By Pierre A. Deymier
This entire booklet provides all facets of acoustic metamaterials and phononic crystals. The emphasis is on acoustic wave propagation phenomena at interfaces corresponding to refraction, specially strange refractive homes and destructive refraction. an intensive dialogue of the mechanisms resulting in such refractive phenomena comprises neighborhood resonances in metamaterials and scattering in phononic crystals.
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12. 96) The dynamical matrix is composed of three separate blocks corresponding to the three uncoupled regions of the cleaved system of Fig. ” Similarly, the Green’s function and the surface operators are also block diagonal matrices. A. Deymier and L. Dobrzynski Fig. 98) To apply the universal equation of the IRT, we need the block “2” of the surface operator matrix in the space of the corresponding perturbed sites M E ½1; L, that is ! À1 t AS2 ð1; 1Þ AS2 ð1; LÞ AS2 ðM; MÞ ¼ ¼ AS2 ðL; 1Þ AS2 ðL; LÞ t þ 1 tL $ !
41) These two solutions are periodic in wave number, k, with a period of pa . These solutions areÂ represented graphically in the band structure of Fig. 7 over the Ã p . This interval is the smallest interval, the so-called irreducible interval, k E 0; 2a Brillouin zone, for representing the band structure. The complete band structure is reconstructed by mirror symmetry with respect to a vertical line passing though the origin. A. Deymier and L. Dobrzynski Fig. 7 Schematic representation of the band structure of the 1-D diatomic harmonic crystal in the irreducible Brillouin zone qﬃﬃﬃﬃ qﬃﬃﬃﬃ The frequencies o1 ; o2 and o3 are given by o1 ¼ m2b1 ; o2 ¼ m2b2 , and rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ o3 ¼ 2b m11 þ m12 if one chooses m1 >m2 .
B 0 0 .. .. Àg b 0 .. .. b Àg b .. .. 0 b Àg .. .. 0 0 b .. .. 0 0 0 .. 32 3 2 3 .. . 76 . 7 6 .. 7 76 unÀ1 7 6 0 7 7 6 7 6 . . 76 un 7 7 ¼ 6 0 7; 7 7 7 6 . . 54 unþ1 5 6 405 .. .. . 22) ! where g ¼ 2b À mo2 . 22), and ~ u is the vector whose components are the displacements of the masses in the crystal. With this notation, the Green’s function, ! G0 , associated with H0 is defined by the relation ! 23) $ In this equation, I is the identity matrix. 24) Here, we have used the Kroenecker symbol d0nn to represent the components !
Acoustic Metamaterials and Phononic Crystals by Pierre A. Deymier