Time evolution of density perturbations in accelerating stratified fluids

We consider $N$ superimposed fluids, having arbitary initial perturbations at their $N-1\mathrm{}$ interfaces, undergoing a Rayleigh-Taylor instability with constant acceleration. The time evolution of these perturbations is described by a sum over normal modes and in general is a combination of oscillation and exponential rise. We derive the equation governing their growth, discuss the case $N=3\mathrm{}$ analytically, and give two numerical examples where we plot the time evolution of the perturbations at the four interfaces of five superimposed fluids. Our examples illustrate how the evolution depends on the density profile, on the wavelength of the perturbations, and on the initial conditions.