CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1219 



Y^,„)[n] = ioo / eCaCgrad T(,„)) • (flux T[n]) dS, 

 Js 



Y[m](H) = ?co / eeslflux T [,„])• (grad T(„)) d*S', 



F[,„][„] = to / eesCgrad T [„,])• (grad T[n]) dS + F,„, [,„][„]. 



It may be noted that if the guide were straight, so that 63 = 1, and 

 if /i and e were constant over the cross section, the F's and Z's would all 

 be zero except for those having equal subscripts. If the curvature of the 

 guide is gentle, and if /x and e do not vary much over the cross section 

 (or if they vary extensively only in a small part of the cross section), 

 then the F's and Z's with unequal subscripts will be small, and we can 

 obtain approximate solutions of (23) based upon the smallness of the 

 coupling. 



We shall now compute first-order approximations to the impedance 

 and admittance coefficients under the following assumptions: 



M = Mo , 



(25) 

 e = eo(l + 5), 



where /xo and eo are constants (usually but not necessarily the per- 

 meability and permittivity of free space), and 6 is a dimensionless func- 

 tion of position. No mathematical difficulties would follow from assum- 

 ing fi as well as e to be variable, but since the case of varying permeability 

 is not of such immediate practical interest we shall omit this slight addi- 

 tional complication. In order that the coupling per unit length due to 

 curvature and to inhomogeneity of the dielectric be small, we further 

 assume that 



\^\^a/b« 1, 



(26) 



I f \ 6 \ dS « 1, 



where, as ireual, S is the cross-sectional area of the guide. 



From (21), first-order approximations to Z„, [,„][„] and F„.,(,„^(n) are 



Zw,[m][n] — l(jiljLo[8,„n — ^[m][n]], 



(27) 



Fu,,(m)(„) — iueo[8mn + 5(m)(n) ~ ^(m)(n)], 



