Growing Waves Due to Transverse 



Velocities 



By J. R. PIERCE and L. R. WALKER 



(Manuscript received March 30, 1955) 



This paper treats propagation of slow waves in two-dimensional neu- 

 tralized electron floiv in which all electrons have the same velocity in the 

 direction of propagation hut in which there are streams of two or more veloci- 

 ties normal to the direction of propagation. In a finite beam in which 

 ' electrons are reflected elastically at the boundaries and in which equal dc 

 currents are carried by electrons with transverse velocities -\-Ui and — Wi , 

 there is an antisi/mmetrical growing ivave if 



Up ~ {rUi/Wf 



and a symmetrical growing wave if 



y- 



i{Tu,/wy 



Here cop is plasma frequency for the total charge density and W is beam 

 width. 



INTKODUCTION 



i It is well-known that there can be growing waves in electron flow when 

 the flow is composed of several streams of electrons having different 

 velocities in the direction of propagation of the waves. ' While Birdsall 

 considers the case of growing waves in electron flow consisting of streams 

 which cross one another, the growing waves which he finds apparently 

 occur when two streams have different components of velocity in the 

 direction of propagation. 



This paper shows that there can be growing waves in electron flow 

 consisting of two or more streams with the same component of velocity 

 in the direction of wave propagation but with different components of 

 velocity transverse to the direction of propagation. Such growing Avaves 

 can exist when the electric field varies in strength across the flow. Such 

 waves could result in the amplification of noise fluctuations in electron 



' flow. They could also be used to amplify signals. 



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