具有非均匀振荡系数的wave equation的均匀化
Homogenization of the wave equation with non-uniformly oscillating coefficients
摘要：
本文的工作重点是描述非均匀介质中低频波动的色散的二阶有效模型（如~功能梯度）介质。对于这类准周期介质的变化，我们追求均匀化的标量波动方程在$\mathbb{R}^d$，$d\geqslant 1$多尺度扩展的框架内。当采用$d=1$或$d=2$时，该模型问题与弹性固体中（反平面）剪切波的描述直接相关。通过采用微观介质波动的长度尺度作为微扰参数，我们通过一个四阶微分方程（具有平滑变化的系数）来控制介质中的平均波动，其中微观非均匀性的影响通过所谓的细胞功能被放大，从而合成了密切相关的低频行为。为了证明我们的分析解决边值问题（被认为是最均匀化研究的最终目标）的相关性，我们还开发了有效的边界条件，到二阶渐近近似，适用于一维（1D）剪切波运动在宏观非均匀固体周期性微观结构。我们通过考虑（i）低频波色散，（ii）有限域中传播的剪切波的平均场均匀化描述，以及（iii）其全场均匀化描述，在1D中数值地说明了分析。与（i）整体波频散似乎可以由首阶模型很好地描述相反，（ii）和（iii）中的结果表明，高阶校正在近似准周期介质中的实际波形时可能具有关键作用。
英文摘要：
The focus of our work is dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g.~functionally-graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in $\mathbb{R}^d$, $d\geqslant 1$ within the framework of multiple scales expansion. When either $d=1$ or $d=2$, this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in 1D by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
