ / \ / \ / / \ / \ l\ 1 FK c TH1 TO ex: SP3 JURE 5( IN T] ] COAT] THE C( =ANSIO] IERICA] ) RATK IE FIE] ]D SPB3 )effic: \F FOR : J RADI/ ) OF C( D EXPi IRCTCAI ENTS 8 ]RE UN( ltor w: )effic: INS ION RADL i IN '. :8ated [TH TH CENTS, FOR ITOR [HE 3 SA1 ffi exc: ^TING f [TATIOl CHICKHI I- (g = r/\) 2SS .IX. € ! = 4_ 1 ' / / CO. i v^ I 1 1 7L 2 1 4 ' J > I £ ) s t It 3 / / / 2 I z FIGURE 51 3 4 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERICAL RADIATOR TO THE COEFFICIENTS^ ,IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. ( g = r/x). COATING THICKNESS .1\. € = k j i l r A — / \ o 1 / / \ \ / \ \ / \ / \ -- \ ^ / \ j \ =3e± L% 1 1 1 1 J \ 1 / / 1 / 1 / 1 <> 1 / \ 1 \ 1 / X, ^ C\~ "r >. T^ y \ / \ / 1 j ^JO A \ / \ \ / / / \ / \/ A / ( / FIGURE 52 R I F R ftTIO JJ THE DR THE M)IATQ ? COEF FIELD COATE =i TO T ilXPANS D SPHE re m- CS,c [ON n tfCAL 1 I i jt FFICIE fUtfSION PHERIC IE SAM S-r/ JICKNE WS,a FOR 9 \L RAD 3 EXCI 0- c 3S .IX ,IN TE ffi UNC EATOR 1 DATION )ATING e = 3 EX- ITED 7ITH k 4 f F V f S \ / T t x / T \ / \ 1 1 / n ■ i ' > " 5 ^ 1 £ ( / i 3 S 5 i \ 1 1 ^ i 3 |< } IS ^25 FIGl RE 55 RADIO PANS I TO TH THE U SAME THICK OF CO DN FOR 3 COEF \fCOATE SXCITA >IESS . 2FFICI THE C *ICIEN D SPHE riON. L\. € ENTS, DATED rs,a ^ RICAE (g = = h 2 , IN THE F TTT.D EJ DIATOI HON F( [ THE ig ;- iv tf IN THE RADIAT EXPAND DR WIT] COATIl « , \ \ 1 1 \ \ \ / \ \ / &* 1 \ jR I /Afc — f / to / \ > < / *x \ / 7+3m / / ■^ q- \ 1 / \ \ 1 ^k 1 \ / \ i \ \ 1 ^ y / \ \ 1 \ 1 \ *r~* 1 A to / N \ \ \ \ i i \ ^""-J t \ t \ IZ \ \ 1 i ft \ I | 1 < i 1 \ \ 4 t ^ r < 5 ; * -X -)653 9 ' 1 1 A 2 /: n — / tf - i I /* f,T \ 126 1 \ (n A 1 1 i \ \ V- /■ p , V —^ -»-ua — Lv ^ a 1 C^ h / i / 1 / 1 s / / i n / i 1 / \ / / | \ / I > / t / f j \ / I = .6 r^ i \ ' J j \i \ ! / V / 1. K / 3 \ y / X / i \ / i I p 1 l i 0.1 1 2 3 * t 5 " 6 7 3 n 10 FIGURE $k RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERdCAL RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE §AME EXCITATION. ( g = r/\). COATING THICKNESS .1\. € = 6.25 A 127 '« 33 \ / \ \ / \ \ tc? / \ \ \ 9- C» £> =r- 3-* TrH-Hx » — - 1 / / 3 e I \ / 1 \ / 10 1 1 \ / / 1 / i V / j \ ' / / \ \ / ^\ a 1 ' / X ^ 7 1 \ / x \ 1 1 \ / \l \ / ^ \ FIG URE 55 RATI pSns COAT] RADII EFFI EXPA ) OF C( CN THE [ON FO] CD SPH LTOR T( :ients ^SION ; )effic: FIELD \ THE SRICAI ) THE < a IN 70$ TH HKTS, EX- :o- THE A | l\ \ i I \ f VIED SJPHERIG \TOR WITH TH 2 SAME r/\). 3 .IX. \ 4' 1 \ / RADL \ \ i EXCI EATIOm (g = ING THICKNES; \ \ / COAT \ i . € = 6.25 1 \ | I \ \ 1 -^ I t J 4 i <5 ' 6 ' 5 > It ) // /< 3 128 / 10: 10 \ V / — /- \ \ \ s. Re i — ^ — ^^ " / s = 1.0 / c > ft / n / / / < * / / 1 V 4 / X \ \ \ \ \ 1 \ \ | \ 1 I'm * \ q = Lf \ , 1 \ 1 J u t / v 1 \ f > s , / / V J / \l \ \ \ / / / s* 1 1 1 i \ 1 ♦ i i 1 » \ 1 1 \ \ \ i 1 1 I in. > O^B & 10 12 13 FIGURE 56 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERdCAL RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. ( g = r/\). COATING THICKNESS .1\. £ = 6.25 9 = 1.4/ Pe 130 * / 8( >s~ tf / / / / u. / A / / / ^ A 4- < / • \ Ito / i / \ i \ 1 1 / \ ,0 i » . \ ! i \ \ ' \ / 1 4 \\ \ / \ I V \ \ i i \ \ \ < \ / v l\ \ \/ \ \ / / .. i \ \ \ r 1 t J- \ \ 1 / > \ 1 / \ 1 1 \ 1 \ V 1 ~n Co 5 10 12. '3 / + FIGURE 58 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHE- RICAL RADIATOR TO THE COEFFICIENTS a IN THE EXPANSION FOR THE UN- COATED SPHERICAL RADIATOR WITH THE SA$E EXCITATION. ( g = r/\). COATING THICKNESS .1\. £= 6.25 131 10 LEG] ;nd A t .100 H 00 REAL IMAG. 'ART PART /. v i \ , .-* J I 1 1 1 , I K \ J \ i N !\ 1 1 / / ] l\ 1 \ \ , \ \ 1 \ \ 1 \ \ 1 < 1 V .0+ YV II 13 15 17 19 FIGURE 59 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERICAL RADIATOR TO THE COEFFICIENTS, a , IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR 1 WITH THE SAME EXCITATION. €= 2.25, £ = 1*6. THICKNESS t/\ VARIED IN SUCCESSIVE GRAPHS 132 FIGURE 60 RATIO OF COEFFICIENTS, c , IN THE FIELD EXPANSION FOR THE COATED SPHERICAL RADIATOR TO THE COEFFICIENTS, a , IN THE EXPANSION FOR THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. € = 2.25, £ _ 1.6. THICKNESS t/\ VARIED IN SUCCESSIVE GRAPHS \ ~ 133 LEGENI REAL PAET LMA.G. PART 100 'IGURlj 61 ratio of coefficients, C , IN THE FIELD E^- | PMSION FOR T HE COATED SPHERICAL RADIATOR iTO THE-TOEFFieiENTST ,. ~ IN THE EXPANSION FoS " THE UNCOATED SPHERICAL RADIATOR WITH THE SAME EXCITATION. € = 2.25, I = l.d. THICKNESS ' i/\ VARIED IN SUCCESS J IVE GRAPHS i - 05 r / 13*« LEGEN > REAL I IMAG. ATIO C IN THE FOR TIj RADIAI CIENTS PANSIC 3PHERI ""HF SA t \RT X PART F COEF FIELD a COAT 3R TO > V M FOR CAL RA W, EXC +.300 + 00 i °- , .-© .._ 135 n ICIEN EXPAN 3D SPH CHE CO EN THE [HE UN DIATOR ETA^'IO i v c FIGURE ] 63 I PS, c 5I0N n ]RICAL SFFI- EX- jOATED WITH 1 ,; ^o / no | i € = 2 IHICKK 3UCCES .25, r X ESS t/ 3IVE G = 1.6 v. VARI 1APHS : SD IN i f ■ | I 1 i\ f p I r 7 © / ^P l / 1 If 1 V ■ \ 1 3 i I / / r \ ■ 1 1 ; / \ / i ! 1 1 i \ \ 1 \ 1 V T \ , ( ; ■ / © 7 J / I + i If \ 1 \T \ 1 \ (4 1 ' lr 1 1 \ i T I I \ | \ i Wl 1 ' \f I- 1 ' 1 \tc 406 Is ;5 $92 Ml '3 17 *9 , 136 REFERENCES 1. Scharfman, H. J. A, P., 23, Nov. 195^. 2. Aden, A.L. and M. Kerker, J. A. P., p. 121+2, Oct. 1951c 3. Elliot, R.S. IRE Trans. PGAP, Vol. XP-k, July 1956. k. St rat ton, J. A. Electromagnetic Theory , p„ 50, McGraw-Hill Book Co. New York, 19^1, 5. Ince, E.L. Ordinary Differential Equations , p. 223ff. Dover Publications, Inc., 1927 (reprint). 6. Churchill, R.V. Fourier Series and Boundary Value Problems , p. k6ff. f McGraw-Hill Book Co*, 19^1. 7. Jahnke, E, and F« Emde, Table of Functions , p. 1^6, Dover Publications Inc., New York, 19^5. 8. Schmid, H.L. Math. Nachr, 1, p, 377-398 (19^8)3 £, p«35-^ (19^9), 9. Wells, C.P. Mathematical Theory of Antenna Radiation , Final Report 00R Project 1013* Dept. of Mathematics, Michigan State University. 10. Stratton, J. A., P. Morse, L. Chu, J. Little, F. Corbato, Spheroidal Wave Functions , John Wiley and Sons, Inc s New York, 1956. 11* Marcuwitz, N. Waveguide Handbook , McGraw-Hill Book Co., New York, 1951- 12. Felsen, L„ Spherical Transmission Line Theory , Report R-253-51 PIB 19^ Microwave Research Institute, Polytechnic Institute of Brooklyn, Jan* 1952. SUPPLEMENTARY BIBLIOGRAPHY BOOKS 1. Flammer, C. Spheroidal Wave Functions , Stanford University Press, Stanford, California, 1957 . 137 2. Morse, P., and H. Feshbach, Methods of Theoretical Physics , Vol. II, McGraw-Hill Book Company, 1953 . 3. Schelkunoff, S.A., Advanced Antenna Theory , John Wiley and Sons, New York, 1952. PERIODICALS 1. Page, L. and N. Adams, Phys. Rev. _53, p. 819-31, May 15, 1938. 2. Chu, L. and J.S. Stratton, J.A.P. 12, p. 2kl-k8, March 19*1-1. 3. Ryder, R., J.A.P. 13, 327-3^3, May 19^2. k. Page, L., Phys. Rev. 65, p. 98-117, Feb. I9H. 5. Bailin, L. and S. Silver, IRE Trans. PGAP Vol. AP-4, p. 5-16, Jan. I956. 6. Myers, H., IRE Trans. PGAP Vol. AP-4 p. 58-64, Jan. 1956. 7. Andreasen, M. IRE Trans. PGAP Vol. AP-5, p. 267-270, July 1957« 8. Wells, C.P., IRE Trans. PGAP Vol. AP-6, p. 125-129, Jan. 1958. 138 APPENDIX A Determination of the Eigenvalues of Unfamiliar Eigenfunctions The purpose of this appendix is to outline a general method for the calculation of the eigenvalues which are associated with a second order differential equation with boundary conditions, in the event that the functions which satisfy the differential equations are unknown. This problem is different from the problem of eigenvalue determination as it is usually encountered in engineering problems in that the functions which represent the solutions are known only in the form of some kind of infinite series, and the dilemma is that the coefficients in the infinite series depend on the eigenvalues (which are as yet unknown) The method discussed here is essentially a generalization of the nfethod described in Section 2.2.3 for the calculation of the eigenvalues of the spheroidal functions, (it is assumed that the reader is familiar with the idea that orthogonal functions are generated by differential equations. This is discussed briefly by Churchill and thoroughly by Ince . ) The method is probably well known to mathematicians, but it is not presented specifically in any of the books into which the engineer ordinarily looks for help in solving his problems . A.l Consider a general second order linear differential equation of the Sturm-Liouville type p(x)y + q(x)y + (r(x) + k k s(x))y = (Al) 139 with the boundary conditions ^y(a) + p.,/ (a) = (A2) a 2 y(h) + fi 2 y'(-b) = The problem is to find the eigenvalues, k, , in the event that the functions y are unfamiliar. To find these, expand the functions y k k in a series of known functions which form a complete orthogonal set and satisfy the equation P^x)^" + q 1 (x)^ n / + (^(x) + \ nSl (x)) n =0 (A5) and identical boundary conditions (A2) . (That is, the eigenvalues of (f) are known or can be readily found.) Thus, assuming that term by n term differentiation is possible, the series ^ ■ I C kn* n=l is substituted into (Al) with the result L Vn W) n=l *< x > Y C kn< + *« I C knV + < r < x) " V«> }_ C lA " ° . (A5) n n n To generate a system of linear equations having the form y C. L =0 kn nm multiply through by d> and integrate over the range. The result, m after interchanging the order of summation and integration and grouping the coefficients of c ~> is kn ltei i 'kn P(x)^ n <^ m dx + / q^ty^dx + rUW^dx+kJ s^^dx Ja Ja ^ (A6) =0 The quantity in brackets is an array of numbers which will be designated L . The eigenvalues are those numbers which make the determinant of this array equal to zero, since this is the condition for consistency of the set of equations. As is shown in Section A. 2, the values obtained from a finite size determinant will be good approximations since the representation has stationary properties. The determination of the elements of L can be simplified, and nm * ' the rate of convergence improved, by the judicious choice of the functions .To show this, multiply each member of the set of Eqs. A3 by the corresponding member of the set of constants, C. equations together to form kn , and add these n n n Now subtract this equation from (A5), multiply successively by the set (A7) of functions, <$> , and integrate over the range of orthogonality to obtain the result I— „b dx Y_ C kn | <*<*> " PiM^V* + / ((l(x) " ^Vn^W n J a a favorable set of functions, namely, they should satisfy a differential equation which is as nearly like the given equation as possible. The best possible condition is that all except one of the following hold: p(x) = p x (x) , q(x) = q x (x) , r(x) = r^x) , s(x) = s^x) . Otherwise, should be selected so as to make the integrations as simple as possible. We note also from (A8) the condition for symmetry of the matrix |l = L ): nm mn J (p-p^/c^dx + (c L -q 1 )^ i / n dx. J a (A9) Thus, if p-p and q = q > there is no question of the symmetry of the matrix, t . An integration by parts permits (A9) to be nm replaced by the condition r L' £_(p-P 1 )-(q-q 1 ) dx = [^(P-Pi)^^! 4> 4 ' TxFm dx, (A10) This is a sufficient, but perhaps not a necessary condition. Ik2 There are a few more comments which can he made ahout (A6) and (A8). If the values of the elements so calculated were placed in an infinite determinant, and the condition I L 1=0 enforced, then J nm exact eigenvalues would he obtained . But the vital question is whether finite determinants can he employed to obtain usefully accurate values. The use of such a finite determinant can he regarded as equivalent to the replacement of the infinite series (A^-) hy a finite sum of N terms as follows : N y k = ^ C kn*n ' n=l The practical questions are: Can the eigenvalues, k_ , he obtained to arbitrary accuracy with a finite N , and if so, how large must n be to obtain a specified accuracy? The answer, of course, is inseparably connected to the matter of how closely the functions y, and 0, resemble each other. In any case, however, as is shown in Section A .2, (A6) and (A8) are results which have stationary properties (that is, they are the same results as obtained by the application of a variational procedure). Thus, the end result is N insensitive to small differences in the quantities y, and 2__ C kn^n n=l Moreover, the assumption of uniform convergence is necessary in order to justify the term by term differentiation to arrive at (A5)» Such uniform convergence means exactly that N I'k" Z C *n'>'n! -' € n=l 1*3 independent of x , for some finite N . This implies that C = > M>? N , so that the accuracy of the th km eigenvalue can be determined to an accuracy set by the value of € selected. In practice, the error analysis is difficult, so that with a high speed computing machine available, the question is perhaps best answered by repeated trials in which the size of the matrix is varied. It is interesting that in the case of the spheroidal functions (Section 2.2.3); even the value N = 1 gives a correction to the eigenvalue of the trial function P (lowest eigenvalue). The idea of replacing the infinite sum (A^) by a finite sum suggests one other technique which is useful in practice in the calculation of the higher order eigenvalues. For with the higher order functions, a good approximation may be as follows : N y k / kn n , that is, the lower order coefficients may also be approximately zero. This means that the accuracy of the higher order eigenvalues can be improved, with a given size matrix, by taking a section "out of the middle" of the infinite matrix (that is, dropping out some of the lower order coefficients as well as some of the higher order) . (Of course, the coefficients for all orders of the functions satisfy the same set of (infinite) equations, so the solution of a single matrix gives an approximation for as many eigenvalues as coefficients included — however, the accuracy of the individual eigenvalues so determined Ikk depends primarily on how well the finite sum for each order approximates the function of that order. A. 2 It will now he shown that a variational method leads to Eq. A6 and hence to the same determinant as the one ohtained earlier. Consider the differential equation in operational form Ly + *y = • Multiply this equation by a trial function, *\> , and integrate over the complete range: / if Lydx + \ j \\> ydx = • Now perform a first variation JS^ Lydx + S \ft ydx + \ W ydx and set £ \ = to obtain the result J (Ly + \y)j> dx = . In the present problem, Ly + \y is of course just the right hand side of Eq. Al, which implies J p(x)y" + q(x)y' + (r(x) + k k s(x)y)^ dx = Now assume y = ) C <$> . take^ = ) C (so/ = ) / C ) * /_ n n / mm / ° m m n m m and find p(x) Y_*X + 4(X) L C A' + < r « + V W] [ C aJ X [i C .e 1*5 or Jj C m[ (p(x) ^' C A* + q(x) Z *' + (r(x) + k k s(x)) Z C nt )( ^rH =0 ' m O a n n n But the variations in C > £ C , must be arbitrary, so b Cp(x)^ C n 0/ + q(x)2_ C n 0/ + (r(x) + k k s(x))^c n n > m dx = L n p(x) m + (r(x) + VCxJ^^ dx = , n *^a which is the result stated in Eq.. A6. Thus we have demonstrated that (A6) has stationary properties. ll+6 APPENDIX B Orthogonality Properties of the Vector Functions of Chapter 3 In the problem of the spherical antenna excited by an arbitrary slot configuration , it is convenient to represent the transverse electric field, E.-r , by means of an expansion in orthogonal functions. In order that this can be done^ the following set of othogonality relations is required: /e\ . e\ dS = L . 1 iV le je ij J it H/ 2 p dS le I j r=r n e. . e. dS = d , . J e. J ie je ij J i< e io • V 3 = <* i J e l< r=r l I (Bl) f C-^^ijC* 3 < b2 > j e "ie • S Jo« " ° < B5 > e; . ev dS = (B^) (B5) 2 ds (B6) e ie • •to* = ° (B7) 1^+7 in which e 'e = V T « 1 1 '-" , e.e = v T.e X r . The relations (Bj5) and (B7)> which involve products of the even and odd vector functions, follow almost immediately since the integrals will all involve the product of a sine function and a cpsine function integrated over the range to 2n The relationship (B^) can be shown in general by employing the divergence theorem and some well known vector identities (listed for example in Stratton, p.60^) as follows: J V* Adv = J A*nds put A = T.(r xv T.) . Then ■*■ J V • A =VT. • (r xVT.) and J A . ndS =J T.ffixVT.) • rdS VT. s (rxyT )dS = e. . e. dS = The relation (Bl) can be demonstrated as follows^- The terms ofyT. m a ie are Q d n cos m "the r i& && 2 ". 2 n X r sin © XkQ After the $> -integration, the results are integrals of the type , dP ™ dP„ 2 _ m „ m x . ^ a ( n I + m P P ) sin#d0 15" TT "5 n * sin6> This integral is zero unless n = I > and constant otherwise (Stratton, p. ^17). The demonstration of (B2) is completely analogous. Furthermore, the results (B5) and (B6) follow from the same type of argument since the operation £ x ^p e (which is the operation required to i o transform £ e into i e ) merely interchanges the O i° j© and -components. ERRATA Page iv In title of Fig. 22, 23, replace € = JZ by € iv In title of Fig. 28, replace —~. ^ — by kl 48 6o 69 93, 9 k 112- 130 f '(p f Ih'O f n r' I n r I iv-vlii Add one to the page number for j>ig. 29 and- beyond. Page 100 is note explaining the symbols in Fig. 29 and those beyond. 16 Line 8 insert (Columbia University Press 19^+7) after Tables of Spherical Bessel Functions . 18 Next to last line; replace (33) by (23). A 23 All quantities Z should read Z . 2k Last line; replace "the lower order ratio" by "the next lower order ratio". a' 26 Subscript n left off N in two equations. kl Line 21; replace appraicably by appreciably. Line 22 j replace magnitude by magnitudes. Line 17; replace / d, 2n(n+l) by l kn 2n < n+1 ) 2n+l 10 10 Center block; replace ) i P by ) cL P n=l n=l Third box from top, on right, replace V 7 d 2n(n+l) by V-2 Ls 2n + l n Line 3; replace (62) by (63). / Equation 89 replace d g by d % . First line after equation 89. Insert e. at the end of the line. / a Fig. 22, 23. Replace €^ = ■= by €^ Ordinate of Figures kO-^d should all be lower case, / n ERRATA (continued) ]>3 Line 9. Replace P "by P- 1^J+ Next to last line; replace 6 = > 6 C by 6^ = / 6C m m 1^5 First equation; replace q(x) }