lOH 



BELL SYSTFAf TECHNICAL JOURNAL 



Since, by assumption, only the dominant mode is freely propagated, k and 

 7m for w > are real and positive. 



We imagine an incident wave of unit amplitude coming down from Z5 

 in the upper left portion of Fig. 1. WTiat arc the amplitudes of the reflected 

 wave traveling back toward Zs and the transmitted wave traveling outward 

 to the- right towards Z3? Our task is to find a Q{x, y), satisfying the wave 

 equation (1-4) and the boundary condition (1-5), which represents a 

 disturbance of the assumed type. 



Z5 = oo e 



L{7r-2a) 



Zn = 00 



v = -t 



1Z4 



v = t 

 Fig. 2 



"T" 



77- 



■e-TT 



V *■ CD 



61 = 



The first step is to find the conformal transformation 



z = x+ iy = f{v + id) = f{w) (2-2) 



which carries the bent guide (shown in Fig. 1) in the (;v, y) plane over into 

 the straight guide (shown in Fig. 2) in the (v, 6) plane. This may be done 

 by the Schwarz-Christoffel method discussed in Appendix I. This trans- 

 formation carries the wave equation (1-4) and the boundary condition 

 (1-5) into 



9r dd- 



^ = at ^ = and = tt 



(2-3) 

 (2-4) 



