011778-1-F 11778-1-F = RL-2251 COLLEGE OF EDNRN DEPARTMENT OF EL~RCLAND COMPUTEImR ENGINEERING Radiation Laboraor MATJIEMTICS ~OF 0fDE WAVE PRIOPAGATION Ml OPF* SihC%~E by Chiao-Mia Chu Final 8ottcReport May 1974 AF01R Great No, 73. 2538 Contract With: Air 'OQOfieOf aSo' Air orce6 e iwbm 0mn4 KMAF 1400WUDoad Arl:z Vrina2a209 An n Arbor, Micbla

011778-1-F ABSTRACT The propagation of electromagnetic waves along open structures whose transverse cross section are composed of two symmetrically placed circular sectors is investigated. The problem is first formulated in the form of a dual summation equation. By introducing a new set of basis functions to represent the surface field, the dispersion relation for such a structure is developed, systematic techniques of finding the approximate solutions of the dispersion relation are formulated, and numerical solutions for the first two orders of approximation are carried out to illustrate the feasibility of using the technique. In general, the scattering and resonance problem of open structures of cylindrical, spherical or other configurations may be formulated in the form of dual summation equations. The approach introduced in this work, i.e., the use of appropriate basis funeLions and the numericai solution oi these equations, therefore, may open a new avenue in the solution of a wide class of problems. i

011778-1-F TABLE OF CONTENTS ABSTRACT i I STATEMENT OF THE PROBLEM 1 II CONSTRUCTION OF BASIC FUNCTIONS 6 mfl THE DISPERSION RELATION 16 IV NUMERICAL COMPUTATION OF ya V CONCLUSIONS AND RECOMMENDATIONS 30 VI REFERENCES 32 APPENDIX A SEARCHING AND ITERATING PROCEDURE 33 APPENDIX B REDUCTION OF THE DISPERSION RELATION 37 ii

I STATEMENT OF THE PROBLEM In this study, the propagation of electromagnetic waves along a class of open structures is investigated by introducing a new set of basic functions in the representation of the surface field. To illustrate this approach, let us consider the waves guided along a curved transmission line composed of infinitely long, symmetrically located conducting circular sectors as shown in Figure 1. Assuming that the time dependence and z-variation of the field to be ejWt - jpz (P is the unknown propagation constant to be determined), all the field components may be generated from a single field component E (for TM modes) or H (for TE modes). Explicitly, for TM modes, we Z Z denote, a Z (y r) E (r - ) An (a) cs (2n+l) (1) n = 0 2n+l1 where 2 W 2 (2) c The Z 's are Bessel functions, representing Hankel functions of the second n kind H )for r > a and the Bessel functions J for r < a. The unknown n n propagation constant is then determined from the boundary conditions as r — a. For perfectly conducting structures, these conditions are listed below: (a) The vanishing of the tangential component of electric field on the conductors yields: a E (a,) ): A cos(2n+l) 0= (3) n= n 1

y Figure: Curved transmission line u x Figure 1: Curved transmission line 2

and 7n= 0 E (a, ~)= - Z (2n+l) A sin (2n+1) p = 0 for j e L1, where L1 denotes the range of 0 such that -40<0<0j, and r - o < 0 < r + o (b) The discontinuity of the tangential component of magnetic field across the conductors yields, 2wc0 a A J (a, ----- cos (2n+1) = 0 (4) z 7r2a 0 H(2 (-ya) J (ya) 2n+l 2n+1 for 0 e L2, where E L2 denotes the range of j such that 0o < 0 < 00 - o, T + 00 < 0 < 27 - 0 Similarly, for TE modes, we denote H (r ) B Z2n+l (Tr). z ('0n Z1 2) sin (2n+1) (5) The boundary conditions are: (a) E,(a, ) = in 2n+ 0 (6) n=0 for e L1, a (2n+l) B (b) J (a, )= -2/ 3 - - - z 3 a(2) (Ta) J' (a) cos (2n+l) -y a n=0 H a) J (a) 2n+l 2n+1 and (7) a~ B n 0, ya H n+l (ya) J' (+a) sin (2n+1) for en L n+ for 0 E L2 3

It is interesting to note that in both cases, the boundary conditions may be expressed in the following general form: a J(0) = C cos(n+ 2 0 = 0 L. (8) n 20 and E(0)= C G (-Ca) sin(n++ 2 = 0 L1. (9) n n 2 n = Here, except for a constant factor J (0) denotes J (a, 0), and E(0) denotes E (a, 0). G (ya) are known functions of (ya), for the TM case, we use G (Ta) = '(2n+1) H2l (7 a) Jn (ya) (10) n ]2n+1 2n+1 while for the TE case, we use (ya) (2) X J2n+l ( 2a) 2+l 2 (11) G (T a)- (a(11) n (2n+l) (Ta) Mathematically, therefore, our problem of finding the propagation constants for the transmission line illustrated in Figure 1 is equivalent to that of finding the set of values of y ( or ya ) such that nontrivial solutions exist for the homogeneous "dual summation equations" given by Eqs. (8) and (9). Although detailed analysis concerning the existence of solutions of such' a problem is difficult to carry out, we assert that solutions do exist, based on the following These particular forms of G (ya) were chosen to simplify the expressions developed later for approximate solutions of ya. 4

two physical arguments: (a) It is known that the structure illustrated in Figure 1 supports the TEM mode of propagation. In this case, i = -, and y = O, thus C we know the "dual summation equation" has at least one solution - namely ya = 0. (b) In the published works on wave propagation along slotted cylindrical waveguide, such as the works of Harrington (1943) and Goldstone and Oliner (1961), experimental evidence of high order propagation (with imaginary part of 3 < 0) was reported. On physical ground, therefore, one may suspect higher order modes also exist in the present structure which may be considered as a cylindrical waveguide with two slots. Our primary objective, therefore, is to develop a systematic approach, with the help of computers, to determine numerically the solution set (7ya) for the system of dual summation equations. Based on the known solutions given by the TEM mode of propagation, we first define sets of functions as the basis of approximating the field components. These sets of functions, including members that are discontinuous and unbounded, may be used to accurately represent the dominant components (from the well known edge condition) of the surface fields. Using this basis for the field representation, an infinite system of equations is obtained for the propagation constant. Since the dominant components of the fields are included in the first few terms of the field representation, truncation of the infinite system appears to be reasonable. Procedures of approximate evaluation of the higher order propagation constants by including N (arbitrary) equations of the infinite set are then developed. 5

I CONSTRUCTION OF BASIC FUNCTIONS It is well known that the structure illustrated in Figure 1 supports the TEM mode of propagation. For this mode, the field components may be represented by - z E=e VV(x, y) (12) and -H = z 1 c A H= e zxVV(x, y) 0 (13) where V (x, y) satisfies the two dimensional Laplace equation and the boundary conditions. By obtaining V through the use of conformal transformation, it is easily verified that, except for multiplicative constants, Eo (a, 0) = 0 1 e L1 A E L2 (14).. and J (a, 0 = z 1 cos 2 - cos 2 00 E L1 E L2 (15) 0 On the otherhand, in the dual summation equations see that for y= 0, G (oya) = 1 n (8) and (9), we 6

Thus Equation (9) and (10) are reduced to: )= Ccos(n + 2 0 = 0e L (16) n=0 n E E(0) = Z C sin (n+)2= 0 e L. (17) n=0 Comparison of (14), (15) with (16) and (17) reminds one of the DirichletMehler relations involving associated Legendre functions. From the Dirichlet - 1Iehler relations, we have (Erdelyi, et al, 1953) for 7r >A> 0, a r( + m) ) P- (cos A) cos (n + ) x n=0 sinm- A (cosx-cosA) - 1/2 0 <x <A = \ (18) 0 O A<x<w And, if we replace x by 7r - x and A by r =- A, r (2+ m) ) Pn (-cos A) (-l)n sin (n + 1)x 2 n=0 2 0 0 <x<A - j9 (19) [ (-I) sinm A(cos A- cos x )ml/2 <x < r. 2f. 7

In the above, if we let m = 0, x = 2 O, and A = 2 0, we have n = P (cos 2 00) cos (n + ) 2 2 n u 2 and = os20-cos 2 00 a 2 1 P (cos2 0) sin (n + ) 2 n= 1 1 2cos 2 - cos 2 0<2 <2 0 2 00 <2 0 < (20) 0 <2 0 < 2 2 o <2 < 7 (21) From Equations (20) and (21), and their periodic extensions, it is evident that Equations (16) and (17), i.e., the dual summation equations (8) and (9) for the case of y = 0, admit the non-trivial solution: C = P (cos 2 f0) Moreover, this solution yield values for E (0) and J (0) that agree with the solution obtained by using the conformal transform techniques. In general, since dominant components of the fields ( J (0) and E (p)) are the same as that of the TEM case, it is logical to expand the fields in a. set of functions including those given by (20) and (21). We therefore introduce 8

a new set of basis functions that may be used to represent J(0). These are: 0( fm( A x)= E m Am A1 sinm A p-m (cos A) cos (n + ) x n 2 (22) For all ranges of real x, the functions f (A, x) are sketched in Figure 2. m The dominant features of this set of functions are: a) In the interval L2: (2(k-1) lr + A <x < 2k7r- A), f (Ax) = m (23) b) In the complementary interval L': (2k7r - <x < 2k7r + A. f (Ax)=+ m - 1 / (cos x - cos A) P (m+ ) (24) where the positive sign holds for even k and the negative sign for odd k. c) f is discontinous, unbounded, and at the end points of LI 0 1 f (Ax) ^ 1 0o /cos x - cos A (25) d) f is continuous, but has discontinuous derivative; f2, together with its first derivative is continuous. In general, f E C n n-l.,. Note that for A = 2 0, x = 2 0, L and Lp are L1 d L2 together with their periodic extensions. 9

k I /f \ \/ / 'f -37r 7 - 0 I P. X 93 7 IT LI / / / I I I \ f0 I Figure 2: Sketches of fm 10

These features illustrate the advantage of using the f function as the basis of expansion for J (0). Another set of functions, "dual" to the f 's, can be obtained n from Equation (22) by replacing A and x respectively by 7r - A and 7 - x. This set of functions is defined by g( A, x) = f ( -A, 7r - x) wm 00 (-l)n sm A-m (-cos A) sin (n + ) x (26) n 2 n = 0 n=0 For real values of x, the functions gm (A, x) are sketched in Figure 3. Again, the following dominant features are obvious: a) In the interval L1 2k7r - A< x < 2k7 +, g ( A, x) =0. (27) b) In the interval L2(2(k-1) r + A <x < 2kw - A\, 1 g ( A, x) c- oA cos x)m (28) P (m+ ) c) go is discontinuous and unbounded. At the end points of L', 1 9 AJ* X) _(29) go ( ) cosA - cos (29) d) In general, gn e C 1. 11

-37, 01 91 -2wj Njir ( 7 r 27r-Z 19 Figure 3: Sketches of g 27r+LA 2w-, /g I I 12

This set appears to be appropriate in using as basis of expansion for E (0). A set of relations, useful in solving the type of dual summation equations of interest, are noted here. From the relations 1 t t t (1-x2)2 P (x)... P(x)dx (30) x xx and P (x)= —). (31) n 2 n. dxn We see that for n > m, P-m (-x) = (-1)n+m p-m (x) (32) n n where for n < m, P (x) and P (-x) are linearly independent. Using n n Equation (31), it is easy to deduce from Equation (22) that, (-f sinm A P-m (-cos A) cos (n + ) x = n 2 n = 0 ( 1m ( )-(-l l sina [P cos4-(-) PV (-cos Acos (n +) m n=0 (33) Similarly, from Equation (26), we have c) Esin APn (cos A) sin (n +2 ) x = n n n=0 m=l =(-l)mg (Ax)+ sinm m(co -(-)n+m (-os sAsin (n +)x n=O (34) 13

We shall use these new basis functions to deduce the approximate dispersion relation of the curved transission in the next section. 14

m THE DISPERSION RELATION By denoting x = 2 ~ A =2 0 the problem of finding the propagation constants of the higher order modes of the open transmission line illustrated in Figure 1 may be stated as follows: Find the set of -y such that non-trivial solutions of the following dual summation equations exist: J(0) =Z cos(n+ )x=0 xEL (35) n=O n 2 2 Co E(0) C G (7 a) si(n+) x E L. (36) n nn 1 n=0 Moreover, due to the well known edge conditions, we require, as x - 0 in 1 J ( 0). 1,(37) v/ cos x - cos 37) and as x - in L' 2 1 )(38) e fo facos a - cos x n In order to derive an equation from which 7ya ( or B3) may be solved, the the following facts are to be noted. 15

(a) Any member of f (x, A) satisfies Equation (35) (c.f. Eq. (23)), In particular, f (x, A) also satisfies the edge condition, given by Equation (37). Thus, we may represent Co ( fm (x, A). (39) m =0 (b) As n --, n H (x) J (z) - r + 2. (40) n n Thus, for the TM case, G (ya) j (2n+1) H ( ) 1a(2 ( -, 1 + (41) ' 2n+l 2n+l' 2 4 (2n+l) Similarly, as n -> oo (2) H') (z) J' (z) 2 n n + z n n (1+ ). (42) z 4n Thus, for the TE case, H' (eya) J' (ya) 2 2n+1 2n+1 ( ya), (3) a (ya)43) n j (2n+l) 2 < 4(2n+1). In either case, we may write G (ya) 1 + S (ya) (44) n n a The completeness of such a representation can be easily demonstrated and will not be considered here. 16

and note that as n -. Co, 2 S (-ya) - ( (45) n 4 (2n+l) Equation (45) justifies our truncation procedure introduced later that S (ya) may be neglected for any particular oya such that (7ya) < < (2n +1) (c) For Sn (ya) = 0, Gn (ya) = 1, we have ya = 0, corresponding to the TEM mode of propagation. (d) For S (ya) = -1, G (ya) = 0, then, ya are the zeros of J 1(-a) for the case ad he zcros of J + (7a) for the TE case. 2n+l 2n+1l (e) In terms of S (ya), we may rewrite Equation (36) in the form n2 Here, the second summation in Equation (36) is of higher order in -2 in comparison to the first summation, and may be neglected (at least partially for large n) in numerical computations. Based on the above facts, let us deduce in steps the approximate dispersion relations for ya. In the zeroth approximation, let us neglect all the S 's except S and take only one term of Equation (39) in representing J (). Thus we let, 17

(0) = a f (A, x). (47) o o From Equation (22), we have, C = a P (cos A). (48) n o n Substitute into Equation (46), and use Equation (34), we have, E (0) = g(A, x) + a P (cos A) S (ya) sin (n + - ) x = 0 o o n o 2 x E L'. (49) Since g ( A, x) is always 0 for x e L'1, Equation (49) is satisfied only when, S (ya) = 0. This is the TEM solution, i.e., 7a = 0 In the first approximation, let us assume that S and S are non-zero, and use two terms in Equation (39). Thus, J(0) a f (A, x)+ f (A x) (50) 0 0 11 This means -1 C =a P (cos A) +a sinA P (cos A). (51) n o n 1 n 18

Substituting Equation (51) into Equation (46), and using Equation (39), yields E ( 0) =aogo(, x) - a1gl (A x) + + sin a S P(cosA) +a S sinAP (csn P1 (cosA)+ P 1(-cosA) + + sin x a SPn(cosA)+Ssin P ) = x L. (52) 2 ol n 1 1 n (52) This is possible only if +a S P (cosA)+)+a SsinAPn (cosA)+sinA P (cosA)+P (-cosA)tO oo n l n LOo (53) -l0 a S1P (cosA)+1S1 sin AP (cosA)0. (54) o 1 n 1 1 n For non-trivial solution of a and a, the dispersion relation takes the form: o 1' S S (1 - cos A)2-2 cos A = (55) The condition S = 0 yields y = 0, the TEM propagating condition, while the -1 other condition is 4 cos A S (ya) = 0 (56) (1 - cos A) The interpretation of Equation (56) is interesting. For A = 180, = 900, corresponding to a closed circular gyide, S (ya)= -1. 19

Thus, in the limiting case, the solution of Equation (56) yields the TM1 or TE1 modes of propagation. For the lowest mode, TE1l, if we assume that ya for the open structure is close to that of a closed circular guide, i.e., 7a = 1.84, then S (ya) 2 =.03 <<1 2 4 x (5) The approximation introduced, i.e., neglecting S (-ya) for n > 2 appears to be reasonable. For the TM case, and for high order modes, however, higher approximation is necessary. For the second order approximation, we assume that S (ya) 0 for n > 3, and represent J(0) =a f (x, A)+1 fl (x, A) +a f2 (x A) (57) 00 11 a ( The resulting approximate dispersion relation is, 0 I S P (cosA):Sin [s Sin P (cos s + P (cos A)+ -PP (-ccos A) 0 0 S1P1(cos A) I Sin ( S Pc (cos ) in P2 (os + P2 (cos A) + P -cos A) -- -'"' -- - L - -.. - - - - - - - - - - - - ( i 2 -2 S2P2 (cos A) Sin A S2 P2 (cos A) ISin AS2 P2 (cos A) (58) 20

Explicitly, Equation (58) may be written as S2 [ 1 3 y3-15y2 + 32y - 24] S - (3 -2y) So+ (2-6y+3y = 0 (59) where, for simplicity, we denote y =(1- cos). (60) In the limiting case of cos A= - 1, y = 2, Equation (59) is reduced to: S2 (ya) [S1 (a) + 1] [S a) (a)+ 1 = (61) The three factors in Equation (26) yield the propagation constants (i.. e, ya) for TEM, TM1 (or TE ) and TM3 (or TE3) modes of propagation respectively. In general, of course, we should represent 00 J(0) a f (x, A) (62) m = 0 and an infinite determinant is obtained. However, if we are computing -ya numerically, we may truncate the determinate to ( N + 1 ) x (N + 1) order, i.e., represent J() f (x,). (63) m= m m tm = 0m Any solution of the truncated determinant satisfying the criterion that (c. f. Equation (45)) 21

2 ya <<1 4 (2N+1) should yield numerically acceptable propagation constants. It is to be noted that with our particular choice of basis functions in representing the surface fields (J (0) or E (0)), the truncation appears to be reasonable since, (a) The first few terms of the series represent the dominant components of E (0) and J (0) correctly, (b) The edge conditions are automatically satisfied in this representation, and (c) It has been shown that results of the first few orders of approximation yield exact results in the limiting case of A = 90. This fact is easily shown O to be true for any order of truncation. For the N-th approrximtion when J (P) is approximated by Equation (64) we have N C = a sin A Pnm (cos A).(64) m = The truncated determinant, i.e., the approximate dispersion relation, takes the form Det K n = 0 (65) n, m where K =S (ya) sinm A -m (cos A ) n,m n n + sinM A P- (cos A ) - (-1)m+ P- (-cos A) (66) n n n, m = 0, 1,...... N. 22

Some reduction of the determinant, and the numerical scheme for computing ya from the determinantal equations are discussed in the next section. 23

IV NUMERICAL COMPUTATION OF ya Although in principle, the solution for the set of 'ya satisfying the dispersion relation for any order (N) of approximation appears to be a straightforward mathematical problem, the actual numerical computation of 7ya is far from trivial due to the complicated functions (Hankel functions for complex arguments) involved in these equations. For the first order approximation, the dispersion relation given by Equation (57) are rewritten as: F (7ya) = S0 (7a) = 4 cos A (68) (1 - cos A) For a given A = 2 P0, the solution of Equation (68) for complex y a can be carried out by Newton-Raphson's iterative method. The convergence in this case appears to be good. For TM modes, some of the numerical results are tabulated in Table 1. For large 00, the solutions are close to the zero's of Bessel functions; for the smaller 00, the deviation from the zero's of Bessel functions becomes greater. For the second order approximation, Equation (60) is again written as F (7ya) = S0(^ya) S1(ya) +- (3y3-15y +32y-24) S1(^ya) y 16 32 2 - (3 - 2y) S0 (ya) +6 (2-6y + 3 y) = 0 (69) y y where y = (1 - cos A). 24

Table 1. ya From First Approximatioh TM Mode First Set Re of Roots Im Second Set of Roots Re Im 0 87.5~ 750 60~ 50O 3.8316950 3.8416062 4.0425478 4.3667455 0.0 0.0000934 0.0474268 0.4574768 7.0155994 7.0337080 7.3553023 7.6335519 0.0 0.0603267 0.1318178 0.6966593 25

For any given A, Newton- Raphson's method is again used to compute the complex roots. In this case the convergence is very poor, and no numerical acceptable roots were obtained after 10 iterations even when we start with the initial guess predicted from the first approximation. A modified conjugate gradient program was also tried in an attempt to solve |F( ya)l =0 (70) without success. After considerable amount of numerical experimentation, we have developed a scheme combining the searching and iterating technique in computing ya. The convergence is greatly improved in using this scheme. Since the numerical problem of computing complex roots of complex equations involving transcendental equations is known to be a very difficult task, our new scheme appears to epreesent a significant contribution in solving such problems. A detailed description of this searching and iterating scheme is given in Appendix A. For TM modes of propagation, some numerical results are given in Table 2. It is to be noted that not all of the roots in Table 2 satisfy the criterion 'a|2 << 4x(2 N+ 1)2 = 176 They are tabulated, however, to illustrate the feasibility of evaluating all the complex roots. For any higher order approximation, the dispersion relation given by Equation (66) may be written in the form, F(ya) = det[K]= 0 (71) 26

Table 2. (hya) From Second Approximation TM Mode First Set of Roots Second Set of Roots j8 Re Im Re Im 87.50 3.9685815 0.4489306 6.0104261 0.6061281 750 600 500 4.1029203 3.9995496 2.8350118 1.5604187 1.4339417 1.5225492 5.7068481 5.6891943 5.9726308 1.2732071 1.8720847 0.4743322 27

where K is a matrix, with elements K (ya) = S (ya) Sin A pm (cos A) n, m n n + sinm A p (cosA) - (-l)mp- (-cosA) In n n, m = 0, 1, 2,.........N. (72' Explicit expression for the dispersion relation using a Laplace development of the determinant appears to be unnecessary since we are interested only in the numerical solutions. For numerical computation, however, since each of the matrix elements K are functions of (ya) and A, they n, m must be calculated for each a and every iteration. To simplify the computation, we have succeeded in reducing Equation (70) to the following form: F(-ya) =det( [S] + [Q] ) = 0 (73 ) ) where so (a) 0 [s]= S l(a) SN(a)] (74) is a diagonal matrix, and is independent of A. On the otherhand, [Q] is a matrix depending only on A = 2 00. For any A, if we denote y = 1 - cos A, 28

the elements of [Q] may be computed by using the following equations: )A) 1 - +J+1 (23 - )t 2)t (2j-t —1), a) AY = (-1) (I (j-t- i-i1)' t'. (j+i-t)'. i>j (75) A.. =o i<j b) C (n+k+r) 1 1) ( 76) n),k: (2k+r)'. (n-k-r)~ ( (76) n=O c)Q =A~- (77) ) QiN AiN (77) d) Q. A. Q N Cc, (78) sN-i -Ai N-i-Qis A. IN 1( N e) Qi, k = Ai. k Qij Cj,k (79) For a fixed N and A, the matrix elements Qi k can be computed first, and in the iteration solution for ya, only the elements of S matrix need to be computed in each iteration. The detailed derivation of Equation (73) is given in Appendix B. Computation of ya, using this scheme appears to be feasible if N is not too large. Actual computation to date, however, has been completed for N = 2 only. 29

v CONCLUSIONS AND RECOMMENDATIONS The modal analysis of the guided wave propagation along an open structure such as illustrated in Figure 1 yields a set of "dual summation equations" for the propagation constants. By introducing new sets of basis functions in the expansions of surface fields, the approximate dispersion relation (for any order N of approximation) have been developed. Numerical scheme for obtaining the complex roots of the dispersion relation were developed. Although the actual computation was carried out only for N = 2, there appears to be no doubt that the scheme is applicable for moderate values of N. Future work concerning this problem should probably include (a) more calculations for higher values of N to investigate numerically the effect of truncation, and (b) more detailed analysis to determine the behavior of the matrLx elements Q.. which, physically may be interpreted as "coupling coefficients" between different modes. The basic mathematical scheme developed in this study may in principle be modified and extended to include the solutions of the following problems: (a) The propagation of waves along a circular waveguide with one longitudinal slot. Although others have performed a theoretical analysis of propagation along slotted cylinders previously, for example, in the works of Goldstone and Oliner (1961), Harrington (1959) and Chen (1973), theoretical analysis and numerical computation of the propagation constants for higher order modes are still lacking. (b) The complex resonant frequency of spherical resonators. The modal analysis of such an open structure yield also a set of dual summation equations involving associated Legendre functions P (cos 0). The solution n 30

of the dual summation equations by introducing the proper basis function to represent the surface field should be worth investigating. (c) The modal analysis of scattering problems involving open structures such as slotted cylinder and slotted spheres yields a set of inhomogeneous dual summation equations. A systematic approach for solving such scattering problems by introducing proper basis functions for the representation of the surface fields should also be tried. The scattering problem of the sources by slotted cylinders has been investigated by Hayashi (1966) by using singular integral equations. It appears that the basic advantage of using singular integral equations is to obtain the dominant component of the surface fields which is the first term of our representation. The computation of other higher order terms, however, is more involved in the singular integral formulation. 31

VI REFERENCES Chen, H. C. "Radial mode analysis of electromagnetic wave propagation on slotted cylindrical structures", IEEE Trans. AP-21, pp. 314 -320, 1973. Erdelyi, A., et al., Higher Transcendental Functions, Vol. 1, McGrawHill Book Co., Inc., New York, N.Y. 1953. Goldstone, L. 0. and A.H.Oliner, "Leaky wave antennas II: Circular Guides", IRE Trans. AP-9, pp. 280-291, 1961. Harrington, R. F. "Propagation along a slotted cylinder", J.A.P. 20, pp. 1366-1371, 1953. Hayashi, Y., "A singular integral equation approach to electromagentic fields for circular boundaries with slots", J. App. Sci. Res. B-12, pp. 331-359, 1966. Hobson, E. W-., "The theory of spherical and ellipsoidal harmonics", Chelsea Publishing Co., New York, Y.Y. 1955. 32

APPENDIX A SEARCHING AND ITERATING PROCEDURE Let us consider the problem of finding a complex root z = x + j y satisying the equation f(z) =0 (A.1) where f (z) is a complex function, involving transcendental functions such as Hankel functions. When the standard Newton-Raphson's method of finding roots of Equation (A. 1) is not successful, a searching and iterating procedure may be used in improving the convergence. In illustrating this procedure, we assume that near any zero of f (z), the function is analytic and the derivative of f (z) may be computed. We shall denote f (z) = f (x + jy) = U (x, y) + j V (x, y) (A.2) f'(z) A f(z) = U (x, y)+iV (x, y) Az x x =+V (x, y)- iU (x, y) (A.3) Y Y and assume that given x and y, U, V and their partial derivitives may be evaluated. Our suggested procedure for finding the complex roots are illustrated schematically in Figure A-1. The procedure may be described in the following steps: 33

(a) Given any initial y, compute U (x, yo) and V (x, yo), and scan x coarsely over a chosen range. As illustrated in Figure A-1, there exists points A (x, y) such that U (x and (x ) such that 0 0 0 0 0 O V (x, y ) = 0. From the coarse searching, in general for simple roots, a and b a b U changes sign at x and V changes sign at x. Values of x and x may be computed more exactly, from the results of coarse searching and Newton-Raphson iterative procedure. To improve convergence, if the difa b ference between x and x are too large, another value of y may be 0 0 chosen and the coarse searching repeated. a b (b) From x, y, and x, yo, we determine y1, corresponding to the y coordinate of C, which is the intersection of the two tangent lines AC and BC. To determine y1, we first computed U = U (x, y ) X X0 O U =U (xa, y ) y y o o b V (xb Vb =V (xb, y) y y o' o From these partial derivatives, it is easily seen that Ua Vb a- - (A 4)V Y1 = (x - U a Vb x x /xa ua xb Vb /Ua Vb \ 1 0 O X / X X 1(A.5) x =VyU - bV ]/ Uy a - Ux (A5) y y / y 34 a

y v=o U=o > (X1 A(x,yo) B(x,y) Figure A-1: llustration of Searching and iterating Procedure. 35

(c) Start from (x1, Y1), i.e., the point C. The process may be repeated, until for any y, |x a - x, x b -x m m m m are less than some preset criterion of convergence. (d) In some cases, this procedure may fail. For these cases in general x - X| > | Xm-l xb If this happens, the role of x and y should be interchanged in order to obtain convergence.

APPENDIX B REDUCTION OF THE DISPERSION RELATION The dispersion relation for the N-th approximation, given by Equation (66), indicates that the determinant of matrix [K] should be zero. If we denote y =(1- cos ) P (y) = sinm A P-r (cos A), n, m n nH Sinm A [pm(cos A) - (-1)+ P-m(-os A)] n., m n a (B. 1) (B.2) =P y) m(y-) T- (2-y)m nmnv m m (B. 3) the elements of LK] matrix are given by K, m= (ya) P (y)+Hn m(y) nm, nm Thus, the [K] matrix may be written as [K] = [S][p] + [H where (B.4) S O S1 [s]= SNJJ (B.5) a 37

and the matrices [p] and [H]have elements p, and Hn m respectively. Since S depends only on (ya) and p and H depends only on y, the roots (ya) of the dispersion relation are the same as that of the equation, det ([S]+ [H][p] ) =0. (B.6) Therefore, the matrix [Q] in equation (73) is given by [Q] = [H][p]' (B.7) or [Q[p] [H]. (B.8) The scheme for evaluating the elements Q.i of the matrix LQ given in Equation (75) through Equation (79) can be derived from the expression of p (c.f. Hobson, 1955). n, m n Pn,m (y)ym (n+r)' 1 (-1) ( )r (B. 9) PnS m = n=0 (n-r) r' (rer)' ()r The derivation makes use of a discontinuous summation formula which we shall state now. This relation is given by n' r (a+r): L=(n-r): r: - (b+r)! (a-b)! ()n a! a >(a-b) >n (a-b-n)0. o(b+n):w! ban b>a (b. 10) otherwis(b+n)e. 0 otherwise. 38

The proof of this relation is straightforward. One starts from the binomial expansion n (1 - U)= (a r) (B. 11) r=0 Multiply the above by Ua, differentiate both sides of the produce (a - b) times and let U —, 1 thus obtaining the first part of Equation (B. 10). Similarly, if we integrate both sides of the product (b - a) times, and let U -. 1, the second part of Equation (B. 10) is obtained. The third part of Equation (B. 10) becomes clear after the first two parts have been determined. Based on Equation (B. 10), one finds that n nm-Y m E (n+r)' 1 (1)r r r = 0(n-r): r: (r+r) 21 -r - 2m (- 1) (Y y)S (n+r)' (-1) 1m ) (B. 12) s= 2 s (n-r)l r s=m r=s-m n+ _ (n+r) (1)r 1 +=2mS rm (n-r) rC (m-s+r). In Equation (B. 12) the series in the square brackets may be expressed in closed form by using Equation (B. 11). Therefore, we have, m 2,m+n Er (-1)r (n+r)! r+m s = 2.= n =2 ((r+m) (n-r) r, 2 r=O n-mn+ fm s(-1)s )(m-s-) ( i n (B. 13) + nS.2y s: 2 (m-n-s-1). (n+m-s)! 39

where the second series is nonvanishing only if m > n. Thus, we have rm-n-1 2 (1)m+n+ (1)s y )s (m-s-l)': s! 2 (m-n-s-l) (m+n-s): H (y) m > n n, m 0 m<n. (B.14) In other words, the matrix [H] is an upper triangular matrix with diagonal terms zero. In order to find [Q, we would like to find the inverse matrix [p]i. However, since the order of matrix N is kept arbitrary, one finds that a relation independent of N, which reduces the [p] matrix into triangular form may be more suitable for computational purposes. ThIis relation is given by: [p] [a] = [c] (B.15) where [a] is a lower triangular matrix while C] is an upper triangular matrix. The diagonal elements of [C] are unity, and the element of [a] are *.. y i+j where a.. are constants (independent of y) and satisfying a =0 a >. (B.16) 40

The quantities acij and Ci.. can again be deduced by using Equation (B. 10). By matrix multiplication, we see that the elements of the [C] matrix is given by, n c = rn 1 n t y t p a r, s sn y s +n r (r - t) (-)t ( 2 ) = ( r - t) 2).- ( n a sn (s + t) '. (B. 17) Since C = 1, r = n, and C = 0 r < n, the set rn rn ac satisfy sn a. sn (s + t). 0 t <n (- 1) n! n (2n)! t = n (B. 18) Using Equation (B. 10), it is evident that n+m m! 2 m (n+m-1)! n., m (m-1)'! n! (m-n)! n>m (B. 19) and n-s n (n+r+s)! )r - (-1 Cn, s r (n-r-s)' r=O t1 r) r. 2 (2s+r)! (B. 20) Now, if we multiply both sides of Equation (B. 8), by [a] we have ] LC] [H] (B.21) 41

The elements of the A matrix is easily shown to be: A.. =- z Hik yk+j 1.3 k=i+l _____ i1 2 2j =I () - (- 1) ( -) (j-1)1. y Z (1)t( - t=0 I(j-t-1-)! t t~) k (k+j-lP. 1 k=j-t 1 Again, use Equation (B. 10), we have A. = (-1) I+1 ii 2' 2j-i- 1 t= t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I__ _ _ _ _ _ (B. 22) Fromi 'Eiquation (B. 2 1) and (B. 22), the expressions fo-r the el'Ae m ent-.ss then easily deduced. Q.i are 42

UNCLASSIFIED - - Secumntv ( ai;sific- ati n DOCUMENT CONTROL DATA R & D (Sectiritv clioxsl ir/ti)rt ott tfMlt, on/1 of ctI n Ittr'li( /mil Imde xlil'f i itti orllf/n nhesN Ae Jir In fered Whien Ilan avornil ren port l. r I- n lfA/if/ )., ORIGINATING ACTIVITY (Crt'porlftle Itlahior) 2l:. 'fCPORT SCCU, 'Y CLAS5FICA TION The University of MRichigan Radiation Laboratory UNCLASSIFIED 2216 Space Research Bldg., North Campus 2b. GROUP Ann Arbor, Michigan 48105 N/A 3. REPORT TITLE F MATHEMATICS OF GUIDED WAVE PROPAGATION IN OPEN STRUCTURES 4. DESCRIPTIVE NOTES (rypo of tcport iend Incluslvo datoa) Technical, Scientific, Final 5 AU THOR(S) (First nIme, middlo Itlltial, iast namo) Chiao-Min Chu 6. REPORT DATE 7t. TOTAL NO. OF PAGES 7b. NO. OF REFS May, 1974 42 6 68. CONTRACT OR GRANT NO. O. ORIGINATOR'S REPORT NUMBEIER(S AFOSR Grant No. 73-2538 b. PROJECT NO. 011778-1-F c. I-h. OTHER REPORT NO(S) (Any ol/hcr,rIlnIcrr Ithot mary I,he ssifiend this report) d. 10. DISTRIBUTION STATEMENT - SUPPLEMENTAR NOTES II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Office of Scientific Research, Air Force System Command, USAF, 1400 Wilson Blvd., Arlington, Virginia 22209 3. ABSTRACT. The propagation of electromagnetic waves along open structures whose transverse cross section are composed of two symmetrically placed circular sectors is investigated. The problem is first formulated in the form of a dual summation equation. By introducing a new set of basis functions to represent the surface field, the dispersion relation for such a structure is developed, systematic techniques of finding the approximate solutions of the dispersion relation are formulated, and numerical solutions for the first two orders of approximation are carried out to illustrate the feasibility of using the technique. In general, the scattering and resonance problem of open structures of cylindrical, spherical or other configurations may be formulated in the form of dual summation equations. The approach introduced in this work, i.e., the use of appropriate basis functions and the numerical solution of these equations, therefore, may open a new avenue in the solution of a wide class of problems. l n FORM 4 A "7 E L}/ I NOV o, 1 -t /1. UNCLASSIFIED 'dt& WarilV ('.~, ~ I, I I,iio

KEY WOHDS Wave Propagation Dual Summation Equation Dispersion Relation.UUNC LASSIFIED SL I E Dlv <'-.l-.,-..;.-,->- -" X*>v~slr'N 'I."I,' %., i I..|}I