416 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



to a small value of b, 



Vi = 



f 



M 



P'2 = k 



f -h M - 1 

 f - h 



Kz 



- 1 



P — fh — g- co-fiuE yi 



2 - /X - k2 



(11a) 



(lib) 



Discarding the z dependence in equation (10) and dropping a constant 

 multiplier, the two characteristic electric field solutions become: 



.(1) 



(,mTlb)n iy 



(12) 



E''' ^ 



'0' 





(mrlb)y 



(13) 



Equations (12) and (13) are parallel plane solutions obtained for 

 some arbitrary direction, y, transverse to the magnetic field. This direc- 

 tion need not be intrinsically real; mathematically, it simply satisfies 

 Maxwell's equations. We may transform to a desired waveguide frame 

 of reference by rotations ^i and ^2 , corresponding to pi and p-2 , about 

 the z axis, where these rotations may possibly be made through complex 

 angles. We then have for the electric fields in the new space: 



^(1) 



1 - 



M 



COS (fl + i sin ^1 



1 - 



M 



sin <pi + i cos cpi 



lyL 



{mTrlb)ii }(j/cos(f i+rsinvJi) 



(14) ., 



I 



.(2) 



I sm (f-i 

 i cos (p2 

 i 



^(.mirlb) (t/cosii!>2+-''sirn?2) 



(15) 



The new 7 axis of the transformeil coordinates is now considered the 

 longitudinal axis of the waveguide. 



The partial wave fields of (14) and (15) may be joined to form a single 



I 



