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):20220124. doi: 10.1098/rspa.2022.0124. Epub 2022 Jun 22.
We present a novel multipole formulation for computing the band structures of two-dimensional arrays of cylindrical Helmholtz resonators. This formulation is derived by combining existing multipole methods for arrays of ideal cylinders with the method of matched asymptotic expansions. We construct asymptotically close representations for the dispersion equations of the first band surface, correcting and extending an established lowest-order (isotropic) result in the literature for thin-walled resonator arrays. The descriptions we obtain for the first band are accurate over a relatively broad frequency and Bloch vector range and not simply in the long-wavelength and low-frequency regime, as is the case in many classical treatments. Crucially, we are able to capture features of the first band, such as low-frequency anisotropy, over a broad range of filling fractions, wall thicknesses and aperture angles. In addition to describing the first band we use our formulation to compute the first band gap for both thin- and thick-walled resonators, and find that thicker resonator walls correspond to both a narrowing of the first band gap and an increase in the central band gap frequency.
我们提出了一种用于计算圆柱形亥姆霍兹谐振器二维阵列能带结构的新型多极公式。该公式是通过将现有的理想圆柱阵列多极方法与匹配渐近展开法相结合而推导出来的。我们为第一能带表面的色散方程构建了渐近近似表示，修正并扩展了文献中关于薄壁谐振器阵列的已确立的最低阶（各向同性）结果。我们得到的第一能带描述在相对较宽的频率和布洛赫矢量范围内是准确的，而不像许多经典处理那样仅在长波长和低频区域准确。至关重要的是，我们能够在很宽的填充率、壁厚和孔径角范围内捕捉第一能带的特征，比如低频各向异性。除了描述第一能带，我们还用我们的公式计算了薄壁和厚壁谐振器的第一带隙，发现较厚的谐振器壁对应着第一带隙变窄以及中心带隙频率增加。
