596 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



minus the same expression with the suffixes interchanged. Hence 



(Ai - Ao)QEe{ro) 



To 



+ hipE + ph) - 



(Ai - A,)m>H r-^ + pAl - Ph') 

 \VeVh 



, / 1 2 i3^ \\ XiroJn'ixiro) 

 AiVh \l - Ph - I > ^—. r— 



(24) 



-<Kpe + Ph) - A.PhII 



2 _ /3" \\ X2roJn'{X2ro) 



VeVh/} JniXiro) J' 



where use has been made of the relation AiA2 = —1. Therefore the 

 characteristic equation for jS is obtained by equating the term in square 

 brackets to zero. Because of the quadratic relation satisfied by A and 

 the relation between A and x, it is possible to write the characteristic 

 equation in a great variety of ways. It will be convenient to introduce a 

 function 



Fnix) = F.nix) = 



Xjn'jx) 

 Jnix) 



(25) 



Using the /^-function and replacing the A's by x's the characteristic 

 equation may be written :* 



2 



/ 2 2\ 



nvH\X2 — Xl ) 



_VeVh 



+ Pe{1 — Ph ) 



_ I3(pe + Ph) 



= ^ i^n(xiro) 

 A2 



2 



— -^ Fn{x2ro). 

 Ai 



(26) 



The asymmetry of this equation between pn , vh and pe , ve arises from 

 the fact that the boundary conditions involve electric field components 

 alone. 



It may be noted that if the basic solution had been taken to vary as 

 cos nd or sin nB, the expression for Eg would have been a linear combina- 

 tion of sin nd and cos nd that could not have vanished at the walls for 

 alU. 



In passing we remark that for a guide of arbitrary cross-section, the 



* The characteristic equations given in Reference 4 were specializations to the 

 ferrite and plasma cases of the form in square brackets. They have also been de- 

 rived by Kales* and Gamo^. These authors have given expressions for some, though 

 not all, of the varieties of cut-off point derived in this paper and classified them 

 as TE or TM according to the field configuration at cut-off. By contrast, they are 

 classified here by their association with quasi -TE or quasi -T]\I limit modes which 

 reduce to the usual TE and TM modes in the unmagnetized medium. 



