REFLECTIONS FROM CIRCULAR BENDS 333 



B in the same manner as V is associated with A, now replaces V. Thus 

 our expression (3.3-13) for the reflection coefficient becomes 



/^i = ^ = - [^i(sinh 2croi)/2 + ^2] (4.4-1) 



where we have neglected the difference in Toi and 70 and where 



00 



Ai = 2w,, + (t? - rSi)r^i' - E 



Tr Om TF,nO + 2 _2 o 



V^ cosh 2croi - e~^'^^^ 

 A2 = Z^ 



=1 2r^iroi7r2(z-2m2 



From TFoo = 1 + ^oo and the asymptotic expressions (Al-19) it follows 

 that, for w = 1, 3, 5, • • • 



Woo = t/^2 



-2 2 (^•4-^) 



TFo. = 2^w V ', Wm, = ^m \-' 



For even values of m, Wom and Wmo are 0(^2) Substitution of these values 

 together with those for the F's given by (4.3-5), using the sums (4.1-7) 

 and expression (4.3-6) for 70 — Foi leads to 



Ai = ^2/12 



for m odd. 



Thus the reflection coefficient for the dominant mode when E lies in the 

 plane of a gentle bend of length 2c is approximately 



,- I'sinh 2croi , ^'4roi V cosh 2croi - e-''"^"^' , . . ., 

 doi = 7^ + —-T- 2^ z;t^ (4.4-4) 



where F^i is given by (4.3-1) and b > a. 



PART V 



NUMERICAL CALCULATIONS 



5.1 Bend in Plane of Magnetic Vector 



Let a/b = 2.25 and XoA = 1.400 where Xq is the free-space wavelength 

 • of the dominant wave striking the bend. The propagation constant 



