GROWING WAVES DUE TO TRANSVERSE VELOCITIES 111 



Let US define 



dx = i(co - ^u,) + u,D (1.3) 



do = ./(w - i8wo) - uj) (1.4) 



We can then rewrite (1.1) and (1.2) as 



f/iPi = Poi-Diji + j(3zi) (1.5) 



dopi = Pi^{ — Dy2 + .7/3i2) (1.0) 



We will assume that we are dealing ^^•ith slow waves and can use a po- 

 tential V to describe the field. We can thus write the linearized equations 

 of motion in the form 



r/iii = -j-^F (1.7) 



m 



d2h = -j-^V (1.8) 



m 



drlji = - DV (1.9) 



m 



d,y, = 1 DV (1.10) 



w 



From (1.5) to (1.10) we obtain 



^m = ~ PoiD' - ^')V (1.11) 



m 



d'p2= --poiD'- ^')V (1.12) 



m 



Now, Poisson's equation is 



{D' - ^')V = _^L±£! (1.13) 



From (1.11) to (1.13) we obtain 



{D' - /3^)y = - Kco/ (^1 + ^^ (D' - /3^)7 (1.14) 



9 ^ 

 — Z— po 



2 m 



Wp = 



e 



Here Wp is the plasma frequency for the charge of both beams. 



(1.15) 



