CON FORMAL TRANSFORMATION 121 



which is taken over 3, 5, 7 • • ■ in the expression (7-7) for /?// \ Moreover, 

 if we make use of 



E Mn - 1)"' = 1 



n=2.4.6,--- 



we see that the contribution of the last term within the square brackets of 

 (7-10) cancels the remaining terms in Rli\ Only the contribution of the 

 first term in (7-10) remains and it gives 



Ri,^> = -U^'c-' E n('^ - ly'C + 0(/3') (7-11) 



71=2.4.6.- • ■ 



The relative simplicity of this result indicates that there may be another 

 method of derivation which avoids the lengthy algebra of our method. 



Recently approximate expressions for the reflection coefficient of gentle 

 circular bends have been published^. In our present notation these may be 

 written as 



Rg ~ — ib" pi 



sm " i L y^ cos ti — e 

 24 m=i.3.5.-- ir^m'^ym 



[-uijc 2 n 



sm M r. '^^ cos u — e n 



Sir-C- n-2.4.6.--- T*C8„ {ll' — 1)'J 



where /Stt is the angle of the bend, pi is the radius of curvature of the center 

 line of the guide and u is 2-k times the length of the center line in the bend 

 divided by the wavelength in the guide: 



u = ^Trkp\/b = ^TTcpi/a 



The first expression for u is to be used in Re and the second in Rh . If we 

 now let /3 -^ 0, keeping pi fixed, then ii -^ 0. The trigonometric and expo- 

 nential terms may be approximated by the first few terms in their power 

 series expansions, and part of the series which make their appearance may be 

 replaced by their sums given, for example, by equations (4.1-7) and (4. 1-8) 

 of reference^. After some cancellation, the above expression for Re and Rh , 

 which hold for gentle circular bends, reduce to (7-5) and (7-11), respectively, 

 which hold for the sharp corners. In other words, the reflection coefficients 

 for both the sharp and the circular bends approach zero as /? -^ 0, and 

 furthermore their ratio approaches unity. 



We shall merely outline the derivation of the approximation i?^ for a 

 truncated corner. Instead of (7-1) we have from (2-6), 



g{v, e) = e.xp M - 1 = a<p + aVV2! + 0(a'), (7-12) 



^ = 2 \og[chv 4- cos e] - log [ch{y - t) - cos 6] - log [ch{v + /) - cos 6] 



