404 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



The same result, (27), might have been obtained by stating the 

 equations of motion in terms of Eulerian coordinates, in which the per- 

 turbed variables are the components of fluid velocity at any fixed point. 

 In this procedure, the "material" or total time derivative would l)e used 

 in the expressions for acceleration. 



The ac space-charge density p is found with the aid of the continuity 

 equation: 



^ (po + p) = -div [(po + p){v, -f i')] (28) 



at 



JPo 



div V (29) 



CO — yuo 



From (23)-(25) and (27), the ac velocity may be written: 



V = ^"^ {Er , 0, E;) (30) 



CO — yuo 



Combining these with Poisson's equation, we find: 



""'' dwE= ^ ^'^ -P (31) 



(co — yuoY (co — yuoy 



There are two possible solutions to (31): 



(co — yuo) = cop (32) 



P = (33) 



Solution (32) represents two longitudinal space-charge waves of arbi- 

 trary amplitude distribution, with plasma-frequency oscillations about 

 the average beam velocity: 



7 = - ± '^ (34) 



Wo Wo 



The second solution, (33), however, permits us to evaluate the compo- 

 nents of the ac convection current density J, and thereby solve the field 

 equations (1) — (8): 



J = Pov -\- pvo (35) 



Jr = PoV,- = —. T -T- (36) 



7(co — 7Wo) or 

 /fl = (37) 



J, = p,v, = _ -li^ E. (38) 



CO — 7Wo 



