Stability of low oscillations in a two-layer inviscid fluid by vertical moving in gravity

The development of small oscillations in a system formed by a heavy layer of an ideal incompressible fluid covering a half-space of an ideal fluid with another density is studied. The entire system is moved as a rigid entity in the vertical direction according to some prescribed law. The apparatus of linearization of the equations and the boundary conditions is used. The investigation of the characteristic equations shows that the gravitational stability depends on the condition on the upper boundary of the layer, namely, the presence of free boundary of the layer or the condition of impenetrability through the rectilinear boundary. The qualitative estimates for the growth or decay of the initial perturbations are presented analytically.