REFLECTIONS FROM CIRCULAR BENDS 



315 



{y)l - Tl) k, = 



(1.3-9) 



and the elements kij, ^2/, • • • of the modal column kj are the ones appearing 

 as coefficients in (1.2-13). 



Equation (1.3-9) and the methods of matrix analysis lead to 



tI = k[y% k-\ Ta= k Md k-' (1.3-10) 



where k is the square matrix whose^ column is kj and [7 ] d, [y] d are diagonal 

 matrices having7/, 7; as thej* elements in their principal diagonals. The 

 representation (1.3-10) certainly holds for the rectangular guide since in 

 this case no repeated roots occur. 



In analogy with the expression (1.3-1) for A we shall henceforth deal 

 with B in the form 



B = sin {irly/h) ^ ^m({z) cos {irmx/a) 



(1.3-11) 



where /has one of the values 1 , 2, 3, • • • . In much the same way as before it 

 may be shown that for a wave traveling along the bend in the positive 

 direction the ^m^(z)'s in (1.3-11) are given by 



^(2) = e-'^' d^ (1.3-12) 



where J+ is a column of arbitrary constants and 



r| = Q-' {tI + U) (1.3-13) 



In (1.3-12) and (1.3-13) 



To = 



Q = 



Kz) = 





u = 



t/01 f/02 

 Un Un 

 U21 ' 



(1.3-14) 



where the elements of To, Q and U are given by equations (1.1-5), (1.2-15) 

 and (1.2-16), respectively. 



1.4 Continuity Conditions at Junction of Straight and Curved Rectangular 

 Guides 



Electromagnetic theory requires that Ex, Ey, Hx and Hy be continuous in 

 crossing a plane s = constant which marks the junction of a straight and a 



