We consider the Rayleigh-Taylor problem for two compressible, immiscible,
inviscid, barotropic fluids evolving with a free interface in the presence of a
uniform gravitational field. After constructing Rayleigh-Taylor steady-state
solutions with a denser fluid lying above the free interface with the second
fluid, we turn to an analysis of the equations obtained from linearizing around
such a steady state. By a natural variational approach, we construct normal
mode solutions that grow exponentially in time with rate like $e^{t
\sqrt{\abs{\xi}}}$, where $\xi$ is the spatial frequency of the normal mode. A
Fourier synthesis of these normal mode solutions allows us to construct
solutions that grow arbitrarily quickly in the Sobolev space $H^k$, which leads
to an ill-posedness result for the linearized problem. Using these pathological
solutions, we then demonstrate ill-posedness for the original non-linear
problem in an appropriate sense. More precisely, we use a contradiction
argument to show that the non-linear problem does not admit reasonable
estimates of solutions for small time in terms of the initial data.