  Classical Cherenkov radiation is a celebrated physics phenomenon of
electromagnetic (EM) radiation stimulated by an electric charge moving with
constant velocity in a three dimensional dielectric medium. Cherenkov radiation
has a wide spectrum and a particular distribution in space similar to the Mach
cone created by a supersonic source. It is also characterized by the energy
transfer from the charge's kinetic energy to the EM radiation. In the case of
an electron beam passing through the middle of a an EM waveguide, the radiation
is manifested as collective Cherenkov radiation. In this case the electron beam
can be viewed as a one-dimensional non-neutral plasma whereas the waveguide can
be viewed as a slow wave structure (SWS). This collective radiation occurs in
particular in traveling wave tubes (TWTs), and it features the energy transfer
from the electron beam to the EM radiation in the waveguide. Based on a
Lagrangian field theory, we develop a convincing argument that the collective
Cherenkov effect in TWTs is, in fact, a convective instability, that is,
amplification. We also derive, for the first time, expressions identifying low-
and high-frequency cutoffs for amplification in TWTs
