Bennetts L G, Peter M A, Craster R V
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia.
Institute of Mathematics, University of Augsburg, 86135, Augsburg, Germany.
Philos Trans A Math Phys Eng Sci. 2019 Oct 21;377(2156):20190104. doi: 10.1098/rsta.2019.0104. Epub 2019 Sep 2.
Energy amplification in square-lattice arrays of C-shaped low-frequency resonators, where the resonator radii are graded with distance, is investigated in the two-dimensional linear acoustics setting for both infinite (in one dimension) and finite arrays. Large amplifications of the incident energy are shown in certain array locations. The phenomenon is analysed using: (i) band diagrams for doubly-periodic arrays; (ii) numerical simulations for infinite and finite arrays; and (iii) eigenvalue analysis of transfer matrices operating over individual columns of the array. It is shown that the locations of the large amplifications are predicted by propagation cut-offs in the modes associated with the transfer-matrix eigenvalues. For the infinite array, the eigenvalues form a countable set, and for the low frequencies considered, only a single propagating mode exists for a given incident wave, which cuts off within the array, leading to predictive capabilities for the amplification location. For the finite array, it is shown that (in addition to a continuous spectrum of modes) multiple discrete propagating modes can be excited, with the grading generating new modes, as well as cutting others off, leading to complicated amplification patterns. The numerical simulations reveal that the largest amplifications are achieved for a single row array, with amplifications an order of magnitude smaller for the corresponding infinite array. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.
研究了C形低频谐振器的方形晶格阵列中的能量放大问题,其中谐振器半径随距离渐变,在二维线性声学环境中对无限(一维)和有限阵列进行了研究。在某些阵列位置显示出对入射能量的大幅放大。使用以下方法对该现象进行了分析:(i)双周期阵列的能带图;(ii)无限和有限阵列的数值模拟;(iii)对阵列各列操作的传递矩阵的特征值分析。结果表明,大幅放大的位置由与传递矩阵特征值相关的模式中的传播截止来预测。对于无限阵列,特征值形成一个可数集,对于所考虑的低频,给定入射波仅存在一个传播模式,该模式在阵列内截止,从而实现了对放大位置的预测能力。对于有限阵列,结果表明(除了连续的模式谱之外)可以激发多个离散传播模式,渐变会产生新的模式,同时也会截断其他模式,从而导致复杂的放大模式。数值模拟表明,单行阵列实现了最大放大,而相应的无限阵列的放大倍数小一个数量级。本文是主题为“结构化介质中动态现象和局域化的建模(第1部分)”的一部分。