1382 THE BELL SYSTEM TECHNICAL JOURN.IL, NOVEMBER 1956 



It follows from (A3b), using the zero-order approximations (A2), that 



1 nha/r2a \ « \ I3,a{e' - ie'f'^ \ 

 so the characteristic equation finally takes the approximate form 



m 



(A4) 



f2a 



[(fsa tan 4^)' + (MV - ie')] 



Now let 



f itt = p + .r, I a; I <<C 1 



h\ 



where a: is a small complex number. The normalized propagation con- 

 stant becomes, to first order, 



iha = [(fia)' - (MT 



= 2M(1 - vy" - ivx{l - vT'" 



= aa + i(l3nma + A^a) 



where /3„„i is the phase constant of the mode in a perfectly conducting 

 guide, and the perturbation terms are 



aa + iA^a = — 



ivx 



(1 - v2)l/2 



(A5) 



For the TM„„i mode, let p be the //;*'' root of J„ ; then from Tajdor's 

 series, to first order in x. 



Jni^ia) = J nip + .r) = .Vjn(p) 



(A6) 



Substituting (A6) into (A4), neglecting the first term on the left side of 

 (A4), and replacing everything on the right side by its zero approxima- 

 tion according to (A2), one obtains 



(/3oa)^ ipMU — ■2"e" - 1 + i'^) tan" \p + (e — ie')] 



X 



{e' - ie" - 1 + v'~y 



12 



or 



(A7) 



