316 BELL SYSTEM TECHNICAL JOURNAL 



curved wave guide (of the same cross-section). Comparison of the first 

 equation in (1.1-7) with the first equation in (1.2-5) shows that Ex is con- 

 tinuous if (1)^ is continuous and (2) if dB/dz in the straight portion is equal 

 (the equality being taken at the junction) to (pi/p) dB/dz in the curved 

 portion. Examination of the expressions for the remaining field com- 

 ponents shows that all the continuity conditions are satisfied if, at the 

 junction, 



[A in straight portion] = [A in bend] 



and likewise for B. 



Let A in the bend be given by (1.3-1) and let a{z) denote the column 

 matrix of coefficients shown in (1.3-2). A in the straight portion may be 

 represented in the same way except that a(z) has a simpler form as explained 

 in Section 1.1. When these expressions for A are inserted in (1.4-1), both 

 sides multiplied by {2fa)s\n{irpx/a) after cancelling out the cos {irly/b), and 

 the results integrated with respect to x from to a we obtain relations which 

 may be expressed as the matrix equations 



[a{z) in straight portion] = [01(2) in bend] 



e square matrix wh 

 1,2,3, .-Ois 



where V is the square matrix whose p^^ row and rn^ column {p, m = 



V vm = (2pi/o) / ^\n {irpx/ a) sin {irmx/ a) dx/p, (1.4-3) 



p being equal to pi-{- x — a/ 2. 



By using expression (1.3-11) for B in the continuity conditions, it may be 

 shown in much the same way that the column matrix /3(2) given by (1.3-14) 

 rflust satisfy the relations 



[^{z) in straight portion] = [/3(z) in bend] 



W- ]-^*'^H "■"' 



where W is the infinite square matrix 



PFoo PToi 

 W = PTio Wii • (1.4-5) 



