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)'.
Keywords: graded diffraction gratings; rainbow trapping; wave scattering.