  We present results of a numerical integration of the hydrodynamical equations
governing the self-similar, two-dimensional gas flow behind the bow shock of an
FRII radio source embedded in an IGM with a power law density profile. The
model predicts pressure gradients within the cocoons consistent with modest
backflow. For very steep external density profiles sources may well not expand
in a self-similar fashion and in this case the model is not self-consistent.
The assumption of ram pressure confinement of the cocoons perpendicular to the
jet axis is found to overestimate the ratio of the pressure in front of the
radio hot spots and that in the cocoons. Based on the properties of the gas
between bow shock and cocoon we calculate the X-ray surface brightness of the
flow. This emission is found to be a good tracer of the density distribution
within the flow and varies significantly with the properties of the unshocked
IGM. The cooling-time of the shocked IGM is found to be comparable to, or
greater than, the Hubble time. The influence of a radio source on the evolution
of its gaseous surroundings therefore extends well beyond the limited life time
of the source itself. We compare our results with the X-ray map of Cygnus A and
find some evidence for cold, dense gas clumps in the surroundings of this
object. The extended X-ray emission observed around 3C 356 may also be caused
by the bow shock of this radio source. We also present an empirical model for
the X-ray emission of the shocked IGM due to thermal bremsstrahlung.
