CONFORM A L TRANSFORMATION 113 



The second approximation, obtained by substituting (4-1) in (3-5), may then 

 be written as 



00 



Q^-\v,, do) = e^''"" + k'2-' X y^' cos mdo 



(4-3) 



L 



The ;/th approximation i?i"^ to the reflection coefficient (when the 

 electric vector lies in the plane of the bend) is defined in terms of Q^"^ by 



Limit Q'^'iv, d) = e-'" + R'.-'e"" (4-4) 



V— » — 00 



Re"^ is also equal to the integral obtained by replacing Q in (3-6) by (2^"~". 

 We have 



7?y> = 0, Rf = -ik2^' r ao(T)e-''''' dv, 



J— cc 



rT =R';-' -ik'j:(^yme^r' (4-5) 



• / dvoam(io) I dvam(v)( 



J— 00 *J — oo 



-ik(v+VQ)—lv—vo\ym 



where jm is given by (2-1). 



The results of this section have the same generality as the integral 

 equation (3-5) in that they are not restricted to corners. 



5. Truncated Corner — Magnetic Vector in Plane of Bend 



When the magnetic vector lies in the plane of the bend the reflection 

 may be calculated by a similar procedure. The wide dmaension a of the 

 wave guide now replaces the narrow dimension h in Fig. 1. We shall call 

 the result of making this change the "modified Fig. 1". We again assume 

 the frequency to be such that only the dominant mode is propagated without 

 attenuation. In place of equations (1-3, 4, 5) involving Q we have those of 

 (1-6, 7, 8) involving P. 



The conformal transformation which carries the modified Fig. 1 into 

 Fig. 2 leads to 



^ + ^ + [1 + ^0', ^)]'<^P = 



9r- dd- (:>-l) 



P = at ^ = and 6 ^ ir 



