44 BELL SYSTEM TECHNICAL JOURNAL 



In order to use these curves it is necessary to express the parameters 

 bo}/(i)e , \e and (a/bXeY in terms of convenient physical quantities. We obtain 



6co/coe = .545 X 10-'' bV"'u,/Jl'^ 



\e = 2.04 X \0-WT/jT (35) 



{a/bKY = 767 h/bW^''' 



Here /o is current in amperes and Jo is in amperes / cm. 



The broadness of the frequency response curves of Fig. 5 is comparable 

 to that of curves for helix-type traveling-wave tubes. 



It is interesting to note that the maximum value of G/{(/\e) varies little 

 for a considerable range of the parameter a/b\e , approaching a constant 

 for large values of the parameter. This means that, with a beam of given 

 length, velocity and charge density, one can obtain almost the same opti- 

 mum gain over a wide range of frequencies simply by adjusting the velocity- 

 separation parameter b. 



4. Concluding Remarks 



There is a great deal of room for extension of the theory of double-stream 

 amplifiers. This paper has not dealt with the setting up of the increasing 

 wave, nor with other geometries than that of a cylindrical beam in a very 

 remote tube, nor with the effect of physical separation of the electron streams 

 of two velocities nor with streams of many velocities or streams with con- 

 tinuous velocity distributions. 



This last is an interesting subject in that it may provide a means for deal- 

 ing with problems of noise in multivelocity electron streams. Indeed, it 

 was while attempting such a treatment that the writers were distracted by 

 the idea of double-stream amplification. 



APPENDIX 



Derivation of Results Used in Section 3 



Consider a double-stream electron beam whose axis coincides with the 

 z-axis of a system of cylindrical coordinates (r, (p, z) and which is subject to 

 an infinite, longitudinal, d-c. magnetic field. The radius of the beam is a 

 and each of the streams is characterized by d-c. velocities, Ui and W2 , which 

 are vectors in the positive s direction, and d-c. space charge densities, po2 

 and po2 . All d-c. quantities are assumed to be independent of the coor- 

 dinates and time, except, of course, for the discontinuities at the surface of 

 the beam. Small a-c. disturbances are superimposed upon these d-c. quanti- 

 ties and they are small enough so that their cross products can be neglected 

 compared with the products of d-c. quantities and a-c. quantities. It is 



