TRAVELING-WAVE TUBES 



25 



components of magnetic tield in the \f/ direction must be the same inside and 

 outside of the sheet. When these boundary conditions are imposed, one can 

 solve for the propagation constant and E^l^-P can then be obtained by 

 integrating the Poynting vector. 



The hehcally conducting sheet is treated mathematically in Appendix II. 

 The results of this analysis will be presented here. 



2.2 

 2.0 



1.8 



12 3 4 5 6 



y3o a coT^ 



Fig. 3.2— The radial propagation constant is 7- = {^^ — /3o)^'^. Here (/So/t) cot ^ is 

 plotted vs /Sofl cot i/', a quantity proportional to frequency. For slow waves the ordinate is 

 roughly the ratio of the wave velocity to the velocity the wave would have if it traveled 

 along the helically conducting sheet with the speed of light in the direction of conduction. 



3.1a The Phase Velocity 



The results for the helically conducting sheet are expressed in terms of 

 three phase or propagation constants. These are 



/So = oi/c, jS = 03/v 

 7 = /^Vl - {v/cy 



(3.1) 

 (3.2) 



Here c is the velocity of light and v is the phase velocity of the wave. /3o is 

 the phase constant of a wave traveling with the speed of light, which would 

 vary with distance in the s direction as exp(— j/Sos). The actual axial phase 

 constant is /3, and the fields vary with distance as exp(— j/Ss). 



7 is the radial propagation constant. Various field components vary as 

 modified Bessel functions of argument 7r, where r is the radius. Particularly, 

 the longitudinal electric field, which interacts with the electrons, varies 

 as h{yr). 



I For the phase velocities usually used, 7 is very nearly equal to ji, as may 

 he seen from the following table of accelerating voltages Vq (to give an elec- 



I Iron the velocity v), v/c and 7/jS. 



