Smith Michael J A, Abrahams I David
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK.
Proc Math Phys Eng Sci. 2022 Jun;478(2262):20220125. doi: 10.1098/rspa.2022.0125. Epub 2022 Jun 22.
We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications.
我们提出了一种求解方法，该方法将匹配渐近展开技术与多极展开方法相结合，以确定二维圆柱亥姆霍兹谐振器阵列的能带结构。当壁厚与孔径宽度相比非常大时（ 极限情况），考虑谐振器的几何形状。在这种情况下，发现第一部分（Smith & Abrahams，2022年，定制声学超材料。第一部分。薄壁和厚壁亥姆霍兹谐振器阵列）中现有的处理方法，在更新参数后，会返回虚假的频谱行为。我们推导了一个正则化系统来克服这个问题，并在此设置下推导了低频色散方程的紧凑渐近描述。我们发现，当适当地截断时，匹配渐近系统能够轻松地在整个布里渊区恢复前几个能带。概述了一种均匀化处理方法，用于描述所有壁厚情况下谐振器阵列的有效体积模量和有效密度张量。我们证明谐振器能够实现极低的亥姆霍兹共振频率，并给出了明确确定这些频率的闭式表达式。我们预计，这里概述的解析表达式和公式可能对设计用于工业和其他应用的超材料有用。
