312 BELL SYSTEM TECHNICAL JOURNAL 



The evaluation of these integrals is discussed in Appendix I. Thus (1.2-9) 

 is the p^^ equation of a set of differential equations to be solved simul- 

 taneously for ciil{z), a2t{z), • • • . 



The customary method of solving a set of equations such as (1.2-9) is to 

 assume that all the a's vary as e^' so that for each ocnliz) we may write 

 e^'grnl. This leads to a set of simultaneous homogeneous linear equations 

 for the constants gmt. In order that these equations may have a solution 

 the determinant of the coefficients must vanish. Since the only derivative 

 of oLm({z) contained in (1.2-9) is the second, 7 appears in the determinant 

 only as 7 . Let 71 j 72 , 73 , • • • be the values of 7 which cause the deter- 

 minant to vanish and let kij , hj , - " be the values of g\l , git, • • • cor- 

 responding to 7 = 7;. The ^'s are determined to within an arbitrary 

 multiplying constant which, for the sake of convenience, is chosen so that 

 ku = 1. 



Thus one solution of the differential equation (1.2-6) is 



00 



A — e"^^' cos {-Klylh) 2 ^w, sin (7rm:«;/a) . (1.2-13) 



w»=l 



This particular solution corresponds to the j*^ one of the modes (traveling in 

 the positive z direction) for which A is proportional to cos {irty/b). 



In much the same way it may be shown that the series (1.2-8) assumed 

 for 5 is a solution of equation (1.2-6) (with A replaced by B) provided the 

 coefficients /3mn(s) satisfy the set of equations 



00 



-V^M^) + S [QvmCtiz) - Upm^mdz)] = (1.2-14) 



for /> = 0, 1, 2, • • • and / = 1, 2, 3, • • • . Here 



Qpm = i^p/a) I (pi/p^) cos {irpx/a) cos (Tmx/a) dx (1.2-15) 

 Jo 



Upm = irmepaT^ I cos {irpx/a) sin (Tmx/a) dx/p (1.2-16) 



Jo 



where €o = 1 and €p = 2 for /> > 0. These integrals are discussed in 

 Appendix I. 



The problem of determining the reflection from a bend in a wave guide 

 involves considerable manipulation of equations (1.2-9) and (1.2-14). The 

 introduction of matrix notation in the manner suggested by the work of 

 Section 1.1 for straight guides simplifies this work. Although Omniz) and 

 /3mn(z) are no longer the simple exponential functions given by (1.1-9), it 

 turns out that the column matrices a(z) and ^(z) are still given by (for a 

 wave traveling in the positive z direction) by the matrix expression (1.1-10). 

 As mentioned earlier, Ta and r^ are no longer simple diagonal matrices. 



