Solutions of first-order nonlinear hyperbolic conservation laws typically develop shocks in finite time even with smooth initial conditions. However, in heterogeneous media with rapid spatial variation, shock formation may be delayed or avoided. When shocks do form in such media, their speed of propagation depends on the material structure. We investigate conditions for shock formation and propagation in heterogeneous media. We focus on the propagation of plane waves in two-dimensional periodic media with material variation in only one direction. We propose an estimate for the speed of the shocks that is based on the Rankine-Hugoniot conditions applied to a leading-order homogenized (constant coefficient) system. We verify this estimate via numerical simulations using different nonlinear constitutive relations and layered and smoothly varying periodic media. In addition, we discuss conditions and regimes under which shocks form in this type of media.

中文翻译：

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周期介质中冲击波的有效 Rankine-Hugoniot 条件

一阶非线性双曲线守恒定律的解通常会在有限时间内产生冲击，即使初始条件是平滑的。然而，在空间变化迅速的异质介质中，可能会延迟或避免激波的形成。当冲击确实在这种介质中形成时，它们的传播速度取决于材料结构。我们研究了异质介质中冲击形成和传播的条件。我们专注于平面波在二维周期介质中的传播，材料仅在一个方向上发生变化。我们建议对冲击速度进行估计，该估计基于应用于领先阶均质化（常数系数）系统的 Rankine-Hugoniot 条件。我们通过使用不同非线性本构关系和分层且平滑变化的周期性介质的数值模拟来验证这一估计。此外，我们还讨论了在此类媒体中形成冲击的条件和机制。