GUIDED WAVE PROPAGATION' TIIliOT'GII GYUOMAGNKTIC MKDI A . 1 1 <)7 I 



(e) j(^ei Kiicxir) 



Y 



ai /vo(air) 



Inside the helix, we require the solution for an isotropic region which 

 remains regular as r -^ 0. Accordingly 



^ (i) _ joco hjaor) . /— — 



w/xo Iiiocor) ' 



?o = w juoeo 



and 



„ (i) _ . weo /i(aoro) 



-» TM —J 



ao /o(aoro) ' 



where eo , mo are the dielectric constant, and permeability of vacuum. 

 Combining these expressions in (34), we obtain after slight rearrange- 

 ment 



hiaoVo) . ooei Kiiain) t ^ .. \ a 





(35) 

 tan ^ 



/o(aoro) aieoKo{airo) ]_ Iiiaon) ixa^ (2% - 1) TFx,i(2a2r)_ 



which determines /3. 



A complete solution of equation (35) is out of the question. However, 

 as in the planar case, for the slow waves used in travehng wave tube 

 work, the equation may be simpUfied so that solutions may be com- 

 puted rather easily. For electron velocities usually employed the result- 

 ant jS must be about 10/3o . Therefore in equation (35) it wU be permis- 

 sible to neglect all the quantities ^q, ^1 , ^2", w nei , in comparison with 

 13^, except in the narrow ranges of magnetic field such that n or 

 fi(l — pb) becomes very large. This will occur near dzco where o-q = 

 — p/2 -f ■\/p-/4: -{- 1, and near o- = 1. A solution obtained by as- 

 suming a large ^ must be self-consistent; that is, it can be credited only 

 in regions where it does, in fact, predict large |S. However, in Section 

 2.3 it was shown for the plane hehx that in any practical case the ranges 

 of magnetic fields so excluded are very narrow, even in the loss-free 

 case, and one may suppose that this is true also in the cylindrical case. 



For slow waves, each of the a's reduces to ] jS | ; the absolute value 

 sign derives from the fact that the positive square root is implied in the 

 definition of the a's. Therefore 



Now the suffix x of the "Whittaker functions no longer depends on the 



