  We study the equations obtained from linearizing the compressible
Navier-Stokes equations around a steady-state profile with a heavier fluid
lying above a lighter fluid along a planar interface, i.e. a Rayleigh-Taylor
instability. We consider the equations with or without surface tension, with
the viscosity allowed to depend on the density, and in both periodic and
non-periodic settings. In the presence of viscosity there is no natural
variational framework for constructing growing mode solutions to the linearized
problem. We develop a general method of studying a family of modified
variational problems in order to produce maximal growing modes. Using these
growing modes, we construct smooth (when restricted to each fluid domain)
solutions to the linear equations that grow exponentially in time in Sobolev
spaces. We then prove an estimate for arbitrary solutions to the linearized
equations in terms of the fastest possible growth rate for the growing modes.
In the periodic setting, we show that sufficiently small periodicity avoids
instability in the presence of surface tension.
