A soluble model of the development of the linear pertubations about a time‐varying state of a compressible medium is presented. A Lagrangian description is employed to rederive the equations for the self‐similar motion of an ideal fluid and to obtain the linearized equations of motion for pertubations about a general time‐varying basic state. The resulting formalism is applied in cylindrical geometry to calculate the growth of flute‐like modes associated with a similarity solution modeling the implosion and expansion of a fluid liner. A complete solution is obtained for the perturbed motion. The only modes for which the perturbation amplitudes grow faster than the unperturbed liner radius during both implosion and expansion are divergence‐ and curl‐free. Numerical and analytical results are obtained for these and shown to reduce, in the short‐wavelength limit, to the Rayleigh–Taylor instability found previously for incompressible time‐independent basic states. In addition, a new kind of instability is found: a class of overstable internal modes (sound waves), which are ’’pumped up’’ in amplitude during implosion, but decay during the expansion phase.