We consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space using wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the “mean” wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of “wavenumber quadrants” featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the “mean” wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points – apexes of locally conical dispersion surfaces – that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Dirac-like system of equations that generates “blunted” conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in ${\mathbb{R}}^2$ that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency- and wavenumber-dependent phonon behavior.

我们考虑在频谱奇点（例如布里渊区的拐角和狄拉克点）处的周期性有效连续运动，该周期性连续被与例如砖石和破裂材料有关的顺应性界面拦截。我们以标量波动方程的Bloch波形式（描述反平面剪切波）为出发点，并以波数分离为摄动参数，在波数-频率空间中寻找参考点的渐近展开。利用破损的Sobolev空间的概念来满足运动学上的不连续性，我们接下来通过Bloch波与周期单元的本征函数（在指定的波数和频率下）的内积定义“平均”波运动。通过这种投影扩展方法，我们得到了一个有效的场方程，对于任意的色散分支，第一个布里渊区具有“波峰象限”的顶点。为了完整起见，我们研究具有（a）孤立，（b）重复和（c）附近特征值的渐近结构。在重复特征值的情况下，我们发现“平均”波动受波动方程和狄拉克方程组的控制，波动方程的大小由特征值多重性给出，其结构由参与的特征函数确定，隶属单元功能，以及波数摄动的方向。这些结构之一显示了描述所谓的Dirac点-局部圆锥形分散表面的顶点-与拓扑受保护的波的产生有关。在具有紧密间隔的特征值簇的情况下，发现有效模型需要一个类似狄拉克的方程组，该系统可以生成“钝头”圆锥形分散面。我们通过数值模拟说明了两个周期构型的分析 ${\mathbb{[R}}^{\mathrm{2个}}$ 从（i）波频散，（ii）强迫波运动以及（iii）依赖于频率和波数的声子行为方面展示了渐近的发展。