Fastest growing linear Rayleigh-Taylor modes at solid∕fluid and solid∕solid interfaces

Previous linear stability analyses of the Rayleigh-Taylor instability problem for elastic solids have been restricted to calculating the cutoff wavelength lambda(c) (zero growth rate) in the limit of Atwood number A of unity. Here, we rigorously derive the dispersion relations for solid/fluid and solid/solid interfaces and perform a systematic investigation to compute the most unstable modes (maximum growth rate) for all A. After rationalizing the dispersion relations into multivariable polynomials, we compute the physically meaningful wavelength lambda and growth rate sigma for all unstable disturbances as a function of the mechanical properties of the participating media (shear moduli, dynamic viscosity, and density contrast) and acceleration. It is shown that at these interfaces, the onset of instability can only arise via monotonically growing disturbances. For solid/fluid and solid/solid interfaces, the locus of the most unstable wavelength lambda(m) and growth rate sigma(m) pairs are calculated to cover the entire range of behavior in dimensionless space. We find that under certain conditions, at solid/fluid interfaces, two configurations with distinct A can have the same lambda(m) (a behavior that does not occur at solid/solid interfaces). In terms of estimating sigma(m), lambda(m), and lambda(c), the applicability of our results extends to layers of finite thickness h provided h > lambda/2. We suggest a plausible mechanism to explain the wavelength selection process in nominally smooth magnetically imploded liners observed in recent experiments.