D'yakov-Kontorovich instability of expanding shock waves

In the range of $h_c<h<1+2 M_2$, where $h$ is the D'yakov-Kontorovich parameter, $h_c$ is its critical value corresponding to the onset of the spontaneous acoustic emission, and $M_2$ is the downstream Mach number, the classic analysis predicts a special form of the instability of isolated steady planar shock waves: non-decaying oscillations of shock-front ripples. For spherically and cylindrically expanding steady shock waves, we demonstrate instead an instability in a literal sense, a power-law growth of shock-front perturbations with time. As the parameter $h$ increases from $h_c$ to $1+2 M_2$, the instability power index grows from zero to infinity. Shock divergence is a stabilizing factor, and instability is found for high angular mode numbers.