Intense sheet beam stability properties for uniform phase-space density

A self-consistent one-dimensional waterbag equilibrium f/sub b//sup 0/(x,p/sub x/) for a sheet beam propagating through a smooth focusing field is shown to be exactly solvable for the beam density n/sub b//sup 0/(x) and space-charge potential /spl phi//sup 0/(x). A closed Schrodinger-like eigenvalue equation is derived for small-amplitude perturbations, and the WKB approximation is employed to determine the eigenfrequency spectrum as a function of the normalized beam intensity s/sub b/ = /spl omega//spl I.cap//sub pb//sup 2///spl gamma//sup 2//sub b//spl omega//sup 2//sub /spl beta//spl perp//, where /spl omega//spl I.cap//sub pb//sup 2/ = 4/spl pi/n/spl I.cap//sub b/e/sub b//sup 2///spl gamma//sub b/m/sub b/ is the relativistic plasma frequency-squared and n/spl I.cap//sub b/ = n/sub b/(x = 0) is the on-axis number density of beam particles.