C^i) 



(32) 



CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1221 



where 



H(m)(„) = / ^(gi-ad r(,„))-(grad T^) dS, 



H(.)W = f s^ (grad Tco) • (flux Tt,]) (/>S, 



H[,„](«) = /^(flux T[,„])-(grad T^) dS, 



H[,«][«] = / ^(grad r[„])-(grad T[n]) dS, 



and 



A(^)(„) = / 5(grad T (,„))• (grad T^n)) dS, 

 •Is 



A(m)M = f digrsid r(,„))-(flux Tm) dS, 

 •Is 



A[m](n) = f d{nuxT{,nj)-(grsidT^n)) dS, 

 •Is 



^Min] = / 5(grad T^) • (grad T[n]) dS. 

 -Is 



The H's are zero unless the angular indices of the two modes involved 

 differ by exactly unity. 



1.2 Representation in Terms of Coupled Traveling Waves 



From now on Ave shall assume that the distribution of dielectric over 

 the cross section of the curved guide is independent of distance along 

 the guide, so that the impedance and admittance coefficients are con- 

 stants independent of z. (We shall henceforth designate the coordinates 

 by (p, <p, z), instead of the {u, v, w) of the preceding section.) The general- 

 ized telegraphist's equations now represent an infinite set of coupled, 

 uniform transmission lines, and their solution would be equivalent to 

 the solution of an infinite system of linear algebraic equations and the 

 corresponding characteristic equation. 



For our purposes it is convenient to write the transmission-line equa- 

 tions not in terms of currents and voltages, but in terms of the ampli- 

 tudes of forward and backward traveling waves, assumed to exist in a 

 straight guide filled with a homogeneous medium. Thus let a and h be 

 the amplitudes of the forward and backward waves of a typical mode at 



