138 BELL SYSTEM TECHNICAL JOURNAL 



where m = 1, 2, 3, • • • , /%„ = cPF,n/dv", and the prime on ^ indicates that 

 the term ;/ — m is to be omitted. In groui)ing the terms we have assumed 

 that Fo is the major part of Q. 



The principal problem is to solve equations (1-4) and (1-5) when the 

 fundamental mode Fq is of the form 



(1-6) 



Fo = Te{v), 1- ^ +0C 



in which Re is a constant and Te(v) represents a wave traveling towards 

 I! = 00 . At z' = ±20 Fi , F2 , ■ ■ ■ have the form of waves traveling (or 

 being attenuated) away from the region around v = 0. As before, we shall 

 be mainly interested in determining the reflection coeflicient R. 



It is assumed that only the dominant mode is propagated without attenu- 

 ation in the straight wave guide far to the left and hence Fi , Fo , ■ ■ ■ all 

 become zero as i' — > — =0 . 



2. Differential Equations when Magnetic Vector is in (x, y) Plane 



The partial differential equation is now given by equation (5-1) of the 

 companion paper'^ 



^+^+ n + dv,e)U'P = ^) (2-1) 



dv^ dd- 



where the dimension of the system normal to the (.v, y) plane is now b, a is 

 the dimension (in the (x, y) plane) of the straight guide at the far left and 



p = at e = and 6 ^ ir 



1 -f g{v, ^) = 1 + E a« cos ne (2-2) 



n = l 



K = 2c/Ao , Xo = free space wavelength 



C = (k2_ 1)1/2 



Since P = at ^ = and 6 = ir we assume 



P = J^Fn sin nd (2-3) 



n = l 



where the F's are functions of v to be determined by the equations 



2 « 

 Fi + [k(\ + flo - 02/2) - l]Fi + ^ E ('/«-i - «n+.)/^„ = (2-4) 





F'J, + [k(\ +Co - a2m/2) — m'\Fm + 2 ('^'"-1 ~ am+i)Fi 



2 « 



-f •;r E' (a\m-n\ " flm+„)Fn = 

 -i n=2 



(2-5) 



