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Analysis of deaf bands using the complex band structure calculated using the EPWE. (a) Complex band structure for a SC with square periodicity which filling fraction is ff = πr 2/a ≃ 60%, being a the lattice constant. Grey area represents the BG. Green area represents the attenuation range produced by the presence of a deaf band. (b) Red continuous line represents the IL evaluated using the structure shown in the inset at point (a, 0). Other coloured lines shows the IL for several structures with less number of rows at the ΓM direction of incidence.
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(a) Acoustic field calculated with the EPWE, corresponding to the mode in the deaf band marked in Figure 1(a) with a red square (Ψ = 1.14, Re(K) = 0.39). (b) Acoustic field calculated with the EPWE, corresponding to the mode in the deaf band marked in Figure 1(a) with a blue square (Ψ = 1.14, Im(K) = 0.19). (c) Acoustic field predicted using the MST at frequency Ψ = 1.14. (d) Blue continuous line represent the values of |p| inside the periodic structure along the white dashed line in (c). Red dashed line represents the exponential decay predicted using the EWPE.
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The properties of sonic crystals (SC) are theoretically investigated in this work by solving the inverse problemk(ω) using the extended plane wave expansion (EPWE). The solution of the resulting eigenvalue problem gives the complex band structure which takes into account both the propagating and the evanescent modes. In this work we show the complete mathematical formulation of the EPWE for SC and the supercell approximation for its use in both a complete SC and a SC with defects. As an example we show a novel interpretation of the deaf bands in a complete SC in good agreement with multiple scattering simulations.
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