Analytical model of the linear Richtmyer–Meshkov instability when a rarefaction is reflected

The Richtmyer–Meshkov instability (RMI) develops when a planar shock front hits a rippled contact surface separating two different fluids. After the incident shock refraction, a transmitted shock is always formed, and another shock or a rarefaction is reflected back. In this work, we concentrate on the case when a rarefaction is reflected from the contact surface. The pressure/entropy/vorticity fields generated by the rippled wavefronts are responsible for the generation of hydrodynamic perturbations in both fluids. In linear theory, the contact surface ripple starts to grow after shock wave refraction and reaches an asymptotic normal velocity which is dependent on the incident shock Mach number, fluids density ratio, and compressibilities. In this work we only deal with the situations in which a rarefaction is reflected. Our main goal is to show an explicit, closed form expression of the asymptotic linear velocity of the corrugation at the contact surface, valid for arbitrary Mach number, fluids compressibilities, and pre-shock density ratios. An explicit formula (closed form expression) is presented that works quite well in both limits: weak and strong incident shocks. The results of this work show reasonable agreement with existing linear simulations, i.e., those reported by Yang et al. [Phys. Fluids 6, 1856 (1994)], with our previous linear calculations using a series of Bessel functions reported by Wouchuk and Nishihara [Phys. Plasmas 3, 3761 (1996)], and with the iterative calculation shown by Wouchuk [Phys. Plasmas 8, 2890 (2001)]. This work presents a faster analytical evaluation of the terminal velocity, implementing the ideas developed by Wouchuk [Phys. Rev. E 111, 035102 (2025); J. Fluid Mech. 1022, A29 (2025)]. Besides, an explicit calculation of the spatial profile of the vorticity field generated by the transmitted shock is given.