G52 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



where 



TM modes show no first order variation with a. 



(b) small q , a" not near unity. Here ^a and Va are the actual propaga- 

 tion constant and radius, without scaling factors : 

 TEiTO mode: 



2 



Pa = (1 — g JW MofO — — ;• 



2 aq' I 2 \ , ^^4^ 



1 — (r2 Vwm^ - 1 



(71) 



TMim modes: 



/3„^ = (1 - g^)coVoeo - %' - co^o T^ (l " "T^, ) • (72) 

 (c) Approximation for large o- ; q not near unity. 



TEi„ mode: 



1 uj' 2q 



1 - g2 ^„2 ^(1 _ ^2) (^^2 _ 1) 



(73) 



All formulae in (a), (b), (c) apply to both positive and negative a (right 

 and left circular waves). Formulae 70 and 73 show that the first order 

 changes in /3 , whether due to very large or very small a, have coefficients 

 that differ only in sign. 



The TEn mode near resonance {q < 1) 



Near cr = +1, 



q' \i^' " V (74) 



/3 = - 



1 - g2 0- _ 1 



