  We develop a universal framework which allows quickly solve a wide class of
problems for longitudinal space charge effects in beams and plasmas in
cylindrical geometry. We introduce the longitudinal dielectric permittivity for
the beam of charged particles, which describes its collective space charge
response. The analyis yields an effective plasma frequency, which depends on
the transverse geometry of the system. This dielectric permittivity mirrors the
dielectric permittivity of plasma and matches the one dimensional (1D)
expression once the transverse size of the beam is large. Several particle
species can be included as additive terms describing susceptibility of each
specie. The developed approach allows to study stability criteria for
collective beam-beam and beam-plasma instabilities for arbitrary transverse
distributions in particle densities.
