  A Double-Double Radio Galaxy (DDRG) is defined as consisting of a pair of
double radio sources with a common centre. In this paper we present an
analytical model in which the peculiar radio structure of DDRGs is caused by an
interruption of the jet flow in the central AGN. The new jets emerging from the
restarted AGN give rise to an inner source structure within the region of the
old, outer cocoon. Standard models of the evolution of FRII sources predict gas
densities within the region of the old cocoon that are insufficient to explain
the observed properties of the inner source structure. Therefore, additional
material must have passed from the environment of the source through the bow
shock surrounding the outer source structure into the cocoon. We propose that
this material is warm clouds ($\sim\!10^4$ K) of gas embedded in the hot IGM
which are eventually dispersed over the cocoon volume by surface instabilities
induced by the passage of cocoon material. The derived lower limits for the
volume filling factors of these clouds are in good agreement with results
obtained from optical observations. The long time scales for the dispersion of
the clouds ($\sim\!10^7$ yr) are consistent with the apparently exclusive
occurrence of the DDRG phenomenon in large ($\ga 700$ kpc) radio sources and
with the observed correlation of the strength of the optical/UV alignment
effect in $z\!\sim\!1$ FRII sources with their linear size.
