II B RAR.Y OF THE UN IVLRSITY Of ILLINOIS 6Z\.3€5 IZG55te no.2-H cop.3 Digitized by the Internet Archive in 2013 http://archive.org/details/asymmetricallyex03hans THE ASYMMETRICALLY EXCITED SPHERICAL ANTENNA Contract No. AF33(616)-310 RDO No. R-112-110 SR-6f2 TECHNICAL REPORT NO. 3 by Robert C. Hansen Research Associate 30 April 1955 THE LIBRARY OF THE AUG 11 1955 UNIVERSITY OF ILLINOIS ANTENNA SECTION ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA. ILLINOIS IV TABLE OF CONTENTS (Cont.) 1- The Experimental Facility 7.1 The Swing Frame Ground Screen 7.2 Control and Electronic Equipment 8 Conclusions Bibliography Appendix I. Calculation of the S m and T m Series Appendix II. A Short Summary of the Illiac Order Code Appendix III Series Involving Spherical Bessel Functions Appendix IV. Watson's Transformation Appendix V. The Z^ Calculation Appendix VI. The Associated Legendre Code Page 96 96 100 103 104 108 118 122 126 135 139 ILLUSTRATIONS Figure p age Num.be r 1 Spherical Coordinate System 5 2„ Cross Section of Antenna with Radial Line 14 3 Parallel Plate Region 15 4 Two Monopole Array 21 5 Sum by Rows 24 6 Sum by Columns 24 7 Real Part of S(m,a) 38 8 Imaginary Part of S(m,a) 38 9 Real Part of T(m,a) 39 10 Imaginary Part of T(m,a) 40 11 Real Part of Z m (ka) 41 12 Imaginary Part of Z^(ka) 42 13 Addition Theorem Coordinates 44 14 Theoretical Patterns for 0„ 4 Offset 49 15 Theoretical Patterns for 0.4 Offset 50 16 Theoretical Patterns for 0.4 Offset 51 17 Patterns for 1.0 Offset 52 18 Patterns for 1,0 Offset 53 19 Patterns for 1.0 Offset 54 20 Patterns for 1.0 Offset 55 21 Pattern as a Function of Offset 57 22 Pattern as a Function of Offset 58 V I ILLUSTRATIONS (Cont.) 23 Sector Dielectric Loading 66 24 Two Sector Guide 68 25 Radial Line with Twenty Sectors 71 26 Real Part of W m (ka) 73 27 Imaginary Part of W m (ka) 74 28 Real Part of W m (ka) 75 29 Imaginary Part of W m (ka) 76 30 Real Part of W m (ka) 77 31 Imaginary Part of W m (ka) 78 32 Real Part of W m (ka) 79 33 Cardioid Patterns of Spherical Antenna 87 34 Elevation Pattern of Spherical Antenna 88 35 Tchebyscheff T 5 Polynomial versus cp 92 36 Fan Beam Antenna Pattern 93 37 Elevation Pattern of Fan Beam Antenna 94 38 Fan Beam for Different Elevation Angles 95 39 Ground Screen from Above 97 40 Underside of Aperture 99 41 Equipment Racks 101 I 1 S m Block Diagram 111 III 1 Block Diagram 124 IV 1 Paths of Integration 128 VI 1 Block Diagram of NALP Code 141 V I I ACKNOWLEDGMENTS The author is indebted to his adviser, Professor Edward C. Jordan, for his help and inspiration. He is also indebted to Dr. Cleve C. Nash who offered initial encouragement in graduate study, to Professor V. H. Rumsey and Dr. R. H. DuHamel of the Antenna Laboratory who made the work possible, to Lynn Yarbrough, who performed hand calculations, to Roger Trapp. who engineered the experimental ground screen, and lastly to his wife, Dorothy, for her patience and understanding. GLOSSARY OF SYMBOLS V I I I p m V 6(x) e n 'm X m m T fa) ?r 'Vim a a n a i Bm Eigenvalue of z equation, p. 15 Argument, p 131 Gap width, p. 23 Kronecker delta, p, 11 Dirac delta function, p. 19 Dielectric constant, p. 64 Neumann number, p. 11 Constant, p. 70 Intrinsic impedance, p. 5 Wavelength Permeability Complex zeroes, p. 129 Argument, p. 131 Coefficient, p„ 25 Transverse Scalar, p. 18 Unit vector in r direction Spherical coefficient, p. 7 Cylindrical coefficient, p. 16 Radius of sphere Coefficient, p. 29 Coefficient, p. 45 Spherical coefficient, p 7 Cylindrical coefficient, p. 16 IX GLOSSARY OF SYMBOLS (Cont. ) b n p. 29 b m Coefficient, p. 81 C nm Spherical coefficient, p. 7 C^ Cylindrical coefficient, p. 16 Cj^ Tchebyscheff coefficient, p. 89 c Radius of feed wire, p. 43 D nm Spherical coefficient, p, 7 D m Cylindrical coefficient, p. 16 d Offset, p. 44 E Electric field vector of (x,y,z,t) Eg Assumed gap field £ x (0,qp) Transverse vector mode function, p. 7 e Naperian base f Scalar mode function, p„ 6 g Scalar mode function, p. 6 H Magnetic field vector of (x,y,z,t) H Magnetic field vector of (x,y,z) h n ^'(kr) Spherical Hankel function of second kind, p. 6 H n ^ , '(kr) Hankel function of second kind, p. 6 Hoo Wire coefficient, p. 45 j -^T J m (kr) Bessel function of first kind, p. 25 k Propagation constant, p. 3 GLOSSARY OF SYMBOLS (Cont.) M N N m (kr) P n m (cos nm Rr (rh n )< S B F S(m, a) s 1 s 2 s m T(m, a) T 1 m T N (z) L n m 1 V^kr) I m Vector solution, p. 5 Eigenvalue of qp equation, p. 5 Start of summation, p. 24 Vector solution, p. 5 Eigenvalue of 9 equation, p, 5 Outward normal unit vector Bessel function of second kind, p. 25 >) Associated Legendre polynomial, p. 6 Poynting vector for (n,m)th. mode, p. 59 Radiation resistance, p. 60 Longitudinal derivative, p. 7 Spherical Bessel function Series, p. 24 Real Part, p„ 26 Imaginary part, p. 26 Series, p. 29 Series, p. 24 Series, p 29 Tchebyscheff polynomial, p„ 86 Normalizing factor, p. 11 Pattern function, p 81 Mode vol tage, p 7 Series, p 72 XI GLOSSARY OF SYMBOLS (Cont. ) V m (kr) Mode voltage, p. 18 z Unit z vector Z Bessel combination, p. 25 Z£ Wire coefficient, p. 43 1. INTRODUCTION The dipole antenna, one of the most useful and simplest of all an- tennas, has been and is still an object of interest to antenna theore- ticians. In all but the elementary geometrical shapes, this antenna poses a cryptic problem. Recently the fat stub antenna, a dipole or monopole whose transverse dimensions are comparable with the wavelength, has aroused some interest as a possible instrument for radiating energy over large azimuthal angles. By varying the shape and size of the mono- pole and by changing the feed geometry, wide variations in radiation patterns can be obtained. Electrically rotated search beams are another possibility as will be mentioned later. Spherical antennas have received much attention in the literature. Stratton and Chu^-'-'' and Schelkunof f ^' have treated the uniformly excited or rotationally symmetric case, finding solutions for the pattern and radiation conductance. A spherical surface with external wires has been studied by Feld'"^'. DuHamel'^' has treated a system of currents flowing on the sphere, from the point of view of synthesizing patterns. A novel continuation of Stratton and Chu's work has recently been made by Karr'^) ] who considers a sphere split along a circle of latitude not the equator, and uniformly excited. All of these, however, yield far field distri- butions that are invariable about the azimuthal angle and involve only a simple set of modes. (1) See numbered reference in the Bibliography. All other such numbers used throughout this manuscript also refer to numbered references in the Bibliography. 2 It is the purpose of this paper to examine in detail the most prosaic prototype of the fat stub antenna, the spherical dipole with asymmetric feed. One may think of this antenna as being made by taking a metallic sphere (hollow), removing a narrow strip of metal around the equator to divide the sphere into two hemispheres, and then applying an electric field in the narrow gap. If the electric field is a function of the azimuthal angle, then the excitation is said to be rotationally unsym- metric or just asymmetric. 2. THE GENERAL ANTENNA PROBLEM 2.1 Introduction In attempting exact solutions of antenna problems, Maxwell's equa- tions in differential or integral form are usually used as a starting point. The former are for a homogeneous and isotropic medium m 9E curl E = -u. — , curl H ■ e r-. (2-1) The divergence equations have been omitted. These equations are equiva- lent to the vector wave equation in E (or H) obtained by the elimination of H (or E) in Eq. 2-1 3 2 E - curl curl E + ^e — — = . (2-2) at It is often easier to deal with the vector Helmholtz equation obtained from Eq. 2=2 by assuming a monochromatic time dependence eJ wt : curl curl E + k*E = (2-3) where k = w [ie . A vector in space is determined by its three scalar components and clearly the problem would be immeasurably simplified if the solutions to the scalar Helmholtz equation, about which a great deal is known could some- how be used. Since Maxwell's curl equations include div E - 0, E can be written as a curl, i.e., as a solenoidal (transverse) vector. The latter is determined by two scalars only as the div E = condition imposes an additional linear relationship. In certain cases, the vector Helmholtz 4 solutions are found by taking the curl and curl curl, respectively, of a suitable unit vector times the scalar Helmholtz solutions. The cases in which this is possible are six: the six coordinate systems which are listed below^o). j n three other cases, (spheroidal coordinates and rotational parabolic coordinates) if the cp variation is absent, (m = 0) the vector equation can be reduced to a single scalar equation which can be solved. For other systems, solutions to the equation might be found but fitting the boundary conditions is almost impossible. The separable systems are: cartesian, with any axis the preferred (longitudinal) axis, spherical and conical, with the radius the preferred axis; circular, elliptic, and parabolic cylinder, with the z axis preferred.* Solution of the vector equation is then reduced to finding scalar solutions and operating on these scalar solutions to give the vector solutions. It is well known that the scalar Helmholtz equation can be solved in any co- ordinate system which allows separation of the partial differential equation into three ordinary differential equations, each depending on only one coordinate. There are eleven such systems^'' and the six systems mentioned above are among these eleven. 2 2 The Spherical Antenna The system of interest is the spherical coordinate system, since the boundary conditions are to be applied over a spherical surface. The orientation is shown in Fig. 1. If the scalar Helmholtz equation is written in the form V 2 f ♦ k 2 f - or V 2 g + k 2 g - 0, (2-4) Figure I. Spherical Coordinate System two independent vector solutions of Eq. 2-3 are given by (8) Mr = curl a rf, N_ = f- curl curl a rg _1 _x _g b _I (2-5) The components of these vector solutions along the coordinate axes are: N. M r " n(n + D< k I W J 91 N< r sin 9 3qp = JL % kr 3r39 N n - 1 9£ r 80 JL (2-6) kr sin 3r3cp For one set of fields let E = Mf; then H = j/r\ Nf with n, a wu./k = -T\jje The other set is taken as E - N„ and H = j/r\ M„. These are seen to be transverse electric and transverse magnetic, respectively. The scalar solution has two separation constants or eigenvalues n and m and is of the form g(r,9 f (p) = h(2)(kr) t^U^P^Ccos 9) cos mcp (2-7) + B nmUn m Pn m ( cos 9 > sin m( P^ or f(r,e,cp) - h(, 2 )(kr) [C nin U n in P n ,n (co S 9) cos mcp ! " D nm U n mp n m ( cos 9 ) sin m ^ > where h^^^kr) is a spherical Hankel function of the second kind, repre- senting an outward travelling wave; P m (cos 9) is the associated Legendre polynomial of degree n and order m. General references on spherical Bessel functions are Stratton' "' , and Erdelyi' ^' . For associated Legendre functions, see Hobson^*', Erdelyi^"^, MacRobert^^', and Whittaker and Watson'^"*) Excellent tables are given by the National Bureau of Standards^^, *•"'.. The spherical Hankel function is related to the more common cylindrical Hankel function by h n (2) (kr) . -2-^Vy 2 (kr) Ttk] (2 8) n n.kr That these functions are simple may be seen from the expressions for the first two, which are h <2)( x ) = ?i e -Jx h ,(2) (x) ■ z2 e jx (1 . J } 7XX Similarly the associated Legendre function is related to the ordinary I • "endre function by P n m(co. 9) - sin" 9 ^ P n (cOS 6) d (cos 9) m (2-9) U n ra is a normalizing factor which will be defined later. The transverse components of electric and magnetic field, called E T , ftp, can now be written in terms of vector normal mode functions e K (0,qp) and mode voltages V x (kr): h - Y_v* (kr) e K (0, sin **] (2-15) where frhj' - f [rh^Ckr)] (2-16) or It can be shown that for TE modes, V'(kr) = *'V ' V(kr) (rh^ while for TM modes, V'(kr) -k 2 rh ? V(kr) 6 2.3 Evaluation of the Mode Voltages From the Uniqueness Theorem^ 1 ' ' for electromagnetic fields, the exterior field is completely specified by tangential E over the surface of the sphere, Over the conducting portion, the tangential electric field is zero. Thus, specifying tangential E over the gap or aperture uniquely determines the field outside the sphere. The basic method here is to assume a suitable distribution of E tan in the aperture, to express the external and internal fields in terms of the assumed distribution, and then to make the two forms for H tan agree at the gap. For a narrow gap, the actual distribution of electric field (except for the distribution with respect to qp) in the gap does not appear to affect the performance of a dipole antenna much. It will be assumed that the electric field in the gap does not vary with z, and that it has only a z component. For narrow equatorial gaps, E z - Eq. Broadly speaking, the external and internal fields are found in terms of the gap field by equating the assumed distribution to the general series form (Eq 2-10), multiplying both sides by one mode, integrating, and using the orthogonality of modes to reduce. This gives a result of the form V q (kr) - P^| (2-17) where q is one mode of x. This procedure will be followed in detail in the next few pages. Call the assumed gap field Eg = Eq(cp). A convenient method uses the Lorentz Reciprocity Theorem^ "''. This theorem relates two electromagnetic fields E 1 , Hi and E 2 , H 2 of the same frequency in space by an integral over the surface enclosing the source free region, G. / G Ei * H 2 -n dS = ^G?2 x Ht'n dS . (2-18) Here the region is taken as that enclosed by the antenna surface of radius a and a sphere of large radius r, r - * 00 . The fields must satisfy the radiation condition^^' at infinity Lim r[n * H + 1 E] = (2-19) r -oo - - n - Let the antenna surface be S x and the surface of large radius be S 2 . Then the integral of (Ei * H - E 2 x Hi ) over Si_ must equal that over S 2 . Now let r vary. Since the integral over Si has not changed, that over S 2 must also be a constant. If the limit of the integral over S 2 is taken as r-°°, and the radiation condition used, zero results. Thus, the constant value is zero. This leaves the integral over the surface of the antenna plus gap; however, due to the perfect conductivity of the sphere the integral over the gap surface, S, alone remains. Let the field corresponding to the assumed distribution of tangential E be subscripted 1, whereas that subscripted 2 will be one TE (or TM) 10 mode. The single mode field may be further split into even and odd terms because the sin mqp, cos mqp factors are orthogonal. H ± is unknown: thus, the general series expression (2-11) must be used^ u '. Four cases are to be considered and each case should yield one set of mode voltages. One of these cases will be examined in detail. Case 1 E 2 , H 2 are TM even. h = V mn£nm> H 2 = i V^ a r x e nm (2-20) with the fields being evaluated at r = a. The left hand side of Eq. 2-18 is 2 which reduces to . 2 ^ " ^ V nm C J e *pQ ^'fnm sin ed9d ^ (2-21) The right hand side of Eq 2 18 is R" 5 = a2 C ! o V nm4 !nm x Z Z (a r X !ji^r Vji sin edGdcp which simplifies to RHS " — V nm CC II V ji £jx £nm sin 9d0d * . (2-22) nk j j It is now possible to interchange the order of summation and inte- gration since the series of Eq. 2-22 is uniformly convergent in 9 and qp. 11 It will be shown that the vector mode functions are orthogonal: f o ; o^i°fnm dS = r 4 ; o V[U n P n mcos "»] "VtUj^cos i The normalizing factor U_ m is defined as the square root of the normal ization factor for the associated Legendre polynomials and is U n m (2n + l)(n -m)! (2 _ 26) 2 (n + m) ! 12 This factor has been used because of convenience in the computer sub- routine for calculating the polynomials; this code will be discussed later. The result of these manipulations is RHS = — V V ' 2nn(n + 1) (2-27) -nh nnrnm P K£ - *•*■ ' MK fc m with the final form for the mode voltages Vnm(ka) = 2 rtn(n m r 1) ^W^ !e'!n m sin 0d ^ • (2-28) Case 2 The remaining cases are identical in result (Eq. 2-28). The differences appear when the values for the mode functions and voltages from Eqs= 2-12 through 2-15 are inserted in Eq. 2-28. To recapitulate, the electric and magnetic fields in the exterior region were written as a summation of mode functions and voltages in Eqs . 2-10,11, and the mode voltages have been found in terms of an assumed distribution of tangential E in the gap, Eq. 2-28. Next, the interior fields will be handled in a similar manner. 3. THE RADIAL LINE 3 I The Radial Line Exciter The spherical antenna of the last chapter was excited by an electric field distribution in the equatorial gap. It is the purpose of this sec- tion to consider methods of producing various voltage distributions in the gap. Since it is usually desirable to feed practical antennas with a coaxial cable whose diameter is small with respect to wavelength, some sort of transmission line or waveguide is needed to transform from the cable to the large diameter gap. A number of cables might be used to feed the gap directly but this is an unwieldy solution. Eiconical and parallel plate transmission lines appear to be two feasible types. The biconical line is simple of solution only as long as the apex is central with the sphere. For this reason the parallel plate line was chosen to handle more complicated feed geometries. The parallel plate line consists of two perfectly conducting parallel plates, spaced a distance apart equal to the gap distance, located interior to the sphere and connected to the sphere at the gap. Thus any radial component of current flowing on the parallel plate line can flow through the gap out onto the surface of the split sphere. If the line is excited at one point, it may be called a radial transmission line. A cross section of sphere and radial line is depicted in Fig. 2. 13 14 Figure 2= Cross Section of Antenna with Radial Line 3.2 The Field Solutions Consider a radial transmission line with the orientation of Fig. 3. Using cylindrical coordinates and the techniques of the previous chapter, a vector wave equation solution is constructed from the scalar wave equa tion solution. Let the scalar solutions be f and g The most convenient vector solutions are then given by the following: *f curl a f -z N - 1/k curl curl a g -g -■ (3 1) M r - 1/k curl N, -f -f The components of these are 1/ 9f 1/r — dtp 9f dr M (3-2) N - 1/k '-f- N 1/kr f^f N -1/kr 3 -(r|S) dr dr -1/r ,d!g (3-3) II'- r <• is used as constant vector the z unit vector a , In general, how- ever, the divergence of unit vectors is not zero, and thus unit vectors ■>"■ not fixed, e g , in spherical coordinates div a - 2/r — r 15 A more usable form of the last is N z = 1/k [k 2 g + -£] , (3-4) The scalar solution is well known: f ~ Z (/k - 3 r) [£ cos (3z + C sin (3z] [A cosmcp + B sin mcp] where Z is a linear combination of J and N Bessel functions. m Z=b Z=0 r ? Figure 3= Parallel Plate Regi on The separation constants are 3 and m. The complete solution is a double summation over the possible values of (B and m„ The two vector solutions give rise to two types of fields; TE and TVL For the TM modes, let E - 1/k-N , H = -J- M • "g - IK g while for the TE modes E ■ M f) H = - N . - n -f The f and g scalars differ only in the constants which are as yet unspee ified. The boundary conditions require that the tangential electric 16 field be zero at the two metal plates. So for z^0,bE (p ^0 = E For the TM case this means that (£ cos |3z - £ sin (3z) = |z = 0,b From this it is seen that a C = 0, sin 3b - 0, and 3 - ~, I = 0,1,2,3, (3-5) b The other constants are not determined yet so B, may be put equal to 1 with- out any restrictive effect. The TE case is analogous, giving (£ cos Bz - C sin 3z) | = z - 0,b and £, = 0, C - 1, and 6 is the same as above. These values, together with the equations (3-3,4) give the field ex- pressions for the permissible modes in the parallel plate line. In full form these expressions are TM. 8 3 E_ = — — — Z_ sin 3z [A cos m

/ C/2; this criterion is the same as that for a plane wave between parallel planes. If i> = there is propagation for 2n;r/X > m, and it may be noted that this con- dition involves the radius; a wave may be evanescent close to the source and propagating farther out. These modes vary with the azimuthal angle. For the assumed distribution of tangential E in the gap, it can be seen from Eqs. 3-6 to 3-15 that (3 = 0; the equations reduce to a simple 18 triplet. The transverse fields (transverse to the radius) can be written in terms of mode voltages: E z = 2 V m (kr) $( Zqo of the lines, the source voltage, and the self and mutual impedances of the antennas (which are, of course, independent of the currents), find the base currents. The input impedance of each element depends on the current flowing 21 k Figure 4 Two Monopole Array in the other element Vi li L X x + I2 ^12 V 2 li Z 12 + 1 2 Z . (3=25) The base voltages and currents obtained from the transmission line equations are given by V = Vi cosh y^i ~~ li Z01 sinh y^i V = V 2 cosh y^2 ~~ I2 Z 2 sinh y^2, (3-26) Eliminating voltages to get an equation in the currents V - [Ix Z n + I 2 Z 12 ] cosh y^i - li Z 01 sinh y^i V = [Ix Z t , 2 + I 2 Z 22 ] cosh y^2 ~~ I2 Z 2 sinh y^2 (3-27) 22 These equations are solved simultaneously for the base currents which are used to calculate the radiation pattern. The solution of Eq. 3-7 is straightforward but tedious. In summary, the impedance of each element must be equated to the impedance seen into the line, and the equations then solved simultaneously. The next chapter will be concerned with matching the antenna impedance to the line impedance, or, what is equivalent, matching the magnetic fields at the aperture or gap. BOUNDARY CONDITIONS I.I The Transverse Magnetic Field The boundary conditions at the gap require that the transverse magnetic field be continuous across the aperture With the assumed distribution of tangential E, the radial line transverse magnetic field has only a cp com ponent. Hence, the boundary situation will be approximated by matching this component to the cp component of the antenna field From Eq , 2-11, the cp component of external field is J H, V( ka > a * £nm( 0( P) (41) Inserting the values for the mode functions and voltages gives cd n (UJTz: H cp = - J/^ /_ — m [\a cos m(p " B - sin m ^ n=l m=0 n(n + X) ni ka Ka P A m (Q) PA" 1 ^) rn 2 (rhj; P n m (0) P n m (A) ka h, (rh n ) a A - cos 6 at the gap edge. na (4-2) Modes here involve both n and m but the radial line field has only m modes Consequently, it is desirable to reorganize the double sum with the m sum outermost so that a term-by-term comparison could be made with the single series of the line. The sum is detailed in Fig 5 23 24 H, nm n=l m= 12 3 4 Figure 5 Sum By Rows This summation is equivalent to the following summation by columns. Note that this does not constitute a rearrangement, as the elements in any one row (or column) have not been rearranged. 00 03 H n l S - S u nm '-'oo m-U n=m » ► / VL ii \ ^ / — C / i / / / Figure 6, Sum By Columns To signify that the n = 0, m = term is not included, the notation used is 00 00 «• nm m-0 n=m where n starts at 1 if m = 0. 25 The magnetic field is then CO 00 Hq, - -j/n ZmlAn cos m( P + B m sin m ^ (U n m ) -D L _* n(n + 1) n = m ka h na P^CQ) p^(A) _ m 2 (rh n ); P n m (0) P n m (A) ka h ( rh n)a na (4 3) The inner series contains two terms; the summation of the first will be designated S(m,a), the second, T(m,a) The value of these series depends on a and m alone. Examination of the convergence and values of these will occur later. Equation 4-3 becomes oo H cp = ~JM / Z m ( S ~ T ) f\i cos m( P + B m sin m ^ • m=D Equating this to the radial line field (3-18) yields oo 3 = ) [Z m (S - T) -1/k ;£- ZJ [A,,, cos mcp + F^ sin mcp] (4-4) nFD which must hold for arbitrary radii a and coefficients A m , B m . This is possible only if each term vanishes. Thus k Z m [S(m.a) - T(m,a)] = §- Z (4-5) for each m = 0, 1, 2, ... It will be recalled that Z m is a linear combination of Bessel functions; since the A, B coefficients are yet unassigned, only one constant need be used in 7^, the ratio is the quantity of importance. Let Z m (ka) - w m J m (ka) + N m (ka). (4-6) 26 1.2 The Impedance Matching Function Equation 4-5 allows the TJ m coefficients to be found for given a and m. It may be recalled from Eq. 4-3 that the S(m,a), T(m,a) are, in general, complex. Calling the real and imaginary parts of (S - T) by S , S and the real and imaginary parts of TJ m by HT , w : S 1 = Re [S(m,a> -T(m,a)] S 2 = Im [S(m,a) -T(m,a)] (4-7) TS' = Re TJ m ^ - Im -0^ . (4-8) The solutions for -®~ are r » _ (SX- Jm)(SX-N m ) MS 2 ) 2 J m N n (S 1 ^- J^) 1 + (s 2 J m ) 2 (4-9) 2 S 2 [J:N m - J m N'] t^' = — -i — " m m a m 2 2- ■ (4-10) m (s 1 ^- Ji)" ^ (S"J n ) a It may be easier to find tsl in terms of THL . ' , m m yi. gl \-t -V s % ■ (4-ii) The quantity of main interest is Z which turns out to be Z (ka) - [J " (Sl " jS2) " J m^ J mN m ~ J m N m ) { ^ u) (s 1 ^ - J m ) 2 ♦ (s 2 J m ) 2 After replacing the Wronskian by 2/rcka, the final form is obtained. Z m ■■ -2- V Sl "J S 2 ) - Jm . (4 . 13) nka (s 1 ^- J m ) a + (s 2 J m ) 2 Primed Bessel functions indicate derivatives with respect to argume nt. I 27 The Z function is related to the relative proportion of radiation from one m mode, assuming that all modes are excited in equal amounts. The actual amount of each mode excited is dependent on the feed arrangement; i.e., on the A, B's. It is next necessary to arrive at some numerical values for the S(m,a) and T(m,a) series which are repeated here, oo S(m>a) . V (2n + l)(n-m)! ka h na P n "(0) P^ n^,* 2n(n + l)(n + m) ! (rh n ) a (A) (4-14) 00 T(m,a) I n=m (2n + l)(n - m)!m 2 (rhj' P n m (0) P n m (A) n"a n 2n(n + l)(n + m)! ka h n; * (4-15) The rapidity of convergence may be examined through the use of asymptotic formulas for large n. From Stirling's formula^'', 2m (2n + l)(n-m)! for large n. (4-16) 2n(n + l)(n + m) ! n 2m + 1 From Morse and Feshbach'™) > the asymptotic form of the Hankel function for large order may be found. I^ (2) (x) _2_ f2jl]n (e) -n + &c + Y 2 KJ >l7in x (4-17) and h n (2)(x) = [^H n ?i/ 2 (x) so that 42 h (2) (x) . 1 lTr2n]n (e) -n + fee + ft* (4 _ lg) x nI e [ x J Thus ka h na h (2) na < rh n>a ka h (2) n - 1, a - n (2) 4ia ka n (4-19) 28 The asymptotic formulas for the associated Legendre functions for large lues of index n are, from Magnus and Oberhettinger^ ^' : va (-l) m+1 2- n m+Y2 sin[(n + V 2 ) 9 + S& - H] S- P n m (cos 0) i5 2 4_ (4 . 20 ) 36 n e ^ sin ^ 9 (-l) m f2~n m - l/2 cos[(n + %) 9 + ^ - ?] >J tc 2 4 P n m (cos 6) (4-21) e "^ sin^ 9 These approximations are for n — °° and for 9 near n/2. Insertion of the foregoing asymptotic forms into Eqs . 4-14 and 4-15 for the series shows that the n tn term of both varies as 1/n plus a slow sign change; the latter produces convergence. These series depend on both a and m and therefore many computations of tedious nature would be needed. An algorithm is needed to ease this problem. An attempt was made to find a closed form for the series. The general term, involving the product of two associated Legendre functions bears resemblance to the Addition Theorem for those functions. The latter, however, contains a summation on m whereas the series at hand involves n. Another possibility sug- gested was the method of replacing the sum by a contour integral where each term is the residue of the function at a pole on the real axis. The trick is to replace this contour by one including the other singu- larities of the function than the poles on the axis. Often this pro- duces a much more rapidly converging series. This technique is elabo- rated i ii Appendix V ■ 29 4 3 Increasing the Rapidity of Convergence Another technique that is often useful in situations of this type is that of adding and subtracting a suitable convergent series to the series under study. The new series is added term by term, and subtracted as a sum. This new series is chosen to make the resulting series converge more quickly. To this end two series are defined. oo m n=m (2n + l)(n - m)! P n m (0) P n m (A) 2n 2 (n + l)(n + m)! 00 (4-22) n=m 00 00 y (2n + i)(n i m)! p n m(o) p n m ( A ) . y L ^~* 2(n + l)(n + m)' *—=- * n-m ^ v " L >\" »!/• n = m (4-23) Subtracting and adding these to Eqs. 4-14 and 4-15, respectively, produces S(m, a) = -ka J S m / nh na + , (4-24) T(m,a) = (r h n>a , - nh na (4-25) From Eq. 4-19 it can be seen that as n increases, the factor nh r *na (r 1 ka 1 5 i 6 i Figure 7. Real Part of S(m,a) \y + 3 ~+2 ^ £/ ^ + 1 / 1 ' ko 5 j 1 1 ] i Figure 8. Imaginary Part of S(m,a) 39 I 1 r i \. 1 L,^>- ^ 1 )\£- 4 i 2 5 ta « 5 6 41 - 1 m=0 1 m=( n L=2 m=3 f \ \ 1 i \ \ \ % k. "0 \ x- -T V \/ r\ 1 J <*# /// f - 1 / 1 1 <£ 1 Figure II Real Part of Z m (ka) I • ^ i \ £ ( 42 0.5 i I 1 / sp \ * \ — i i i j V_ \ i \ \ ! I i i \ \ i \ \ i i i 1 1 1 imc2 n *k - ■ \J .9 1 1 ' m>i -1 m.o -1 C ) 1 2 : i A I > 6 ko Figure 1 1 2. Imagir lary Pa rt of Z m (ka ! 5. OFFSET FEEDS 5. I Wire Feeds It is the purpose here to examine a simple wire feed for the radial transmission line and to determine the effects of locating the feed eccen- trically. Let the feed be a small wire of radius c carrying a current, I. Since both the wire diameter and length are small compared with wavelength; one may assume that the current is uniform along the length and around the periphery of the wire. Furthur, assume perfect conductivity of the wire so that the internal field is zero. Then^^' I = 2nc K (5-1) where the wire is centrally positioned. The general radial line field was given by Eq. 3-18. oo H^ = -j/n 2_ Z m [A m cos mcp + Bm sin m ^ ' (3 " 20) m=0 ^ is now a known function; the A, B coefficients are determined by com- paring these two equations for H™. It is evident that for this trivial case all EL are zero and also all A m except A Q . The latter becomes jr\l , „, d Z — where Z 00 ° 2nc Z^o uu d ki (5-2) Again the prime on the Z indicates differentiation with respect to argu- ment. From the above, the radiated field may be readily found. 43 44 5.2 An Offset Wire Moving the wire of Section 5.1 off center appears to be a promising type of feed. Let the wire be situated a distance, d, from the center. (See Fig. 13.) d Figure 13, Addition Theorem Coordinates The coordinates of the wire are now (r^qPi). The Bessel function Addition Theorem^ ' is useful for expanding the field in the coordinates (r,qp). The field of interest is that at r - a because the antenna impedance boundary condition must be satisfied at the gap. In this region, the condition for convergence of the first form of the Addition Theorem, namely, r > d cos cp is always satisfied and the series converges. This form gives the field as a sum of modes or waves originating from the center. Note that the origin is, and must be, excluded from the region of analyticity of the function as a spherical wave may come from the direction of the center but cannot start at the center, i.e., the Hankel functions have a singularity at r - 0. The first form of the Addition Theorem is oo /. n (krj) cos n

^ / \ /// XX / \ *^ x^ \. / /J\ / // y /iS jmS^ ka=2.05 ka=2.20 ka=2.52 q) patterns. polarization Figure 18. Patterns for LO Offset 54 / \ * * "«» f X / x /" \ x \ / >-"" /* "N * x \ / '*" X f I x^"*^ \ / ' \1 1/ * \ / y aA C^i \. — **■ / \ — ?jf \ yf >A / \ /* 1 \/ i \ V\ / 1 \ / Nw \ y // y // x^ \^s\\. ^£^^ kar2.83 ka=3.l4 ka = 3.50 ka = 3.76 q) patterns 9 polarization Figure 19 Patterns for ! Offset 55 ko = q 7i feM5.34 ka= 5 80 ka = 6.28 cp patterns, polarization Figure 20 Patterns for 1.0 Offset 56 Experimental measurements were conducted using the equipment and setup described in Chapter 8. The antenna was a hemisphere of spun copper and six inch diameter with a brass plate soldered across the base, The feed was a standard BNC fitting, with a 1/16 inch gap. Measurements were taken from 1 to 4 kmc and are the dashed curves of Figs. 18 and 19, Several factors limit the correspondence of these with the theoretical patterns. Several gap thicknesses were tried and it was found that at the higher frequencies the patterns varied appreciably with the gap dimension. Ideal ly ; this dimension would be as near zero as possible but the smaller the gap the more critical the antenna is to errors in spacing around the antenna base. Thus the aperture plate and antenna base must be perfectly flat and parallel, a difficult condition to produce. Due to the finite size of the receiving horns, the boom angle could not be 90°, the closest value obtained was 83°. This also yields a slight divergence. From the patterns taken, it appears that the gap introduces a fre- quency shift of about five percent. That is, the theoretical patterns corresponded most closely to those measured at a lower frequency. Variation of a typical pattern with the offset is shown in Figs. 21 and 22 where patterns are given for offsets of 0.0, 0,2, 0.4, 0.6, 0.8, and 1 times the radius. A smooth transition from the circular pattern at the center to the multi-lobed pattern at the edge may be seen- 5.4 Radiation Resistance The radiation resistance for the antenna configuration of the last sections is readily found by inserting the far field expressions into the 57 kd = kd=0.2 ka kd=0.4 ka kd = 0.6ka cp patterns, 9 polarization, 9 = 90°, ka = 2.83 Figure 21 Pattern as a Functicr. of Offset 58 kd-0.8 ko l (5-15) E * H 9 = Tiki m L 8 8 A ^ H " (rh n )(U n m ) 2[P n^ 2 ««' »* ■ The power flow integral is then p . r[h n (rh n ) A nm | ;2^ (Un m )2 {[ p n m, ]2 cos * m(p + _dC [P n m ] 2 sin 2 m Bessel function parts are equal, term by term 67 Unconnected equations result which are readily solved With a radial boundary, on the other hand, the cosine factors are fixed whereas the Eessel function factors differ in eigenvalue and cannot be individually equal; perforce, one must consider the series in toto Here then is the set of simultaneous transcendental equations 6 3 Approximate Solutions Since the problem cannot at present be exactly solved,, approxi mate methods are in order. One such method, valid for coaxial or radial lines whose ratio of outer to inner diameter is nearly unity, has been obtained by considering the transmission region to be a rectangular waveguide wrapped around laterally ^5) Another attack is that of Schelkunof f ^o; wn i c h replaces the Maxwell equations by a set of ordinary differential equations. If the parameters of the medium can be linearly approximated, then the set of differential equations reduces to a set of linear algebraic equations, a situa- tion not nearly so formidible„ It would probably be possible to attain a good answer through use of variational methods or Rumsey's reaction concept. The author leaves this as a worthy problem, A crude approximation which gives usable answers assumes the field in each sector to be that which would exist in a radial line completely filled with the dielectric of that sector. This is tantamount to neglecting the reflected waves at the discontinuity surfaces, and might be termed an optics or short wavelength approxi- mation The radial guide with two 180° sectors of dielectric will exemplify this approach Refer to Fig. 24. 68 Figure 2H Two Sector Guide The homogeneous field in region I would be, from Eq. 5-2 J zl ZTtC Zjqq (6-2) with the current I flowing in the feed post of radius c. For small c (with respect to A) the behavior of Z o will be inspected. From Chapter 4, Z ' - - 'TJToJi - Ni (6-3) 1 2 From that same chapter, W is given in terms of S , S These in turn go back to the S(m,a), T(m,a) series which must be evaluated for small radius, c With kc «1, h (2) (kc) ^ j (2n)! 2 n n! (kc) n+1 (rh n )^- j (2n)! 2 n (n-1)! (kc) n+1 (6-4) (6-5) 69 which allows nc ~-l . (6-6) (rhn)6 When these are inserted in Eq. 4 24, the series reduce to S(0,c) = -kc So , T(0,c) = . Thus S and S are, respectively, S 1 - -kc S , S 2 = . (6-7) The equations for U, (Eqs. 4-9, 4-10) now give the value s'No + Ni -GTo 1 = ~ "Ob 2 = (6-8) S'Jq + Ji Using Eq. 6-3 to get Z o in terms of Bessel functions and S's: S 1 (J.No - JoNj Z-oo ~ — (o-y) S'Jo + J a For small argument, Jo ^ 1, J! ^Y 2 kc, N x ~ ~ 2 jkc and Zq< S (4 + TtkV N ) n kc (2 S - 1) 2 2 For wires of small but finite size, K k c N <<; 4 and Zoo - 4 S Akc(4S -l) . 70 The constant part may be called zeta, 4S C C = , Z 00 = • (6-10) kc(4S - 1) rckc The electric field formula becomes jrukjl Zo(kjr) E z2 " ' 2£ (6 11) Since nk = wu., the e dependence enters only in the Z argument. The two fields are then given by E jri k I ZoCkjr) E = JT) k I Z (k 2 r) ^zl = "-"zl • 2 t 2 C These step function components can be expanded in a cosine Fourier series of the form oo e z = Y_ Zm(kr) ^ ° os m (3 ~ i8) m=0 with the coefficients A, Z m - ^ M Sin( ^ } [Z (k ia )^(l- [-1]-) Z (k 2 a)]. (6-12) me m £ 6 ^ A Twenty Sector Example. To offer an example that could yield practical answers, one can choose any large number of sectors in which to divide the circle, A convenient number is twenty: since the loading was to be symmetrical, thil ivs ten variables. (See Fig 25.) The plane of symmetry is represented by

n m (6-15) where the A nm , D nm are related to the P^ by Eqs . 2-14, 3-20, amd 3-33 Insertion of these into Eq. 6-15 yields, after rearrangement, (U n m ) cos mcp 2 2 - n m n(n + 1) m 2 h n [P n m (0)j na y„ f ka (rh n )' , [P n m (0)] kr (rh n )' (6-16) This summation may, like the summation of Chapter 4, be rewritten with the m sum outermost. Then the n sum, a function of m and a only, is called W for convenience,, » (U n m ) 2 fka (rh n )' w - y — ^— — — tp n m (o)] 2 - m Z^* n(n + 1)1 kr (rh n ) ' a n-m m 2 h n [P n m (0)] 2 na . V \ Z m W m cos m ® (6-17) (6-18) m=0 The radiation field (that field at great distance) is the only field of interest here soW m can be simplified by using large argument approxi- mations for the Bessel functions. (See Eq. 5 18). In the form given below, the constant factor kr eJ has been deleted. *m ' l_ (U n m ) 2 (-j) n fka[P n m (0)] 2 m 2 [P n m (0)] + J" n~~m* r '( n + 1) (6-19) ( rh n); na W m h.i , been computed using techniques already described, lor several ka and is shown in the set of graphs, Figs. 26 through 32.. For a given antenna size, ka, and for a given set of dielectric sectors, the A m Z m 73 / / / ?V V -" . -■"■ \ CJI e' to ii / / / <■*■ / 1 s > \ \ \^ y / 1 f \ v > X \ X s \ (0 E 3= / V J o It «J Qu <0 o Q£ to CM A '/ / . \ / CD U_ \l \ tf> in lO CM <£> to ro CO f T CVJ 1 to i sr in i to i &. 74 C\j i T CM 1 T i m i fc 75 / / / 2 m=5 / / / • 1 / s / / Q X / / / ^m=7 '\ j » / m=4/ -'i / / / -z \ \ \ / / ■I. -■> \ \ 1 / "3 \ \ \ / / 1 \ \ t / / -A ^_ / ) 3 4 5 6 Figure ; 28 Real f >art o1 f W m (kc 76 \ \ II t/ \ \ \ / / \ \ \ \ /- y i 1 i \ \ \ / / / / / ■ \ 1 / 1 / / / - 1 / / / / / / / / / / // ro CM i CM I ro i I in ir. >> c to M CM CD i_ =3 cn CM CO I 77 \ s \ \ N <0 it \ \ \ \ /CO \ \ \ \ \ \ \ \ \ v N \ \ N \ \ \ \ \ \ \ ^ / / -\ft- N \ 1 / / E 2 \ \ * \ \ v o L. a. \ \ L. c en E r<\ 1 CO - Csl- a) k. ro CVJ E i 79 o -"* II *"*~ *»» II (£> "■»» \ V m \ \ \ * IO *>i o (0 <0 <0 O Q2 CM CO -1 mc. ran 9mn (Not^a) [sin -sin 0] + • * •+ N (k l0 a) [sin sin mrc] } 10 10 (6-24) It will be necessary to adjust the constant to scale the solutions into the range of the No(kj L a) function which, for all practical cases, will be less than one in absolute value. The £• or |i^ are found from the k^ need- ed to fit the N Q to the values found from the Eq. 6-24 this set of equa- tions must be solved simultaneously for the ten N With high speed com- putational aids available, this is exceedingly simple; without such help the synthesis technique may not be possible The bracketed sine coeffi- cients of Kq 6-24 are listed in TabJe 8 Some latitude exists in the choice of antenna radius a As in all types of arrays, a given pattern will require more complex excitation (widely varying dielectric constant) as a decreases, resulting eventually in a supergain phenomenon where rapid large changes in e versus qp take placi A ten sector configuration would not approximate such a situation Probably a good starting value would be the radius of a circular 83 m = 3 .15451 . 14695 .13483 .13938 09082 .02367 .11061 00000 -.10701 .07102 -.09082 -. 14947 .02447 -. 14695 -06870 .02447 -.14695 .06870 .07102 -.09082 14947 .11061 .00000 .10701 .13938 .09082 -.02367 15451 .14695 -.13483 11888 . 10000 .07925 04541 -. 10000 -.12824 14695 -.10000 .00000 04541 . 10000 . 12824 11888 . 10000 -.07925 11888 -.10000 -.07925 04541 -. 10000 . 12824 14695 . 10000 . 00000 04541 . 10000 -.12824 11888 -.10000 .07925 m - 7 m - m - 9 .05779 .03674 .01717 .12572 -.09618 -.04982 .09000 .11888 .07760 .01991 -.09618 -.09778 .11341 .03673 .10839 .11341 .03673 -.10839 .01991 -.09618 09778 .09000 .11888 -.07760 .12572 -.09618 .04982 .05779 .03673 -.01716 Table 8. 1 miK sin 2m 10 Dielectric Sector Coefficients m(i-l )k sin 10 , i * 1, 2, , 10 84 array which produces a comparable pattern^ 4 ' Two examples will be worked out. The first is a cardioid, the sec ond an optimum fan h eam 6 6 The Cardioid. The cardioid pattern, |E| - 1 + cos cp, offers an elementary example. It is already written as a Fourier series of two terms, so immediately b = 1, bi - 1, b - for m > 1 Equation 6 24 becomes -V ImV.' -VfieWo * ic = -{N (k ia ) — + ••- + N (k 10 a) — } |W ' 2 10 10 ■ctq Im W s - T«J"o Re W i n = -%{N (k a a) [sin — -sin 0] + Iw-J 2 10 9TC + IM (k 10 a) [sin -sin Tt] } 10 27t 18Tt a {No(k.a) [sin sin 0] +••« + N (k 10 a) [sin sin 2k] } 10 10 (6-25) The desired pattern has been analyzed into Fourier cosine coefficients and these have been inserted into Eq 6 24 Next a normalized radius, selected such that the decrease of W m with increasing m is commen with the corresponding decrease of the b m In this example, 85 ka - 2 83 was chosen. Values of VV are then put into Eq 6 25 and the set solved with the answers of N (kia) - const x - 1106 N (k 2 a) = const x - 0975 N (k 3 a)- const. x -.0727 N (k 4 a) - const, x -.0386 N (k 5 a) - const, x 0014 N (k 6 a) = const x 0436 N (k 7 a) = const,, x 0837 N (k e a) = const x H78 N (k 9 a) = const , x 1426 N ik 10 a) - const, x 1556 (6-26) A suitable scale factor or constant is 3,81 The values calculated for 6^ are listed in Table 9- It would be possible to produce sectors with the proper dielectric constant by loading paraffin with titanium dioxide Sector Solutions k^a N Q (k^a) /e^ e^ i to 6 25 1 -.1106 4 89 -290 1 73 3 00 2 -.0975 4,73 -.256 1.67 2, 80 3 -.0727 4 49 -191 1.58 2.50 4 - 0386 4 22 -.101 1 49 2 22 5 0014 3 95 0037 1 39 1 94 6 0436 3 68 115 130 1 69 7 0837 3 42 220 121 1 47 8 1178 3 19 309 1 13 1 28 9 .1426 3 01 374 1 06 1 12 10 .1556 2 90 408 1 02 1,04 Ta^le 9 Dielectric Sectors for Cardioid powder ; or by perforating a suitable material with an array of small holes to adjust the dielectric constant However, satisfactory results can be obtained from a simpler arrangement Measurements were taken (see Chapter 7 for a description of the experimental apparatus) using a 180° sector of 86 polystyrene, dielectric constant of 2 5 This approximates five of the sectors of Table 9 by 2 5 and the remaining five sectors by 10 The average dielectric constant was then 1 75 which made the average ka equal to i 1 75 ka or 3 75 This corresponded to a model frequency of 2340 mc In Figure 33 is depicted the theoretical curve together with the measured curve Excellent agreement is noted except in the rear direction; prob- ably a more accurate mockup of the sector dielectric constants would improve this condition The 6 pattern, for qp - is shown in Figure 34 6 7 Fan Beams. For an example of shaped beams, an optimum fan beam with equal level side lobes will be worked out This pattern is a Tchebyscheff polynomial in the azimuthal plane and a fan shape in elevation Much space has been devoted in the literature to array synthesis and to the use of Tchebyscheff polynomials to optimize certain patterns References 4 and 38 contain ex- tensive and up to-date bibliographies Tchebyscheff polynomials are de fined by (39) Tj\(z) - cos (N arc cos - ) , \ z | <_ 1 T N (z) -- cosh (N arc cosh z) |z| > 1 (6-27) where it is convenient to let z c cos cp + d The amplitude of the func- t ion varies from H to 1 and back to H as cp goes from to 2k Oscillation • between the range + 1 which gives the function a "main beam" plus ,i ouaber of equal level "side lobes" For a given side lobe level 1/R " 1 >' the ratio of main beam to side lobes) the range of argument z is found ^ V / / magnitude of Eq is plotted versus solid line: theoretical pattern dashed line: measured pattern Figure 33. Cardioid Patterns of Spherical Antenna agnitude of £q is plotted versus 9 Figure 34. Elevation Pattern of Spherical Antenna 89 from z ~ cosh ( 1/N arc cosh R); c and d are found from c - % Uo + 1) , d - % (z - 1) . (6 28) The Tchebyscheff polynomials can be expanded in an exact finite Fourier series^'' of the form N T N (cp) = Y C N X cos icp . (6-29) Duliamel gives the coefficients for N up to eight. For the example, take N = 5. The C's are C 5 ■- 5d - 20d 3 + 16d 5 - 30c 2 d + 8c 2 d 3 + 30c^d Ci 5 = 5c - 60cd 2 + 80cd 4 - 15c 3 + 120c 3 d 2 + 10c 5 C 2 6 = -30c 2 d + 80c 2 d 3 + 40c 4 d C 3 5 - -5c 3 + 40c 3 d 2 + 5c 5 C 4 5 =- 10c 4 d C 5 6 = c 5 (6-30) Let R - 5 which corresponds to -14 db side lobes. Then z = 1-10695898 c = 1.05347949 d - 0.05347949 and the values calculated for the series coefficients, calling them b m now, are the following- 90 b = 0.46124418 b t = 0.92687461 b 2 = 0.86782692 b 3 = 0.77574232 b 4 = 0.65870596 b 5 ■ 1.29756889- (6-31) The pattern calculated from these b's is that of Figure 35 where the main beam to side lobe ratio of five is seen. The b values were inserted into Eq. 6-24 along with the functions for ka = 5.80 which was taken for a typical example. The answers and the calculations for £j are given in Table 10. Again rather than attempting to produce the exact dielectric constants, polystyrene and air were used. It was thought that the values of Table 10 might be approximated by four sectors of polystyrene, five of air, and one of polystyrene, giving again an average dielectric constant of 1.75 with the corresponding average ka of 7.66 and model frequency of 4800 mc. Since polystyrene has a dielectric constant considerably higher than that needed here, smaller sectors (in angle) are indicated. Measured patterns, using seven sectors of polystyrene in the cp = 0° direction and two sectors of polystyrene in the cp : 180° out of a total of 20 sectors, are shown in Figs 36 and 37. The beam has a ratio of 4 8 which is very satisfactory. Additional patterns for different elevation angles are also shown.. It is interesting to note that the theory demands dielectric loading in both the main beam direction and in the rear direction and that this requirement is borne out by the measurements., 91 Sector Solutions kia N (k ia ) i to Eq. 6-24 1 1609 7 86 199 1.36 1.85 2 .0972 7,35 079 1 27 1 62 3 0594 7.25 048 1.25 1 56 4 0883 7.33 071 1.26 1.59 5 0972 7.35 .079 1.27 1 62 6 -.0144 6,69 -018 1.15 1.33 7 -.1927 6.25 -.238 1.08 1.17 8 -.2566 5. 80 -.318 1 00 100 9 -. 1270 6 39 - 103 1 10 1,21 10 .0428 7.27 .053 1.25 1.56 Table 10 Dielectric Sectors for Fan Beam Tchebyscheff T 5 polynomial plotted versus cp Figure 35. Tchebysct.eff Fan Beam curve is measured value of magnitude of En versus 9;

o Si E o L. o o l_ o CO ■o c o u CD C7> CO CD en 98 24 inch D, , two inches thick and two inches deep, which is secured to the underside of the brass ring. The aluminum anulus is angle ironed to a B-29 gun turret ring and bearing assembly. The latter has two concentric formed rings, about a yard in diameter, with one ring supported from the other by a number of radial and thrust ball bearing races Figure 40 shows the underside of the screen, looking up at the ring assembly This type of construction allows the access space be- low the aperture plate to be completely clear Drive is provided by a General Radio 1/15 hp direct current motor through a Boston gear re duction box, using the large ring gear of the gun turret assembly for traction Position data are provided by a 1:1 geared Selsyn. Another aperture plate contains a square recess to accommodate aperture plates 14 inches square, Both plates fasten by flat head screws about 1 1/4 inch apart. The top surface of the screen and aper ture is flat and smooth A boom or swing frame carries the receiving horn. As may be noted from the photograph, the boom is pivoted at two gear boxes located on opposite corners of the screen. Vertical members are 2" by 4" lumber while the horizontal crosspiece is a composite hollow four inch square, glued and dowelled A universal mounting bracket allows quick inter changeability of antennas The gear boxes each contain a large worm •r to which the 2 by 4 is bolted, and a worm drive; one box also contains a Selsyn takeoff These gear boxes are connected by a shaft train which La offset to avoid fouling the aperture access, and which is 'irivr-n near its center Ijy another DC motor through a gear reduction box. 100 Ball bearing pillow blocks were used throughout to reduce friction To insure reliable operation in damp and cold weather, the motors and Selsyns were encased in sheet aluminum boxes, each heated by a light bulb, Adjustable limit switches prevent the boom and antenna from hitting the ground plane The crosspiece is adjustable in distance from the aperture in six inch steps with a minimum of four feet and a maximum of seven feet The control equipment will next be described. 7 2 Control and Electronic Equipment Power for the drive motors is supplied by a five ampere Accurate Engineering Co, dry rectifier for the armatures and a one ampere supply of the same type for the fields These supplies are also used by another pattern range A main control panel in the equipment shack (see Fig 41) contains speed adjustments (variable resistance in arma- ture circuit) and control switches. In this photograph, the racks on the left hand side belong to another pattern range Motors may be started and reversed from three locations main control panel, recorder console, and a switchbox on the ground screen, with relays providing the necessary interlocking Selsyn information from the ring or boom may be switched to a dial indicator, or to the polar recorder A Styroflex (low loss) cable and several other coaxial cables connect the screen and the equipment racks These racks contain several illators of the cavity type covering the range 300 to 3000 mc as well ;is Klystron power supplies used to pipe voltages over a special cable to Klystrons that might be located at the antenna terminals. 102 To date it has been possible to cover the entire range with adequate power using the Hewlett-Packard line of standard Klystron signal generators. A versatile switching arrangement allows any oscillator to be connected to a wavemeter or to the aperture. The received signal is intercepted by one of a set of horns and corner reflectors that cover the range 1 to 18 kmc . A crystal or bolometer de tector may be used, feeding the 1000 cycle signal to a PRD bolometer ampli- fier via a coaxial switch panel . A signal is taken from this amplifier to operate an external portable meter, and also the recorder. This latter signal is rectified by a synchronously-driven chopper and the resultant DC is fed to the recorder circuits. Both linear and square root response are available in this recorder (Leeds and Northrup), an unusual feature of which is that the servo amplifier gain is adjusted (on square root) by a potentiometer ganged to the pen shaft so as to maintain the servo loop gain at the low end of the scale This reduces the sluggishness of the pen near zero. To obtain patterns from an antenna mounted on either the square or round aperture plates, the procedure is simple. The plate is bolted in, then switches are set to connect a suitable oscillator. Next the proper horn and detector is mounted on the boom. Maximum signal is then obtained tuning the matching stubs, using the external meter. The aperture may be set to a desired

e (-D m^l 2 n| 71 4sin G (1-3) The approximation with respect to n is poor for n as large as 10 whereas the approximation is good for 9 near rc/2. A good approximation for both is gained by combining Eq 1-2 and Eq. T-3 to get p m (I-*) ±I!_ -v ( 1)^1 M35 (n 4 m ) in [(n + 1/)Q + m .. aj 12 4-6... (n - m - 1) 2 4 108 109 A typical case would have a gap angle of 1° or A = 1°. Then n+3m+l P n m '(A) - (-1) 2 l|3-5 (n + m) cQs (n + % )Q 1 2*4"6. . . (n - m - 1) (1-5) The general term of the series is then (2n + l)(n - m)! 2n 2 (n + l)(n + m) ! 1-1'3-S... (n + m) 1-2-4-6.. . (n - m - 1) cos (n + y 2 )° (1-6) where (n + m) is odd. For large n this becomes ^ cos n. If a zero gap had been assumed, the cos n factor would be absent and the series would diverge. The cos nA factor produces a slow sign change or alternation; clearly, the larger the gap the faster the sign change. A one degree gap corresponds to a sign change every 180 terms. Since (n + m) is odd> two cases may be treated. Case 1 n odd, m even. Let then a = (n ± % ) n (n + 1) 1-3-5... (n - 2 ) 1-2-4.. . (n - 1) F m cos (n + y 2 )° , m m ± l)(n - m + 3)...(n - l)(n + 2)(n + 4) (n - m + l)(n - m + 2) . . . (n + m) (1-7) < n + m) ^ (1-8) F = 1 F = (n - l)(n + 2) 2 n(n + 1) F = (n - 3)(n - Uln + 21lB + 4) (n - 2)n(n + l)(n + 3) etc. 110 ! a s e 2 Let th en m n even, m odd a = (n + Y ? ) n (n + 1) M-3-S. .. (n - 1) 1*2'4'6... (n) F m cos (n + Y 2 ) v.\o Ji + 1)(n + 3) ..(n + mUn - m + 1 Hn - m + g 3K . .tnll (n - m + l)(n - m 4 2)...(n + m)n (1-9) (1-10) Fi = iljL-L F„ = (n - 2)(n t D(n + 3) etr (n - l)(n)(n + 2) The general scheme for performing the calculation is to: 1) set up a table of cos (n + l A) in the memory, 2) form an iterative loop for com- puting ' 1-1-3-5. .. (n - 2) 1-2-4-6... (n - 1) using previous values. 3) set m, 4) form a loop for finding F m for the given m, 5) change m and repeat. The program for one of these calcu- lations is shown in the next several pages and in the block diagram of Fig. II. There is a possibility of sign change every 90 terms; the signs alternate in pairs, +,-,-, + , + ,-, etc. The terms have been combined in sets of two, one plus, one minus. Each set then contains 180 terms. The sets form an alternating series so the truncation error can be re- duced by adding only half of the last set. Using the ILLIAC, five sets (900 terms) were evaluated giving an accuracy of four significant fig- ures with the Las! figure in doubt. Cumulative error is believed to be than the truncation error o ._ & • 5 • g S o g o o m 9 **> s a. «- - ° E CO • u> ° • «. « E CM wo »1 >> « O ■O •o 1 o z M (A (A O <■> = CVJ OT3 >> c o a ( , O z N- >« « 00 2 N "8 *' 112 The code prints the first 25 terms (for later use) and the partial sums for each set, Table 1-1 shows the results. m 'in 1.6276 1 1.0558 1.1362 2 .8251 .9991 3 .6868 .9028 4 5896 .8283 5 5150 .7677 6 .4549 .7163 7 .4046 .6719 8 .3616 .6326 9 .3241 .5973 10 .2910 .5653 Table 1-1. Values of Sm and T m T m Series The T series was CD T i?" + }w" " m \\ P n m (Q) P n m (A) • (I'll) Z__* 2(n + l)(n + m)S n n m Following the same steps used before, a good approximation for both n and 6 is, for a one degree gap P n m (A) n - 3m (-1)~~Z~ 113 5— (n + m - If cos ( n + i/)0 12 4 6... (n - m) (1-12) The general term is, where (n + m) must be even. b i2!LJ l) (n - m) l ri 2(n l)(n + m)! 1 1 3 ■■ 5... (n + m - 1) 1 2 4 6... (n - m) cos (n + l A)° . (113) 113 This has been separated into two cases and computed in the same fashion as the earlier series. Results of this are also shown in Table 1-1. Program for S m Series - m even. 114 main code 0010K addresses 21 22 23 24 ~2?~ 26^ 27 28 29 30 31 32 33 34 36 37^ 38 "3^ 40 41 00 F 00 F counter 00 IF 00 F 00 90F 00 F 40F00 058505360000J 41 80L set switch B 50 25L 26 100F to Tl 50 26L 26 108F to Tn 51 82L 75 OF 00 IF 40 82L L5 79L 32 31L switch A LI 82L 40 82L 15 82L 10 3F L4 83L 40 83L Sn L5 81L L0 76L 32 35L is n > 53? 26 39L L5 81L L0 84L 22 44L 22 44L 15 84L L4 77L 40 84L inc print test L5 81L 10 29F JO 3F 50 40L 26 140F print n L5 82L 115 42 50 50 43 26 92 44, 92 %S 45 36 L5 46 L0 36 47 L5 LO 48 46 26 ^9*L5 LO 50 32 L5 51 L4 46 52^ 26 T.5 53 LO 46 54 I L4 55 40 22 "56 T5 L4 57 40 92 58 92 L5 59 10 26 60 JO 50 61 26 L5 62 50 50 63 26 22 -64J2 L5 65 L4 40 6F 66 L5 81L 42L L4 71L 170F print Tn 67 40 81L 129F LO 72L 513F 68 36 69L 80L 22 "22 26L 49L switch B 69* 69L 73L 22 OF 28L 70 00 512F 56L is add. = 200? 00 F 28L 71 00 1024F 78L 00 F 28L 72 00 901F 66 L 00 F 28L 73 75 200F 74L 00 IF 52L is add. = 289? 74 75 289F 28L 00 IF 78L 75 80 F 28L 00 F 66L 76 00 53F 28L 00 F 22L 77 00 4F 28L 00 F 79L 78 00 2F 75L flip switch A 00 F 79L 64L 28L first term code 22L 00100K 28L 133F 22 L 513F S5 IF 81L 1 L4 L 29F 42 6L 60L 2 LI 97F 3F 36 4L 60L 3 41 93F 140F print n 22 6L 83L 4 ""19 4F 6F 40 F 62L 5 00 IF 170F print Sn L4 F 64L 6 40 93F 513F 22 OF 80L 75L 80L flip switch B 116 ger iera 1 term 00108K 22 L S5 IF 1 L4 L 42 31L 2 L5 91F L0 97F 3 32 4L 41 92F _4J6 L5 28L 91F 5 L4 80F 40 2F 6 19 10F L4 91F 7 66 2F 7J 95F 8 40 IF 50 IF 9 7J 95F 40 92F 10 41 96F L5 91F 11 40 F LO 80F 12 40 IF L5 2F 13 L4 80F 40 14 L5 3F 96F LO 97F 15 36 28L L5 96F 16 L4 81F 40 96F 17 L5 IF 66 F 18 7J 92F 40 92F 19 L5 3F 10 IF 20 66 2F 7J 92F 21 00 IF 26 22L 22 40 92F L5 F 23 LO 81F 40 F 24 LO 80F 40 IF 25 L5 2F L4 81F 26 40 2F L4 80F 27 40 3F .26 14L 28 L5 91F L4 80F 29 40 IF LO 80F 30 66 IF 7J 95F 31 40 95F 22 OF outer loop 1 30400K 92 137F 92 512F 1 L5 86F 00 9F 2 40 86F L5 82F 3 00 9F 40 82F 4 L5 98F 00 9F 5 40 98F 41 97 F s '92 259F 92 706F 7 92 967F 92 706F 8 92 194F 92 258F 9 92 514F 92 194F 10 92 706F 92 97 5F 11 92 643F 92 967 F 12 92 51 4F 92 706F 117 13 92 967F 92 707F 14 L5 97F 10 29F 15 JO 2F 50 15L 16 26 140F 92 133F T792 513F 50 17L 18 26 10F 92 145F 19 92 513F 26 20L 20 L5 97F LO 98F 21 32 23L L5 97F 22 L4 81F 40 97F 23 26 6L OF F tape layout; 1„ Decimal Order Input 2. 0010K mam code 3. 00100K first term code 4<, 00108K general term code 5. 00140K P4 print code 6. 00170K P2 print code 7. 00300K Tl sin-cos code 8. 00400K outer loop code 9. 24 999N input par. tape parameter tape: 1. 0098K OOmF OOF 2„ 24 400N start program APPENDIX II A SHORT SUMMARY OF THE ILL I AC ORDER CODE The machine contains an accumulator, A, which is a double shifting register, a mul tipl ier - quotient register, Q, which is also a shifting register, and an order register. Registers A and Q are coupled together so that a number may be shifted any number of places from one to the other. In addition, one starts with a number in the accumulator and ends the same way. Multiplication and division use both registers the multiplier is placed in Q and the product (which has 79 digits) appears in AQ with the most significant portion in A the dividend is placed in A, the quotient appears in Q. The input to the machine is by standard five hole teletype tape with a photoelectric tape reader. The sexadecimal number system is used, for orders and numbers on the tape Output is by the same tape via a high speed punch or by a cathode ray tube console; a special program is provided for use with the latter. The high speed memory contains 1024 locations which are numbered starting at zero with each spot capable of storing 40 binary digits The registers also store a 40 bit number Orders are treated in the machine as numbers, hence, arithmetic operations may be performed on the orders This fact and the single ;i'l 0, same as corresponding 2 orders if A ^ 0, go on to next order replace number in memory at n by number in A replace location n and A by zero replace address of RH order at n by address of RII order in A same as 42 except both LH addresses replace Q by number in n, number in n is N(n) divide A + 2 39 Q by N(n) put N(n) times Q + 2~ 39 A into AQ multiply N(n) by Q rounded product of q and N(n) 00 n OF 10 n 19 n 20 n 22 n 24 n 26 n 30 n 32 n 34 D 36 n 40 n 41 I! 42 D 46 n 50 n 66 n 74 ii 75 ii 7.J n 121 80 n input n,: 4 sexadecimal characters from tape 82 n punch n/4 sexadecimal characters on tape K5 n put Q + 2~~ 39 into A S5 n put Q into A JO n replace Q by logical product of Q and N(n) F5 n put N(n) + 2~ 39 into A L0 n subtract N(n) from A LI n put - N(n) into A L2 n subtract |N(n) | from A L3 n put - |N(n) | into A L4 n add N(n) to A L5 n put N(n) into A L6 n add |N(n) | to A L7 n put |N(n) | into A A number of 92 output orders for printing special teletype characters are available, a few will be listed All letters and punctuation marks can be printed 92 963 space 93 131 carriage return and line feed 92 515 delay 92 259 letters shift 92 707 numbers shift 92 387 A or ) etc. APPENDIX III SERIES INVOLVING SPHERICAL BESSEL FUNCTIONS The two series to be calculated are oo *■ 00 K{^^\ (in-2) where the a , b are constants that were obtained in Appendix I, and are different for each n and m. Because the second calculation is almost identical to the first, only the latter will be described. If the spherical Bessel function of the first kind is denoted by j and that of the second kind by n , then the Hankel is h n (2) (kr) = j n (kr) - j n n (kr) (III-3) where the j without subscript is the operator of the Argand diagram. The real and imaginary parts of the spherical Bessel function fraction are Re nh na ir,(ka i_ i - n i_) + n.(ka n„ i - n n_ ) J -Hl JlLrl Jjil nl D-L_ ni (HI-4) [rh n ]M (ka j n _ 1 - n j n ) 2 + (ka n n _ 1 - n n n ) l m f " h na . ka ff (in .. 5) \ L rh n J a / ^ ka Jn-1 ~ n Jn^ + (ka n n-l " n n n' All spherical Bessel functions are evaluated at the argument ka. 122 123 Scaling is a serious problem here as the spherical Bessel functions cover a wide range of values. To obviate this, use has been made of a floating decimal library program. This routine is an interpretive code, i e , the routine in effect forms a new machine that handles numbers in floating decimal form. It has a floating accumulator (in the memory) and a new order code The problem is coded as usual except that the new order code is used and control is transferred to the subroutine The subroutine interprets each new order and follows the proper sequence of regular orders to convert from floating decimal to binary, add, subtract, scale, reconvert, etc. The limitation of these codes is their intrinsic slowness, by a factor of about 20 (50 for addition and 10 for division), Coding, on the other hand, is greatly simplified because the interpretive subroutine has internal provisions for handling red tape or bookkeeping processes. No library routines to calculate spherical Bessel functions were available thus, values were read from tables and punched on tape to be read into the machine The first step of the calculation, after the numbers have been read in and stored, is to compute Eqs III-4 and III - 5 These are printed as they are calculated After this has occurred for n up to ten, the terms of the series are formed and the series summed for each m These results are also printed out A block diagram is shown in Fig III- 1 and a typical output in Table III - 1 124 o ii E "w o o BE Q- a o o c Ii Q. O O i a>cvj M 4- 0)X> 32 « E c CO «0 (/i i_ en J* O O Ol ka * + 628 + 01 125 +999341 +00 +163271 +00 +995193 +00 +344473 +00 +980108 +00 +565257 +00 +927879 +00 +866987 +00 +724807 +00 +132496 +01 -893408 -01 +181255 +01 -114535 +01 +985087 +00 -817478 +00 +175458 +00 -486961 +00 +228330 -01 -325720 +00 +249976 -02 -270211 +00 +260340 -03 +00 -64 +895596 +00 +506870 +00 +20 +01 +109263 +00 +349327 +00 +40 +01 -753936 -01 +196348 +00 +60 +01 -109853 +00 + 560569 -01 +80 +01 -357072 -01 +892486 -03 +10 +02 -126242 -01 +751196 -05 +10 +01 +354415 -00 +448749 +00 +30 +01 +167808 -01 +290973 +00 +50 +01 -846435 -01 +137149 +00 +70 +01 -599906 -01 +816438 -02 +90 +01 -168156 -01 +832884 -04 Table III- . Typica il Printed Results APPENDIX IV WATSON'S TRANSFORMATION This material represents an exploratory investigation of Watson's Transformation, which is a method for summing slowly convergent series. This method, first used for problems involving propagation around the earth, consists of three separate steps. First, the series is replaced by a contour integral taken around a set of simple poles on the positive real axis. The residue of the integrand at each pole gives a term of the series. Second, the contour about the poles on the real axis is replaced by an exactly equivalent contour about a set of complex order poles, so-called because they occur for certain complex orders of the functions involved (here spherical Bessel functions), these poles are in the upper right half plane. Third, the contour integral is replaced by the sum of the residues at the new complex order poles. The resultant series is much more rapidly convergent. The method was used by Watson ^4JJ £ n 1917 or earlier, and later by Sommerfeld ^^^' However Bremmer appears to have been the first to make numerical use of the technique . ^o, 44.) Recently it has been used by Sensiper and Lucke . ' ^' The series of interest is S(m,a) given by Eq„ 4-14 and the simple case of m :: may be taken Then S(0 fi n = l (2n ' 1) ka h na P n '(0) P n '(A) 2n(n + 1) [rh n ] a ' (JV-1) 126 127 It will be shown that this series is exactly equal to the following contour integral, taken around the path of Fig IV-1 S = i / v ka h ^ P ^ (0) P ^ (A) (TV 9) J Ci 2{v - X)(v + %) cos vn [rh vJ/2 ] a UV "*' Sec vtt has simple poles on the real axis at vk = rc/2 371/2, . . . , or v - 1/2, 3/2, 5/2, 7/2,... Next the residue of the integrand at each of these must be found The appropriate Lemma from complex variable theory is Lemma Let a function G(z) have simple zeroes at z 0: z lj z 2/ etc... and let F(z) be finite at each of these Then the residue of F(z)/G(z) at any of these poles is simply F(z )/G'(z) , The proof follows immediately from the method for finding residues when only simple poles are involved Thus Residue of F(z)/G(z) = Lim [(z - z ] F/G] = F(z )/G'(z) L (IV-2) z-'Zo ' ° at Zn It can now be observed that the residue of Eq IV-2 times 2rtj yields one term of the series (IV-1) The sign change (-l) v ~' 2 has been incorporated into P^ [cos (71 - 0)] ~- P V ^(A) Next the different and unrelated contour C2 of Fig IV-1 will be considered, the integral around C2 is zero This path is along the imagi- nary axis and around a semicircle of radius R, the semicircle has center at the origin and the imaginary axis is the diameter R tends to infinity A brief outline of the proof will be given First the integrand of 128 v- plane Re v Figure U-l Paths of integration 129 Eq IV-2 must have no singularities on C2 This is true because for Re (argument) > 0, both the Legendre function and the spherical Bessel functions are entire Second, the integrand except for v is even. From the hypergeometric function representation of P v> P v _i/ (x) = P_j,_y (x), so the Legendre function here is an even function of v. Cos vn is clearly also even. For the spherical Bessel function term, let h„ 1/ h, F (v) - ^ = ^ _d_ dx d- [xh^] xh V-3/ 2 (V - Y 2 ) \l v J£ Now for any v, ^ly\y 2 (x) = e'3 vK h^f^ (x) which gives f(-v) ^ i^V^ H F(v) xh_ v _ 3 / 2 + (v + K) h. v . X e'J^[xh v _3/ 2 - (v - K) h v „i /z] and thus this factor is also even. A recursion formula was used in the last step. All the integrand is even except v which makes it all odd Contributions over paths A and B will then cancel. From the asymptotic expansions for large order of Legendre functions and spherical Bessel functions, one may easily show that as the radius of the semicircle of C2 tends to infinity, the integrand tends to zero. Then by Jordan's Lemma, the integral over the semicircle tends also to zero, The integral is now zero over path C2 It was mentioned earlier that the spherical Bessel function factor in the denominator is zero for certain complex orders; hence the inte- grand is singular at these v Note that the argument of the spherical Bessel function is fixed, and the zeroes (simple) occur for complex 130 order. Now the integral around C2 is zero so the sum of residues on the real axis must be exactly the negative of the sum of residues at the complex order poles, The transformation has now been effected, The resulting series is V" v ka h v u PI i/(0) P' i/,(A) S = -2ti / v n v /2 — v " ■ ■. (IV-4) V 2(v - Y 2 )(v + %) cos vk $- {[ T U i/]'} v m 3v v /2 a This series is exact; no approximations have yet been made. Some ap- proximation will have to be made though, because at present no means exist for locating the complex zeroes and for evaluating the derivative, with respect to v, at these zeroes. The single exception is that of large argument ka, If ka >> 0, then one of the several asymptotic formulas may be used. Bremmer has shown that zeroes occur only for large v which means that an expansion for large argument and order must be employed Watson and Sommerfeld used the Debye type approximation for large ka and v which is obtained by the method of Steepest Descents. In this method, the Hankel function is written as an exponential contour integral with the contour passing over two saddle points (two must be used for v near ka). Next the path is made that of the steepest descent so that only the region of the saddle top is of importance., The exponent is expanded into a three term Taylor's series about the point, and the integral can readily be evaluated The range of validity is |v-ka|< (3) ntfj O) (IV-6) Bremmer has shown that near the zeroes, the second term is slowly varying and the location of them is thereby controlled by the factor Hkr' ((3). '73 The first term of course is very small In the notation above, v - x y. 3 = 1/3(2t) 3 /^ e -J< 3 */ 2) (IV-7) (IV-8) .(2) (46) The H 2 / factor is zero for the following values, v * o; where Hi) m (IV-9) 132 m is a m is 0.6855 x m is 0.808 e J (n / 3) 1 3.903 2.577 e J (7l/3) 2 7.055 3.824 e J (Tl / 3) 3 10.200 4.892 e J (n/3) 4 13.345 5.851 e J (n/3) 5 16.488 6.737 e J (rc/3) Now that the zeroes have been located, the residue at each must be evaluated. Recourse must here be made to the Debye approximation; now that the more accurate Hankel approximation has been used to locate the zeroes, the derivative may be evaluated by the less accurate formulas. The result is 9. j hK-%K \ a -77-^" (IV- 10) dv |ka h„^(ka)J < ka > /s ' UV i0) The spherical harmonic P can be dealt with from the simple asymptotic form: d< r < mi ^ cos [y(rc - 9) - n/4] , Tlfll * ?l_y 7 [cos (rt - 0)] ~ . (IV-11) •FH sin 9 For positive Im (v) t in the cosine term the second exponential is pre- dominant. Similarly for cos vk. P' „ (A) - rZ e ~J(vn/2) - j(n/4) + jvA cos un ~ %e'i v1t . (IV- 12) 133 So COS VTC K The negative real part of the exponent means that the values of v near the origin are the most important These are inserted into Eq IV-4 to give ^~~ P' i/(0) P' i/(A) n \ 2 / 3 ^~~ e J2>rn S = -Tt ) *V^u; v v _W*) ^ _. (ka) ) f . (iv-14) for the final series Numerical values are easily obtained, using six terms and ka = 6 28 gives S = -3 2 - j 16 which is not close to the result obtained by summing the original series, S - -4 6 + j 32 Due to the small gap angle, the terms here do not de- crease rapidly, the sixth term is about five percent of the total This is a considerable improvement over the original series though, as hundreds of terms were needed there The reason for poor agreement is the value of ka of 6 28 This was the largest diameter sphere of interest here but the asymptotic formulas are just not accurate for this small an argument, especially when the zero point is of importance This points up the limitations and usefulness of the method The physical interpretation of Watson's Transformation is of inter- est Papas ^ 4 '' has shown that rigorous solutions to the Maxwell equations and the wave equation can be constructed from complex order wave functions 134 of the type h!, 2) (kr) P v (cos 9) m m where the v m are the complex order roots of f [rh v (kr)] a - or m These solutions satisfy the Sommerfeld radiation condition as they must. (2) The complex wave functions are orthogonal in an unusual way - the h v (ka) m are orthogonal over the various v m ! The P v (cos 9) are no longer orthogo- nal. These complex order wave functions are just the terms of the residue series of the Watson Transformation. Whereas the classical solution h n 2) (kr) P n (cos 9) n = 1, 2, 3, . . . represents a travelling wave in the r direction and a standing wave in the 9 direction, the complex order solution represents a travelling wave in the 9 direction (with Green's function source at 9 - 0), and a wave going in the r direction that is reflected at infinity; in other words a standing wave in the r direction at infinity. In some physical problems, the reflection coefficient at infinity may be zero, Note that only (2) ttjj (kr) can be used here in the classical solution as j n (kr) and n n (kr) do not satisfy the radiation condition. To avoid the circuitous path of Watson, a problem might be set up directly ID complex order wave functions A worthy project might be the calculation of the zeroes u for various ka APPENDIX V THE Z m CALCULATION From Eq- 4-13 it is seen that Z m is a simple function of S S , J m , J m ' The program to perform the calculation uses the floating decimal subroutine, first reading in the S functions, then the J functions After the results are printed out, a new set of functions may be read in and the calculation repeated. Several tables of Bessel functions were found usef ul ^° ' ^' » **" ' ^ ' Results are tabulated in Tables V-l and V- 2 16 -2184-02 -3718+01 -3319+02 -6711+03 31 -5677+01 -1910+01 -8851 01 -9250+02 47 -2380-01 -1240+01 -3927-01 -2684-02 63 -1277-01 -9025-00 -2155+01 -1112+02 78 -8193+00 -7042-00 -1412+01 -5900+01 1 05 -4842-00 -4798-00 -7868-00 -2438-01 1 25 -4003+00 -3733+00 -5977-00 -1481+01 1 s- 7 -3586+00 -2661-00 -3583+00 -7720+00 1 88 -1340-01 -2165+00 -2507-00 -4677-00 2 05 +7602-00 -2063+00 -2110+00 -3694+00 2 20 +8499+00 -2045-00 -1830+00 -3056+00 2 40 +5164 00 -2100+00 -1558+00 -2426-00 2 52 +3822+00 -2172+00 -1432+00 -2135^00 2 83 +2068+00 -2529+00 -1225+00 -158400 3 14 +1384+00 -3480 J 00 -1151+00 -1225+00 3 50 +1053+00 -5448+00 -1156+00 -9666-01 3 76 +9411-01 +5501-00 -1216-00 -8592-01 4 71 +9973-01 +9451-01 -3869-00 -7845-01 5 34 -2790+00 +6844-01 + 1909^00 -7177-01 5 80 -1359+00 +6883-01 +9203-01 -2081+00 6 28 -6249-01 +9459-01 +6265-01 -6273+00 Table V-l Real Part of Z m 135 m 4 b b I ka 0,16 -2164+05 -9644+06 0.31 -1539+04 -3540+05 -1043+07 0.47 -2937+03 -4449+04 -8654+05 -2427+06 0.63 -9112+02 -1031+04 -1493+05 -2639+06 0.78 -3906+02 -3581+03 -4170+04 -5947+05 1.05 -1209+02 -8197+02 -7109+03 -7528+04 1.25 -6114+01 -3477+02 -2532+03 -2249+04 1 57 -2534+01 -1145-02 -6621+02 -4674+03 1,88 -1266+01 -4817+01 -2318+02 1363+03 2.05 -9280+00 -3191+01 -1407+02 -7570+02 2,20 -7076+00 -2205+01 -9384+01 -4699+02 2.40 -5236+00 -1529+01 -5574+01 -2621+02 2.52 -4405+00 -1223+01 -4346+01 -1894+02 2.83 -2949+00 -7243+00 -2281+01 -8800+01 3.14 -2084+00 -4604+00 -1295+00 -4476+01 3.50 -1488-00 -2239+00 -4914+00 -1374+00 3.76 -1179+00 -2181+00 -5039+00 -1433+01 4.71 -6640+01 -9387-01 -1705+00 -3764+00 5.34 -5828-01 -6317-01 -9908-01 -1901+00 5 80 -5839+01 -5254-01 -7148-01 -1252+00 6.28 -6620-01 -4780-01 -5420-01 -8616-01 ka 10 16 - - - 31 - - - 47 - - - 63 - - - 78 -1004 07 - - 1.05 -9424+0 5 -1364-07 - 1 25 -2363-0 5 -2865+06 - 1 57 -3900+04 -3761-04 -4113+06 1 88 -9378+03 -7616+04 -6940+05 2.05 -4818+03 -3546+04 -2961+05 2 20 -2783+03 -1906*04 -1510+05 2 40 -1419+03 -8901+03 -6327+04 2 52 -9748-02 -5814+03 -3933+04 2.83 -4017+02 -2127+03 -1277+04 3.14 -1833+02 -8708+02 -4696+03 3.50 -457 5+01 -1769+02 -1464+03 3 76 -4842 01 -1902+02 -8490+02 4 71 -1029^01 -3039*01 -1063+02 r > 34 4301+00 -1148 01 -3489+01 5 BO -2567-00 -6214+00 -1716*01 6 28 -1601 +00 -3495*00 -8310+00 Table V-l Real Part of Z„ (conto ) 137 16 -5955-01 +2605-02 - - 0.31 1165+00 +8950-02 +2728-03 _ 0.47 -1757-00 +1754-01 +9565-03 - 63 -2341+00 +2533-01 +2094-02 +1244-03 0,78 -2872+00 +3029-01 +3635-02 +2841-03 1,05 -3811+00 +2989-01 +7143-02 +8605-03 1.25 -4586+00 +2185-01 +9802-02 +1587 02 1 57 -6665+00 -1683-02 +1268-01 +3192-02 1,88 -1171+01 -2873-01 +1252-01 ^5043-02 2.05 1013+01 -4112-01 +1092-01 +6003-02 2.20 -3343+00 -4955-01 +8537-02 +6713-02 2,40 -2947-02 -5767-01 +4296-02 +7353-02 2 52 +4399-01 -6185-01 +1245-02 +7516-02 2 83 +6625-01 -7773-01 -6719-02 +7010-02 3 14 +6420-01 -1359+00 -1215-01 +4998-02 3.50 +5840-01 -1173+01 -1444-01 +1077-02 3.76 +5500-01 -6163-01 -1581-01 1914-02 4,71 +8560-01 +3165-01 -1788+00 -5642-02 5 34 +5765+00 +2886-01 +2520-01 -6502-02 5.80 -5775-01 +3687-01 +2080-01 -5931-01 6.28 -3583-01 +6886-01 +1791-01 -1191+00 m 4 5 6 7 ka 16 _ „ „ =. 0.31 - - - - 0,47 - - - - 0,63 - - - - 0.78 - - - 1.05 +7221-04 - - - 1.25 +1726-03 - - - 1.57 +5088-03 +7241-04 ■ — 1.88 +1042-02 +1806 03 - - 2 05 +1483-02 +2590-03 - - 2.20 +1872-02 +3490-03 +6287-04 - 2.40 +2572-02 +5871-03 +1183-03 - 2.52 +2986-02 +7297-03 +1320-03 - 2,83 +4017-02 +1229-02 +2696-03 +6019-04 3 14 +4814-02 +1860-02 +4922-03 +1011-03 3 50 +5220-02 +2167-02 +5697-03 +1274-03 3 76 +4724-02 +3218-02 +1211-02 +3428-03 4 71 -5109-03 +3520-02 ^2683-02 +1180-02 5 34 -2336-02 +1403-02 +305402 +1971-02 5 80 -2443-02 -4051-03 +2559-02 +2437-02 6.28 -5356-02 -1160 02 +1297-02 +2589-02 Table V-2 Imaginary r Part of 1„ 138 10 2.83 - - - 3.14 - - - 3.50 +3219-04 - - 3.76 +7179-04 - - 4.71 +4183-03 +9496-04 - 5.34 +8261-03 +2713-03 +7981-04 5.80 +1261-02 +4916-03 +1551-03 6.28 +1738-02 +7974-03 +2545=03 Table V-2. Imaginary Part of Z m (Cont ) APPENDIX VI THE ASSOCIATED LEGENDRE CODE The ILLIAC program to calculate associated Legendre polynomials (and the auxiliary program to calculate the derivative) is arranged as a stand- ard closed subroutine so that it can be used by others. Entry is made with the parameter n in the link and the angle 0° * 2~ *■* (as a left hand address) in the accumulator. The library sine-*cosine and square root codes must be appended to this code. Also 1 £ n ± 10 due to scaling difficulties. The routine is vacated with e m /4 times the normalized associated Legendre polynomials P n °(cos 9), P n i (cos 9), .... P n (cos 9) in the preset locations S3, 1S3, . . . , nS3. Scaled this way, the functions are always less than unity. To avoid singularity difficulties at 9 0, calculation is first made of the R functions, defined here: P n m (cos 9) = sin m 9 R n m (cos 9) . (VI-1) Initial values for the R function are first calculated, then used in the three term recursion formulas to generate successive functions. The formulas are r m = y 2 | (n + l A)(n -%)... (Yi) (VI-2) >n-l Hg- 1 *M B n „! cos (n + %)(n - X)...(X)n (yi _ 3) 2n! U1 6) 139 140 R-2 . £ j (m-_l) pos 9Bg-l _ l/2 sin2 r (n+ m )(n-m + U _ or ^ m - 2 M(n +m-l)(n-m + 2) V (n + m - 1) (n - m + 2) ta (VI-4) After the set of R' s has been stored, the P's are found by multiplying by sin m 9. These new values then replace the R' s in the store. The auxilliary program calculates the derivative functions, times e m- 1^2 • Utilizing the P n m code, it permits formation of the set of R's, then finds the derivative from well known recursion formulas involving the P n m , finally returns to the original subroutine and stores the P n m . The equations involved are gp o -g- = -2~ 5 sin 6 ^n(n + 1) R^ 39 ^ = 2- 4 sxn- 1 39 -^ — nI (n + m) (n e m-l A block diagram will be found in the next pages along with the codes themselves . Normalized Associated Legendre Polynomial Code 40 F S5 IF 1 L4 L 42 98L link 2 10 20F 42 102L 3 22 3L 51 F 4 00 12F 75 1061. 5 00 IF V2 9 50 5L 6 26 112L to sin-cos 40 107L 7 L5 F ^40 108L 8 L5 105L L4 103L 9 42 91L set add. 42 93L 10 L5 105L L4 102L 11 L0 103L 42 35L f - w ^ M O I UJ / II ™ a> a> o o 5 •S c V) - il CM CV-. ^ + C N i c II • i N + *■§ o ^ + >T3 M T5 O o i li o M z CO N ._ s Id ^. 0) o ■*■ >- IO + cu ._ «n "> S — w — w O M £ u « c c > ) i ,i ' a> -|N - + M r 0-. "* H II CO c UJ £ E >- o «- CO ■s o in « <^ i . O 1 z ' 1 CD «" !?. co 8 » o r code e sin- addre O 1 E-r c " « o,._ c «~s 8mm IO (A 1 1 1 V) Id 1 > 6 tr c | in a>£ a> £ — w "O *<3 <\~ J + §£ a. c u 1? " 8° .e o z 2§? o o = ■ o ° i i XJ O C_> O H 142 12 42 65L L0 103L 13 42 46L 42 73L 14 LO 103L 42 79L 15 L5 102L 42 30L 16 L5 103L 40 IF 17^ 40 2F "51 2F 18 75 IF 00 39F 19 40 2F L5 IF 20 L4 40 10 4L IF 21 10 IF LO 102L 22 36 23L ~23 22 "L5 17L 103L 40 IF 24 40 F "51 IF 25 75 F 00 39 F 26 40 IF i L5 F 27 L4 103L 40 F 28 LO 102L 32 29L 29 22 "L5 24L IF 30 00 2F 00 (n J ' 31 40 IF L5 2F 32 66 IF 41 F 33 S5 F 50 33L 34 26 142L 40 L11L 35 10 • I 40 (nS3 His + 3/ 21 38 r(K) n!2 39 DF n!2 n-36 to square root %\\r 36 L5 102L L0 103L 37 22 37L 50 37L 38 26 142L % U/2 00 13F 39 40 IF 51 IF 40 75 107L 00 IF 41 40 IF 51 IF 42 7J 111L 40 IF 43 L5 104L L0 102L 44 36 46L E n~l L5 IF 45 00 IF 40 IF 4