REFLECTIONS FROM CIRCULAR BENDS 317 



whose elemenst are 



Wpm = {€ppi/a) I cos {irpx/ a) cos (irmx/ a) dx/p 



^0 (1.4--6) 



€0=1, ep = 2 for p > 

 Both Vpm and Wpm are discussed in Appendix I. 



PART II 

 THEORY FOR A GENERAL WAVE GUIDE 



2.1 Matrix Propagation Constant for a Curved Wave Guide of Arbitrary Cross- 

 Section 



In Section 1 .3 it has been shown that for a curved rectangular wave guide 

 there exists a square matrix Ta (or Tp) which plays the same role in the 

 propagation of a wave consisting of many modes as does the propagation 

 constant in a simple transmission line. There Ta was obtained from a 

 special form of the wave equation which is suited to bends in rectangular 

 guides. Here we adopt a different approach with the idea of showing that a 

 matrix propagation constant T exists under more general conditions. 



The general theory of wave propagation in tubes shows that a wave 

 traveling in the positive z direction may often be represented as 



^ = 12c,e-'y^^{x,y) (2.1-1) 



where $ is some quantity associated with the field and is analogous to the 

 functions A and B of Part I. In (2.1-1) x and y are transverse coordinates 

 and z a longitudinal coordinate, jj is the propagation constant for the f^ 

 mode and <Pj{x, y) the corresponding eigenf unction. For a circular bend in a 

 rectangular wave guide (pj{x, y) is a combination of trigonometric and Bessel 

 functions and 7; is proportional to the order of the Bessel functions. 



We assume that we may find a set of functions Bm{Xj y)^m = \,2,?), . . . 

 such that every (pj{x, y) may be represented as 



00 



^;(^, y) = ^ kmidmix, y) (2.1-2) 



m=l 



The usefulness of this procedure depends upon our ability to pick a system 

 of dm{x, yYs which is appreciably simpler than the system of (pj{x, yYs. In 

 the work of Part I dm{x, y) was taken to be the eigenf unction of the typical 

 mode of propagation in a straight guide, i.e. the product of a sine and a 

 cosine. 

 We assume further in (2.1-2) that the square matrix k~^ exists where kmj is 



