1274 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



With these expressions the functions (36) are approximated by: 



Vn\X, pX) ,2 2\ I fs(f2\ 



■f=- ^ = b\x - n ) + 0(6 ), 



Z„(.r, px) 



and for n = especially: 



1 Fo(a;, px) _ 1 U\{x, px) 

 px~ Zo{x, px) X Ri(x, px) 



(39) 



-8 



l+l + sM^o; 



a---i 



(40) 



+ (5^) 



Since for 5 = the roots of Jn{pxi) = and Jn(pxi) = represent the 

 solutions of (1) for the TM and TE waves respectively, we expand the 

 Bessel functions of the argument pXi in series around these roots: 



pxi = (1 - 8)(pnm + Ax). 



The result for TM waves is 



JniPnm) - 0: :^4^ = ^ \^ ; (41) 



Jn{pXi) Ax — 8pnm 



for TE waves: 



//(p..) = : -If^ = '^^^ (X - 6p.„.) ; 

 and for TEom waves especially : 

 Joipom) = 0: 



'Jo {pXlJ 1 / . 2 , -2 2\ 5 /r, 1 >-. 2\ 

 f-7 T- = Ax — dpom — [AX -\- 8 Pom) — -7 Pom{o + 2po,n ). 



Introducing (39 • • • 42) into the characteristic equation (1) and neglect-| 

 ing higher order terms in 5 and Ax we get the following approximations 

 for (1): 



TMnm waves: Ax = ( pnm — ) 8 



2 2 2 ^ 



rrjTTt A X2 Pnm 'I' ^ 



Thnm waves: Ax = — - — ^— ^ ■ 5 



n^O Pnm " ^ ^Pnm 



(42) ' 



