Waves in Electron Streams and Circuits 



By J. R. PIERCE 



(Manuscript Received Jan. 9, 1951) 



This paper reviews some of the assumptions made and some of the general 

 problems involved in analyzing the behavior of electron streams coupled to cir- 

 cuits. It explains why a wave approach is used. The propagation constant of the 

 wave is obtained in terms of the properties of the electron stream and the im- 

 pedance of the circuit. Some general properties of waves are discussed. The im- 

 portance of fitting boundary conditions in the solution of an actual problem is 

 discussed, and examples, including that of "backward-gaining" waves, are dis- 

 cussed. 



Introduction 



Of recent years, a good deal of work has appeared concerning small 

 linear perturbations of uniform clouds of electrons and ions.*^~^ A number 

 of questions can be raised concerning the physical interpretations of such 

 mathematical labors. 



First of all, for there to be a very direct physical interpretation, the 

 unperturbed state must exist at some time or place and then be modified 

 in the manner described by the perturbation. This condition is satisfied, for 

 instance, in the case of an electron stream of moderate current shot into a 

 long metal tube and confined by a longitudinal magnetic field. However, if 

 the current is made large enough, the uniform flow becomes unstable^ • ^ and 

 the method of perturbations can be used only to establish such instabiUty 

 and not to determine what form the flow will assume. I feel some misgivings 

 about drawing physical interpretations from perturbations of uniform d-c. 

 plasmas and infinitely extending clouds of charge unless these unperturbed 

 states can be shown to exist physically, or unless the results can be shown 



* A few late references only are given; others are quoted in those cited. 



^ D. Bohm and E. P. Gross, Theory of Plasma Oscillations: A. Origin of Medium- 

 Like Behavior, Phys. Rev., Vol. 75, pp. 1851-1864 (1949); B. Excitation and Damping 

 of Oscillations, Phys. Rev., Vol. 75, pp. 1864-1876 (1949). Effects of Plasma Boundaries 

 in Plasma Oscillations, Phys. Rev., Vol. 79, pp. 992-1001 (1950). 



2 J. A. Roberts, "Wave Amplification by Interaction with a Stream of Electrons," 

 Phys. Rev., Vol. 76, pp. 340-344 (1949). 



' V. A. Bailey, "The Growth of Circularly Polarized Waves in the Sun's Atmosphere 

 and Their Escape into Space," Phys. Rev., Vol. 78, pp. 428-443 (1950). 



* "Traveling Wave Tubes," J. R. Pierce, Van Nostrand, 1950. 



» A. V. Haeff, "Space-Charge Effects in Electron Beams," Proc. I.R.E., Vol. 27, pp. 

 586-602 (1939). 



* J. R. Pierce, "Limiting Stable Current in Electron Beams in the Presence of Ions," 

 Jour. A pp. Phys., Vol. 15, pp. 721-726 (1944); and "Note on Stability of Electron Flow 

 in the Presence of Positive Ions," JoUr. A pp. Phys., Vol. 21, p. 1063, Oct. 1950. 



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