  Observations of both gamma-ray bursts (GRBs) and active galactic nuclei
(AGNs) point to the idea that some relativistic jets are suffocated by their
environment before we observe them. In these "choked" jets, all the jet's
kinetic energy is transferred into a hot and narrow cocoon of near-uniform
pressure. We consider the evolution of an elongated, axisymmetric cocoon formed
by a choked jet as it expands into a cold power-law ambient medium $\rho
\propto R^{-\alpha}$, in the case where the shock is decelerating ($\alpha<3$).
The evolution proceeds in three stages, with two breaks in behaviour: the first
occurs once the outflow has doubled its initial width, and the second once it
has doubled its initial height. Using the Kompaneets approximation, we derive
analytical formulae for the shape of the cocoon shock, and obtain approximate
expressions for the height and width of the outflow versus time in each of the
three dynamical regimes. The asymptotic behaviour is different for flat
($\alpha \le 2$) and steep ($2 < \alpha < 3$) density profiles. Comparing the
analytical model to numerical simulations, we find agreement to within $\sim15$
per cent out to 45 degrees from the axis, but discrepancies of a factor of 2-3
near the equator. The shape of the cocoon shock can be measured directly in
AGNs, and is also expected to affect the early light from failed GRB jets.
Observational constraints on the shock geometry provide a useful diagnostic of
the jet properties, even long after jet activity ceases.
