1366 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



The upper sign in (A15) gives the phase constant of the slow wave, a 

 wave with a phase velocity less than that of the electrons. The lower 

 sign gives that of the fast wave, a wave with a phase velocity faster than 

 that of the electrons. 



From (A15) and (A6) we note that 



i= ±- pov (A16) 



The upper sign holds for the slow wave; the lower sign for the fast wave. 

 It has been convenient to use — /o and i as current densities and — po 

 and p as charge densities. In subsequent work and in the text, — /o and 

 i will be used as beam current and — po and p as charge per unit length. 

 All the relations of this appendix except (A11)-(A13) will hold if the 

 quantities are so interpreted. 



Appendix B 

 power flow in space charge waves 



The purpose of this appendix is to justify the expression for power flow 

 in the beam. 



Consider that the electron beam is acted on over a short distance by 

 an ac voltage. Imagine, for instance, that the beam passes through two 

 very closely spaced grids which form a part of a resonator, and that a 

 voltage AV appears between the grids. What does the voltage do to the 

 beam? 



The voltage AV changes the velocity of the electrons but it does not 

 change the convection current. To find out how much the velocity is 

 changed we need only consider the case in which the beam has no ac 

 velocity on reaching the grids, smce in a linear system the change will 

 be the same in all other cases. The total velocity ^lo + t^ is given in terms 

 of the total accelerating voltage F + AF by 



uo+ V = J 2 £ (Fo + AF) (Bl) 



We assume A F to be small, so that 



AF - AF ^ 



^ = -^ (B2) 



2 ^ Fo ^° 



/ 



m 



