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.
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