CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1215 



modes may be assumed to be numbered in order of increasing cutoff 

 frequency; later when we have occasion to refer to specific modes we 

 shall replace the subscript n by the customary double subscript notation 

 for TE and TM modes in a circular guide. 



The 7'-functions are assumed to be so normalized that 



f (grad r) ■ (grad T) dS = f (flux T) • (flux 7') dS 

 Js Js 



(7) 



^x" f T'dS ^ 1, 



where S is the cross section of the guide. The gradient and flux of a 

 scalar point-function W are transverse vectors with the following com- 

 ponents: 



dW dW 



grad„ W = ——, grad„ W = -—-, 

 eidu e20V 



(8) 

 dW dW 



flux„ W = ^^, flux„ W = — ^^. 



e2dv eidu 



Various orthogonality relationships exist among the T-functions corre- 

 sponding to different modes of the guide, and among their gradients and 

 fluxes. These relationships have been listed by Schelkunoff.^^ 



The transverse components of the fields in the curved guide may be 

 derived from potential and stream functions, U and ^ for TE waves 

 and V and n for TM waves. Thus 



Et = —grad V — flux^, 



(9) 

 Ht = flux n -grad U. 



We now assume series expansions for the potential and stream functions 

 in terms of the functions T{u, v), with coefficients depending on w. Let 



V = -JLVMiw) Tm{u, v), n = -T.hn){w) Tm{u, v), 



n n 



^ (10) 



n n 



The /'s and F's have the dimensions and properties of transmission-line 

 currents and voltages. If we substitute (10) into (9) and expand in 



