Elastic Wave Phononic Band Gap (1D)

A periodic bilayer medium creates forbidden frequency bands for elastic waves. Transfer matrix method reveals dispersion and band structure.

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Transfer matrix for unit cell (layers A+B):
M = M_B · M_A
M_j = [[cos(k_j d_j), sin(k_j d_j)/Z_j], [−Z_j sin(k_j d_j), cos(k_j d_j)]]

Band gap condition: |½ Tr(M)| > 1 → evanescent modes → forbidden band.

Bloch wavenumber K: cos(Ka) = ½ Tr(M)

Gap opens at k = nπ/a (Bragg condition). Gap width ∝ impedance contrast.
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