Abstract
We analyse the propagation of airborne pressure waves through a three-dimensional array of rigid coated spheres (shells) in air. When we dig a channel terminated by an air cavity in each rigid shell we observe the appearance of a low frequency stop band. Each shell with a hole acts as a Helmholtz resonator supporting a low frequency localized mode. Isofrequency surfaces and contours reveal the strong anisotropy of the periodic structure at the edge of the stop band. A simple mechanical model of springs and masses allows for asymptotic estimates of the low frequency stop band for elongated channels. Increasing the radius of an air channel shifts up the position, and enlarges, the low frequency stop band. Adding holes in shells also shifts up the frequency of the stop band, and embedded shells lead to additional stop bands. Localization effect induced by a large defect in a periodic macrocell of Helmholtz resonators is finally investigated.
Highlights
ACOUSTIC METAMATERIALSIn the tracks of photonic crystals, phononic crystals (Dowling, 2008) have provided a fillip for research in acoustic stop band structures (Kushwaha et al, 1993; Kafesaki and Economou, 1999) within which light or sound is prohibited to propagate due to multiple scattering between periodically spaced inclusions
We have seen that it is possible to sculpt the Bloch spectrum of three-dimensional phononic crystals almost ad libitum by digging some holes and adding cavities in rigid spheres periodically arranged along a cubic lattice
One of the main achievements of our numerical study is the appearance of ultra-low frequency stop bands at frequencies predicted quantitalively by an asymptotic model, that can be viewed as a 3D counterpart of (Movchan and Guenneau, 2004)
Summary
In the tracks of photonic crystals, phononic crystals (Dowling, 2008) have provided a fillip for research in acoustic stop band structures (Kushwaha et al, 1993; Kafesaki and Economou, 1999) within which light or sound is prohibited to propagate due to multiple scattering between periodically spaced inclusions. In 2000, Liu et al provided the first numerical and experimental evidence of frequency dispersive elastic parameters of locally resonant structures for elastic waves in three-dimensional arrays of thin coated spheres (Liu et al, 2000): The effective parameters were shown to turn negative where low frequency stop bands occur. Compared to problems of linear elasticity, the present study does not deal with dynamic degeneracies at low frequencies, which may occur for certain type of geometries of elastic systems, resulting in a group of very small eigenvalues being separated from the remaining spectrum
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