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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



and its phase at y is 



<p(y,<po) = oj f- - tj 



i 



The velocity of the electron is expressed as 



dz 



dt 



= Wo[l + Cw{ij, ip^)] 



where Uo is the average velocity of the electrons, and, Cuow(y, tpo) as men- 

 tioned earlier, is the ac velocity of the electron when it is at the position 

 y. The electron charge density near an electron which has an initial phase 

 cpo and which is now at y, can be computed by the equation of conserva- 



tion of charge, it is 



p(y, <Po) = - 



Wo 



d(po 



d(p{y, <po) 



1 



1 + Cw(y, ifo) 



(13) 



One should recall here that h is the dc beam current and has been de- 

 fined as a positive quantity. When several electrons with different initial 

 phases are present at y simultaneously, a summation of 



d<po 



of these electrons should be used in (13). From (13), the fundamental 

 component of the electron charge density is 



pMt) = --- 



sm 





d<po 



sin (fiy, <po) 

 1 + Cw{y, <pq) 



r^" , cos <p{y, <po) 

 + cos <p I d(po 

 Jo 



(14) 



1 + Cw(y, ifo)/ 



These are important relations given by Nordsieck and should be kept 

 in mind in connection with later work. In addition, we shall frequently 

 use the transformation 



I = t s = ^"(' + ^'"(^-» 1^ 



which is written following the motion of the electron. Let us start from 

 the forward wave. It is computed by means of (6). After substituting 

 (8) and (14) into (6), we obtain by equating the sin <p and the cos v' 



