REFLECTIONS FROM CIRCULAR BENDS 309 



1.2 Propagation in a Circular Bend 



In dealing with a circular bend we choose cylindrical coordinates (p, (p, y) 

 as shown in Fig. 1. With these coordinates we associate new coordinates, 

 shown in Figs. 1 and 2, (x, y, z) which have approximately the same signifi- 

 cance as in the straight guide, z is the distance measured along the axis of 



z=o 



Fig. 1 



Fig. 2 



the guide (defined as the locus of the centers of gravity of the transverse 

 cross-sections of the guide), and x and y are the transverse coordinates. 



Let p = pi = (p2 + p3)/2 = P2 + a/ 2 be the radius of curvature of the 

 guide axis, and let the origin of the polar coordinates be taken at the center 

 of curvature. Then s is equal to —pup where the minus sign is necessary to 

 make {x, y, z) a right-handed coordinate system. Since the vertical (in 



