The Character of the Equilibrium of a Compressible, Inviscid Fluid of Varying Density.

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Abstract

This paper is devoted to an investigation of the initial growth of an infinitesimal perturbation of the static state of a compressible, inviscid fluid arranged in horizontal layers, with the density a function of the vertical co-ordinate only. The problem of determining the rate of growth of a perturbation periodic in the horizontal planes is reduced to one in characteristic values of a second-order differential equation A variational principle is shown to exist for this problem. The special case of two uniform fluids of different densities separated by a common boundary is investigated in some detail. There exist for this case modes of two distinct kinds: those resulting mainly from the discontinuity of the density (Rayleigh's modes) and those resulting mainly from the compressibility (sound waves and convective instability). The rates of growth of Rayleigh's modes in the unstable case that the upper fluid is the more dense are determined; under certain conditions, Rayleigh's modes do not exist for sufficiently small values of the horizontal wave number of the perturbation. The frequencies of oscillation of Rayleigh's modes in case the lower fluid is the more dense are determined The behavior of Rayleigh's modes is compared with their behavior in case the two fluids are incompressible, and the modification of Rayleigh's modes by compressibility is explained in terms of the physical mechanism giving rise to the propagation of sound waves and the onset of convection.

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