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Bloch waves in bubbly crystal near the first band gap: a high-frequency homogenization approach. (English) Zbl 1408.35220

This work is a follow up to the work [J. Differ. Equations 263, No. 9, 5610–5629 (2017; Zbl 1401.35331)] where the authors studied the band structure of a bubbly phononic crystal consisting of periodically arranged bubbles in a homogeneous fluid. In that article, the authors prove there exists a subwavelength band gap. In the present article, the authors continue the previous analysis using homogenization theory near the frequency where the band gap opens. Using asymptotic analysis and layer potentials, the authors show that the first Bloch eigenvalue attains its maximum at the corner of the Brillouin zone. Further, it is shown that the Bloch eigenfunctions can be decomposed into a slowly varying part which satisfies a homogenized equation, and a periodic part which varies. The analysis completed by the authors confirms the band gap opening near and above the critical frequency.

MSC:

35R30 Inverse problems for PDEs
35C20 Asymptotic expansions of solutions to PDEs

Citations:

Zbl 1401.35331
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Full Text: DOI arXiv

References:

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