1214 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



written in the form:^ 



I 

 1 



(5) 



In these equations the permeabiHty ju and the permittivity e may be 

 functions of position. If there is dissipation in the medium, either e or /x 

 or both may be complex. 



To convert Maxwell's equations into generalized telegraphist's equa- 

 tions, we introduce the field distributions characteristic of the normal 

 modes of a straight, cylindrical guide filled with a homogeneous dielec- 

 tric. The derivation follows very closely that given by Schelkunoff" for 

 an inhomogeneously-fiUed straight guide. Each mode is described by a 

 transverse field distribution pattern T(u, v), where T{u, v) satisfies 



6162 



du \ei du / dv \e2 dv , 



xT, 



(6) 



and X is a separation constant which takes on discrete values for the 

 various TE and TM modes. We shall denote the function corresponding 

 to the nth TE mode by 7'[„](m, v), and the .separation constant by X[«i , 

 with the subscript in brackets. The normal derivative of T[„] vanishes 

 on the perfectly conducting waveguide boundary. Similarly the function 

 corresponding to the nth TM mode is denoted by T^n){^i', v), and the sepa- 

 ration constant by xcn) , with the subscript in parentheses. The function 

 T(^n) vanishes on the boundary of the waveguide. For the present the 



