Shock Dynamics in Layered Periodic Media
For researchers in hyperbolic PDEs and wave propagation, this work provides insight into shock dynamics in heterogeneous media, though it is limited to 1D periodic structures.
This paper investigates shock formation in one-dimensional periodic layered media, finding that such media inhibit shock formation for small initial conditions and large impedance variation, and that weak shocks are dynamically unstable. A generalized Lax entropy condition is proposed to predict shock formation.
Solutions of constant-coefficient nonlinear hyperbolic PDEs generically develop shocks, even if the initial data is smooth. Solutions of hyperbolic PDEs with variable coefficients can behave very differently. We investigate formation and stability of shock waves in a one-dimensional periodic layered medium by computational study of time-reversibility and entropy evolution. We find that periodic layered media tend to inhibit shock formation. For small initial conditions and large impedance variation, no shock formation is detected even after times much greater than the time of shock formation in a homogeneous medium. Furthermore, weak shocks are observed to be dynamically unstable in the sense that they do not lead to significant long-term entropy decay. We propose a characteristic condition for admissibility of shocks in heterogeneous media that generalizes the classical Lax entropy condition and accurately predicts the formation or absence of shocks in these media.