The problem of hydrodynamic stability is important for inertial confinement fusion (ICF) systems based upon high compression of fuel before its ignition. This problem for the case of complicated multilayer foils has been studied here by a new approach describing the development of Rayleigh-Taylor or interchange instability in compressible media with inhomogeneous distribution of “entropy”s = ρ/ρk, ∂ where K = (∂ In ρ/∂ In ρ)s is an adiabatic derivative taken in the local hydrostatic values of ρ and ρ. Inhomogeneous distribution of s simulates the dynamics of development of perturbations of multilayer flyer foils and shells. Besides instability, the same approach has been used for analysis of ID pulsations of a levitated foil. The problem of pulsations is real in the case of foils. Indeed, (1) an ablative acceleration is equivalent to an effective gravity field, which causes the appearance of an atmospheric-type distribution of thermodynamic functions, (2) the duration of ablative flight of foil is at least several times larger than the time that is necessary for an acoustic wave to travel from one side of the foil to another side, and (3) there is a strong initial impulse that initiates the motion of foil. This impulse together with (1, 2) is a reason for the powerful pulsations of foils. The period of pulsations is defined by the velocity of sound in the foil material, which is dependent on the derivatives of an equation of state (EOS). The check of the derivatives gives us finer information concerning the current state of matter and the EOS than the usual measurements of material velocity and pressure that are rougher measures. Therefore, an analysis of pulsations seems to be a promising tool for tracking the dynamics of flyer foil and for the definition of thermodynamic properties of matter.