By Pierre A. Deymier
This entire booklet offers all elements of acoustic metamaterials and phononic crystals. The emphasis is on acoustic wave propagation phenomena at interfaces resembling refraction, particularly 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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Extra info for Acoustic Metamaterials and Phononic Crystals
This formalism is based on the Green’s function approach called the Interface Response Theory. 1 Fundamental Equations of the Interface response Theory In this section, we review the fundamental equations of the Interface Response Theory (IRT) for discrete systems . This formalism allows the calculation of the Green’s function of a perturbed system in terms of Green’s functions ! of unperturbed systems. 22) defining the Green’s function, G0 , by ! H0 G0 ¼ I: The operator H0 is the infinite tridiagonal dynamic matrix: 2 ..
95) to obtain the displacement field. For this we also need to assume a form for the reference displacement UðDÞ ¼ tn. This displacement corresponds to a wave propagating in the infinite crystal and launched from n ¼ À1. The displacement inside the side crystal is also assumed to be equal to zero. The displacement in the space M takes the form UðMÞ ¼ ðUð0Þ; Uð10 Þ; Uðp2 Þ; Uð100 Þ; . . ; U ðpNc Þ; Uð1Nc Þ ¼ ð1; 0; tp2 ; 0; . . ; tpNc ; 0Þ: ð2:119Þ 2 Discrete One-Dimensional Phononic and Resonant Crystals 43 Fig.
The 0 0 renormalized displacement then reduces to uðn0 Þ ¼ tn þ tÀn þ1 . This expression is that of the displacement of standing waves in the finite crystal. 4 One-Dimensional Monoatomic Crystal with One Side Branch The calculation of the displacement in a system composed of a 1-D monoatomic crystal with a finite crystal branch coupled to its side via a spring with constant, bI , as illustrated in Fig. 10, begins with the block matrix describing the Green’s function of the uncoupled system ðbI ¼ 0Þ $ GS ¼ $ $ G0 $ 0 $ 0 $ g0 S2 !