■ Warn itii I WW' HI HHHMHHmaRMi L I B RAR.Y OF THE UNIVERSITY OF ILLINOIS 621. 365 I i G55te no. 50-51 cop. 2 Digitized by the Internet Archive in 2013 http://archive.org/details/studyofnonunifor53mitt Westinghouse Electric Corporation Air Arms Division Attn: Librarian (Antenna Lab) P„ 0. Box 746 Wheeler Laboratories Attm Librarian (Antenna Lab) Box 561 Smithtown, New York Electrical Engineering Research Laboratory University of Texas Box 8026,. Univ„ Station Austin, Texas University of Michigan Research Institute Electronic Defense Group Attn; Dr. J, A„ M, Lyons Ann Arbor. Michigan Dr„ Harry Letaw, Jr„ Raytheon Company Surface Radar and Navigation Operations State Road West Wayland, Massachusetts Dr c Frank Fu Fang IBM Research Laboratory Poughkeepsie, New York Mr. Dwight Isbell 1422 11th West Seattle 99, Washington Dr. A K. Chatterjee Vice Principal Birla Engineering College Pilani, Rajasthan India New Mexico State University Head Antenna Department Physical Science Laboratory University Park, New Mexico Bell Telephone Laboratories, Inc. Whippany Laboratory Whippany, New Jersey Attn: Technical Reports Librarian Room 2A-165 Robert o Hansen Aerospace Corporation Box 95085 Los Angeles 45, California Dr„ D„ E„ Royal Ramo-Wooldridge, a division of Thompson Ramo Wooldridge Inc„ 8433 Fall brook Avenue Canoga Park^ California Dr u S Das gup t a Government Engineering College Jabalpur, M o P I ndi a Dr. Richard C „ Becker 10829 Berkshire Westchester, Illinois ANTENNA LABORATORY Technical Report No. 53 A STUDY OF THE NON-UNIFORM CONVERGENCE OF THE INVERSE OF A DOUBLY - INFINITE MATRIX ASSOCIATED WITH A BOUNDARY VALUE PROBLEM IN A WAVEGUIDE by R. Mittra Contract AF33 (616) -6079 Project No. 9-Q3-6278) Task 40572 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois ENGINEhKING LIBRARY ACKNOWLEDGMENT The author wishes to acknowledge the helpful discussions with his col leagues in the Antenna Laboratory of the University of Illinois and the sponsorship of Aeronautical Systems Division, Wright-Patterson Air Force Base, Ohio, through the research grant AF33 (616) -6079. CONTENTS 1 . Introduction 2. Development of the Infinite Set of Equations for H-Plane Bifurcation 3. Solution of Truncated Set of Equations 4. Numerical Calculations of Solutions of Finite and Infinite Set of Equations 5. Asymptotic Behavior of Higher Order Coefficients 6. The Edge Condition and the Proper Choice of C 7. Further Work Page 1 3 6 10 14 20 21 1. INTRODUCTION In formulating electromagnetic boundary value problems we are often led to an infinite set of equations. In most cases it is not possible to invert the infinite matrix and we are forced to resort to truncating the above to a finite size. We then solve the equations for a number of increasing sizes of the matrix and study the convergence of the solution. If at least the leading members of the unknown coefficients (which are usually of primary interest) tend to converge, we feel satisfied and assume that we have obtained a reason- ably good approximation for the leading coefficients. It is the purpose of this paper to show in the first instance that for a particular doubly infinite set of equations associated with the bifurcation problem in a waveguide, there is a conditional convergence of the solution, meaning that the solution converges to a different set of answers for every different choice of the ratio C = P/Q. P and Q are the numbers of equations from the first and second set, respectively, out of the doubly infinite set of equations. These equations for the bifurcation problem have been obtained by Hurd and Gruenberg , who have also presented an exact solution of the infinite set through the use of calculus of residues , The above set, because it has a known solution, is particularly suited for our purpose which is to demonstrate by inverting several finite size matrices that there results a conditional convergence when P -> oo and Q -> °o. Furthermore, we are also able to find the ratio of P/Q which yeilds the correct answer in the limit. In the second part of this paper we present a theoretical basis for choosing the correct ratio of P/Q when working with a truncated set. With this choice, when the size of the set is increased indefinitely while keeping the 2 ratio C constant, the solution does converge to the correct answer. It is shown through the study of the asymptotic behavior of the higher order unknown coefficients and the application of the edge condition, why only an unique choice of the ratio would make the solution asymptotically tend to the correct one and why otherwise an incorrect solution will result. In a separate paper to be published in the near future we shall apply the ideas developed here to other problems in a waveguide which cannot be solved exactly. We will again show that there is non-uniform convergence of the doubly infinite set and how using the ideas developed here one is able to choose the correct ratio of P/Q for the problem under consideration. 2. DEVELOPMENT OF THE INFINITE SET OF EQUATIONS FOR II-PLANE BIFURCATION The infinite set of equations which will be discussed here in connection with the problem of H-plane bifurcation in a rectangular waveguide have been derived by Hurd . We shall therefore skip the details and merely outline the procedure for their derivation. © ^0 X=b X = Q Figure 1. H-Plane Bifurcation of a Rectangular Waveguide The geometry under consideration is shown in Figure 1. Assume that the incident wave from the negative z-direction is a TE mode with the electric vector parallel to the edge of the septum. It can be easily shown that the only non-zero field components are E , II and H and that they can all be expressed in terms of a scalar function (ft = E and its partial derivatives. The problem can be stated in terms of the equation V xz 4> + ^ 4>= ° (1) and the following boundary conditions on (ftt a) (ft and V$ are finite everywhere in the region concerned except at the edge of the bifuraction at z = 0, x = b where \J (ft becomes infinite. b) (ft and SJ(ft are continuous in the subregions and at z = 0, c) (^behaves as an outgoing wave at large (z) apart from the incident field. 4 d)

0. 1/2 e) (j) satisfies the adge condition and hence goes to zero at the edge as r where r is the distance from the edge. V 0. It is easily verified that (j) and (f) expressed in the following equations satisfy the equation (1) and the conditions (a), (c) and (d) . The expressions are: _. a z , 7Tx< °S, . n7Tx n ,_ . d> = A sin ■ — + S A sin — e (2a) ^A a n=1 n a d> . = Z B sin —— e ^n (2b) ^B n b n=l , £ „ n7Tx ~^ n z ,„ , (h = 2 C sin — r- e n (2c) ^C , n jc n=i where A = amplitude of the incident field a =[( 2V - k 2 ] 1/2 n l a J P ft «[(^) 2 -k 2 ] V2 n77 2 2 1/2 277 k = '— r-, X = free space wave length. a ^ (3 and y' s are the mode propagation constants in the three regions. By applying the continuity conditions at z = 0, and subsequently equating the Fourier components of the resulting equations in the range < x < b and 5 b < x < c one arrives at the following doubly infinite set of equations after some manipulations (for details see Hurd): nTTb (-) P 2p77 S (A 6' +■ A ) sin — r— n=l n n B = ^ a 2 -P 2 n P n7Tb . 7Tb ^ A sin A sin — * a - p = a + (3 P-l, ....-oo (3a) n=l n p 1 p n7Tb . 7Tb ^ A sin A sin — 2 -2_ _L- = a q = 1, ....oo (3b) The set of infinite equations (3) has been solved exactly by Hurd. We shall however^ concentrate on the solution of the truncated set of equations for various combinations of P and Q where these are the number of equations from the set (3a) and (3b) respectively. We proceed to do this in the follow- ing section. 3. SOLUTION OF TRUNCATED SET OF EQUATIONS Consider the solution of the turncated set of equations n77b . 7Tb P+Q A sin — — A sin — 2 n a o - P n=l n p a i + Pp p = 1, 2, . ... P (4a) ntfb . 7Tb P+Q A sin — — A sn,n ■ — y, _n ?L_.— a a - v a. + v n=l n Y q 1 T q q = 1, 2, . ... Q (4b) In attempting to solve the set of equations (4) we recognize first of all that the determinant of the equation is of a particular kind which is called a double alternant. Written explicitly the determinant is A = 1 i 1 > y 1 1 a -P > 1 1 . 1 a i-e 2 ' 2 r 2 a 3" P 2 a -6 P+Q ^2 1 1 1 ) ) 1 1 a -6 ' 2 HP 1 ) a P+Q" (3 P 1 a rV a &i a p+Q _Y l a i"V 2 Y Q a P+Q~ >l 'Q (5) It is to be noted that we have grouped the factors sin n7TJj/a with the unknowns A . n It is observed that each element of the determinant is a reciprocal of the difference of two quantities, a and (3 or -y in general, only one of which, vi; 7 the a, changes as one goes along the columns whereas only (3 or y changes as one goes down the rows. Hence it' a > a , where r and n are two different subscripts, then it is obvious that the determinant A -> and that the zero is a simple one. This is effective to saying that (a - a ) for various combinations of r and n are the factors in the continued product expansion of the determinant. In a similar manner we observe that (S - (3 ) , (v - v ) and r n ' "r 'n ((3 - -y ) are also to be included in the product expansion of the determinant. We have to make sure, however, that all such factors are included in the product and that none is repeated. It is also obvious that the denominator of this expression for A must contain the factors (a - (3 ) and (a - -y ) for various combinations of the subscripts. The expression for the A developed on the basis of above arguments, is M,M-1 P, P-l Q,Q-1 Q,P n " (a - a ) IT ((3 - {3 ) II (v -y ) II (p - y ) yj , ,» , m n . . r m r n Y m Y n r m T n y m=(nntl),n=l m=n-f-l , n=l m=n+l , n=l m=l ; n=l A= (_1) M,P M,Q n (a -(3 ) n (a -v ) m n , m "n m= 1 , ( n= 1 m=i , n= 1 where M = P + Q; v is an even or odd integer depending on P and Q and as yet undertermined, although we shall not really need to find v. This is because in order to calculate the coefficients A we will have to find only the ratio r J of the determinants A /A where A is the determinant obtained by replacing the rth column of A by the column representing the right hand side of (4). [t is fairly straightforward to see that A r = A A(a r ^ - a ± ) (7) 'here A(a -> - a, ) is the determinant A with a replaced by - cl . We can ' r 1 r 1 therefore obtain through the ase of (6) and (7) an expression for A in the product form, which is sin (■ — -) ACa^ -> - a 1 ) A - — — — "Str- = A r — r— r , 77b. A sm ( ■ — ) P (a - P ) Q (a - v ) r-1 (-a_ -a ) M (a + a ) = A n * ?\ n . r \ • n ., - 1 " n n 1 _ (-a n -(3 ) (-CL-.-V ) _ ( a -a ) (a ■ n=l 1 n n=l 1 "n n=l r n n=l n a ) r P+Q P (a r~ p n ) Q (a r" V (1) ( r n ^n n (9) We also quote below for comparison and reference, the result arrived at by Hurd as a solution of the infinite set of equations. His expression is n (1) (-a ljtt ) n (a JL ,|3) Jl (a pY ) J1 U) (a r a);ri ( Wl ,p) n*(-a 1 ,y) x exp [ (2a/70 1 - a + bin(a/b) + cin(a/c) 1 ] (10) where for instance 00 *: d7T n (cj a) = n (a - <*>)(=-) e p " p=l P P?7 and the superscript (1) implies as before that the factor corresponding to p ; = 1 is to be omitted. 10 4. NUMERICAL CALCULATIONS OF SOLUTIONS OF FINITE AND INFINITE SET OF EQUATIONS In this section we shall present and compare the numerical values of the reflection coefficient R for certain choices of b/c, b/a (note these fix c/a) v etc. We shall see that even when (P + Q) is very large the expression (9) for a reflection coefficient yields different values for different ratios of P/Q. We shall then compare the answer for a particular choice of P/Q and see that it indeed converges to the exact value of R calculated through the use of (10). For the purposes of numerical calculation: we shall consider the particular case when all the mode propagation constants excepting a are real and a is purely imaginary. This is done merely for the ease of numerical work, and the conclusions regarding the convergence phenomenon reached in this case will still be applicable to the general case when the above condition is not true. For this case it is obvious from (9) that R=e i.e. I r| = 1. The angle = argument R is given by = arg R S tan F n=l K n It - 16 1j tan — n=l ^n P+Q 3 tan" n=l 1 6 (11) where a = j6 But since (3 2 2 / 2 9 (nTT/b) - k and 6 = ^ k (7T/a) , one can write .-16 • -l tan p— = sin b5_ 77 (n - — ) After similarly expressing the other arctans in terms of arc sines, one can rewrite (11) as 11 a& b6 c6 e P ; Q . -i -? ; , -i -j l . -i i - = 1] sin — S sin - S sin 2 n=l v/V - 1 n=l /2 7b.2 n=l / 2 ,c N 2 /n -(-) /n - (-) (12) For the purposes of numerical calculation (12) is more suitable than (8) A similar expression for the exact 9 has been derived from (10) and is given in the following | (1 . ^ in a _ c in a 5 7T a b a c 2 1' where ■^1.0, -8,(^,0, S (u ; v 0) = 2 [sin N m , 2 K 2 ^ 1 / 2 n n=N (n - b ) Calculations were done for 9/2 for the following of dimensions of the waveguide bifurcation a = 2.286 b/a = 0.313 hence c/a = 0.687 b/c = 0.456 and for \ = 3 cms: The results for 9, calculated using (11) are shown in the following table for various combinations for P and Q and are calculated to the nearest degree. The 9 values are compared with 9 calculated from (13). It is seen that the values of 9 converge to different numbers for different values of P/Q but that the value of 9 is not too sensitive for slight changes in P/Q as may be seen from the last four lines of the table in which 9 is seen to agree with 9 . It is also observed that there is a considerable deviation from the correct value for some choices of P/Q. The correct choice for P/Q TABLE 1 12 p Q P/Q (radians) (calculated) 8 (radians) ex (exact) 10 10 1 -42° 20 20 1 -42° 20 10 2 -18° 30 15 2 -18° -50° 8 17 0.471 -50° 10 22 0.455 -50° 20 44 0.455 -50° 10 20 0.5 -50° 13 seems to be in the vicinity of P/Q = 0.48 and to find it more accurately through numerical means one has to calculate the results to a higher degree of accuracy. In the following section we give an explanation for the behavior of the results displayed in Table 1 and a theoretical basis for t^e correct choice of P/Q. It will be shown that the criterion for the proper choice of P/Q is based on the asymptotic behavior of the higher order mode coefficients A } for (P + Q) large. We develop the necessary formulas for the study of the asymp- totic behaviors in the following section. 14 5. ASYMPTOTIC BEHAVIOR OF HIGHER ORDER COEFFICIENTS In this section starting from (8) we shall develop an asymptotic expres- sion for A /A for large P and Q. Let us rewrite (8) as a' = n r n=l p (1 - pr> q (1 - ^ p+q n n (i + -i-) (i) (1 + p-) (1 + — ) r n "n (1 -_) (14) where r7Tb A' = A r r 7Tb Since n£ b n77 c and n77 for large n, it is convenient to introduce some additional factors in the numerator and denominator of (14) and rearrange it as 1 n n=l a 1 - *r> n (1 - — ) (1 ) p -' V n ' P+ Q (r) a b , a c n x < n n=1 n * % (i + g-) r n n n=l f~> P + Q (1 + f* — n ■ -2- n=l (1 a a a b a c a ) n/i--f-> V 1 ' r n=l n=l 77 P+Q n (i n=l (1 P+Q -) n (i p n (i + :1 a x b Q ) n (i n=l 15 Now notice first of all that the factors inside the first two curly brackets tend asymptotically to constants as P and Q are increased indefinitely for a given r. This is because the factors like => 1 for large n a a - Q -i> and similar reasoning holds for other factors appearing inside the curly brackets. It will therefore be sufficient to study the behavior of the ratio of the products F, where a b a c p _£_ Q _£_ n (1--I-) n a --L-) V P = ^— 55L-. (i --JL-) (16) a a r P+Q -T- n (i L_) T n n=l md of a similar ratio of products with - a replacing a Let us rewrite (16) as a b a c f ' = " (1 ~- P n£l (1 - ~r- ) Q nSl (1 - -r- > (P + Q) r a a a b/7T a c/7T a b/77 P+Q -|- P r Q r (p+q) r n (i - — _ > n=1 " (17) 2 nd then use the following representation (see Magnus page 2) of l/r(z+l) here T(x) is the Gamma function of argument x. 16 lim m Z n (1 + -) (18) T(z+1) m -> oo _ n to recast (16) into its asymptotic form for large P and Q. We derive using (18) in (17) and letting P/Q = C(constant) a a a a -£_ r oo 77 a b/77 a c/77 X (1 + ^) r (1 + C) r (20) The next step is to study the limit of F as r and hence a becomes very large We have from Stirling's formula, z r (z) ~ -TJo for a lar g e z ( 21 ) z i/2 Using (21) in (20) and letting a -» r77/a we find -.rb.rb/a re re/ a F _* K -1 / • 4?2 (1 + F> rb/a "(l - C) rC/a (r) r r 3/2 C for large r ? P and Q, (22) 17 where K is a function with a limited upper bound lor arbitrarily large r. II is to be noted that we have used the fact that (1 - (a a/7T)/r) /sin a a has the r / r limit r7T/a as a -> rTT/a, Writing r ,rb rc.rb/a + rc/i • = ( — + — ) ,e obtain from (22) ^ (i + i) rb/a (i + o rc/i ^T 7 * a + !) rb/a (i + :V C/1 b c for P/Q = C, P -> oo , Q -> co and r large. Equation (23) gives the desired asymptotic behavior of F. It is not ifficult to show that the ratio of products P+Q — n (i +-1— ) . n=l (23) a--?-) a b a c p JL ' q JL n a ■+-£-) na + -f) n=l n=l = G, say, is the asymptotic behavior lim G . i . ra * ^ ra + ^ (1 + I,-i^ (1 + c,-i^ a a a a C Q-*«> IT r(1 + IT* W (1--4-) (24) 18 and hence tends to a constant as r becomes large. We can sum all this up and arrive at the limit K x (1 + i) rb/a (1 + O rC/a D lim A r ~ "372 * — crb/a „ b,rc/a f ° r large T > (25) P -»oo r (1 + — ) (1 + -) Q->*> be where K is a constant. Suppose now that we pick a C which is different than the ratio of b/c and let C > b/c. Calling X = b/c, we can write (25) after some manipulation as K l lim A ' ^ P -*0o r Q -> oo %$% ■ <§>* rc/a '« i«- '■ < 26 > Without loss of generality we can let X (= b/c) > 1, and show that C X + C X X > X X + X X C, since C > X > 1 and hence (£) X > ( I_±_C) or (i-i-C) X > 1, (28) P -»«w r Q -> 00 where "H < 1 . This shows that under this condition the higher order coefficients have an exponential decay with r unless T| = l which happens when C = X. 19 Similarly it may be shown that if C < X > 1, / 1 m r lim A ~ b for large r p _>co r r °/ 2 Q -* °° where b > 1, i.e.. that in this case the coefficient A has an exponential growth with r unless h = 1, which happens again when C = X. When C = X, then the coefficients A have an algebraic behavior for large 3/2 r and go to zero as 1/r This completes the study of the asymptotic behavior of the higher order coefficients for different ranges of C/X with X > 1. In summarizing we note; (i) A has an exponential growth for large r when X > C and an expo- r nential decay for C > X. (ii) A has an algebraic behavior and is 0(l/r ) for large r when C = X. In the next section we appeal to the edge condition given by condition (e) in Section 2 and show how the proper choice for C = P/Q is to be made. 20 6 . THE EDGE CONDITION AND THE PROPER CHOICE OF C The condition at the edge of septum requires that the field potential X = b/c. the expression for d(f) A/3z I at the edge z = has a bounded sum and hence violates the edge condition. Also when C < X, A 7 has an exponential growth and hence when sub- stituted in (29) j makes d A/9z go to infinity in a much stronger manner than z as z -^ 0. It is only when C = X and A / is 0(l/n ) for large n that -1/2 the sum of the series is 0(z ) as z -> as may be shown by following a method due to Hurd . It may be pointed out that in the exact solution obtained by / 3/2 Hurd A is 0(l/n ) fcr large n, as it of course must, n The conclusion is then that the proper choice of C = P/Q is C = b/c = X; only when such a choice is made and, P and Q are increased indif initely, that the solution converges to the one which satisfies the edge condition, and thus yields an unique answer satisfying the physical condition. This section will be concluded with one further remark which concerns the coefficients A , m = 1.2.... etc., m finite. From (15), (16) and (19) it is nr clear that even when P ->oo and Q -» oo the limits of A A etc. are dependent or C, i.e. j on the choice of the ratio of P and Q. Numerical calculations for A for various values of C have been made using the above equations and the results agree with those presented in Table 1. Only the choice of C - b/c yields the correct answer for A although the result for the leading coefficient A is not very sensitive in the vicinity of the correct choice of C. 21 7. FURTHER WORK Further work has been done along this line in connection with the step discontinuity problem in a waveguide. Although the exact solution of the step problem is not possible, the correct choice of P/Q can be made on the basis of asymptotic behavior of the higher ortjer coefficients. A detailed discussion on this problem will be the subject of a later report. ANTENNA LABORATORY TECHNICAL REPORTS AND MEMORANDA ISSUED Contr act AF33 (616 ) -310 "Synthesis of Aperture Antennas/' Technical Report No. 1, C.T.A. Johnk, October, 1954.* "A Synthesis Method for Broad-band Antenna Impedance Matching Networks," Technical Report No. 2, Nicholas Yaru, 1 February 1955.* "The Asymmetrically Excited Spherical Antenna," Technical Report No . 3, Robert C. Hansen, 30 April 1955.* "Analysis of an Airborne Homing System," Technical Report No. 4, Paul E. Mayes, 1 June 1955 (CONFIDENTIAL). "Coupling of Antenna Elements to a Circular Surface Waveguide," Technical Rep ort N o. 5, H. E. King and R. H. DuHamel, 30 June 1955.* "Axially Excited Surface Wave Antennas," Technical Report No. 7, D. E. Royal, 10 October 1955. * "Homing Antennas for the F-86F Aircraft (450-2500mc), "' Technical Report No. 8, P. E. Mayes, R, F. Hyneman, and R. C. Becker, 20 February 1957, (CONFIDENTIAL) "Ground Screen Pattern Range," Technical Memor andum N o. 1, Roger R. Trapp, 30 July 1955.* Contract AF33 (616) -3220 "Effective Permeability of Spheroidal Shells," Technica l Report No. 9, E. J. Scott and R. H DuHamel, 16 April 1956. An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report No. 10, D. G. Berry and J. B. Kreer, 10 May 1956. A Technique for Controlling the Radiation from Dielectric Rod Waveguides," technical Report No, 11, J. W. Duncan and R. H. DuHamel, 15 July 1956.* Directional Characteristics of a U-Shaped Slot Antenna,"' Technical Report ko. 12, Richard C Becker, 30 September 1956.** | Impedance of Ferrite Loop Antennas," Technical Report No, 13, V. H. Rumsey nd W. L Weeks, 15 October 1956. -- — — Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Repor t EL_Ai> Rich ard F. Hyneman, 20 December 1956. "Distributed Coupling to Surface Wave Antennas, " Technical Report No. 15, Ralph Richard Hodges, Jr., 5 January 1957. "The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical Report No, 16, Robert L. Carrel, 15 January 1957. "On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17, Walter L e Weeks, 10 April 1957.* "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source Antenna," Technical Report No. 18, Walter L. Weeks, 20 April 1957. "Broadband Logarithmically Periodic Antenna Structures," Technical Report No. 3 , R. H. DuHamel and D„ E. Isbell, 1 May 1957. "Frequency Independent Antennas," Technical Report N o. 20, V. H. Rumsey, 25 October 1957. "The Equiangular Spiral Antenna," Technical Report No, 21, J. D. Dyson, 15 September 1957. "Experimental Investigation of the Conical Spiral Antenna," Technical Report No^_22, R. L„ Carrel, 25 May 1957.** "Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technic; Report No. 23, E. J. Scott, 10 August 1957. "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," Technical Report No. 24, J. W. Duncan and R, H, DuHamel, August 1957. "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary Cross Section," Tec hnical Report No . 25, Robert L. Carrel, August 1957. "Cavity -Backed Slot Antennas," Technical Report No. 26, R. J. Tector, 30 October 1957. "Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical Report No. 27, W. L, Weeks, 1 December 1957. "Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," T e chn i c al Rep o r t^ No . _28_, W. L. Weeks, 20 December 1957. "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitral-; Cross Section," Technical Report No. 29, J. D. Dyson, 10 January 1958. "Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. '. . D. W. Isbell, 20 February 1958. "Electromagnetic Fields in Rectangular Slots," Technical Report No. 31, N. J. Kuhn and P. E. Mast, 10 March 1958. "The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder," Technical Report No. 32, J. W„ Duncan, 25 May 1958. "A Unidirectional Equiangular Spiral Antenna," Technic al Report No. 33, J. D. Dyson, 10 July 1958. "Dielectric Coated Spheroidal Radiators," Technical Report No. 34, W. L. Weeks, 12 September 1958 "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report No. 35. P. E. Mast, 12 September 1958. Contract AF33 1616) -6079 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna, " Technical RiP^LL-!^— 3 -^ R - H - MacPhie, 30 April 1959. "On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and Infinite Ranges/"' Technical Report No. 37, Raj Mittra, 15 May 1959. "Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical Report No . 38, W L„ Weeks, 5 June 1959. log Periodic Dipole Arrays," Technical Report No. 39, D. E„ Isbell, 1 June 1959. "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical Report No„ 40, S. Dasgupta and Y. T Lo, 17 July 1959 . 'The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical R gPP2:t_Ng.° _ . 1L K - G - Balmain, 1 August 1959. 'The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem in a Waveguide," Technical Report No. 42, Raj Mittra, 1 October 1959. Impedance Properties of Complementary Multiterminal Planar Structures," Techni cal Report No. 43, G. A. Deschamps, 11 November 1959. "On the Synthesis of Strip Sources," Technical Report No. 44, Raj Mittra, 4 December 1959. Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,' Technical Report No. 45, C„ H Tang and Y. T. Lo, 11 March 1960. ('New Circularly Polarized Frequency Independent Antennas with Conical Beam or Omnidirectional Patterns," Technical Report_No. _46 J) J D. Dyson and P. E. Mayes, June 1960. Logarithmically Periodic Resonant-V Arrays," Technical Report No. 47, P. E. Mayes md R. L, Carrel, 15 July 1960. |A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Technical eport No. 48, Y. T. Lo. "Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems, Techn ical Report No. 49, R ff„ MacPhie, 25 January 1961. I "Synthesis of Antenna Product Patterns Obtained from a Single Array," Technical Report No. 50, R„ H. MacPhie. * Copies available for a three-week loan period. ** Copies no longer available. AF 33(616)-6079 DISTRIBUTION LIST One copy each unless otherwise indicated Armed Services Technical Information I Agency Attn. 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K„ Chatterjee Vice Principal Birla Engineering College Pilani, Rajasthan India New Mexico State University Head Antenna Department Physical Science Laboratory University Park, New Mexico Bell Telephone Laboratories, Inc„ Whippany Laboratory Whippany, New Jersey Attn: Technical Reports Librarian Room 2A-165 Robert C c Hansen Aerospace Corporation Box 95085 Los Angeles 45, California Dr c D„ E„ Royal Ramo-Wooldridge, a division of Thompson Ramo Wooldridge Inc„ 8433 Fall brook Avenue Canoga Park, California Dr S c Dasgupta Government Engineering College Jabalpur, M„P India Dr Richard C „ Becker 10829 Berkshire Westchester., Illinois Antenna Laboratory Technical Report No. 55 AN INVESTIGATION OF THE NEAR FIELDS ON THE CONICAL EQUIANGULAR SPIRAL ANTENNA by 0. L. McClelland Contract AF33 (657)-8460 Project No. 6278, Task No. 40572 May 1962 Sponsored by: AERONAUTICAL SYSTEMS DIVISION Electrical Engineering Research Laboratory Engineering Experiment Station University of Illinois Urbana, Illinois