  A solution to the ultra-relativistic strong explosion problem with a
non-power law density gradient is delineated. We consider a blast wave
expanding into a density profile falling off as a steep radial power-law with
small, spherically symmetric, and log-periodic density perturbations. We find
discretely self-similar solutions to the perturbation equations and compare
them to numerical simulations. These results are then generalized to encompass
small spherically symmetric perturbations with arbitrary profiles.
