  We investigate the diffraction of a slow symmetric TM mode by an open-ended
corrugated cylindrical waveguide with a flange. This mode can be generated, in
particular, by a charged particle bunch moving along the waveguide axis. We
analyze the so-called ''longwave'' range when the wavelengths and the waveguide
radius are much greater than the corrugation period. In this case, the
corrugated waveguide wall can be replaced with a smooth one on which the
so-called equivalent boundary conditions (EBC) are fulfilled. Here, we also
assume that the wall is deeply corrugated, i.e. the corrugation depth is of the
same order as an inverse wavenumber. The approach applied uses the solution of
the corresponding Wiener-Hopf-Fock equation. This solution is in turn utilized
to construct an infinite linear system for reflection coefficients which can be
solved numerically using the reduction technique. The dependences of the
components of the electromagnetic field on the problem parameters are obtained
and analyzed. Typical radiation patterns are presented.
