Rippled shock transmission through slow–fast interfaces

The evolution of a rippled shock transmitted through slow–fast interfaces is investigated, considering the phase-dependent initial conditions generated during transmission. A matched solution is first constructed by mapping the transmitted-shock initial amplitude and growth rate to an equivalent isolated-shock evolution, but it provides accurate predictions only in the early stage because the subsequent shock–interface coupling is not retained. A generalised analytical model is derived by reducing the pressure perturbation dynamics to the Klein–Gordon equation. The resulting Bessel series solution retains the even-order terms that vanish in conventional initial-phase formulations, thereby enabling good prediction of transmitted shocks initiated at arbitrary phases with non-zero initial pressure and normal velocity perturbations. The results show that the transmitted-shock evolution is governed by two coupled mechanisms: phase-controlled initialisation, which determines the transmitted shock’s initial amplitude and growth rate via the pre-impact shock–interface distance, and acoustic feedback from the interface, which modulates the subsequent oscillation. Each slow–fast transmission increases both the attenuation rate and the oscillation frequency of the shock-front perturbation, and the analytical theory can predict the evolutions of successive slow–fast transmissions. Using the explicit dependence of transmitted-shock initial conditions on the shock–interface parameters, we determine the optimal parameter combinations for suppressing corrugations, enabling active control of transmitted-shock perturbations.