We report the mechanism and modeling for the formation of cavitylike structures on a planar interface subjected to a perturbed shock wave. The cavity is distinguished from bubbles and spikes formed in the standard Richtmyer-Meshkov instability (RMI). The two-dimensional direct numerical simulation is conducted at a range of shock Mach numbers and Atwood numbers. We elucidate the effects of the interfacial vorticity and the shock-induced vorticity on the cavity formation. The interfacial vorticity, which is important in the standard RMI, only has a minor influence on the cavity width in the linear stage. Alternatively, the cavity width is determined by the Mach-stem height when the shock accelerates the interface. A pair of vorticity patches connecting the Mach stem, as a part of the shock-induced vorticity, penetrate the interface to form the cavity via strong shear layers generated by slipstreams during shock propagation. Inspired by this mechanism, we develop a model of the Mach-stem height to estimate the cavity width in