The corrugation instability of a piston-driven shock wave

We investigate the dynamics of a shock wave that is driven into an inviscid fluid by the steady motion of a two-dimensional planar piston with small corrugations on its surface. This problem was first considered by Freeman [Proc. Royal Soc. A.~{\bf 228}, 341 (1955)], who showed that piston-driven shocks are unconditionally stable when the medium through which they propagate is an ideal gas. Here, we generalize his work to account for a fluid with an arbitrary equation of state. We find that shocks are stable when $-1 < h < h_c$ , where $h$ is the D'yakov parameter and $h_c$ is a critical value less than unity. For values of $h$ within this range, linear perturbations imparted to the front at time $t = 0$ attenuate asymptotically as $t^{-3/2}$ or $t^{-1/2}$. Outside of this range, they grow --- at first quadratically and later linearly --- with time. Such instabilities are associated with non-equilibrium fluid states and imply a non-unique solution to the hydrodynamic equations. These results may have important implications for driven shocks in laser-fusion and astrophysical environments. As a benchmark of this analysis, we compare our solution with one derived independently by Zaidel' [J.~Appl.~Math.~Mech.~{\bf 24}, 316 (1960)] for stable $h$-values and find excellent agreement.