By Pierre A. Deymier
This accomplished e-book provides all facets of acoustic metamaterials and phononic crystals. The emphasis is on acoustic wave propagation phenomena at interfaces equivalent to refraction, in particular strange refractive homes and unfavorable 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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Additional info for Acoustic Metamaterials and Phononic Crystals
The complete band structure of the 1-D diatomic harmonic crystal is illustrated schematically in Fig. 8. 4 Monoatomic Crystal with a Mass Defect To shed additional light on the origin of the band gap in the band structure of the diatomic harmonic crystal, we investigate the propagation of waves in a 1-D monoatomic harmonic crystal with a single mass defect. This is accomplished by substituting one atom with mass m by another atom with mass m0 . The diatomic crystal may subsequently be created as a periodic substitution of atoms with different masses.
23) then states that ! À1 $ ! the denominator) of the Green’s function are the eigenvalues of the operator, H 0 0 . 30) This condition is met when t ¼ eika ¼ cos ka þ i sin ka. In the case, oE½0; o0 ; t ¼ 2 , x þ ið1 À x2 Þ1=2 if À 1 x 1. We can subsequently write cos ka ¼ x ¼ 1 À 2o o2 0 which, using trigonometric relation, reduces to the dispersion relation of propagating waves in the crystal (Eq. 5)). , a super-cell representation of the crystal. This system is represented in Fig. 4. We will solve the equation of motion of the mass, “l” in the first super-cell, that is, l E ½0; N À 1.
The displacement is 2 Discrete One-Dimensional Phononic and Resonant Crystals 37 0 If we choose Uðn0 Þ ¼ tÀn , corresponding to an incident wave coming from 0 n ¼ þ1, the displacement field in the semi-infinite chain takes the form 0 0 0 0 uðn0 Þ ¼ tÀn þ tn À1 ¼ eÀikn a þ eikðn À1Þa : This is a standing wave resulting from the superposition of an incident wave and a reflected wave. We can also obtain this result by writing the equation of motion at site 1 of the cleaved crystal: À mo2 u1 ¼ bðu2 À u1 Þ: This equation implies that u1 À u0 ¼ 0, where u0 is the displacement of the site 0 taken as a fictive site imposing a zero force boundary condition on site 1.
Acoustic Metamaterials and Phononic Crystals by Pierre A. Deymier