Shock wave stability in steep density gradients

The stability of spherical accelerating shock waves is discussed via the examination of the stability of the new Waxman-Shvarts self-similar solutions to the strong explosion problem with a density profile ρ ∝ r^-ω with ω > 3. We show that accelerating shock waves that diverge in finite time (obtained for ω larger than a critical value ω_c : ω > ω_c) are unstable for small and intermediate wavenumbers, in accordance with the conclusions of Chevalier, who studied the stability of planar shock wave propagating in an exponentially decaying density profile. However, accelerating shock waves that diverge in infinite time (obtained for ω < ω_c) are stable for most wavenumbers. We find that perturbation of small wavenumber grow or decay monotonically in time, while perturbations of intermediate and high wavenumber oscillate in time.