344 BELL SYSTEM TECHNICAL JOURNAL 



tions as restricted by the assumed boundary conditions at the edge of the 

 beam. 



It is found that two waves are possible, one traveUng sHghtly faster than 

 the electron beam and the second traveling slower. A point where the 

 velocity components are in phase will correspond to the input to the beam, 

 while points where the current components are in phase correspond to the 

 desired positions for the output. The propagation constants for these two 

 waves in a simplified special case where the magnetic field strength is 

 infinite are given by Hahn, as well as expressions for the optimum drift 

 tube length. He goes on to consider the case where the magnetic field is 

 zero and finds that for this case the density of the charge does not vary much 

 but instead the beam swells in and out so that instead of being lumps of 

 charge with spaces between, the lumps appear in the outer boundary. Hahn 

 has extended his general method of analysis to consider the modulation 

 coefficient of gaps through which the beam must pass. His results are a 

 great deal more general than those we have presented. 



Ramo has reformulated Hahn's theory by means of retarded potentials 

 for the most important case. This results in some simplification of the 

 theory. He computes the more important design constants for a velocity 

 modulated tube, such as the optimum drift tube length and the amount and 

 phase of the transconductance. Those of you who are particularly inter- 

 ested are referred to the original paper. An interesting aspect brought 

 out rather forcibly by Ramo's analysis is the existence of higher-order waves 

 on the beam, always occurring in pairs, one faster and the other slower than 

 the beam velocity. 



The Magnetron 



In what time remains I want to say just a very few words about the mag- 

 netron. This is a very complicated subject and one which cannot be ade- 

 quately dealt with in an entire evening, and certainly not in the time 

 remaining. 



As you all know, the magnetron was invented and named by Dr. A. W. 

 Hull. Habann, Zacek, Okabe and others pioneered in the use of the mag- 

 netron as an ultra-high-frequency oscillator. As envisioned today a 

 magnetron is a two-element device, usually cylindrical with a centrally 

 located cathode and a surrounding anode. The anode may be continuous 

 or it may be split into a number of segments as suggested by Okabe, and 

 these segments joined together either externally or internally by resonant 

 circuits. 



The basic ballistic problems of the magnetron, and hence the only prob- 

 lems which directly concern us at this time are (1) that of determining the 



