  During its early evolution, a pulsar wind nebula (PWN) sweeps the inner part
of the supernova ejecta and forms a thin massive shell. Later on, when the
shell has been reached by the reverse shock of the supernova remnant, the
evolution becomes more complex, in most cases reverting the expansion into a
compression: this later phase is called "reverberation". Computations done so
far to understand this phase have been mostly performed in the thin-shell
approximation, where the evolution of the PWN radius is assimilated to that of
the swept-up shell under the effect of both the inner pressure from the PWN,
and the outer pressure from the supernova remnant. Despite the thin-shell
approach seems rather justifiable, its implementations have so far been
inaccurate, and its correctness, never tested. The outer pressure was naively
assumed to be scaled according to the Sedov solution (or a constant fraction of
it) along the entire evolution. The thin-shell assumption itself fails along
the process, being the shell no longer thin in comparison with the size of the
PWN. Here, through a combination of numerical models, dimensional arguments,
and analytic approximations, we present a detailed analysis of the interaction
of the PWN with the supernova remnant. We provide a new analytic approximation
of the outer pressure, beyond the Sedov solution, and a revised "thin-shell"
able to reproduce results from numerical simulations. Finally, we compute the
efficiency by which the PWN is compressed during reverberation over a wide
population of sources.
