(2.2- 13) 



(2.2-14) 



REFLECTIONS FROM CIRCULAR BENDS 321 



Since /~ and /+ specify the reflected and transmitted waves, respectively, 

 they give the answer which we are seeking. 



If the curved guide should extend from z=— octoz = and the straight 

 guide from s = to s = oo the response to an incident wave e''^ h coming in 

 along the curved guide would be 



^(2) = e~'^ h-\- e"" r, z < 



m(2) = e-^'V\ 2 > 



A procedure similar to that used above shows that 



/-= -(ro+ Fr)-i(ro- w)h 

 /+ = (Fo + Fr)-i 2Wh 

 where, instead of condition (2.2-9), we have used 



2.3 Reflection Due to a Bend 



Let the guide be straight for — oc < z < —c and for t < s < co , and let 

 these two portions be connected by a curved portion in which the longi- 

 tudinal coordinate z runs from — c to -f^. As in Section 2.2 we take the 

 matrix propagation constants for the straight and curved portions to be the 

 square matrices To and F, respectively, and assume an incident wave, 

 specified by the column matrix h, to come in from z = — oc . 



The column matrix /Lt(2) whose tn^ element appears as the coefficient of 

 Bm{x, y) in the representation (2.2-3) for $ is now given by 



^(2) = e-'^'h-{- /^V~ z< -c 



y.{z) = (cosh sr)^ + (sinh zT)q, -c < z < c (2.3-1) 



^(z) = .-^'V^ c<z 



In these expressions /~, /+, p, q are column matrices which may be de- 

 termined as functions of the known matrices To, T, V and h by substituting 

 (2.3-1) in the conditions (2.2-7) which must hold at the junctions z = —c 

 and z = c. 



By straightforw^ard algebra similar to that used for the analogous problem 

 in transmission line theory we obtain 



-cTo^ + €-'''' f"^ = [-/ + 2(Fr tanh cT + To)"' To] e'^' h 



(2.3-2) 

 -cT,j- _ ^-cToj+ ^ [_j _^ 2(VT coth cT + To)"' To] e*^ ° h 



