We study waves governed by the planar Helmholtz equation, propagating in an infinite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength. Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved by a generalised Fourier series, whose singular asymptotics in the vicinities of scatterers yield the dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed. Characterising the latter is useful in interpreting and potentially designing the dispersion diagrams of such lattices. The method presented, that simplifies and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of the operating wavelength.
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机译:我们研究由平面Helmholtz方程控制的波,这些波在亚波长Dirichlet散射体的无限晶格中传播,其周期性与波长相当。应用匹配渐近展开法,散射体被渐近点约束有效取代。广义傅里叶级数解决了由此产生的粗粒度布洛赫波频散问题,该广义傅里叶级数的散射点附近的奇异渐近性产生了色散关系控制模式,这些模式被不受散射点存在的平面波解强烈干扰。也有只有微弱扰动的空晶格波。表征后者有助于解释和潜在设计此类晶格的色散图。提出的方法可以简化和扩展Krynkin&McIver [Waves Random Complex,19 347 2009],将来可以用于研究更复杂的设计,这些谐振子需要在工作波长上具有周期性的,分布在晶格上的谐振子波长元素。
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