By Pierre A. Deymier

ISBN-10: 3642312314

ISBN-13: 9783642312311

This entire publication offers all features of acoustic metamaterials and phononic crystals. The emphasis is on acoustic wave propagation phenomena at interfaces similar to refraction, specially strange refractive homes and detrimental refraction. a radical dialogue of the mechanisms resulting in such refractive phenomena contains neighborhood resonances in metamaterials and scattering in phononic crystals.

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M6 6... 4 .. .. 0 0 0 .. .. 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.

This condition may be rewritten as tL À tÀL ¼ 0 . For angular frequencies, o E ½0; o0 Š; t ¼ eika and the modes of the finite crystal are given by eikLa À eÀikLa ¼ sin kLa ¼ 0. These modes correspond to standing waves with wave number conditioned by k ¼ pp La with p being an integer. 95). 103) gives 0 0 uðn0 Þ ¼ tn þ tn 2Lþ2 2Lþ1 t t2L 0 t 0 0 t þ tÀn þ tn þ tÀn : 2L 2L 2L 1Àt 1Àt 1Àt 1 À t2L This expression diverges when 1 À t2L ¼ 0 . It is therefore necessary to obtain a finite displacement by renormalizing the previous expression by W.

B 0 G0 ðp3 p2 Þ 0 G0 ðp3 p3 Þ 0 GS ðMMÞ ¼ B G0 ðp3 p1 Þ B 0 0 0 0 0 gs ð1ð3Þ 1ð3Þ Þ B B B .. .. .. B . . . B B 0 G0 ðpNc p2 Þ 0 G0 ðpNc p3 Þ 0 @ G0 ðpNc p1 Þ 0 0 0 0 0 0 ... ... G0 ðp1 pNc Þ 0 G0 ðp2 pNc Þ 0 0 0 ... ... . 0 G0 ðp3 pNc Þ 0 .. 0 0 0 .. 1 . . G0 ðpNc pNc Þ 0 À Á ... 116) In this matrix, the odd entries (rows or columns) correspond to locations along the infinite crystal in M and the even entries correspond to the position of the first atom of the finite crystals (also in the space M).

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Acoustic Metamaterials and Phononic Crystals by Pierre A. Deymier


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