  We study an electromagnetic radiation of a small bunch having a variable
charge value and crossing a flat interface between two media. Both media are
homogeneous, stationary, and isotropic. They may have frequency dispersion but
no spatial dispersion. Cherenkov radiation can be generated in the second
medium only. It is assumed that the bunch charge value decreases exponentially
starting from some time moment after the charge enters the second medium. This
means that we take into account the scatter of the particle path lengths,
connected with a statistical nature of energy losses of particles. It is taken
into account that the filamentous trace consisting of immobile charges is
formed in the second medium.
  We obtain the general solution of the problem. The asymptotic study for the
wave zone is carried out. We obtain expressions for the spherical wave as well
as for the cylindrical wave generated in the second medium if the charge
velocity is sufficiently high. The main properties of radiation are described.
If the process of the bunch decay starts from the boundary between media, then
the angular distribution of the radiation energy has the single maximum in each
from two regions. In the second medium, the radiation, as a rule, is greater
than in the vacuum area, and this difference increases with increasing the
charge velocity. If the charge velocity is higher than the speed of light in
the second medium, then, along with appearance of Cherenkov radiation, the
properties of spherical wave change radically. In particular, the main maximum
of the angular distribution of the radiation energy increases sharply. If the
distance from the charge entry point to the region of the bunch decay is
sufficiently large, then the complex interference pattern with many extrema
take place.
