408 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



A.t r = a, the boundary conditions are: 



E^ + Ee' cot 1/^ = 



cot i/' = 



Er - e!" = 



Er + Er 



Hj' + He'' cot yp 



Hz — Ha 



cot \p 







(57) 



Inasmuch as cot rp ~ t/A-, the contribution of Ee to the field at the 

 helix can conceivably be comparable to that of Ez . The TE fields are 

 coupled to the TM group, in addition, through the angular surface cur- 

 rent Ge , which depends on Ez . AllS equations must therefore be solved 

 simultaneously. 



The procedure follows that of Chu and Jackson" for the field solution 



Vr 



3.4 

 3.2 

 3.0 

 2.8 

 2.6 

 2.4 

 2.2 

 2.0 

 1.8 

 1.6 



/b 



Fig. 1 — Space-charge wavelength X, for cylindrical beam -n-ith Brillouin flow, 

 in a concentric drift tube. Here 6 and a are the beam and tube radii, respectively; 

 R^'^ is a dimensionless parameter; and the waves propagate as exp j{cot — yz). To 

 compute Xs c^ 'ZwR^'-flSp , use I3pb = 174P"2, where P is the beam perveance. The 

 abscissae are approximately given by 7 '^ |3e = oj/uo . (Equations 52-55.) 



