波在时变超颖表面中传播的界面条件
Interface Conditions for Wave Propagation Through a Time-varying Metasurface
AI总结：
本研究针对时变薄异质层的波传播问题，推广薄层多尺度收敛方法，推导了含非标准界面跃变条件的有效波传播模型，并证明了模型的唯一性。
AI中文摘要：
我们研究波在时间调制的薄异质层中的传播问题。假设该层厚度为ε阶（ε≪1），且其材料特性在多个空间和时间尺度上呈现快速振荡。我们的目标是严格推导波通过极限界面传播的有效模型及相应的界面条件。\n 为了同时开展均匀化和降维处理，我们将Neuss-Radu和Jäger（2007）提出的薄层双尺度收敛概念推广到多时空尺度框架。均匀化分析中出现的主要解析难点之一是，通常情况下，对于含时变系数的波动方程，无法得到一致能量估计。因此，我们确定了两类具有物理相关性的系数，针对它们可以推导一致能量界，其中包括特性行波型调制的系数。\n 利用能量估计和薄层多尺度收敛概念，我们推导出了有效模型，该模型由体域中的线性波动方程组成，这些方程通过界面处的非标准跃变条件耦合。该跃变条件由一类波动型动力学方程控制，其有效宏观系数由合适的椭圆型和双曲型胞元问题确定。最后，我们讨论了有效模型的唯一性。
英文摘要：
We study wave propagation through a time-modulated thin heterogeneous layer. The layer is assumed to have a thickness of order $\es\ll 1$ and its material properties exhibit rapid oscillations on multiple spatial and temporal scales. We aim to rigorously derive the effective model and the corresponding interface conditions for wave propagation through the limiting interface. To perform the homogenization together with dimension reduction, we generalize the notion of two-scale convergence for thin layer introduced by Neuss-Radu and Jäger (2007) to a multiple space-time scale framework. One of the main analytical difficulties arising in the homogenization analysis is that, in general, a uniform energy estimate cannot be obtained for a wave equation with time-varying coefficients. Therefore, we identify two physically relevant classes of coefficients for which we can derive a uniform energy bound. These include coefficients with traveling wave-type modulations of their properties. Using the energy estimates and the multiscale convergence for thin layer concept, we derive the effective model, which consists of linear wave equations in the bulk domains coupled through a nonstandard jump condition at the interface. This jump condition is governed by a wave-type dynamical equation with effective macroscopic coefficients determined by suitable cell problems of elliptic and hyperbolic type. Finally, we discuss the uniqueness of the effective model.
