Response to “Comment on `Instability of isolated planar shock waves'” [Phys. Fluids 20, 029101 (2008)]

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In a recent article [J. W. Bates, “Instability of isolated planar shock waves,” Phys. Fluids 19, 094102 (2007)], we derived linear instability criteria for an isolated, planar, two-dimensional shock wave propagating through an inviscid fluid with an arbitrary equation of state. The basis for this analysis was a novel solution for the time-dependent Fourier amplitude of a single-mode perturbation on the front, which was expressed in the form of a Volterra equation. In the comment by Tumin [“Comment on `Instability of isolated planar shock waves',” Phys. Fluids 20, 029101 (2008)], the author demonstrated the consistency of our results with those of Erpenbeck, whose mathematical approach avoided the derivation of an integral equation in the time domain, but required a complicated, inverse Laplace-transform operation to ascertain the temporal evolution of disturbances at the shock's surface. Here, we emphasize that such information is obtained more readily from a direct solution of the aforementioned Volterra equation using modern numerical techniques.