318 BELL SYSTEM TECHNICAL JOURNAL 



For the //-wave, Ez is zero everywhere, 



Eb ~ Jn (Xp) { . „ 



I Sin w^ J 



and the possible waves are determined by the transcendental equation 



/„'(Xa) = 0, (7) 



where 



\1 = y-i J^ ^2J^2 



and Jn(z) = {d/dz)J„(z). These values of X and consequently of 7 

 will, of course, differ from those characterizing the 7^^- waves. Similarly, 

 however, there will be a doubly terminating series of possible com- 

 ponents, Hnm- 



Hence for both types of wave the hollow conducting cylinder consti- 

 tutes a high-pass wave-filter. The critical frequency /„„ of the Enm- 

 wave is given by 



fnm = rnm(c/2Tra-y[eil), (8) 



where r„m is the mth root of /„(Xa) = or r,,^ = X,„„a. Similarly for 

 the //nm-wave, the critical frequency is 



fnm' = rnm'icllira^), (8)' 



where 



rnm' is the mth root of Jnm'(}<(i) = 0. 



The propagation constant jnm is then 



to} c _ toi 



(9) 



where the ratio cjvnm of the velocity of light in air to the phase velocity 

 of the wave in response to any frequency/ is given by 



Civnm' = V^Vl - (f.Jfy 



-^0 when /-^/„^, (10) 



— > ^^ when / -^ oc 



for the £-wave and 



Cl%J = VcmVI - (fnJIf)' 



for the //-wave. 



For the £-wave we have 



^01, ro2, ••• = 2.405, 5.52, • • • 

 ru, ri2, ■ ■ • = 3.8«1, 7.02, ••• 



