Summary We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in $\mathbb{R}^d$, $d \geqslant 2$, and (b) exclusions, that is, ‘voids’ that are subject to homogeneous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch-wave expansion, we pursue this goal via asymptotic ansatz featuring the ‘spectral distance’ from a given wavenumber-eigenfrequency pair (situated anywhere within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion via projection(s) of the scalar wavefield onto the Bloch eigenfunction(s) for the unit cell of periodicity, evaluated at the origin of a spectral neighborhood. For generality, we account for the presence of the source term in the wave equation and we consider—at a given wavenumber—generic cases of isolated, repeated, and nearby eigenvalues. In this way, we obtain a palette of effective models, featuring both wave- and Dirac-type behaviors, whose applicability is controlled by the local band structure and eigenfunction basis. In all spectral regimes, we pursue the homogenized description up to at least first order of expansion, featuring asymptotic corrections of the homogenized Bloch-wave operator and the homogenized source term. Inherently, such framework provides a convenient platform for the synthesis of a wide range of intriguing wave phenomena, including negative refraction and topologically protected states in metamaterials and phononic crystals. The proposed homogenization framework is illustrated by approximating asymptotically the dispersion relationships for (i) Kagome lattice featuring hexagonal Neumann exclusions and (ii) square lattice of circular Dirichlet exclusions. We complete the numerical portrayal of analytical developments by studying the response of a Kagome lattice due to a dipole-like source term acting near the edge of a band gap.

中文翻译：

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基于狄利克雷或诺依曼排除的周期性介质中波的谱渐近

总结 我们考虑周期性介质中标量波动方程在有限波数和频率下的均匀化，重点是连续体，其特征是： (a) $\mathbb{R}^d$, $d \geqslant 2$ 中的任意 Bravais 晶格， (b) 排除项，即受齐次（Neumann 或 Dirichlet）边界条件影响的“空隙”。利用布洛赫波展开，我们通过渐近分析来实现这一目标，该分析以给定波数-特征频率对（位于第一布里渊区域内的任何位置）的“光谱距离”作为扰动参数。然后，我们通过将标量波场投影到周期性晶胞的 Bloch 特征函数上来引入有效波动，在光谱邻域的原点进行评估。为了一般性，我们考虑了波动方程中源项的存在，并在给定的波数上考虑了孤立、重复和附近特征值的一般情况。通过这种方式，我们获得了一系列有效模型，具有波型和狄拉克型行为，其适用性由局部能带结构和特征函数基础控制。在所有光谱范围内，我们追求至少一阶扩展的均质描述，具有均质布洛赫波算子和均质源项的渐近校正。本质上，这种框架为合成各种有趣的波现象提供了一个方便的平台，包括超材料和声子晶体中的负折射和拓扑保护状态。所提出的均质化框架通过渐近逼近（i）具有六边形诺依曼排斥的 Kagome 晶格和（ii）圆形 Dirichlet 排斥的方形晶格的色散关系来说明。我们通过研究 Kagome 晶格的响应来完成分析发展的数值描述，这是由于在带隙边缘附近作用的类偶极源项引起的。