We consider the scattering of in-plane waves that interact with an edge of a structured penetrable inertial line defect contained in a triangular lattice, composed of periodically placed masses interconnected by massless elastic rods. The steady state problem for a time-harmonic excitation is converted into a vector Wiener–Hopf equation using the Fourier transform. The matrix Wiener–Hopf kernel of this equation describes the dynamic phenomena engaged in the scattering process, which includes instances where localised interfacial waves can emerge along the structured defect. This information is exploited to identify the dependency of the existence of these waves on the incident wave parameters and the properties of inertial defect. Symmetry in the structure of the scattering medium allows us to convert the vectorial problem into a pair of uncoupled scalar Wiener–Hopf equations posed along the lattice row containing the defect. The solution embodies an exact representation of the scattered field, in terms of a contour integral in the complex plane, that includes the contributions of evanescent and propagating waves. The solution reveals that in the remote lattice, the reflected and transmitted components of incident field are accompanied by dynamic modes from three symmetry classes, which include localised interfacial waves. These classes correspond to tensile modes acting transverse to the defected lattice row, shear modes that act parallel to this row, and wave modes represented as a mixture of these two responses. Benchmark finite element calculations are also provided to validate the results against our semi-analytical solution which involves, in particular, numerical computation of the contour integrals. Graphical illustrations demonstrate special dynamic responses encountered during the wave scattering process, including dynamic anisotropy, negative reflection and negative refraction.

中文翻译：

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面内波与弹性晶格中结构化可穿透线缺陷的相互作用

我们考虑与三角晶格中包含的结构化可穿透惯性线缺陷的边缘相互作用的面内波的散射，该三角晶格由周期性放置的质量组成，通过无质量的弹性杆互连。使用傅里叶变换将时谐激励的稳态问题转换为矢量 Wiener-Hopf 方程。该方程的矩阵维纳-霍普夫核描述了散射过程中的动态现象，其中包括局部界面波沿着结构缺陷出现的情况。利用该信息来识别这些波的存在对入射波参数和惯性缺陷特性的依赖性。散射介质结构的对称性使我们能够将矢量问题转换为沿着包含缺陷的晶格行提出的一对非耦合标量维纳-霍普夫方程。该解以复平面中的轮廓积分的形式体现了散射场的精确表示，其中包括倏逝波和传播波的贡献。该解表明，在远程晶格中，入射场的反射和透射分量伴随着来自三个对称类的动态模式，其中包括局域界面波。这些类别对应于横向于缺陷晶格行作用的拉伸模式、平行于该行作用的剪切模式以及表示为这两种响应的混合的波模式。还提供了基准有限元计算，以根据我们的半解析解决方案验证结果，其中特别涉及轮廓积分的数值计算。图形说明展示了波散射过程中遇到的特殊动态响应，包括动态各向异性、负反射和负折射。