We present new solutions to the strong explosion problem in a non-power law density profile .
The unperturbed self-similar solutions discovered by Waxman & Shvarts describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as r ^ { - \omega } , where \omega > 3 ( Type-II solutions ) .
The perturbations we consider are spherically symmetric and log-periodic with respect to the radius .
While the unperturbed solutions are continuously self-similar , the log-periodicity of the density perturbations leads to a discrete self-similarity of the perturbations , i.e .
the solution repeats itself up to a scaling at discrete time intervals .
We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations .
Finally we show that this method can be generalized to treat any small , spherically symmetric density perturbation by employing Fourier decomposition .