56 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX II 



PROPAGATION ON A 

 HELICALLY CONDUCTING CYLINDER 



The circuit parameter important in the operation of traveUng-wave tubes 

 is: 



(eWpY" (1) 



/3 = Wv. (2) 



Here Ez is the peak electric field in the direction of propagation, P is the 

 power flow along the helix, and v is the phase velocity of the wave. The 

 quantity Ez/l3'P has the dimensions of impedance. 



While the problem of propagation along a helix has not been solved, what 

 appears to be a very good approximation has been obtained by replacing 

 the helix with a cylinder of the same mean radius a which is conducting 

 only in a helical direction making an angle ^ with the circumference, and 

 nonconducting in the helical direction normal to this. 



An appropriate solution of the wave equation in cylindrical co-ordinates 

 for a plane wave having circular symmetry and propagating in the z direc- 

 tion with velocity 



' = i' « 



less than the speed of light c, is 



Ez = [Aloiyr) + BK,(yr)W''''-^'' (4) 



where /o and A'o are the modified Bessel functions, and 



y' = ^' - m = ^' - ^0. (5) 



The form of the z (longitudinal) components of an electromagnetic field 

 varying as e' "'^ ' and remaining everywhere finite might therefore be 



Ihx = B,h{yr)e''"'-^'' (6) 



£z3 = B,h{yr)e'''"-^'' (7) 



inside radius a, and 



J(ut-ez) 



//.2 = B,K,{yr)e'"''-'"' (8) 



