Rayleigh–Taylor stability for a normal shock wave–density discontinuity interaction

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Abstract
The solution for the perturbation growth of a shock wave striking a density discontinuity at a material interface is developed. The Laplace transformation of the perturbation results in an equation which has a simple solution for weak shock waves. The solution for strong shock waves may be given by a power series. It is assumed that the equation of state is that of an ideal gas. The four independent parameters of the solution are the ratio of specific heat for each material, the density ratio at the interface, and the incoming shock strength. Properties of the solution which are investigated include the asymptotic behavior at large times of the perturbation velocity at the interface, the vorticity near the interface, and the rate of decay of the solution at large distances from the interface. The last is much weaker than the exponential decay in an incompressible fluid. The asymptotic solution near the interface, in addition to a constant term, consists of a number of slowly decaying discrete frequencies. The number is roughly proportional to the logarithm of the density ratio at the surface for strong shocks, and decreases with shock strength. For weak shocks the solution is compared with results for an incompressible fluid. Only interface perturbation velocities which tend to zero at large times lead to a limited deformation of the interface. It is found that these are possible only for density ratios less than about 1.5.