II B R.AFLY 
 
 OF THE 
 UN IVLRSITY 
 OF 1LLI NOIS 
 
 621.365 
 
 Ii655te 
 
 no. 40-49 
 
 cop. 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/numericalanalysi45tang 
 
ANTENNA LABORATORY 
 
 Technical Report No. 45 
 
 NUMERICAL ANALYSIS OF THE EIGENVALUE PROBLEM 
 OF WAVES IN CYLINDRICAL WAVEGUIDES 
 
 by 
 
 C.H. Tang and Y.T. Lo 
 
 11 March 1960 
 
 Contract AF33 (616) -6079 
 Project No. 9-Q3-6278) Task 40572 
 
 Sponsored by; 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
^n ^^^ 
 
 ACKNOWLEDGMENT 
 
 The authors wish to express their gratitude to Professors G. A. Deschamps 
 P E. Mayes for their helpful suggestions and comments and to Professor 
 r Scott for suggesting this investigation. 
 
CONTENTS 
 
 Page 
 
 1, Introduction 1 
 
 1.1 Equivalent Network Approach 1 
 
 1.2 Numerical Methods 3 
 1.2.1 Variational Method 3 
 
 Finite Difference Method 5 
 
 2.1 Approximations 8 
 
 2.2 Higher Order Formula 9 
 
 2.3 Approximation in Neumann Boundary Condition 11 
 
 2.4 Computational Procedure 12 
 
 2.5 Study of Convergence 15 
 
 2.6 Ridge Waveguide 15 
 
 3. Computations Based upon Various Boundary Condition Approximation 30 
 
 3.1 Results Obtained from Improved Approximation for Wave Equation 30 
 
 3.2 Results Obtained from Improved Approximation for Neumann 
 
 Boundary Condition 30 
 
 4. Conclusion and Discussion 33 
 References 36 
 Appendix A 37 
 
ILLUSTRATIONS 
 
 Figure Pag 
 
 1. Ridged Waveguide and Its Equivalent Network 2 
 
 2. Equivalent Circuit in the Transverse Plane of a Ridged Waveguide 2 
 
 3. Window in a Rectangular Guide and Its Equivalent Circuit 2 
 
 4. Mesh Pattern for Finite Difference 5 
 5 Mesh Pattern with a Curved Boundary 6 
 
 6. Mesh Points in a Given Cross-Section 7 
 
 7. Relation of Interior, Exterior and Boundary Point 11 
 8 Computation Procedure Diagram 12 
 9. Convergence Curves for TE 10 and TE 20 17 
 
 10. Convergence Curves for TEn and TM^ 18 
 
 11 . Comparison between Approximated Cut Off Wavelength (n = 21) 
 and Exact Cut Off Wavelengths of a Rectangular Guide for 
 
 Various TE Modes 21 
 
 12. Comparison between Approximated Cut Off Wavelengths (n = 78) 
 and Exact Cut Off Wavelengths of a Rectangular Guide for 
 Various TE Modes 
 
 13. Mode Designations 
 11. Mode Designations 
 
 15. Field Distribution in Ridged Guide 
 
 16. Field Distribution in Ridged Guide 
 
 17. Field Distribution in Ridged Guide 
 
 18. Comparison between Sets (n = 9 and n = 57) of Approximate 
 Cut Off Wavelengths of a Ridged Guide 
 
 Convergence Curves of Cut Off Wavelength of a Ridged Waveguide 
 
 nvorgence Curves for Neumann Problem with Different Orders 
 ol Approximation on Wave Equation but with Same Order 0(h) on 
 Boundary Condition 
 
 nvergence Curves for Neumann Problem with Different Orders of 
 Approximation on Boundary Condition but with the Same Order on 
 
 < I 1 OILS 
 
 rgence Curves for Various Approximations for Wave Equation 
 and Th.it for Boundary Condition (Neumann Problem) 34 
 
1 . INTRODUCTION 
 
 The waves in a cylindrical waveguide are found from solutions of the two 
 dimensional differential equation 
 
 2 2 
 V u + k u = (1) 
 
 where u is a function of the coordinates in the transverse plane and k is a 
 constant. The Dirichlet boundary condition, u = on the boundary of the cross- 
 section, corresponds to Transverse Magnetic modes and the Neumann boundary 
 condition, 9u/3n = on the boundary, corresponds to Transverse Electric modes. 
 In (1) k is the wave number and the function u, which is independent of the 
 longitudinal coordinate z, represents the component of electric (TM-case) or 
 magnetic (TE-case) field intensity along longitudinal direction. The time 
 dependence is assumed to be of the form e ' in this formulation. 
 
 The permissible values of k are also called eigenvalues of the differential 
 Equation (1) and u is the eigenfunction. Since the wave is confined to a 
 finite region, the spectrum of the eigenvalues jk I is a discrete set. Accord- 
 ing to mode theory, these eigenvalues k determine the cut off frequency of 
 each mode propagating along the waveguide. It is necessary only to find the 
 resonant frequency of the two dimensional problem defined by the guide boundary 
 since there is no axial variation at cut-off; all energy does propagate back 
 and forth in the transverse plane. 
 
 The exact solution of (1) with prescribed boundary condition can be 
 obtained only when the given boundary constitutes a (or a set of) coordinate 
 surface of the separable coordinate system. It has been shown only few regular 
 cross sections can be treated by the method of separation of variables. Wave- 
 guides with odd cross section, yet valuable in practical use (such as folded 
 waveguide and ridged waveguide), demand the result for engineering design. 
 1 .1 Equivalent Network Approach 
 
 Some of the "non-separable" problems can be treated by using the equiva- 
 lent circuit representation of the cross-section . For ridged waveguide, the 
 approximate equivalent circuit is obtained by considering region I (Figure 1) 
 as a capj 
 L A = pib 
 
 as a capacitor of capacity C = £S/h and regions II as inductances of value 
 
 B 
 
f c = 
 
 ar Jc b l a /2 
 
 (2) 
 
 1 
 
 b 
 
 1 
 
 r 
 
 
 E 
 
 i 
 
 
 
 
 - — / — ~\ 
 
 -* 
 
 m f ■» 
 
 
 - a — 
 
 * A, 
 
 ^_ 
 
 U ±c* iL 
 
 •B 3H 
 
 Figure 1. Ridged Waveguide and Its Equivalent Network 
 
 It is seen that it applies only for the case of b » h. 
 
 More accurate equivalent representation for cut off calculation has beei 
 
 2 
 obtained by considering the ridges as two step discontinuities (Figure 2) 
 
 where the step susceptance B is approximated by using that of a capacitive 
 
 window (Figure 3). 
 
 8 = 
 
 Y 
 
 = 3 
 
 X 
 
 Jt 
 
 Figure 2. 
 
 Equivalent Circuit 
 in the Transverse 
 Plane of a Ridged 
 W.-iveguide 
 
 Figure 3 
 
 ±B Y 
 
 Window in a Rectangul: 
 Guide and Its Equivali 
 Circuit 
 
3 
 
 Then by symmetry, the resonant input function at the center is either zero 
 or infinity. Thus we have 
 
 b 3s B 
 cot (3i - - tan -*- - - — = For TE n = odd (3) 
 
 a * c\\ no 
 
 R„ b P S B 
 cot Pi + - cot — - - — = For TE m = even (4) 
 
 d ^ Y mo v ' 
 
 Wh6re Y 01 = ^ i Y 02 = ^ \ 
 
 Numerical solutions of above transcendental Equations (3) and (4) can be 
 obtained by tabulation. It is again restricted to the wavelength range of 
 2bA < 1 for the single ridge and b/\ < 1 for the double ridges. 
 1.2 Numerical Methods 
 
 The approximate solution of (1) with "non-separable" boundary can also 
 
 be obtained from numerical analysis; such as the variational and finite dif- 
 
 4 
 rerence methods or the analogue method of a network analyzer . 
 
 1.2.1 Variational Method 
 
 The approximate eigenvalue of the wave equation can be obtained by using 
 
 the approximated Rayleigh-Ritz formula 
 
 , ,2 y (o) 2 (o) 
 
 (o) _ Sju Vu (5) 
 
 1 T (o) 2 
 
 2j u 
 
 We start with the unperturbed eigenfunction u in the given cross-section 
 (which corresponds to the eigenfunction of the rectangular guide of same aspect 
 
 ratio, but without ridge) . After obtaining the first approximation value of 
 
 (o) 
 k , the wave equation in finite difference form can be used as a formula to 
 
 obtain the higher order value of the eigenfunction u (the formula will be 
 
 derived later) . Hence the iterative process consists of successive corrections 
 
 between the value k and eigenfunction u 
 
 It has been shown that the formula (5) always converges to the lowest 
 
 eigenvalue. For higher order eigenvalue, say the second, the first term of 
 
 the orthogonal expansion 2c u must be eliminated from the assumed function 
 
 n n 
 
u . By using the orthogonality properties of these normal functions, C 
 can be determined, and (u - C u ) is then used for the computation of tb 
 second eigenvalue. 
 
 
 A combination of variation and relaxation methods has been used by Blacl 
 
 3 
 and von Rohr to calculate the cut off wavelength of semicircular ridges in 
 
 2 
 rectangular waveguide. A typical sequence of values for k has been given 
 
 2 2 2 
 
 as ll.l./a , 8.3/a and 7.7/a for nets containing respectively 19, 97 and 42 
 
 pts, (where a is the broad face demension of the guide), with the error clai 
 to be less than 2 percent. 
 
 e 
 
FINITE DIFFERENCE METHOD 
 
 Another way of solving this particular boundary value problem is by the 
 use of finite differences The differential operator is first approximated 
 by a finite difference formula. Then by setting up a finite number of mesh 
 points, we transform, approximately, the wave equation and the boundary con- 
 dition into a matrix eigenvalue problem. From the matrix we get a set of 
 approximate eigenvalues corresponding to a set of cut-off frequencies of the 
 particular waveguide structure. 
 
 The study reported here was initiated to investigate the practicality 
 of using a general purpose, high speed digital computer to perform the calcu- 
 lation of cut-off frequencies for cylindrical waveguides with irregular cross- 
 section. 
 
 2 
 The Laplacian V operating on a function u as in the scalar Helmholtz 
 
 equation, can be replaced by a set of finite difference approximations relat- 
 ing the values of the function at the nodes of a mesh pattern such as shown 
 in Figure 4 . 
 
 -I-Z 
 
 -r- 
 
 h 
 
 o L 
 
 -A 
 
 Figure 4. Mesh Pattern for Finite Difference 
 
 ' The application of the finite difference method to ridge waveguide problems 
 was initiated in this laboratory by Professor E. J'. Scott. 
 
From a Taylor's series expansion we have 
 
 /8u\ u 1 /9 2 u\ 2 1 /9 3 u\ 3 n , 4, 
 
 = u o + (ax)o h + 2\K^2j h + 3\\^2j h + 0(h } 
 
 /8u\ . i /a 2 u\ 2 i /a u \ 3 
 
 adding these two equations and neglecting the terms after the third gives: 
 
 r fl 2 
 o u 
 
 Ldx J 
 
 = u, + u„ - 2u + 0(h ) 
 13 o 
 
 Similarly 
 
 ft 2 ' 
 o u 
 
 a 2 
 Ldv -I 
 
 u n + u. - 2u + 0(h ) 
 2 4 o 
 
 Hence 
 
 2 2 4 
 
 h [V u] = u. + u„ + u_ + u A - 4u + 0(h ) 
 o 1 2 3 4 o 
 
 (6) 
 
 For a curved boundary we have 
 
 h 
 
 2 u„ 2 u_ 2 u„ 2 u 
 
 B 
 
 6d+fe) n( 
 
 U C ; U 3 ; U 4 /2 2N 
 
 i+n) " d+fe) + d+n) U + n/ u ° 
 
 (7) 
 
 Mesh Pattern with a Curved Boundary 
 
Thus for an ordinary point the wave equation becomes 
 
 u + u + u + u + (a-4) u = 
 
 1 *s o 4 o 
 
 (8) 
 
 In terms of mesh pattern, V can be expressed as 
 
 2 1 
 h 
 
 1 
 1-4 1 
 1 
 
 and 
 
 2 2 
 
 o = h k 
 
 (9) 
 
 (10) 
 
 Setting up a suitable number of meshes for a given cross section, and 
 applying the above procedure for each point, we would have as many simultaneous 
 equations as the number of points in the cross section: 
 
 1 
 
 1 
 1 
 
 T" 
 
 l 
 
 1 
 
 i 
 i 
 
 T 
 
 l 
 
 — >- 
 
 1 
 
 i 
 
 1 
 
 — > 
 
 1 
 1 
 
 1 
 
 i 
 
 i 
 i 
 
 I 
 1 
 
 Figure 6. Mesh Points in a Given Cross-Section 
 
The difference equations corresponding to points 1, 2, and 5 with Dirichlet 
 boundary condition are respectively 
 
 (a-4) u + u + u =0 
 
 u 1 + (a-4) u 2 + u g + u =0 
 
 u 2 + u 4 + (a-4) u 5 + u g + u g = (11) 
 
 Point 1 is a corner point, point 2 is an ordinary boundary point and point 5 
 is an interior point. 
 
 Writing in matrix form, we get the general formula 
 
 A u = a ii (12) 
 
 where A is the matrix with its element a. . corresponding to the coefficient 
 
 of u. at ith equation, u is the eigenvector (u n u n . . . u ) . In order to get 
 J _ 1 2 n 
 
 the non-zero eigenvector U, we set 
 
 det (A - a I) = (13) 
 
 where I is the identity matrix. This equation leads to the set of eigenvalues, 
 
 a 
 
 n 
 
 , corresponding to the roots of the nth order polynomial derived from (13 
 The relative cutoff wavelength \ /a, of the ith mode, in terms of the 
 eigenvalues will be 
 
 ^i = 2^- (i4> 
 
 & ; 
 
 a a. 
 v i 
 
 2.1 Approximations 
 
 The approximations involved in the finite difference method are 
 M) Finite mesh size, h 
 
 (2) Truncation error in Taylor series expansion 
 
 ('}) Approximation Ln Neumann boundary condition which will be discussed 
 i n 2 . 3 
 
2.2 Higher Order Formula 
 
 In order to improve the accuracy of the result, especially in the TE case 
 more terms in the Taylor series expansion may be taken into account. The point 
 pattern representation (see Figure 4) of the Laplacian when neglecting the 
 terms after the third is 
 
 2 1 
 
 V = — 
 
 1 
 1-4 1 
 
 1 
 
 (9) 
 
 that for neglecting the terms after the fifth is* 
 
 2 
 V = 
 
 840 h 
 
 - 3 
 
 -16 
 
 - 32 
 
 -16 
 
 - 3 
 
 -16 
 
 176 
 
 800 
 
 176 
 
 -16 
 
 -32 
 
 800 
 
 -3636 
 
 800 
 
 -32 
 
 -16 
 
 176 
 
 800 
 
 176 
 
 -16 
 
 - 3 
 
 -16 
 
 - 32 
 
 -16 
 
 - 3 
 
 (15) 
 
 Due to the complexity of (15) and the fact that it results in an unsymmetrical 
 matrix in the Neumann problem, the derivation of an alternative formula is 
 desired . 
 
 Define the following operators: 
 
 E f(x) = f(x + h) 
 
 D f(x) = f'(x) 
 
 6 f(x) = f(x + |) - f(x - |) 
 
 hence 
 
 f(x) = f(x + nh) 
 
 6 = E a - E" 
 
 2A 
 
 E 2 = (1 + 4 6 ) 
 
By Taylor series expansion 
 
 E f(x) = f(x+h) = 
 
 2 2 3 3 
 
 hD h D h D 
 
 1 + T7 + ^T~ + TT + 
 
 f(x) 
 
 we obtain 
 
 E = e 
 
 hD 
 
 hD = log E = 2 log 
 
 ., 1 .2,2 lc 
 (1 + - 6 ) + -6 
 
 -1 6 
 = 2 sin h — 
 
 2 2 2 2 2 2 
 
 j- 1 K 3 1 -3 j-5 1 -3 -5 ,-7 
 
 - — o + - o + 
 
 2 2 .3 
 
 2 4 -5! 
 
 2 6 -7 
 
 t^ 2 « 1 r R 2 1 A 4 1 A 6 1 A 8 i 
 hence D u = u = — [6 -— 6 + — 6 -— 6 +...]„ 
 
 h 
 
 1 c2 r 1 .2, 1 .2 1 
 h h 1 + - 6 
 
 (16) 
 
 The approximate formula is thus obtained 
 
 h 2 [1 + \- 2 6 2 ] u" = 6 2 u 
 
 (17) 
 
 For the one dimensional wave equation 
 
 u + k u = 
 
 (18) 
 
 we get 
 
 *2 2 2 r 6 2 . 6 2 
 
 6 u = - h k [l + — ] u = - a [l + — ] u 
 
 hence 
 
 u , - 2 u + u 
 
 n + 1 n n-1 
 
 Q r r, 1 
 
 au -- u , - 2 u +u n J 
 n 12 n+1 n n-l J 
 
 oi 
 
 (1 + 75 } u ,.i " (2 " I a > u + (1 + ts) u . = 
 1 <s n+1 6 n 12 n-1 
 
 (19) 
 
Similarly, for a two dimensional wave equation, we get 
 
 11 
 
 24 
 
 1 
 
 4 
 
 1 
 
 4 
 
 -20 
 
 4 
 
 1 
 
 4 
 
 1 
 
 u = -a 
 
 1 
 
 10 
 
 1 
 
 10 
 
 100 
 
 10 
 
 1 
 
 10 
 
 1 
 
 (20) 
 
 The order of approximation of (20) corresponds to that of (15). 
 
 2. 3 Approximation in Neumann Boundary Condition 
 
 In the Neumann boundary condition, several possible approximations are 
 
 considered. Let u be the value of u at a boundary point adjacent to an interior 
 o 
 
 mesh point with value u and an exterior point with value u 
 
 u = u, 
 
 Boundary 
 
 l h ii 
 
 u, = u +hu + — ru + 
 1 o o 2 . o 
 
 u' = 
 o 
 
 u = u + 0(h ) 
 1 o 
 
 Figure 7. Relation of Interior, Exterior and 
 Boundary Point. 
 
 (21) 
 
 b. 
 
 U -l = U l 
 
 2 3 
 
 u, = u +hu + — r u + —j u + 0(h ) 
 1 o o 2 o 3 o 
 
 2 3 
 
 u n =U -hu + — 7 u - -xt u + 0(h ) 
 -1 o o 2 o 3! o 
 
 u' = 
 o 
 
 U l = U -l + ° (h ) 
 
 (22) 
 
 c. In this case, we relate the wave equation to the boundary approxima- 
 
 tion as following: 
 
 hD 
 
 u - u = (E-l) u = E u - u = e u - u 
 lo o oo oo 
 
 , 2„2 
 h D 
 
 = [ 1 + hD + — r + 
 
 1 u - u 
 J o o 
 
 2 2 ,33 
 

 // 2 
 
 hence with the wave equation relation u = - k u we have 
 
 o o 
 
 2 3 
 
 / h // h /// 
 
 u_-u =hu + — ; u +-r-ru + 
 
 1 o o 2.! o 31 o 
 
 v, 2 i 2 
 / h k 
 
 h , 2 / 
 
 = h u - r~ u ~ 7TT k u + 
 
 o 2[ o 3! o 
 
 to. ah / 
 
 = hu - ~ u 2~ u + 
 
 o 2 o 6 o 
 
 Since 
 
 u = 
 o 
 
 i- . a a -, 
 
 U l = [1 -2 + ^i '•• ] U o 
 
 (1 - f) u + 0(h 3 ) 
 
 ^ o 
 
 (23) 
 
 2 , 4 Co mputational Procedure 
 
 The computational procedure used in this finite difference method is as 
 f ol lows 
 
 WAVE EQUATION 
 AND 
 BOUNDARY COND- 
 
 NUMERICAL 
 ANAD5I5 
 
 DET (A-JLl)=0 
 
 ~1 
 
 MATRIX 
 EI6EN VALUE 
 PROBLEM 
 
 COMPUTER PROGRAMING L 
 
 1 
 
 EIGEN VALUES 
 
 AND 
 EIGENVECTORS 
 
 l i ure 8. Computation Procedure Diagram 
 
13 
 The matrix A is first scaled so that its norm is less than 1; i.e. 
 
 2 a. < 1 
 
 tj 1J 
 
 Then by using the ILLIAC, the University of Illinois digital computer, we can 
 determine the eigenvalues a up to a maximum matrix size of 128 X 128 (Library 
 Routine M20-234, Digital Computer Laboratory, University of Illinois) and we 
 can determine both the eigenvalue a and eigenvector U up to a maximum matrix 
 size of 40 X 40 (Library Routine M18-213) . However, in this procedure, we 
 are restricted by the present technique of computer programming to the symmetric 
 matrix, since the possibility of complex eigenvalue for an unsymmetrical matrix 
 makes the programming .much more involved. 
 
 In some cases, the results thus obtained can be very simply improved by 
 
 i 6 
 
 using the Richardson s extrapolation . 
 
 It is reasonable to suppose that the error in the approximation is a func- 
 tion of mesh size h. If this function is expanded into a Taylor series we can 
 write 
 
 X. - \(h) = B h + c h 2 + ... (24) 
 
 o 
 
 where X- = lim \(h) 
 
 ° h^o 
 
 we assume that the term B h is the major part of the error. If we calculate 
 
 two approximate solutions \ and \ with different mesh size h and h respec- 
 
 12 i z 
 
 tively then 
 
 h.(\ -V,) - h(\ -\_) = h 9 (ch 1 2 + ..) - h (ch +...) 
 
 2 O 1 1 O ^ A i. i- & 
 
 solving for X. , we obtain the extrapolation formul 
 
 a 
 
 o 
 
 x o = irV [h 2\ " h i V - ch i h 2 + ••• <25) 
 
 2 1 
 
 If the error function can be expressed as an even function of h 
 
 \ - \(h) = c h 2 + E h 4 + ... (26) 
 
which is the case for our wave equation approximation, we have 
 
 \ = 2 — 71— + 0(h i h i > 
 
 h 2 " \ 
 
 (27) 
 
 For a set of data which increases monotonically 
 
 \ > \ if h 2 > h l 
 
 the extrapolated value \ obtained by dropping the higher order terras in (27), 
 becomes 
 
 .1 *_< 
 
 (28) 
 
 If h — ^o then X. approaches \_ which in turn approaches the exact value \ 
 2 ' e A o 
 
 as seen from (27). 
 
 On the other hand, for a set of data decreasing monotonically 
 
 \>\ 
 
 if h Q < h 
 <& 1 
 
 \ I \ l 2 h 2 ) < \ 
 
 e I 2 , 2 , 2 u 2 2 
 
 h l " h 2 / 
 
 (29) 
 
 As h — ^ o, the same conclusion as before is reached. In many cases, this methoc 
 shows a great improvement with practically no further labor added in the over- 
 all computation. 
 
 By using two data \ , \ , we can eliminate the necessary knowledge of 
 
 4 
 coefficient c; therefore, the result is accurate to 0(h ). By generalizing 
 
 this idea with a set of a data \ X. 
 
 ... \ , it seems that n coefficients 
 be '•! i mi nated and the result will be accurate to 0(h ). This is probably 
 
 ter than repeatedly using the same formula for just a pair of data at a 
 i Lme as has been done later. This may also explain why the extrapolated value 
 II h (such as h = 1/12, and 1/14) is better than that from all the h's 
 iwn 1 a t < ii 
 
15 
 
 2.5 Study of Convergence 
 
 The convergence of the above methods has been tested by the application 
 to a rectangular guide where the exact solution is available for comparison. 
 When the first approximation formula (9) is applied to the wave Equation (1), 
 it shows a better result for the TM case than that for the TE case (Figures 
 9, 10) . It is believed that this is a result of poor approximation in the 
 boundary condition for the TE case. In fact, in the TM case there is no 
 approximation for the boundary condition. 
 
 Table I and Figure 9 and 10 show the results (from ILLIAC) for a particular 
 rectangular waveguide (b/a = 0.5) as compared with exact solution. 
 
 Table II shows results obtained by repeatedly using the extrapolation 
 Formula (27). They show a remarkable improvement over the results of Table I. 
 
 Figures 11 and 12 show the difference between exact relative cut off 
 wavelength and the calculated value where \ /a are obtained from the finite 
 set of eigenvalue < a. ? . 
 
 2.6 Ridge Waveguide 
 
 7 8 
 Many authors ' have shown that the insertion of rectangular ridges have 
 
 the following effects 
 
 1. A decrease in the lowest cut off frequency. 
 
 2. An increase in mode separation. 
 
 3. An increase in attenuation. 
 
 4. A concentration of electric field intensity at corners of the ridges. 
 The results obtained in our calculation have shown good agreement with 
 
 the data in existing literature. 
 A typical example shows, for 
 
 - = 0.625 - = 0.375 — = 0.4 with net point = 22 
 a a b 
 
 mode TE 1Q TE Q1 TB n TE^ TM^ TM^ TM^ TM^ 
 
 \ /a. 2,541 1,059 1.051 1.038 0.675 0.638 0.468 0.466 
 
 c 
 
 where the ridge guide modes are given the same designation as the corresponding 
 modes in the rectangular guide. Figures- 13 and 14 show the method of mode desig- 
 nations for ridge guide where the boundary condition are shown for only 1/4 
 
16 
 
 of the cross section. The eigenfunctions for this particular ridge guide have 
 also been obtained. Figure 15, 16, and 17 indicate that the largest cut-off 
 wavelength (i.e., dominant mode which corresponds to smallest eigenvalue) are 
 increased due to the distortion of the field distribution. 
 
 Sets of higher order mode cut-off wavelength are given in Table III. 
 
 Figure 18 shows the difference between two sets of cut-off wavelength 
 obtained by using different numbers of mesh points, namely 9 points and 57 
 points . 
 
 Figure 19 shows the} convergence curve for a particular ridge size. 
 
17 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 o 
 
 
 
 H 
 
 SI 
 
 
 * 
 
 8S 
 
 
 
 ivl 
 
 h 
 
 ^ 1 
 
 k 
 
 Vi 
 
 
 
 5 
 
 
 I 
 
 
 
 
 v. 
 
 
 
 
 
 
 9 
 
 
 
 
 
 
 
 h. 
 
 
 
 
 54 
 
 (41 
 
 hi 
 
 
 viv. 
 
 
 
 
 P 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 l 
 
 
 ^\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ,yf 
 
 
 2-n- 
 
 
 c 
 
 c 
 
 y HJ.SN31 3AVM J JO j. 
 
 '.no . 
 
 3 ( 
 
 
 
 
 1 
 
 CO 
 
 1 
 
T?Z hUCNW-l WAW1 *J0-Js>3 MJJtiJd 
 
TABLE I 
 Set of Approximated Cut Off Wavelengths 
 For a Rectangular Waveguide 
 
 19 
 
 -4pXq) 
 
 X. /a\ 
 
 c 
 
 
 
 
 
 
 
 
 
 Exact 
 Solution 
 
 \ 
 c 
 
 2 
 
 (3X7) (4x9) (5X11) (6X13) 
 
 a 
 
 Vm 
 
 2 A 2 
 
 +4n 
 
 TE TM 
 
 TE 
 
 TM 
 
 TE 
 
 TM 
 
 TE 
 
 TM 
 
 TE TM 
 
 m, n 
 
 
 OO 
 
 
 OO 
 
 
 CO 
 
 
 CO 
 
 
 OO 
 
 * 
 0,0 
 
 
 
 1.765 
 
 
 1.82 
 
 
 1.84 
 
 
 1.861 
 
 
 2 
 
 1,0 
 
 
 
 0.905 
 
 
 0.919 
 
 
 0.931 
 
 
 0.938 
 
 
 1 
 
 2,0 
 
 0,1 
 
 
 0.785 
 
 0.915 
 
 0.822 
 
 0.908 
 
 0.848 
 
 0.904 
 
 0.867 
 
 0.901 
 
 0.895 
 
 1,1 
 
 
 
 0.718 
 
 0.726 
 
 0.748 
 
 0.72 
 
 0.77 
 
 0.716 
 
 0.787 
 
 0.714 
 
 0.708 
 
 2,1 
 
 
 
 0.63 
 
 
 0.628 
 
 0.631 
 
 
 0.637 
 
 
 0.667 
 
 3,0 
 
 
 
 0.594 
 
 0.583 
 
 0.613 io.573 
 
 0.626 
 
 0.567 
 
 0.633 
 
 0.564 
 
 0.555 
 
 3,1 
 
 
 
 0.503 
 
 
 0.499 1 
 
 0.507 
 
 
 0.511 
 
 
 0.5 
 
 4,0 
 
 0,2 
 
 
 0.491 
 
 0.535 
 
 0.49 0.517 
 
 0.485 
 
 0.507 
 
 0.483 
 
 0.501 
 
 0.488 
 
 1,2 
 
 
 
 0.454 
 
 0.488 
 
 0.445 
 
 0.473 
 
 0.446 
 
 0.465 
 
 0.449 
 
 0.46 
 
 0.448 
 
 2,2 
 
 4,1 
 
 
 0.44 
 
 0.437 
 
 0.433 
 
 0.423 
 
 0„433 
 
 0.416 
 
 0.437 
 
 0.412 
 
 0.4 
 
 3,2 
 
 5,0 
 
 
 0.436 
 
 0.429 
 
 
 0.407 
 
 
 0.396 
 
 
 0.389 
 
 0.372 
 
 5,1 
 
 
 
 0.424 
 
 0.416 
 
 
 0.382 
 
 
 0.371 
 
 
 0.366 
 
 0.354 
 
 4,2 
 
 
 
 0.405 
 
 
 
 
 
 
 
 
 0.333 
 
 6,0 
 
 0,3 
 
 
 0.403 
 
 0.393 
 
 
 0.378 
 
 
 0.365 
 
 
 0.355 
 
 0.329 
 
 1,3 
 
 
 
 0.381 
 
 0.373 
 
 
 0.363 
 
 
 0.349 
 
 
 0.339 
 
 0.316 
 
 2,3 
 
 6,1 
 
 
 0,368 
 
 0.365 
 
 
 
 
 
 
 
 0.312 
 
 5,2 
 
 
 
 0.358 
 
 0.36 
 
 
 
 
 
 
 
 0.298 
 
 3,3 
 
 
 
 0.346 
 
 
 
 
 
 
 
 
 0.285 
 
 7,0 
 
 
 
 0.314 
 
 0.338 
 
 
 
 
 
 
 
 0.278 
 
 4,3 
 
 6,2 
 
 
 0.301 
 
 \ 
 
 0.325 
 t 
 
 \ 
 
 1 
 
 \ 
 
 f 
 
 1 
 
 r 
 
 0.274 
 
 7,1 
 
 
 
 (21) 
 
 (36) 
 
 (55) 
 
 (78) 
 
 
 
 
 * There is no field corresponding to this particular eigenvalue, 
 
 
2C 
 
 TABLE II 
 Results Obtained by Extrapolation 
 
 *• ^lO 
 
 
 Exact 
 \ /a 
 
 value 2 
 
 
 
 
 
 h 
 
 a 
 
 c 
 
 
 
 
 
 
 1/3 
 
 2,0 
 
 1 ,48 
 
 1.84 
 
 
 
 
 
 1/4 
 
 1 .0 
 
 l r 57 
 
 1.925 
 
 1 
 
 .88 
 
 1 
 
 .91 
 
 1/5 
 
 0,585 
 
 1,641 
 
 1.965 
 
 1 
 
 .945 
 
 1 
 
 .942 
 
 1/6 
 
 0,381 
 
 1.695 
 
 1 ,94 
 
 1 
 
 .953 
 
 1 
 
 .97 
 
 1/7 
 
 0,269 
 
 1,73 
 
 2.01 
 
 1 
 
 .975 
 
 2 
 
 .008 
 
 1/8 
 
 0,198 
 
 1.765 
 
 2.04 
 
 2 
 
 ,03 
 
 1 
 
 .994 
 
 1/10 
 
 0.1206 
 
 1 ,82 
 
 1.94 
 
 1 
 
 .99 
 
 1 
 
 .989 
 
 1/12 
 
 0.081 
 
 1,84 
 
 1.987 
 
 1 
 
 .961 
 
 
 
 1/14 
 
 0,0581 
 
 1 .861 
 
 
 
 
 
 
 2 > ^20 
 
 
 Exact 
 \ /a 
 
 value' 1 
 
 
 
 
 
 h 
 
 a 
 
 c 
 
 
 
 
 
 
 1/8 
 
 753 
 
 0,905 
 
 0.977 
 
 
 
 
 
 1/10 
 
 0,468 
 
 0,919 
 
 0.991 
 
 
 
 .983 
 
 
 
 .982 
 
 1/12 
 
 317 
 
 931 
 
 0,98 
 
 
 
 .967 
 
 
 
 1.918 
 
 1.936 
 1.96 1.96 
 
 1.984 1.947 
 
 1.999 ' 1.975 1.966 
 
 1.98 1.977 
 
 1.995 .1.983 
 
 1.987 
 1.992 
 
 1/14 229 ,938 
 
21 
 
 Figure 11 
 
 Comparison between Approximated Cut Off Wavelength 
 (n = 21) and Exact Cut Off Wavelengths of a Rectan- 
 gular Guide for Various TE Modes 
 
 Sequences of Relative Cut- off Wave Length 
 
on- 
 
 SsQUENoe. of Relative Cut- of t Wavelength 
 
£ 
 
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 II 
 
 
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 3C 
 3 COMPITATIONS BASED T PON VARIOUS BO' MDARY CONDITION APPROXIMATION 
 
 3 1 Results Obtained from Improved Approximation for Wave Equation 
 
 When Formula (20) was applied to the rectangular guide (with b/a = 5), 
 
 considerable improvement was obtained for the Dirichlet problem, but the result 
 
 for the Neumann problem becomes even worse than the first approximation (Figure 
 
 20) It seems to indicate that Formula «20) received more propagation error 
 
 from approximation in Neumann boundary condition than that of first approximate 
 
 Formula (9) and it is probably also true that, in our case, the error due to 
 
 boundary approximation predominates over that of wave equation approximation. 
 
 !n short, we can sav that merely improving the wave equation approximation 
 
 but not the boundary condition does not guarantee better results, 
 
 In our first calculation, the wave equation has been approximated with 
 
 3 
 the formula which takes 0(h > into account with the boundary approximation 
 
 only up to 0(h),, while in the second computation a higher order approximation 
 
 5 
 to the wave equation up to 0(h ) is considered but with boundary approximation 
 
 still t 0th' . ft happened in our case, that the higher order approximation 
 
 is only an improvement for the wave equation and is a worse formula when it 
 
 combines with 0'h) Neumann boundary approximation. 
 
 The above argumpnt is ^trengthpned by the fact that for Dirichlet boundary 
 condition u = o. we do get better results for higher order approximation It 
 thus seems likely that the approximation for the wave equation and the boundary 
 condition should be of the same order, 
 3.2 Re 0b1 m ne d f rom Improved Approximation for Neumann Boundary Cond i tior 
 
 Owing to the above undesirable results, the application of the improved 
 jpproxi mat i on for Neumann boundary condition (22), (23) becomes necessary 
 
 ■r'.x obt nn<-d bv using 'he boundary approximation b (see (22)) and wave 
 equation approximation (9> becomes non- symmel ri c, a case which is difficult 
 
 Howevei for a small matrix, computations can be done by a desk cal- 
 Clllator The results are plotted in Figure 21, It is seen that not only is 
 
 ipproximatlon greatly Lmproved bu1 the direction of convergence is changed 
 
31 
 
 3 
 
 <^. oq ^ '^ k> N4~ en rJ -■* <^ 
 
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 +-> 
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 s 
 
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 w +-> 
 
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 +-> >> 
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 OS 
 
 S-i T3 
 
 C 
 
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 s o 
 
 
 lb 
 
 S3 (D 
 
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 5 as 
 
 5 W 
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 as 
 
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 o 
 
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 3 
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'VAy HJ.9H3~1*AW ^QJ/>D 3A/J*~!Zy 
 
33 
 
 4, CONCLUSION AND DISCUSSION 
 
 To connect the differential equation of a boundary value problem to a 
 difference equation one must replace the differential operator by a different 
 operator wherein a truncation of the series representing the operator is 
 involved. Unless the rigorous solution to the problem is known the actual 
 error committed in truncation is unknown, although the upper bound of the 
 truncation error can be determined. A higher order approximation in this 
 process results only in a decreased upper bound of the error, but does not 
 necessarily guarantee lower actual error in a particular computation. 
 
 Improving the approximate formula for the wave equation does not necessarily 
 give better results; it will depend also on the order of approximation for 
 the boundary condition. In the present investigation, it turns out that the 
 later approximation is of even greater importance. 
 
 In our analysis, two different orders of approximations have been con- 
 ' sidered for the wave equation and two for the boundary condition. 
 
 3 
 
 For the wave equation (I) 0(h ) 
 
 (II) 0(h 5 ) 
 
 For the boundary condition (a) 0(h) 
 
 (b) 0(h 2 ) 
 
 It turns out that different combinations give results in the following order 
 of accuracy: lib (best, lb, la, Ila (worst). 
 
 Convergence curves show that the Dirichlet boundary problem converges 
 faster than that of the Neumann boundary problem. Furthermore, they converge 
 in different directions, i.e., in Dirichlet boundary problem, the exact value 
 is the lower bound of the set of approximated results -j (\ /a)^ L where n 
 indicates the number of points used in the approximation while in Neumann 
 boundary problem the exact value is the upper bound of the set of approxima- 
 tion results. 
 
 This method can be applied to both TE and TM cases. The approximate cut- 
 off wavelength and field distribution have been obtained. However, only the 
 data of a limited range for TE case is available in literature. They are found 
 in good agreement with the present results. 
 
3' 
 
35 
 
 This investigation indicates that better results for ridge waveguide might 
 be obtained by the following considerations. 
 
 (a) Apply better approximation for boundary condition. It is probably 
 preferable to use the same order of approximation for the boundary condition 
 as for the wave equation. Since these higher order approximation formula 
 always result in non-symmetric matrix, it is thus desirable to investigate 
 the properties of the eigenvalues of a non-symmetric matrix. In general a 
 non-symmetric matrix has complex eigenvalues; however, the computation which 
 has been done for a few small matrices, as indicated in Figure 22, with a desk 
 calculator shows that they possess only real eigenvalues. Therefore, it may 
 infer that in our present problem the non-symmetric matrices could have only 
 real eigenvalues. If this is the case the computation procedure could be 
 simplified somewhat. 
 
 (b) An alternative method to solve this ridge guide problem may be sug- 
 gested as follows: To consider ridged cross section as composed of a few simple 
 regions where their eigenfunction expansion are known, by matching the eigen- 
 function at the common boundary we may arrive at a set of integral equations. 
 These equations may be solved approximately, 
 
 (c) In one-dimension problems, as demonstrated in the Appendix, the 
 fields at interior mesh points can be expressed in terms of those at boundary 
 points only. It is not known whether it is possible to achieve the same goal 
 in a two-dimensional problem. If it can be done, the problem will be solved. 
 
3( 
 REFERENCES 
 
 1. S, Ramo and J„ R, Whinnery, Fields and Waves in Modern Radio , Second Editic, 
 
 p , 409 , 
 
 2 N„ Marcuvitz, Waveguide Handbook , Radiation Laboratory Series, Vol. 10, 
 p. 399, 
 
 3, J. Van Bladel and 0. von Rohr, Jr„, Semi -Circular Ridges in Rectanguler 
 Waveguides," IRE Transactions , MTT-5 , No 2, April 1957, p. 103. 
 
 4. G, Swenson and T c J Higgins, A Direct-Current Network Analyzes for Solving 
 the Wave-Equation Boundary Value Problem, Journ. Appl . Phys . , Vol. 23, 1952 
 pp. 126-131. 
 
 5 W, E, Milne, Numerical Solution of Differential Equation, John Wiley and 
 Sons, Inc., 1953, pp. 133, 
 
 6 . L Fox, The Numerical Solution of Two-Point Boundary Value Problems , Oxf o] 
 Press, 1957, p. 332. 
 
 7. SB, Cohn, "Properties of Ridge Waveguide, 1 Pro c . IRE, Vol. 35, August 
 1957, p. 783-788. 
 
 8. Sammel Hopfer, "The Design of Ridge Waveguides," IRE Transactions , MTT-3 , 
 No , October 1955, p. 20. 
 
 9 F B, Hi Idebrand, Introduction to Numerical Analysis , McGraw-Hill, 1956. 
 
37 
 
 APPENDIX A 
 Closed Form for One-Dimensional Case 
 
 For the lowest mode of a rectangular guide as discussed previously, the 
 problem is essentially one dimensional. In order to see how the eigenvalue 
 varies with n, the number of mesh points, their difference equations are studied 
 and solved. 
 
 Applying (8), we have the approximate wave equation 
 
 i-1 
 
 i+1 
 
 N 
 
 a-2 
 
 u. n + (a-2) u. + u. = 
 l-l i l+l 
 
 (Al) 
 
 Similarly from (19) we have 
 
 i-1 
 
 i+1 
 
 a 5 a 
 
 1 + 12 6 Q " 2 l + 12 
 
 a-2 
 
 u i-l + ~ a "I "i+1 
 
 u - + u , .i = ° 
 
 1 + 
 
 12 
 
 (A2) 
 
 Equations (Al) and (A2) can be generalized as 
 
 u - 2m u . + u . , = 
 i-1 i i+l 
 
 (A3) 
 
 where 
 
 
 m = 1 - | For (Al) 
 
 m' = 1 — , For (A2) 
 
 *♦! 
 
 (A4) 
 (A5) 
 
 Let m = m , hence 
 
 a 
 
 2 + 
 
 a 
 
3, 
 
 Therefore 
 
 a < a 
 
 Since 
 
 . 27T 27Th 
 
 (A6) 
 
 Va 
 
 we have that \ /a obtained from (Al) is always greater than that from (A2) 
 irrespective of boundary approximation, (21) or (22). This also agrees with 
 the numerical results obtained previously. Therefore, for TM case we have 
 better results by applying improved Formula (20), yet the same formula gives 
 worse results for TE case since they converge to their exact value in different 
 directions as shown in Figure 22. 
 
 The general solution of Equation (A3) is 
 
 u. = a z, + b z„ 
 i 1 2 
 
 (A7) 
 
 where z and z are the roots of the quadratic equation 
 
 z - 2mz +1=0 
 
 (A8) 
 
 Applying the boundary conditions 21, 22, 23 respectively we would have 
 the following cases 
 
 (a) 
 
 o 1 N-l N 
 
 From (7) we obtain 
 
 a + b = a z, +bz 
 1 l 
 
 N-l . N-l N , N 
 a z + b z = a z + b z 
 
 (A9) 
 
 In order to get a non-trivial solution for (29) we set 
 
 (1-z^ 
 
 (l-z 2 ) 
 
 (1- Zi ) z 2 (l-z 2 ) z 2 
 
 = 
 
 (A10) 
 
 
 (1-Z.) ( L-Z.) (z 2 - 2 ) = 
 
39 
 
 The solutions z = z = 1 lead to trivial solution a = for (z N_1 - z N_1 ) = 
 i a z 1 ' ' 
 
 we get 
 
 \ N -\li2fm 
 
 2p7T 
 N-l 
 
 p = 0, 1, 2, ... N-2 
 
 (All) 
 
 From (A8) 
 
 z = m + v m -1 
 
 z 2 = m 
 
 -/^ 
 
 Substituting in (All) and solving for m from 
 
 ,p7T Vl-m 2 
 
 tan (itt) = 
 
 N-l m 
 
 we have 
 
 m = + cos 
 
 pit 
 
 N-l 
 
 Following the same routine, we obtain for the other two boundary approximations 
 (22), (23) 
 
 (b) 
 
 U -l = U l 
 
 U N-1 " U N+1 
 
 m = + cos 
 
 pn 
 
 N 
 
 (c) 
 
 = (1 - 
 
 2 O 
 
 m = + cos 
 
 N-l 
 
 N 
 
 (1 " V u n 
 
 for the lowest normal mode, p = 1, thus we have the lowest cut off wavelength 
 for (21), (22), (23) respectively, 
 
 (a) 
 
 fY ir 
 
 N ]/l - cos 7T/N-1 
 
 (b) 
 
 c 
 a 
 
 \[~2 7T 
 
 N |/l - cos ff/N 
 
 (c) 
 
 2 7T 
 
 a N v/l " cos 7T/N 
 
 It can be shown that the value of cut of wavelength Formula (a) is always less 
 than 2, while those of (22) and (23) are always greater than 2. Again this 
 
 ■— >iMi uij-£L -*- 1 
 
 .W+ r^ •; ~ ^.^4 ««/M.-{/Min1» ^ Pi /rtivi^ Oil 
 
ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 Contract AF33 (616 )-310 
 
 "Synthesis of Aperture Antennas/' Technical Report No. 1 , C.T.A. Johnk 
 October, 1954.* 
 
 ,r 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). 
 
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 No. 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.* 
 
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 P.E. Mayes, R. F. Hyneman, and R. C. Becker, 20 February 1957, (CONFIDENTIAL). 
 
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 Scott and R. H. DuHamel, 16 April 1956. 
 
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 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 
 No. 12 , Richard C. Becker, 30 September 1956.** 
 
 "Impedance of Ferrite Loop Antennas," Technical Report No. 13 , V. H. Rumsey 
 and W. L. Weeks, 15 October 1956. 
 
 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
 No. 14, Richard 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. 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. 19 
 R.H. DuHamel and D.E. Isbell, 1 May 1957. 
 
 "Frequency Independent Antennas," Technical Report No. 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," Technical 
 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," Technical 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," 
 Technical Report No. 28 , W.L. Weeks, 20 December 1957. 
 
 "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary 
 Cross Section," Technical Report No. 29 , J.D. Dyson, 10 January 1958. 
 
 "Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. 30 , 
 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/' Technical Report No. 33 J.D. 
 Dyson, 10 July 1958. 
 
 "Dielectric Coated Spheroidal Radiators," Technical Report N o. 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 (616)-6079 
 
 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Technical 
 Report No. 36, 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 
 Report No. 41 , 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," Technical 
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 M/F Contract AF33 (600) -3324i 
 Buffalo 5, New York 
 
 Bell Telephone Labs., Inc. 
 Attn: R. L. Mattingly 
 
 M/F Contract AF33 (616) -5499 
 Whippany, New Jersey 
 
 Bendix Radio Division 
 Bendix Aviation Corporation 
 Attn: Dr. K. F. Umpleby 
 
 M/F Contract AF33 (600) -35407 
 Towson 4, Maryland 
 
 Commander 
 
 Air Force Missile Development Center 
 
 Attn Technical Library 
 Hoi Ionian Air Force Base, New Mexico 
 
 tander 
 
 80] -. • Ai r DivisJ on (SAC) 
 
 At DCTTTD, Major Hougan 
 Air Force Base, Oh it) 
 
 All Development Center 
 ERA-] M. lJi.ii/ 
 '■. Ll Porci Ba ■ York 
 
 , Lance Dep1 
 
 \ i ■ .i 
 
 m i ' .i i Do< umenl Cen i er 
 r< i 
 
 Boeing Airplane Company 
 Attn: C. Armstrong 
 
 M/F Contract AF33 (600) -3631S 
 7755 Marginal Way 
 Seattle, Washington 
 
 Boeing Airplane Company 
 Attn: Robert Shannon 
 
 M/F Contract AF33 (600) -35992 
 Wichita, Kansas 
 
 Canoga Corporation 
 M/F Contract AF08 (603) -4327 
 5955 Sepulveda Boulevard 
 P.O. Box 550 
 
 Van Nttys, California 
 
AF 33<616)-6079 
 
 page 3 
 
 Dr. C H. Papas 
 
 Department of Electrical Engineering 
 California Institute of Technology 
 Pasadena, California 
 
 Chance-Vought Aircraft Division 
 United Aircraft Corporation 
 
 Attn: R.C. Blaylock 
 
 THRU: BuAer Representative 
 
 M/F Contract NOa(s) 57-187 
 Dallas, Texas 
 
 Collins Radio Company 
 
 Attn: Dr. R. H. DuHamel 
 
 M/F Contract AF33 (600) -37559 
 Cedar Rapids, Iowa 
 
 Consolidated Vultee Aircraft Corp. 
 Fort Worth Division 
 Attn: C.R Curnutt 
 
 M/F Contract AF33 (038) -21117 
 Fart Worth, Texas 
 
 ^onsol idated-Vul tee Aircraft Corp. 
 Attn: Mr. R E. Honer 
 
 M/F Contract AF33 (600) -26530 
 P.O. Box 1950 
 5an Diego 12, California 
 
 :onvair 
 
 Attn: R. Honer 
 
 M/F Contract AF33 (600) -26530 
 >an Diego Division 
 )an Diego 12, California 
 
 50NVAIR 
 
 ■"ort Worth Division 
 Attn: C. R. Curnutt 
 
 M'F Contract AF33 (600) -32841 & 
 AF33 (600) -31625 
 "ort Worth, Texas 
 
 lepartment of Electrical Engineering 
 
 Attn: Dr H. G Booker 
 Cornell University 
 thaca. New York 
 
 niversity of Denver 
 'enver Research Institute 
 niversity Park 
 enver 10. Colorado 
 
 Dalmo Victor Company 
 
 Attn: Engineering Technical Library 
 M/F Contract AF33 (600) -27570 
 1515 Industrial Way 
 Belmont, California 
 
 Dome &, Margolin, Inc. 
 
 M/F Contract AF33 (600) -35992 
 
 30 Sylvester Street 
 
 Westbury 
 
 Long Island, New York 
 
 Douglas Aircraft Co., Inc. 
 Attn: G. 0' Rilley 
 
 M/F Contract AF33 (600) -25669 & 
 AF33 (600) -28368 
 Tulsa, Oklahoma 
 
 Douglas Aircraft Company, Inc. 
 Long Beach Plant 
 
 Attn: J. C. Buchwalter 
 
 M/F Contract AF33 (600) -25669 
 Long Beach 1, California 
 
 Exchange and Gift Division 
 The Library of Congress 
 Washington 25, D.C. 
 
 (2 copies) 
 
 Fairchild Engine & Airplane Corp. 
 Fairchild Aircraft Division 
 Attn: Engineering Library 
 
 S. Rolfe Gregory 
 
 M/F Contract AF33 (038) -18499 
 Hagerstown, Maryland 
 
 Dr. Frank Fu Fang 
 
 Boeing Airplane Company 
 
 Transport Division, Physical Research 
 
 Renton, Washington 
 
 General Electric Company 
 
 Attn: D.H. Kuhn, Electronics Lab. 
 M/F Contract AF30 (635) -12720 
 Building 3, Room 301 
 College Park 
 113 S. Salina Street 
 Syracuse, New York 
 
 General Electronic Laboratories, Inc. 
 Attn: F. Parisi 
 
 M/F Contract AF33 (600) -35796 
 18 Ames Street 
 Cambridge 42, Massachusetts 
 
AF 33l616)-6079 
 
 page 4 
 
 Goodyear Aircraft Corporation 
 Attn: G. Welch 
 
 M'F Contract AF33 (616) -5017 
 Akron, Ohio 
 
 HRB-Singer, Inc. 
 
 Attn: Mr. R. A. Evans 
 Science Park 
 State College, Pa. 
 
 Granger Associates 
 M F Contract AF19 (604) -5509 
 966 Commercial Street 
 Palo Alto. California 
 
 Grumman Aircraft Engineering Corp. 
 Attn: J S. Erickson 
 
 Asst. Chief, Avionics Dept . 
 M F Contract NOa(s) 51-118 
 Bethpage. Long Island, New York 
 
 Gulton Industries, Inc. 
 Attn B Bittner 
 
 M'F Contract AF33 (600) -36869 
 P.O. Box 8345 
 15000 Central, East 
 Albuquerque, New Mexico 
 
 Hal 1 lcraf ters Corporation 
 Attn. D. Herling 
 
 M/F Contract AF33 (604) -21260 
 440 W. Fa fth Avenue 
 Chicago, Illinois 
 
 Technical Reports Collection 
 Attn Mrs. E.L Hufschmidt 
 Librarian 
 303 A. Pierce Hall 
 Harvard University 
 Cambridge 38, Massachusetts 
 
 Huffman Laboratories, Inc. 
 
 Attn. S Varian (for Classified) 
 Technical Library (for 
 
 Unci assi f led ) 
 M F Contract AF33 (604) -17231 
 Los Angeles, California 
 
 Hughes Aircraft Corporation 
 At t n D Ad cock 
 
 M'F Contract AF33(616)-5648 
 1 1 Teale 
 
 Cu i Ca l i i --in i a 
 
 Mr, Dwight Isbell 
 4620 Sunnyside 
 
 ITT Laboratories 
 
 Attn: A. Kandoian 
 
 M/F Contract AF3C 
 500 Washington Avenue 
 Nutley 10, New Jersey 
 
 ITT Laboratories 
 Attn: L. DeRosa 
 
 M/F Contract AF33(616)- 
 500 Washington Avenue 
 Nutley 10, New Jersey 
 
 ITT Laboratories 
 
 A Div. of Int. Tel. & Tel. Corp. 
 
 Attn: G. S. 
 3700 E. Pontiac Street 
 Fort Wayne, Indiana 
 
 Giffin, ECM Lab, 
 
 Jansky and Bailey, Inc. 
 Engineering Building 
 
 Attn: Mr. D. C. Ports 
 1339 Wisconsin Avenue, N.W. 
 Washington, D.C. 
 
 Jasik Laboratories, Inc., 
 100 Shames Drive 
 Westbury, New York 
 
 John Hopkins University 
 Radiation Laboratory 
 Attn: Librarian 
 
 M/F Contract AF33(616)-68 
 1315 St. Paul Street 
 Baltimore 2, Maryland 
 
 Applied Physics Laboratory 
 Johns Hopkins University 
 8621 Georgia Avenue 
 Silver Spring, Maryland 
 
 u. f. Hyneman 
 P B 
 
 c-i r>2 
 600 
 ' ■ lysti ma Group 
 
 ( il lfornla 
 
 Lincoln Laboratories 
 
 Attn: Document Room 
 
 M/F Contract AF19 (122) -458 
 Massachusetts Institute of Technology 
 P.O. Box 73 
 Lexington 73, Massachusetts 
 
AF 33 1616) -6079 
 
 page 5 
 
 Litton Industries, Inc. 
 Maryland Division 
 
 Attn: Engineering Library 
 
 M'F Contract AF33 (600) -37292 
 4900 Calvert Road 
 College Park, Maryland 
 
 Lockheed Aircraft Corporation 
 Attn: C D. Johnson 
 
 M/F Contract NOa(s) 55-172 
 P.O. Box 55 
 Burbank, California 
 
 Lockheed Missiles & Space Division 
 Attn: E. A. Blasi 
 
 M/F Contract AF33 (600) -28692 & 
 AF33 (616) -6022 
 Department 58-15 
 Plant 1, Building 130 
 Sunnyvale, California 
 
 The Martin Company 
 
 Attn: W. A. Kee, Chief Librarian 
 
 M/F Contract AF33 (600) -37705 
 Library & Document Section 
 Baltimore 3, Maryland 
 
 W. L. Maxson Corporation 
 M/F Contract AF33 (600) -31225 
 460 W. 34th Street 
 Mew York 1, New York 
 
 Snnis Kuhlman 
 
 McDonnell Aircraft 
 
 D Box 516 
 
 .ambert Municipal Airport 
 
 it Louis 21, Missouri 
 
 del par, Inc. 
 
 Attn: Technical Library 
 
 M/F Contract AF19 (604) -4988 
 Intenna Laboratory 
 3000 Arlington Blvd. 
 r alls Church, Virginia 
 
 lelville Laboratories 
 l/alt Whitman Road 
 lelville, Long Island, 
 lew York 
 
 University of Michigan 
 Aeronautical Research Center 
 Attn: Dr. K. Seigel 
 
 M/F Contract AF30 (602) -1853 
 Willow Run Airport 
 Ypsilanti, Michigan 
 
 Microwave Radiation Co., Inc. 
 Attn: Dr. M. J. Ehrlich 
 
 M/F Contract AF33 (616) -6528 
 19223 S. Hamilton Street 
 Gardena, California 
 
 Motorola Inc. 
 
 Attn: R. C. Huntington 
 8201 E. McDowell Road 
 Phoenix, Arizona 
 
 Physical Science Lab. 
 
 Attn: R. Dressel 
 New Mexico College of A and MA 
 State College, New Mexico 
 
 North American Aviation, Inc. 
 
 Attn: J. D. Leonard, Eng. Dept. 
 M/F Contract NOa(s) 54-323 
 4300 E. Fifth Avenue 
 Columbus, Ohio 
 
 Autonetics 
 
 North American Aviation, Inc. 
 Attn: S. Kerber 
 
 M/F Contract AF33 (600) -27109 
 9150 E. Imperial Way 
 P. 0. Box "AN" 
 Bellflower, California 
 
 North American Aviation, Inc. 
 Attn: H. A. Storms 
 
 M/F Contract AF33 (600) -36599 
 Department 56 
 International Airport 
 Los Angeles 45, California 
 
 Northrop Aircraft, Inc. 
 
 Attn: Northrop Library, Dept. 2135 
 M/F Contract AF33 (600) -27679 
 Hawthorne, California 
 
AF 33(616)-6079 
 
 page 
 
 Dr. R. E Beam 
 Microwave Laboratory 
 Northwestern University 
 Evanston, Illinois 
 
 Ohio State University Research 
 Foundation 
 
 Attn: Dr T. C. Tice 
 
 M F Contract AF33 (616) -6211 
 1314 Kinnear Road 
 Columbus 8, Ohio 
 
 Dr. D. E. Royal 
 
 Ramo-Wooldridge, a division of 
 Thompson Ramo Wooldridge Inc. 
 8433 Fallbrook Avenue 
 Canoga Park, California 
 
 Rand Corporation 
 Attn: Librarian 
 
 M/F Contract AF18 (600) -1600 
 1700 Main Street 
 Santa Monica, California 
 
 University of Oklahoma Res. Inst. 
 Attn: Prof. C, L. Farrar 
 
 M/F Contract AF33 (616) -5490 
 
 Norman, Oklahoma 
 
 Rantec Corporation 
 
 Attn: R. Krausz 
 M/F Contract AF19 (604) -3467 
 Calabasas, California 
 
 Prof A. A. Oliner 
 Microwave Research Institute 
 Polytechnic Institute of Brooklyn 
 55 Johnson Street - Third Floor 
 Brooklyn, New York 
 
 Philco Corporation 
 
 Government and Industrial Division 
 Attn: Dr Koehler 
 
 MF Contract AF33 (616) -5325 
 4700 Wissachickon Avenue 
 Philadelphia 44. Pennsylvania 
 
 Radiation, Inc. 
 Technical Library Section 
 Attn; Antenna Department 
 
 M F Contract AF33 (600) -36705 
 Melbourne, Florida 
 
 Radio Corporation of America 
 RCA Laboratories Division 
 At tn: Li bran an 
 
 M F Contract AF33 (616) -3920 
 Princeton. New Jersey 
 
 Rad I op] ane Company 
 
 II F Contl .< •' AF33 (600) -23893 
 
 Van Nuys, Cal i I ornla 
 
 IfOOl 'I I i dge 3 ;i d i v i s i on o I 
 □ Wooldridge. inc. 
 Atti TechnJ ca l I n J orma t i on 
 
 i illbi ook Avenue 
 P Bo> L006 
 
 Raytheon Electronics Corp. 
 Attn: H. K. Hudson 
 
 M/F Contract AF33 (604) -1563^ 
 1089 Washington Street 
 
 Newton, Massachusetts 
 
 Raytheon Manufacturing Corp. 
 Attn: Dr. R. Borts 
 
 M/F Contract AF33 (604) -15634 
 Wayland, Massachusetts 
 
 Dr. Harry Letaw, Jr., Manager 
 
 Systems Marketing 
 
 Raytheon Company 
 
 1089 Washington Street 
 
 West Newton, Massachusetts 
 
 Republic Aviation Corporation 
 Attn: Engineering Library 
 
 M/F Contract AF33 (600) -34752 
 Farmingdale 
 Long Island, New York 
 
 Republic Aviation Corporation 
 Guided Missiles Division 
 Attn: J. Shea 
 
 M/F Contract AF33 (616) -5925 
 223 Jericho Turnpike 
 Mineola, Long Island, New York 
 
 Sanders Associates, Inc. 
 95 Canal Street 
 
 Attn; Technical Library 
 Nashua New Hampshire 
 
AF 33 (616) -6079 
 
 page 7 
 
 Smyth Research Associates 
 
 Attn: J. B. Smyth 
 3555 Aero Court 
 San Diego 11, California 
 
 Tamar Electronics, Inc. 
 
 Attn: L. B, McMurren 
 2045 W, Rosecrans Ave. 
 Gardena, California 
 
 Space Technology Labs, Inc. 
 Attn: Dr. R, C. Hansen 
 P.O. Box 95001 
 Los Angeles 45, California 
 M/F Contract AF 04(647) -361 
 
 Sperry Gyroscope Company 
 Attn; B Berkowitz 
 
 M/F Contract AF33 (600) -28107 
 Great Neck 
 Long Island, New York 
 
 Stanford Electronics Laboratory 
 Attn: Applied Electronics Lab. 
 Document Library 
 Stanford Univeristy 
 Stanford, California 
 
 Stanford Research Institute 
 
 Attn: Mary Lou Fields, Acquisitions 
 Documents Center 
 Menl o Park, California 
 
 Stanford Research Institute 
 Aircraft Radiation Systems Lab. 
 Attn: D. Scheuch 
 
 M/F Contract AF33 (616) -5584 
 Menl o Park, California 
 
 Sylvania Electric Products, Inc. 
 Electronic Defense Laboratory 
 VI F Contract DA 36-039-SC-75012 
 P Box 205 
 Mountain View, California 
 
 Mr Roger Battle 
 
 Supervisor^ Technical Liaison 
 
 Sylvania Electric Products, Inc. 
 
 Electronic Systems Division 
 
 P 0. Box 188 
 
 Mountain View. California 
 
 Technical Research Group 
 M/F Contract AF33 (616) -6093 
 2 Aerial Way 
 Syosset, New York 
 
 Temco Aircraft Corporation 
 Attn: G. Cramer 
 
 M/F Contract AF33 (600) -36145 
 Garland, Texas 
 
 Electrical Engineering Res. Lab. 
 University of Texas 
 Box 8026. 
 
 Austin. 
 
 University Station 
 Texas 
 
 A. S. Thomas, Inc. 
 M/F Contract AF04(645)-30 
 161 Devonshire Street 
 Boston 10. Massachusetts 
 
 Westinghouse Electric Corporation 
 Air Arm Division 
 
 Attn: P. D Newhouser 
 
 Development Engineering 
 M/F Contract AF33 (600) -27852 
 Friendship Airport 
 Baltimore, Maryland 
 
 Professor Morris Kline 
 
 Institute of Mathematical Sciences 
 
 New York University 
 
 25 Waverly Place 
 
 New York 3, New York 
 
 Sylvania Electric Products, Inc. 
 Electric Systems Division 
 Attn: C, Faflick 
 
 M/F Contract AF33 (038) -21250 
 100 First Street 
 tfaltham 54, Massachusetts