REFLECTIONS FROM CIRCULAR BENDS 327 



The expression for /p may be obtained in the same way but it is slightly 

 more complicated. 



/t = eHx, - y,) = e''''--'-\\ - A^ + A, (sinh 2cy^)/2 + A,] (3.3-14) 



where 



^, = (1 - r^-')(Tp - KY/«) 



A — \^' -2cTm T/ T/ _l_ * P^ *^ ^P ~ ' mp ^ pm ^^ pm J^ mp 



A4 — Z^ e \ — Vpm Vmp -r »2 ^2 I" /s2 $2x2 



m L Om ~ Op \^0m — Op) J 



_ ^, (g-^T-»»cosh2c7p - 1) 



m 28m8p{dm — 5p) 



^2 rr t:^ ^2 



(81 + 5yFp^F« 



pm^mpOm Vrnp^pmOp "T ^2 ^2 I 



L d,„ — Op J 



There are several points we should mention about these formulas for/p and 

 /pi The summations with respect to m run from 1 to oo with the term 

 m = p omitted. 7y and 8j are the propagation constants of the yth mode 

 in the bend and in the straight portion, respectively. The difference 

 7p — 6p may be expressed in terms of the F's by equation (3.2-2). In the 

 course of obtaining (3.3-13) and (3.3-14) relations of the following sort 

 were used. 



^p(l + ip)'' = ^-''''^'^(sinh 2cyp)/2 



(tm + 4)(1 + ^p)~'(l + 0~' = e-'^'^(cosh 2cTp - e-'''-) 

 The term Az arises when we subtract (1 — tp) (1 + /p)"^ from 

 5p/(5p + 7p/p) - 8ptp/(yp + 8ptp) 



Since 7p — 8p is 0(^^) for a circular bend in a rectangular guide (7p — 5p)^ 

 is Oit) and hence ^3 is negligible in the cases we shall consider. 



The reflected wave set up by an incident wave of unit amplitude and con- 

 taining only the p^^ mode (i.e. the incident wave described at the beginning 

 of this section) is given by the column matrix/" whose elements may be 

 obtained from (3.3-12) and (3.3-13). Likewise, the transmitted wave is 

 given by /+. 



PART IV 



GENTLE CIRCULAR BENDS IN RECTANGULAR WAVE 



GUIDES 



4.1 Propagation of Dominant Mode in a Gentle Bend — H in Plane of Bend 



When the magnetic intensity H lies in the plane of the bend, Hy = 0, 

 and equations (1.2-5) show that B = 0. Thus we have to deal only with 



