We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.

中文翻译：

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诺依曼散射体平面阵列中布洛赫波和拓扑模式控制的渐近近似

我们研究了平面内诺依曼散射体的无限或有限周期阵列引起的波散射的典型问题；散射体的特征长度尺度被认为相对于晶格周期较小。我们利用匹配渐近展开的方法，连同傅立叶级数表示，来创建一种有效且准确的数值方法，用于通过无限阵列的散射体找到与 Floquet-Bloch 波相关的色散曲线。该方法适用于有限阵列的直接散射问题，我们在拓扑波物理学的局部例子上说明了这些渐近表示的灵活性。