CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1217 



and integrating gives 



dV[m 

 dw 



= —lo) 



£ [/(„) f m (grad Tm) ■ (flux T^m{) dS 



+ /[„] / ^lC:i (grad T[m]) • (grad T^) dS 



(15) 



Similarly, from the fourth and fifth equations we get 



dl (m) 



dw 



= — ?co 



E 



l',„) f «, (grad r,..,) • (grad T,.,) dS 



+ 



F[„] f ees (grad r(„o) • (Aux ^[,0) <i*S 



(IG) 



dl 



[m] 



dw 



X[m]-^ w ,[7n] 



= -i^ S 



T\,o f ees (grad r(„)) • (flux T^„,{) dS 

 •Is 



— V[n] I ecz (grad ?'[„,]) • (grad T[n]) dS 



Js 



(17) 



From the third of equations (5) using the fact that the ^-functions 

 satisfy (6), then multiplying through by !'[,„] and integrating over the 

 cross section, we get, 



's ez 



V[m] — — io) 



Z2 ^^v,[n]X[n] I 

 n Js 



dS. 



(18) 



Similarly, from the sixth equation, 



T _ ■ \^ jr f ^T(n)T(m) 



1 im) — —tOl 2^ \ «,,(,0X(n) / 



n Js 63 



These equations may be written in the form 



dS. 



(19) 



V[m\ — — 2_/ 



n 



/(m) = — 2_/ 



^w ,[m] [n] -*«;,[«] 

 X[m] 



V V 



■' to, (to) (n) ' w, (n) 

 X(m) 



(20) 



where we define 



,[TO][n] — ■?WX[m]X[ 



Js €3 



-t«),(TO)(n) — ^WX(ro)X(n) / ttO. 



Js Cz 



(21) 



