L I B R.AFLY 
 
 OF THE 
 
 U N IVERSITY 
 
 Of ILLI NOIS 
 
 621.365 
 
 Ii€55-te 
 
 no. 40-49 
 
 cop. 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/finiterangewiene42mitt 
 
ANTENNA LABORATORY 
 Technical Report No. 42 
 
 THE FINITE RANGE WIENER-HOPF INTEGRAL EQUATION AND A BOUNDARY VALUE 
 
 PROBLEM IN A WAVEGUIDE 
 
 by 
 Raj Mittra 
 
 1 October 1959 
 
 Contract No. AF33 (616)-6079 
 Project No. 9- (13-6278) Task 40572 
 
 Sponsored by: 
 WRIGHT AIR DEVELOPMENT CENTER 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
-<s?n fit /O 
 
 CONTENTS 
 
 L*P> 
 
 i z 
 
 Page 
 
 1. Introduction 2_ 
 
 Formulation of the Problem 2 
 
 3. Solution for the Infinite L Case 7 
 
 -. Calculation of the Unknown Coefficients for L = 10 
 
 5. Solution for Finite L 15 
 
 6. The Complementary Integral Equation 23 
 
 7. The Edge Condition 30 
 
 8. Numerical Calculation 31 
 Conclusion ^2 
 Acknowledgments ^2 
 Appendix y^ 
 References ^Q 
 
1. Introduction 
 
 In this report we shall formulate the problem of a finite bifurcatien in a 
 rectangular waveguide the geometry of which is shown in Figure 1. The formu- 
 lation will he in terms of a finite range Wiener-Hopf integral equation. 
 
 x=^ a 
 
 x= o 
 
 / 
 
 1 
 
 
 i*o 
 
 *1 
 
 ?=L 
 
 FIGUEE 1 THE FINITE BIFURCATION PROBLEM IN A WAVEGUIDE 
 
 We shall subsequently present the solution of this integral equation using an 
 analytic, rather than iterative or approximate means. The solution for the 
 case when L is infinitely large will be a starting point for developing the 
 solution for finite L. The mathematical problem of solving a finite range 
 integral equation of the type 
 
 -p 
 
 9<y) = ] foo kox-y) d* , °<y^D (11) 
 
where ty(y) is prescribed for O <^ y <, D , and where 
 
 KfixO =!C n e' k " M (i.2) 
 
 has been discussed in considerable detail in a technical report . In this 
 paper we shall be concerned only with the particular kernel associated 
 with the present problem which also is a series of the type given in 
 (1.2). However, solutions to other problems, as for instance that of the 
 surface wave propagation on a corrugated surface as shown in Figure 2 can 
 also be obtained by the method outlined here. 
 
 FIGURE 2 CORRUGATED SURFACE 
 
 Formulation of the Problem 
 
 Consider the simplified problem in which the incident electric field 
 in the waveguide is the dominant mode field and is entirely in the ' V 
 
 direction. Let it be assumed that only the dominant mode propagates 
 
 * i i 
 
 in the guide. With no variation of the fields along y , the only field 
 
 components present are Ey, H^ and Hy and the entire field can be 
 
 derived from a scalar field potential A(%2) = t^. The other field 
 
 components are given by the equations, 
 
 * 
 
 This condition can be relaxed without requiring any modification of 
 the results. 
 
Li _ _ Uog "bA 
 
 u iwe ^>A 
 
 k a* 
 
 which are easily verified. 
 
 The problem can be stated as follows 
 is to be found, satisfying the equation 
 
 The function A (V^) 
 
 
 = 
 
 (2.1) 
 
 and the following conditions, 
 
 1. A=0 at X- } a for all 2 . 
 
 2. A-0 at *-*/2 ; 0<T^<L- . 
 
 3- A(*/?) = Incident wave + reflected wave for ^ 
 
 — 00 
 
 Transmitted wave for 
 
 -f-00 
 
 4. A an ^ ^^ are continuous everywhere except at the edges where 
 |VA| becomes infinite as h. (ft. is the radial 
 distance from the edge). 
 In order to derive the integral equation define a Green ' s function G- 
 
 by the equation 
 
 £x 2 "d* 2 
 
 (2.2) 
 
 and the conditions 
 
 (a) Q is continuous except at X* X i 7. - Z where it has 
 
a logarithmic singularity. 
 
 (b) Q is outgoing for |Z| — ^ °° , 
 
 (c) 6 = at X = Oja , 
 
 Using standard techniques such a (t is easily derived and the Green's 
 function is given by 
 
 , 00 -1%, |Z-B | 
 
 Q - . l ^ 1 sin nil* sin tfrrXo £ ( 2 3) 
 
 where 
 
 (A 
 
 y a 5 
 
 The 7^ S are quickly recognized as the mode propagation constant in the 
 guide. 
 
 From (2.1) and (2.2) the following is obtained 
 
 f(GV 2 /\ - AV 2 Gf) dxd* * A (Yo, h) 
 
 (2.4) 
 
 where 
 
 and the area S is the entire region concerned i.e. the region bounded 
 oy the contour Q shown. The surface integral is transformed in a 
 
line integral with the contour shown below and there results after some 
 manipulation, 
 
 1 
 
 — o0<4— 
 
 -tod 
 
 Co*\h>M* C- 
 
 fw-^ )d , = f$-*fyi 
 
 ^ e '.Sinn* + J jG.B(X;Z)j -d 
 
 (2.5) 
 
 2 
 
 where 
 
 ax 
 
 2 
 
 }x 
 
 2 
 
 - di'Sonkinuily of bA 
 
 ' ' dx 
 at y=a/ 2 
 
 The contribution of the line integral comes, only from the plate at 
 
 X =. Q~/2 and from the part of the contour at large negative % 
 
 - 1% 2 
 The latter gives rise to the term Q -SinrTX i n (2.5)' From 
 
 a. 
 (2.4) and (2.5) we obtain, 
 
 -17,2 f L 
 
 (2.6) 
 
 6m - Bfx,2)| y = 
 
 ^7 
 
6 
 Now applying the boundary condition A = for X - ^/z 
 0^2 £L , we get the desired integral equation for the current density 
 distribution on the septum which is simply related to BC2). The 
 equation is 
 
 -tf, Z ° 
 
 e 
 
 
 (2.7) 
 
 y = 4/ z 
 
 P L 
 
 -[y 2 . H i-*- z, 
 
 k ^ 1- .e .B^)dz 
 
 a 4 ,=o -/ 
 
 '2/?+l 
 
 or 
 
 e 
 
 "^ ; f L - , ^-i^/s-zoi 
 
 - i 
 
 
 We note that the kernel KlZ-iJof this integral equation is a series of 
 exponentials and our object is to solve for the function £>(£), 
 
 Before we go on to discuss the case of finite L we shall solve the 
 equation for L =0 ° • The reason for doing this is twofold. First of all, 
 in so doing, we shall be able to point out the modification necessary to 
 go from the infinite to the finite L case. Furthermore, we shall see 
 that the infinite L solutions are useful in developing the expressions 
 for &(2) for a finite |_ * The next section will therefore be devoted 
 to the case when the length |_ is infinite in the positive 2 -direction. 
 
3- Solution for the Infinite L Case 
 
 X = ^L 
 
 x = <V 2 
 
 > VoO 
 
 iU 
 
 x«« 
 
 '2 
 
 FIGURE 3 THE SEMI-INFINITE BIFURCATION 
 
 In this section we discuss the special case of L = 0° . It can 
 be shown without any difficulty that the integral equation in the present 
 case is obtained by putting L = oo in the Equation (2.7) and although 
 conditions are now different for J£ — >? cO j this does not modify 
 
 the Equation (2.7) in any way. 
 
 Hence, we consider the equation 
 
 e - ly >*° i 
 
 , e -i? w |2-2 | (3-1) 
 
 <L J 
 
 
 Assume now that 3 (2) admits of an expansion of the form, 
 
 Br*) = £ C„e"' % " 2 (3. 2 ) 
 
 Such an assumption is justified by the easily recognized fact that 
 ^n S are the mode functions of the half guide. Another explanation 
 
8 
 
 is that /iy)S are the zeros of the Fourier Transform of Kl^l 
 a fact which is stated here without proof but can be verified without too 
 much difficulty. As was explained in the technical report, a series of 
 exponentials of the zeros of the Fourier Transform of the kernel is the 
 proper representation for the unknown function of the integral equation. 
 
 Presently, the problem is to find the unknown coefficients Cj), 
 We discuss in the following a method for obtaining a set of simultaneous 
 
 equations for the coefficients Cr\ . It should be pointed out that this 
 
 2 3 
 problem (for L = oo) has also been worked out by Marcuvltz and Hurd and 
 
 Gruenberg , but by using different methods. 
 
 Going back to the Equations (3«l) and (3.2), we notice that the sub- 
 stitution of (3.2) in (3«l) would involve the evaluation of integral X 
 of the type, 
 
 I = /V v <e- ly - IZ "*''d Z (w) 
 
 for various combinations of 10' and V . 
 
 Upon evaluating the right hand side of (3. 3) there is obtained, 
 
 _ + Usui — _ (3 .4) 
 
 tii„rU) L(iL-K) 
 
 Hence, we conclude the following. The integral in the right hand side of 
 (3«l), when evaluated, yields series of exponentials of the type 
 
Now, in order for (3-l) to be true we must satisfy the following conditions; 
 
 —L'y 2 
 
 1. The coefficient of q ° must agree in both sides, i.e., 
 
 the coefficient of (=>"" t ' in the right hand side must be 
 
 equal to 1. 
 
 « -Winn z o 
 
 2. The coefficients of c must equal zero for all 
 
 3- The coefficients of 6 must equal zero for all Y) • 
 
 The first two conditions yield a set of equations for Cy^. The 
 equations are, 
 
 £ G, — „ a%s* n =o,u-- (3.5) 
 
 7)- 1 rj rJ > 
 
 Condition 3 however yields a set of identities as follows, 
 
 ^7o 7 z -7 2 
 
 '2/Z+l '2H 
 
 (3.6) 
 
 Clearly, the Equations (3.6) do not impose any condition on C^ S 
 Implicitly, however, the equations put a condition on the exponentials which 
 were used to express 3(2) in (3-2). It can be verified that the partic- 
 ular choice of the mode functions / zy) S in the series of exponentials 
 for B(2) instead of any other arbitrary set of constants, indeed does 
 make the Equations (3. 6) to be identically true. Another way of looking 
 at it is the following. / 5 are the zeros of the Fourier Transform 
 °f Kl2l and hence the Equations (3-l) a *"e identically true. 
 
10 
 We therefore should concentrate on solving the set of Equations (3-5) 
 for Cn • It i s seen that we shall accomplish this by calculating 
 
 Z^jbA^ where ^ is the determinant of (3-5) and ^ |p is 
 its cofactor. In the following section we present a method for calculating 
 the Cy» S> • 
 
 k. Calculation of the Unknown Coefficients for L = «o 
 
 To calculate the coefficients Cr\ S it is useful to consider a finite 
 size matrix, i.e., one with finite p 1 and then go to the limit of V) 1 — ^oo 
 The determinant of the system, if a ( btfj? ) one is considered, is, 
 
 A 
 
 v*» y 4 -7, 
 
 7.-7, 
 
 '2. '5 
 
 I 
 
 7-7 
 
 * '2|H 
 
 '4- '3 
 
 
 ^W ' 7. 
 
 %-% 
 
 7 -7 
 
 (*.l) 
 
 The determinant A due to the form of its elements is called a 
 "double alternant" in the language of the theory of determinants. This is 
 because each element of the determinant is a reciprocal of the difference 
 
 [^iy\ ~ y o \) and onl y one index viz., 'n' varies as we go 
 
 along a row and the other index tl' changes (keeping 10 constant) 
 
11 
 
 when we go along the columns. 
 
 Because of its form, the determinant can be simply expressed in terms 
 of difference products as shown below. 
 
 A = l 'l*l ^" (k.2) 
 
 KM 
 
 Notice first of all that the expression for A in (4.2) contains all the 
 factors (ill ~ Aq-fi) i n "the denominator which it must, in view of 
 the nature of the elements of the determinant given by the Equation (4.1). 
 Moreover, if one lets, say, ^ = °/ 4 in (4.1), it is seen that A = 0. 
 Hence A must have combinations of factors of the type i.'lra - 7lt) 
 For similar reasons it also has factors of the type ( ll 2P*i ~ °/ a+i) 
 A little algebra of repeated subtraction of rows from rows and columns from 
 columns would show that (4.2) is indeed true. 
 
 The calculation of the cofactors follows along the same lines. We 
 note for instance that ^it is just a (|^'0 * ('F"' 1 ) determinant 
 of the same type as A, so we have, 
 
 „ui«ca<aTY Of IIUNOIS 
 
A 
 
 M^'> 
 
 2 
 
 
 -^J 
 
 
 12 
 
 (M) 
 
 Essentially then, we get ^||- from A by cancelling out the factors 
 containing 7\ and Tzt- an d using a positive or negative sign depend- 
 ing on whether t is odd or even respectively. Since our primary in- 
 terest is to form ^Ifc/^ we calculate that using (^-.2) and (^-3) 
 obtaining, 
 
 
 Ht 
 
 •±* = (-) 
 
 
 <$=0 
 
 1- 
 
 £#fc 
 
 (*.*) 
 
 I 5 "" 
 
 f- 
 
 t-l 
 
 /} U (tl -*) il (V^D^J 
 
 Noting that the last two factors in the denominator can be combined and 
 rewritten as 
 
13 
 fct«. ^ f (4.5) 
 
 (4.6) 
 
 the ratio ^ /& is simply expressed as . 
 
 ZAib _ |=6 m 
 
 Equation (4.6) gives an expression for ^ l& i- n terms of the 
 mode constants of the guide. Once ^^/^ is known in the explicit fc 
 of (4.6), the set of Equations (3-5) can he considered as formally solved. 
 We have, of course, yet to let the index f go to infinity. Before we do 
 that, however, let us modify slightly the expression in the right hand side 
 of (4.6). Rewrite (4.6) as, 
 
 orm 
 
Ik 
 
 Ait 
 
 I- 1 
 
 ' 1*1 
 
 (VX) 
 
 R (1*,-*) h 
 <- ft 
 
 ( '2(- 4tj 
 
 (*.7) 
 
 and again as, 
 
 At 
 
 TT 
 
 a 
 
 
 
 ■ n iy 
 
 J 
 
 ^ & (iz V> ~ i>] (^ 
 
 (4.8) 
 
 and CK ~ 
 
 The reason for introducing the multiplying factors of the type v^fT 
 ( 2fy\) TT is the following. When b— ^o* the ratio of products 
 inside the curly "brackets can be asymptotically expressed as a ratio of 
 Gamma Functions multiplied by a finite number of terms. This makes the 
 numerical calculation easier , and moreover, assures the convergence of 
 
15 
 
 the expression in (4.8). This point is discussed in more detail in the 
 appendix (Appendix i). 
 
 Using (4.8) in (3-5) we obtain the expressions for Cvj as, 
 
 C 
 
 y\ = 
 
 h 
 
 
 f /JV-HV \ \ fJs-% 
 
 ^ 
 
 
 n 
 
 H 
 
 (^•9) 
 
 ^^T"'^ 
 
 c t 
 
 We have' in the Equation (4.9), in effect the solution of the integral 
 equation for the case of L = 00 . In the following we proceed to show that 
 the idea developed in this section can he extended to the finite L case. 
 
 5« Solution for Finite L 
 
 Let us go hack to Section 2 and to the Equation (2.7) where we have 
 the integral equation, 
 
 -tr,* 
 
 I fc 
 
 fit 
 
 aQ 
 
 -Um\ \?'2o\ 
 
 n^O 0/ 
 11 
 
 .%(z)di , 0^,<l 
 
 24+' 
 
16 
 and the problem is to determine B("?j • For L finite we assume the follow- 
 ing expansion 
 
 "° -l/rn 1 °° ,"v /z-L) 
 
 B(zj = «2- ^e -t Z— ^r» e ,_ „. 
 
 nd yi = i v5-i; 
 
 Physically the choice is dictated by the fact that in this case we have 
 
 to allow for standing waves in the region of 0£ 2- £-*- , and hence 
 
 _ / y / 1 _ 2 ) 
 
 include the terms £ 2M because of the extra discontinuity 
 
 _'w ^ 
 
 introduced at 2=i- . Mathematically the reason is that both 1? 
 
 and £ t ' iy| are now acceptable because the upper limit L is finite, 
 whereas when L is infinite the integrals involving £ do 
 
 not converge. 
 
 When (5-1) is substituted in the integral equation, we get in this case 
 the following four different types of terms after evaluating the integral 
 in that equation. 
 
 li^Zc L ^ 2>1 ( io ~ 1 ^ ~Llzmi &o 1/2/1+1 ^o 
 
 Following the same argument as was used for the case of infinite L one 
 sets the following conditions, 
 
 1. The coefficients .of 6 and £ should equal 
 
 zero for all 'n l . 
 
 n ^'211+1 io 
 
 2. The coefficients of 6 should equal zero for all 'r' and 
 
 of e ~^f^*\ Ho fQr all , r , except r = . 
 Since the left hand side of the integral equation contains only C 
 it is seen with little difficulty that the above conditions must be satisfied 
 in order for the two sides to agree, for all ~£ in the range concerned. 
 
IT 
 
 As in the infinite L case, it is found that the coefficient of 
 Q when equated to zero yields an identity. This 
 
 identity turns out to be the same as in the ]_, = oo case. Furthermore, 
 
 L%vi (Zo-L) 
 
 the coefficient of Q ' equated to zero yeilds the same 
 
 identity. Hence condition one is automatically satisfied. Equating the 
 coefficients of <? ~ u ^+' Z * and e ^ ^° yields the 
 
 following doubly infinite set of equations for the coefficients .Un and 
 
 Dn 
 
 03 
 
 - Fy, e 
 
 - it* i 
 
 '2 VI + '1Q.» | 
 
 a% S> 
 
 I ^0 
 
 (5-2) 
 
 e*9 
 
 11=1 
 
 -inr\i- 
 
 Dn e 
 
 Tavi -t 7 2/l+ 
 
 n 
 
 /-2V1 '3 
 
 20 + 
 
 = 
 
 (5.5) 
 
 There doesn't seem to he a way of expressing the determinant of the 
 set (5.2) and (5-5) in a simple form. However, a couple of steps modify 
 things to a more suitable form. Adding and subtracting (5-2) and (5-5) we 
 get, 
 
 oo 
 
 fDn-F^ 
 
 »rl 
 
 7^-7^+1 
 
 -L72nL 
 
 '2W 4'?+ I - 
 
 = 0-71S2 
 
 (5-M 
 
18 
 
 -i7zni 
 
 2(141 
 
 M S Q * 
 
 (5-5) 
 
 It is seen that (5.4) are independent sets of equations and the determi- 
 ats associated with them are quite similar. We also notice that L = 0° 
 )uld make the determinants of the sets the same and hence would imply that 
 
 Dn" ^y\ = Dn^ P>, or simply F n - > a result which 
 1 should of course expect. We now proceed to develop the expressions for 
 ^determinants of (5. 4) and (5-5), and their cofactors. 
 
 Written explicitly the determinant of (5.4), say A L is, 
 
 A L = 
 
 >-lV,L 
 
 V>5 -Tz-% 
 
 ■kA -\--hl \%-\ -%-% 
 
 -l7 4 L 
 
 j e 
 
 -UhL 
 
 > - 
 
 (5.6) 
 
 It is seen that (5-6) is a determinant each column of which is a sum 
 J terms. One can therefore use the well-known rule for expanding such 
 ■eterminant. Consider for instance the matrix [M] which can be written 
 a s a sum of two matrices [M-J and [M ]. Let 
 
19 
 
 [M] 
 
 a„ +b 
 
 l| T U„ 
 
 4,2. t bn 
 
 CU -4- t> 
 
 2| "+ u 2.l 
 
 a n + (> 
 
 2.1 
 
 (5-7) 
 
 - M + M 
 
 where 
 
 [H,] = 
 
 a». a 
 
 12 
 
 a ; 
 
 a 
 
 22. 
 
 [M 2 ] -. 
 
 L 
 
 >!2 
 
 >22 
 
 Then the determinant A of M can "be expressed as 
 
 A = 
 
 a,, 
 
 $21 
 
 + 
 
 
 a l2 . 
 
 
 -r 
 
 Q.i 
 
 biz 
 
 
 a i2 
 
 
 a 2 , 
 
 kv 
 
 b.l 
 
 a, z 
 
 
 
 
 bzi 
 
 a 22 
 
 
 
 
 
 fc>n bii 
 
 (5.8) 
 
20 
 It is seen that the determinant A can be expressed as a sum of, say., sub- 
 determinants which are formed by picking all possible combinations of the 
 columns of the matrices [mJ] and [m "] but never changing the respective positions 
 of the columns. 
 
 The rule is easily extended to an arbitrary size determinant. Going- 
 tack to (5-6), it is noticed that the matrix associated with it say [N] can 
 also be expressed as a sum of two matrices [N, ] and [N Q ] as follows, 
 
 [u]=M +K) 
 
 (5.9) 
 
 
 + 
 
 
 
 \%\- 
 
 e 
 
 ^3 
 
 Notice now that the first matrix N is the same as the one for L = cO , 
 and we have already calculated its determinant A. Notice also that if the 
 first column [N,] is replaced by the first column of [Ng], the new determi- 
 nant can be expressed as - e" * Af-V) > where ^ (" 7J 1S 
 
 the original determinant A of [N ] with 9^ replaced by - % 
 Thus we have the following possible expansion for the determinant ^ 
 
21 
 
 A L = A 
 
 00 -L7 2 aL 
 
 1- * 
 
 ^ -UTk + TUjL 
 
 (510) 
 
 8" , <*> , °° 
 
 - 2 
 
 _ C(1z.% + % n -*1is)L 
 
 ^il } n=^toS=n + i 
 
 /^-fy-w&K 
 
 4- - 
 
 Since the higher order mode coefficients '/-» S etc are negative imaginary 
 
 numhers and <i _-> _ 2^/71 as q becomes large, it is seen that 
 
 d a 
 the higher order terms tend to zero in an exponential manner. It can be 
 
 verified from the expression of the ratios of the determinants in (5-6) that 
 
 they only have an algebraic rather than exponential behavior. 
 
 The cofactors of the determinant J Say, A^ L can be expressed in a simi- 
 
 lar manner. Take for instance A 
 
 IbL 
 
 It has the expansion 
 
 A 
 
 ItL = 
 
 
 (5.11) 
 
 *," , 
 
 4 
 
 - 1 <VW L 
 
 
 A^-v^A 
 
22 
 The prime on the summation implies that the term corresponding to q = t, 
 r = t, etc. are to be excluded from the summation. ^\t is "the corre- 
 
 sponding cofactor of the determinant A, an expression for which was given 
 earlier in (^-3). The exponential decay of the higher order terms is also 
 present in the expansion (5-11) of A\H . Using (5-10) and (5.11) one 
 has , 
 
 Utr fc - A V (5.12) 
 
 where U and V are the expressions inside the curly brackets in (5. 11) and 
 
 (5-10) respectively. 
 
 From the form of the determinant of the set (5-5)? it is obvious that 
 ( TV ^ p \ can be expressed in a similar manner as, 
 
 (D t + \) -■ Q.% £11 X 
 
 (5-15) 
 
 where , 
 
 Xe j, ,. zS^H^c-w/ 
 
 <*>**/ 
 
 UM+fzn)*- 
 
 **»'"•»" . 1 (5.1^) 
 
 4 -t -- 
 
and. 
 
 Y = 
 
 / .-LTWl 
 + ^- e 
 
 -L (^7 2 a)L 
 
 ■+ - - - + -"- 
 
 Hence from (5-12) and (5.13) 
 
 23 
 
 A (-7 ZV ^)A (5.15) 
 
 D t 
 
 07 ^ib 
 2 ' Z 
 
 
 • F < 
 
 _ a'Vi At 
 
 2 A 
 
 ~* -U 
 - Y v. 
 
 (5.16) 
 
 The formal solution of the finite range integral equation is thus complete . 
 
 6. The Complementary Integral Equation 
 
 In this section we shall formulate the complementary integral equation 
 in terms of the fields outside of the bifurcation region. The advantage 
 of using this equation for the determination of the expressions of the fields 
 outside the bifurcation region will he pointed out. In addition to that the 
 method developed in Section 5 will he shown to he applicable for solving this 
 equation also. It will therefore be of interest to discuss the complementary 
 
 integral equation. 
 
 X- 4. 
 
 ^ 
 
 i 
 1 
 1 
 
 
 z--<- 
 
 -< 
 1 
 
 J 
 
 C 
 
 y=o 
 
 ^ x = a/ 2 
 
 
 
 
 
 FIGURE k 
 
 
2k 
 
 Consider the geometry of the problem again. From the symmetry of the struc- 
 ture and for the dominant mode incidence it is seen that the potential A 
 satisfies the boundary condition, 
 
 M =0 
 
 H <LO 
 
 cvn 
 
 A 2 >L 
 
 in addition to the ones mentioned previously, viz., 
 
 A = o, x = OA h y *" ' z ' aYU * A = °' X °i ' ° a ^ 
 
 Now suppose we consider the region bounded by the contour C shown in 
 Figure ^-. Choosing a Green's function G which satisfies, 
 
 G =0 , x = a , O for o\\ Z 
 
 z 
 
 G — *0 oo ? — * =*> 
 
 the following equation is easily derived 
 
 (6.1) 
 
 Afo, z ) = 
 
 •at? I 
 
 dz 
 
 X=G 
 
 tx 
 
 (6.2) 
 
 /V*,?) 
 
 
 di 
 
 X-A/a 
 
 The Green's Function G satisfying (6.1) is 
 
 ^--""rv,^, Xv) fl- 
 
 CC 
 
 a 
 
 (6.3) 
 
e<t 
 
 uah'ovi 
 
 25 
 
 The Equation (6.2) is the complementary integral since the range of 
 
 A 
 
 integration along Z in (6.2) is complementary of the range of (2.6). 
 Note also that the unknown now is A, or for that matter the Ey field 
 along £ in appropriate ranges rather than the current distribution on the 
 septum. 
 
 = for - ©* <£ 2 <:o and 
 
 Applying the conditions |n 
 
 X = */z 
 
 L i 2 <L°° 
 
 we get 
 
 O = 
 
 A 
 
 lx^a/ 2 
 
 dx c)X 
 
 oO 
 
 - d* 
 
 
 x-£/ 2 3x^x 
 
 da 
 
 (G.h) 
 
 _ cd C 2o<o o^ A i{z e /oO 
 
 Using the expression for ^£ in the appropriate ranges in 
 we obtain writing 7(2)- /\(x, 2) I* -<y 
 
 O ^ 
 
 pr?) 
 
 
 -C7a»| 
 
 7- 2, 
 
 dn 
 
 (6.5) 
 
 + 
 
 P^) . 
 
 2./M) 
 
 T. 
 
 n =i 
 
 
 Hz* {*'**) 
 
 _ oo <£ 2o ^.0 
 
26 
 
 Now assume that PCZ) - A C >r y ^ j ) ^ ^ has the fQrm 
 
 Pr z) . e + Z_^„e ,-<«. (66) 
 
 which is obtained by assuming that J\(x,2) is expanded in terms of 
 
 the appropriate mode functions in the guide for the ranges of 2 concerned. 
 When (6.6) is substituted in (6.5) we get integrals I of the type, 
 
 ii r / e . e cig 
 
 which are similar to the integrals I appearing in (3.3) but with the roles 
 of r f zn and 7i>, interchanged. We also get integrals I , where, 
 
 J-z = J e > e «i 
 
 L. 
 
 When these integrals are evaluated we get the exponentials £ 
 
 -L7j» £0 ^ Using the ideas developed before, we obtain the Equation 
 
 by equating the coefficient of 6 to zero, which condition 
 
 is required for the satisfaction of (6.5). Furthermore, the coefficient of 
 <o " ° turns out to be an identity and hence yields no new 
 
27 
 
 information or equation for A and F . Another set of equations is 
 
 P P 
 
 obtained when the range L < ^o <°° is use( i- The set is 
 
 ^°\ v -^ V - ^' ■'**^ 
 
 (6.8) 
 
 *Ul,2, — ■ 
 
 From (6.7) and (6.8) one may obtain the following equations which 
 prove to he more suitable to work with. 
 
 <r /a \ / 1 e -t^L \ _ou 
 
 £_ ^ h F p ) -L_— ^ £ ' t £ 
 
 *>=o v 7™-'i|»ti -7^,-7^/ 7»-f7»v, 7,-7 ? n (6.9) 
 
 - L 72./) L \ - t ?2» L 
 
 J___ _ £ ) m j 1_. 
 
 h° \ Xn "A^fi ~ /zbti ~7**i ' 7i+%v? 7,- 7 2K1 
 
 A^+fy) 
 
 (6.10) 
 
 fl = 1,2; - OO 
 
 Notice first of all that the matrix of (6.9) is the transpose of 
 (5-5) and the matrix of (6.10) is the transpose of (^.k). Hence, the method 
 of dealing with them will be expected to be similar to the one developed 
 earlier. However, the right hand side of the equations here is different 
 from that obtained in the corresponding case in Chapter 5- Here, the 
 right hand side, treated as a column vector is obtained by replacing 7/ 
 by -Y in the first column of the matrix associated with (6.9) , (6.10) . 
 
28 
 
 Hence , we readily obtain the form of the solution 
 
 (6.11) 
 
 where A is the determinant of (6.9) and Z\ I H ry ) implies 
 
 1 '2/J + l ''/ 
 
 the determinant obtained by replacing n l )V r/ 
 
 /Z^-+! / ~ 'I » 
 
 Similarly, 
 
 (2-) 
 
 
 (6.12) 
 
 (2) 
 
 where A is associated with (6. 10). 
 
 If our primary interest lies in obtaining the Reflection and the 
 
 Transmission Coefficients- A^ and &., it is seen that we have to evalute 
 
 J) 
 
 (6.11) and (6.12) for n = O only. That A and ft are indeed the Reflec- 
 tion and Transmission Coefficients is seen from (^6.6). 
 
 To complete the solution we should present the expression for the 
 ratios of A's. The expressions are developed in the manner developed 
 earlier. Consider for instance A , which written explicitly, is, 
 
 A 
 
 (0 
 
 
 ,-O^L 
 
 -%-Vi 
 
 
 ) —j—>\ 
 
 l*-* 
 
 
 (6.13) 
 
29 
 
 The expansion of Z^ 
 a difference of two, is, 
 
 obtained by treating each of its rows as 
 
 A 
 
 iu _ ^ 
 
 
 (6.1*0 
 
 
 where, 
 
 Z^ - 
 
 
 (I) 
 
 (6.15) 
 
 The expression for A is the same as given in (k.2) , since it is essen- 
 tially the same determinant as appears in (4.1). 
 
 It is seen that the leading term of the expansion of A is A. It 
 will be of interest to derive the expression for the leading term of the 
 
 Reflection Coefficient A . Since ' 
 
 1 L-9o0 
 
 l_-* eO 
 
 Hence, 
 
 From (h.2), 
 
 tivy, A -" A (* ~* ■ rV ' ) 
 
 
 A 
 
 o = 
 
 \\*v\ 
 
 nj ■ r 
 
 M 
 
 (6.16) 
 
30 
 The higher order terms are developed without much difficulty. 
 
 7. The Edge Condition 
 
 As was pointed out in Section 2, the field potential has to satisfy the 
 edge condition, which is in fact a condition on | Sjh\ in the vicinity of 
 the edge. In this section we shall indicate a way of verifying that the 
 edge condition is indeed satisfied by the solution obtained. An alternative 
 way of stating the edge condition is that the current density distribution 
 on the septum tends to infinity as Vj^" 1 as Z—*Ot an( 3- as /ft- 7 ft 
 as 7 — * L_. We will accomplish this if we show that Bfc) defined in 
 connection with the Equation (2.5), has the above behavior. Following Hurd 
 we know that this will follow if in the expansion of B(2) , which is 
 
 Bfz)* 1. ^e ■+ 2_ R, e 
 
 we show that, 
 
 
 F* 
 
 l<i ^ y\ — » °o j Ki= romUvit 
 
 ^ ^ (<a. = ^onstatft 
 
 »* 
 
 It is easily seen that the coefficients D and F determine the behavior 
 . J n n 
 
 of PjfZ) as 2 -*fl and Z — ' L respectively. 
 
 From (5.16), remembering that the leading term of the expansion of both 
 )</Y and U/ v is 1, we see that the behavior of D n as y\ -» *> is 
 
 primarily determined by ^iv> . As shown in Appendix I, ^JJ? indeed 
 
 A ^ 
 
 tends to j£i as /) becomes indefinitely large 
 
31 
 
 Considering F now, we see from (5.l6) that the leading term in F 
 n n 
 
 is P e /£±\*\ where R is a constant. Hence, as ?^L. 
 
 2T at fa* edtyz &L- 
 
 6(2) again goes to infinity as /(i_-2)^ • The integrated current of 
 
 course is much smaller compared to the current density at the edge 2=- o r 
 
 A 
 
 because of the multiplying factor Q~ z (note that Ok is a 
 
 negative imaginary number). 
 
 8. Numerical Calculation 
 
 Only a brief discussion will be given here on the aspect of numerical 
 calculation. We observe first of all that the computation of the mode co- 
 efficients in the various regions involves the calculation of infinite 
 series of terms which contain ratios of infinite products. It was pointed 
 out in earlier sections that only a finite numbers in the infinite series 
 need be taken because of the exponentially decaying nature of the higher 
 order terms in the series. As far as the ratios of infinite products are 
 concerned, they can be expressed asymptotically to ratios of finite products 
 and appropriate Gamma functions as shown in the Appendix. Sample numerical 
 computations have been carried out and they have not been found either 
 difficult or large time consuming. Although the digital computers are 
 helpful, a hand computer is also quite efficient for obtaining answers even 
 when L is of the order of ^9/2 for the dominant mode., or even smaller. 
 
32 
 Conclusion 
 
 The finite bifurcation problem in a rectangular waveguide has been 
 formulated in terms of a finite range Wiener-Hopf integral equation. The 
 solution of this equation has been obtained by analytic means. The case 
 when the septum is semi-infinite has also been included and a connection 
 between the methods of solution for the finite and the infinite case has 
 "been provided, the finite case being an extension of the infinite one. 
 The problem of numerically calculating the quantities of interest, viz., 
 the mode coefficients has been discussed briefly. The technique presented 
 here is general and is applicable to other finite range Wiener-Hopf integral 
 
 equations for which the kernel can be expanded as a series of exponentials 
 
 - % \2-2o\ 
 of the type (=> where 2 and 2.0 are the variable 
 
 coordinates. 
 
 Acknowledgments 
 
 The author wishes to acknowledge the support provided by Wright Field 
 Air Development Center through the Research Grant AF33(6l6)=-6079. It is 
 also a pleasure to acknowledge the facilities of the Antenna Laboratory of 
 the University of Illinois who provided for the preparation of the paper. 
 
33 
 
 APPENDIX 
 
 The purpose of this appendix is to recast the form of ^»fc given in 
 the equation (^-.8) into one involving Gamma functions which enables one 
 to show the convergence of the products and also helps study the asymptotic 
 
 behavior of ^Jt as t becomes large. The technique illustrated here for 
 
 An- & 
 
 — c can also be applied to the ratios of other determinants that appear 
 
 in the text. 
 
 Let us rewrite (^.8) as 
 
 
 - i 
 
 v*)f ¥ 
 
 1 *l" 
 
 
 XJ l H- l% )\ 
 
 >(A-1) 
 
 I (i1 V lf& 
 
 -2 
 
 J 
 
 = !I.P,.R 
 
 I • ' 2 
 
 4, 
 
 where P and P are the ratios of the products in the first and second curly 
 brackets respectively. 
 
 Now P, can be modified and put in the form 
 
 
 
 P 
 
 HOV* J fz 
 
 (A-2) 
 
 =. R,. R 2 - ^3 
 
3h 
 
 where R, , Rg, R, are the three ratios of products enclosed by the big brackets 
 in that order. 
 
 The reason for putting P in the above form will now be explained. Re- 
 membering that l?^ — => 1>il , for q large compared to k, it is observed that 
 the ratio , 
 
 (U\-^)% 
 
 
 for large values ' q'. 
 
 (A-3) 
 
 Hence, instead of going to indefinitely large values of p when comput- 
 ing the first curly bracket R in (A-2), we can only take a finite number 
 as the upper limit. For a given accuracy requirement this upper limit, say 
 P, is easily determined. The same is clearly true for the ratio of the prod- 
 ucts R x . What we must do now is evaluate the ratio of the products appear- 
 ing at the center, or R_. To this end consider the following well known 
 expansion of the inverse of the Gamma function V(oc) 
 
 _ at 
 
 o<r (V) 
 
 
 Y\-\ 
 
 (A-k) 
 
 So for 'large 'p' we can write the limit say Q of R as, 
 
 l l> 
 
 
 1 
 
 n 
 
 
 — \ - 
 
 fri-W 
 
 |( Wl 
 
 V 1 !- 
 
 a 
 
 r' l 
 
 (a-5) 
 
 J*) See for instance Magnus and Oberhetinger "Functions of Mathematical 
 Physics", Chelsea Publishing Company 
 
or 
 
 35 
 
 Q, = 2.. 
 
 
 (a-6) 
 
 Similarly the second curly bracket P p in (A-l) may "be written as 
 
 
 S.,( ( V*)^J 
 
 (I 
 
 ->o0 
 
 ^k 
 
 ^2t 
 
 . ill* 
 
 f 1 
 
 
 ■ — / * 
 
 Y 1 J 
 
 v 
 
 ^2 ' ^ w 1 IT 
 
 / 
 
 v 2 
 
 It ^) 
 
 r7°£) 
 
 s> 
 
 ""•* f'-'fj z f'-^) '7^) 
 
 (A-7) 
 
 Combining all this it is possible to express ^/t with a high degree of 
 
 accuracy as ; 
 
 <4 
 
 CL At 
 
 IT A 
 
 i irtr*) T,^i) r(^) ry^] coht,^ 
 
 — * 
 
 0*^) (1-1%) ^,-rt.j J 
 
 (A-8) 
 
36 
 
 where 
 
 W) ^ { 
 
 V 
 
 
 
 (A-9) 
 
 and 
 
 J = J with the factors q = 1 omitted 
 
 [\U 
 
 
 T 
 
 (A-10) 
 
 
 Since the products J and H have a finite upper limit P, it is much more 
 convenient to use (A-8) for calculating the ratio Ait > rather than the 
 expression in (A-l), which involves the upper limit p tending to infinity. 
 
 Next we go on to consider the asymptotic behavior of 4i£ as t — ^u> . 
 Going hack to the equation (A-8) we see that this amounts to studying the 
 behavior of the function say T, where 
 
 r = 
 
 r i <***) 
 
 Co 
 
 trM 
 
 z 
 
 
 l"Vtf* 
 
 irrtr, 
 
 2 1 2.|T 
 
 i|Ti 
 
 (A-ll) 
 
 
 o 
 
 as 7*fc — * "" l ")£ It is noted that the only other 
 
 factors that involve 't' are 7 (YU ) and hf(Yu) and the y consist of 
 finite products and their limit for large t are constants which are easily 
 obtained. 
 
ar 
 
 37 
 Inserting the asymptotic behavior of the Gamma functions for large 
 guments we get using jcob (lY*t^\ ft IT - ^'Y*t-£ )| ~ * ' ^ ^2t^ -* ^ > 
 
 U IT I — * S J- 
 
 where S is a constant. Hence there follows that, 
 
 (iv^ Z\it __^ — where S Q is another constant (A-12) 
 
 2 
 
 Equation (A-12) gives the asymptotic behavior of ^i£ as t -=? ^ . It also 
 
 tr 
 demonstrates the validity of the statement made in Section 7 of the text 
 
 that ^Mt has the rVu behavior as t ' — *? <*) 
 
 We shall close the appendix with one further remark. For numerical 
 
 calculation, the most convenient expression for the ratio of the products 
 
 <4j£ is the one appearing in (A-8). This Is because the products J and 
 
 H appearing in that expression have a finite upper limit. As pointed out 
 
 in the beginning of this section, other ratios of products appearing in 
 
 the text can also be recast into a similar form. 
 
REFERENCES 
 
 38 
 
 1. Mittra, R., "On the Solution of a Class of Wiener-Hopf Integral 
 Equation in Finite and Infinite Ranges," Technical Report No,. 37 y 
 Antenna Laboratory , University of Illinois. 
 
 2. Marcuvitz, Waveguide Handbook , R. Lab. Series, Vol. 10, 
 McGraw-Hill Co. 
 
 3- Hurd and Gruenberg, "H-Plane Bifurcation of Rectangular Waveguides," 
 Canadian Jour , of Phys . , Vol. 32, No. 11, November, 195^.