792 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



They are assumed to be normalized as follows 



J|(grad T) • (grad T) dn = x' JJt' d^ = 1 , (34) 



where dQ is an elementary solid angle. Hence, equation (14) will again 

 represent the complex power flow in the direction of propagation. 

 The various field components may then be expressed as follows 



^^(.n) . jT dT[n] j^ _ ^-r dT(„) ,, dT{„] 



^ a ' ^["1 ^ ' rJ^i' — y (.n) — — V [n] - , 



Si du 62 dv 62 dv Ci du 



rM„ - 1 („) — -— + 1 [n] —-— , rHu - — ^ (n) — + I [n] —^- , 



Bi du 62 dv e-i dv ex du 



r Er = X(n) Vr,{n) T (n) , T Hr = X[n] /f . [n] T [n] . 



It should be noted that the physical dimensions of Vt.m and Ir,[n\ are 

 not those of voltage and current. Substituting in Maxwell's equations 

 and using transformations similar to those in the case of plane waves, 

 we find 



'^ - - // (* 



dT{m) ^n dT(r, 



62 dv ei du 



dJ^ 



dr 



dV[m] 



">) 1 JO I .. ^-2 



_rB,-^]dn-\- xwr n.c^), 



dr 



dllm] 



dr 



j. K (rD,. "^ + tD. ^') da, 



J J \ ei du e-i dv / 



f[(rBj-^^rB.^-^)dU, (36) 



J J \ Ci du e-z dv / 



= jc f( (- vD. ^ + rD. ^') dfi + X[.ir-^.fH , 

 J J \ 62 dv 6i du/ 



= -jcu 



F[.] = -jo^ ff(r'Br)T[^^dn, Ion) = -jc^ fl{r'Dr)T^,n) 



dQ. 



Returning to the plane wave case and assuming the following general 

 linear relations 



Bu = UuuHu + fJLitvHy + HuzHz , Du = e„„£^„ + euvEv -\- tuzEz , 



Bv = fJivuHu + Hvvtlv -\- UvzHz , Dy = truEu + e^Ev + t^zE , , (37) 



Bz = nziiHu + i^ivH„ + nzzHz , Dz = tzuEu + tzvEv + e33£'2 , 



