HELIX WAVEGUIDE 1381 



APPENDIX 

 APPROXIMATE SOLUTION OF THE CHARACTERISTIC EQUATION 



The characteristic equation (6) of the heUx guide may be written in 

 the dimensionless form 



[i^a tan ^ - -^ ^^.^ - (fta) -j-^^ 



fia tan ^ - -^ ] J^,,^ . — (M' 



(Al) 



If I e' — «" I is sufficiently large, the right side of the equation is large 

 and either J„(fia) or Jn'itio,) is near zero. Let p denote a particular root 

 of Jn or Jn', then to zero order, 



Tia = p 



ha = finma = i8oa(l - v')''' (A2) 



Ua = iSoaie' - U" - 1 - vy 



where 



V = p/^oa 



Henceforth assume that 



I r2a 1 » I (4n' - l)/8 I (A3a) 



and 



I r2a I » I n I (A3b) 



It is convenient to postulate both inequalities, even though the first is 

 more restrictive than the second unless \ n \ = lor|nl =2. 



If (A3a) is satisfied, the Hankel functions may be replaced by the 

 first terms of their asymptotic expressions, and 



Eq. (Al) becomes 



A , , 7ihaY Jniha) /« n2 ^n'(fia) 



ijia 



2 



Ua tan ^p — —- j + (M'U - 'i^") 



