REFLECTIONS FROM CIRCULAR BENDS 



313 



We now turn to the task of expressing (1.2-9) and (1.2-14) in matrix 

 form. 



1.3 Propagation Constant Matrix for Curved Rectangular Guide 



From this point onward in our investigation of propagation in the rec- 

 tangular guide we shall assume A to be proportional to cos (irfy/b). Thus 

 instead of the general expression (1.2-7) for A we shall deal with the more 

 restricted form 



A = cos {irly/h) zl ccml{z) sin {irinx/a) 



(1.3-1) 



where /has one of the values 0, 1, 2, 3, • • • . Since the most general dis- 

 turbance may be obtained by the superposition of disturbances of the form 

 (1.3-1) no real generaUty will be lost. 



The introduction of (1.3-1) is suggested by the fact that the set ai^(z), 

 ci2({z), ' ' ■ may be determined from (1.2-9) (at least to within arbitrary 

 constants of integration) without considering the other OmnizYs, n 9^ t 

 The introduction of (1.3-1) is also suggested by physical reasons. The 

 plane of the bend is the z, x plane and there is nothing in the system tending 

 to change the field distribution in the y direction. 



Equation (1.2-13) is a special case of (1.3-1). Furthermore the most 

 general form of (1.3-1) (corresponding to a wave progressing in the positive 

 z direction) may be obtained by multiplying (1.2-13) by an arbitrary con- 

 stant Cj and summing on j. 



In order to write the set of differential equations (1.2-9) for the amtizYs in 

 matrix form we introduce the infinite matrices 



Fo 



a{z) 



aitiz)' 



0L2l{z) 



azl{z) 



(1.3-2) 



S = 



Sn S12 



»S'21 ►S'22 



where the elements of To are obtained by setting n = I'm equation (1.1-5) 

 for Tmm and the elements of P and Q are given by the integrals (1.2-10) and 

 (1.2-11). The rules of matrix multipHcation then show that (1.2-9) is the 



' element of the matrix equation 



Poc'\z) - (rS -f S)a{z) = 

 Premulti plying by P~^ converts this equation into 

 a"{z) - T\ a{z) = 



(1.3-3) 



(1.3^) 



