TRAVELING-WAVE TUBES 733 



inal exciting current q. Both E, and q vary as (expj(al) (exp — Tz). The 

 relation is 



^' = ''^o*(r^-rS)- • ^^^ 



Here To is the propagation constant of the transmission mode considered and 

 is defined in such a sense that for unattenuated propagation, To = j^o where 

 jSo is a positive number. The quantity \f/o is defined as 



2P 

 ^0 = ;^. (2) 



Here P is complex power transmitted by the mode and Eg is the field asso- 

 ciated with the mode. 



In generalizing (1), let us consider the combination of equations (1) and 

 (2) 



p* = iqE:-r^,. (3) 



1 — 1 



Now, suppose there is motion of the electrons not only in the z direction but 

 in a direction normal to the z direction, which we will call the y direction. 

 We shall have two extra first-order terms of the same general nature as 

 qEg , which contribute to the power, giving 



P* = i-(?,£: + {-Io)y ^ + ?„£.*) j;r3Yl ' ^*^ 



Here qz is the a-c convection current in the z direction, — /o is the d-c con- 

 vection current in the z direction (assumed to be the only d-c convection 

 current), y is a small displacement, qy is the convection current in the y 

 direction and Ey is the field in the y direction. 



We will now speciahze this expression. Suppose we consider a two-dimen- 

 sional transverse magnetic wave propagating in the z direction with a phase 

 velocity v such that v^ <<C c^. Then over a restricted region the electric 

 field can be represented quite accurately as the gradient of a scaler potential 

 of 



F = exp (-Tz)(A exp (jTy) + B exp {-jTy)). (5) 



Here A and B are constants. Using our notation, in which the field is 

 understood to include the factor exp (—Tz), we obtain 



Ez = T{A exp OTy) + B exp (-;Ty)) (6) 



^^ = jr\A exp (jTy) - B exp {- jTy)) (7) 



oy 



Ey = -jTiA exp U^y) - B exp (-;Ty)). (8) 



