Adopting the standard evolutionary picture (Scheuer 1974; Begelman & Cioffi 1989), we assume that the radio jets are enveloped in, and feed, a cocoon of relativistic material which is overpressured with respect to the ambient medium (which may be the ISM of the host galaxy, or the intra-cluster medium (ICM) of the host cluster). This overpressure drives a strong shock into the ambient medium and forms a shell of shocked ISM/ICM surrounding the relativistic cocoon. The expansion velocity of this shell is determined by the ram pressure of the ambient material entering the shock. The cocoon material and the shocked ISM/ICM shell are separated by a contact discontinuity.
While it is clear that the large scale structures of radio sources show
elongation along the jet axis, one rarely observes large axial ratios. For
example, low-frequency (327 MHz) radio maps of the powerful FR-II source
Cygnus A (Carilli, Perley & Harris 1994) reveal a radio emitting cocoon
with an axial ratio of 
, even though its jets are observed to be
collimated to within a few degrees. A similar situation is found for the
smaller CSOs and MSOs. Thus, for the purposes of our simple model, we
shall assume that the cocoon and bow-shock are spherical. We will denote
the radius of the cocoon as 
, and the radius of the bow-shock
as 
( 
). We also denote by 
and 
the volume of the
cocoon and shocked shell, respectively.
We make several further assumptions. Firstly, we assume that at any given
instant the pressure within the shock (of both the cocoon and the shocked
ISM/ICM shell) is spatially uniform with value p(t). See Kaiser &
Alexander (1997) for an explicit justification of this assumption.
Secondly, we suppose that only a small fraction of the total kinetic
luminosity of the source, 
, is radiated. The rest of this energy is
assumed to be fed into the cocoon and thus drive the expansion of the
cocoon/shocked-shell system. Thirdly, the ambient (undisturbed) medium is
assumed to have a density distribution of the form
, where r is the distance from the center
of the radio source.
Given these assumptions, the conservation of energy can be applied to the cocoon and shocked shell to give,
and
where 
and 
are the ratio of specific heat capacities in the
cocoon and shocked-shell, respectively, and the dot denotes differentiation
with respect to time. We close the system of equations with the
ram-pressure condition, 
. This condition will be
valid provided that the expansion of the shocked shell remains highly
supersonic with respect to the ambient medium. If the ambient medium is
identified with the hot component of the ISM/ICM, its sounds speed will be
. Thus, the expansion of the source will be
highly supersonic provided 
.
Once the expansion of the source ceases to be supersonic, the
cocoon/shocked-shell structure will disrupt and dissipate.
In order to relate this model to radio observations, we need a prescription
relating the radio luminosity, Q, to the physical parameters of the
model. To do this, we assume that the radio emission is dominated by the
synchrotron radiation of the relativistic electrons in the cocoon. If we
further suppose that the magnetic field is in equipartition with the
relativistic electrons, standard minimum pressure arguments give 
.