We present a multiple scattering analysis of robust interface states for flexural waves in thin elastic plates. We show that finite clusters of linear arrays of scatterers built on a quasi-periodic arrangement support bounded modes in the two-dimensional space of the plate. The spectrum of these modes plotted against the modulation defining the quasi-periodicity has the shape of a Hofstadter butterfly, which as suggested by previous works might support topologically protected modes. Some interface states appear inside the gaps of the butterfly, which are enhanced when one linear cluster is merged with its mirror reflected version. The robustness of these modes is verified by numerical experiments in which different degrees of disorder are introduced in the scatterers, showing that neither the frequency nor the shape of the modes is altered. Since the modes are at the interface between two one-dimensional arrays of scatterers deposited on a two-dimensional space, these modes are not fully surrounded by bulk gaped materials so that they are more suitable for their excitation by propagating waves. The generality of these results goes beyond flexural waves since similar results are expected for acoustic or electromagnetic waves.

中文翻译：

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散射体准周期线性阵列中弯曲波的边缘模式

我们对薄弹性板中弯曲波的稳健界面状态进行了多重散射分析。我们表明，建立在准周期排列上的散射体线性阵列的有限簇支持板的二维空间中的有界模式。根据定义准周期性的调制绘制的这些模式的频谱具有霍夫施塔特蝴蝶的形状，正如以前的工作所建议的那样，这可能支持拓扑保护模式。一些界面状态出现在蝴蝶的间隙内，当一个线性簇与其镜面反射版本合并时，这些界面状态会得到增强。这些模式的稳健性通过数值实验得到验证，其中在散射体中引入了不同程度的无序，表明模式的频率和形状都没有改变。由于这些模式位于二维空间上沉积的两个一维散射体阵列之间的界面处，这些模式并未完全被块状有隙材料包围，因此它们更适合通过传播波激发它们。这些结果的普遍性超出了弯曲波，因为预期声波或电磁波会产生类似的结果。