REFLECTIONS FROM CIRCULAR BENDS 331 



further approximation of putting r„,o for ym . Since T^o — Jm is 0(^^) no 

 serious error is introduced and we have 



_ _ e sinh 2crio eS ^ (cosh2crio-g-^^'"o) m' ,, , ^, 



glO - op2 2 — 2 Z^ ^ -, 2 r-^ TT^ (4.2-6; 



olio a TT ,„=2.4.6... llOlmOa (w^ — 1)' 



in which 



ar^o = lArn - 1) + a'^5o]^ ^ = a/pi . (4.2-7) 



For frequencies such that only the dominant mode is propagated the 

 ratio of the power in the reflected wave to the power in the incident wave is 

 I gTo 1^. Marshak has given an expression for this ratio w^hich is the same 

 as that obtained from (4.2-6) when the negligible (for his case) terms e~^''^'"° 

 are omitted. 



The corresponding expression for the transmission coefficient derived 

 from (3.3-14) ior ft is not as simple as (4.2-6). 



4.3 Propagation of Dominant Mode in a Gentle Bend — E in Plane of Bend 



WTien the electric intensity E lies in the plane of the bend, Ey = 0, and 

 equations (1.2-5) show that ^ = 0. Here we deal with B in much the 

 same way as w^e dealt with A in Section 4.1. The dominant mode is ob- 

 tained by setting / = 1 in the siniir ty/b) in the formulas pertaining to B 

 in Section 1.3. It is assumed that b > a. 



Examination of the matrices (1.3-14) indicates that, for the sake of con- 

 venience, we should call the top row of our matrices the 0*^ row and the left- 

 most column the 0*^ column. In line with this we call 70 the propagation 

 constant of the dominant mode in the bend. The elements bm of the diagonal 

 matrix t\ are obtained by putting n{= /) = 1 in (1.1-5): 



bl = rL = ^'+ (Trm/af + ir'/b\ /« = 0, 1, 2, • • • (4.3-1) 



When we make the appropriate shift in the subscripts, equation (3.2-2) 

 yields 



00 

 To' = To'i + /^oo + Z Forr.F„.oa\-^m-^ (4.3-2) 



m=l 



in which the elements of the matrix F are to be determined from (1.3-13): 

 F=tI-tI = (Q-' - I)tI -f Q-'U (4.3-3) 



As in (4.1-4) we have, with Q = I -\- T, 



/ °° \ * (4.3-4) 



Fa = ( - Tii + Z Ti„. T„u ) t\, + Uu - Z Ti,. Unu . 



\ m— / mmmO 



