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High frequency homogenization of bubbly crystals

by H. Ammari and H. Lee and H. Zhang

(Report number 2017-40)

Abstract
This paper is concerned with the high frequency homogenization of bubbly phononic crystals. It is a follow-up of the work [H. Ammari et al., Subwavelength phononic bandgap opening in bubbly media, J. Diff. Eq., 263 (2017), 5610--5629] which shows the existence of a subwavelength band gap in such media. We show that the first Bloch eigenvalue achievs its maximum at the corner of the Brillouin zone. By computing the asymptotic of the Bloch eigenfunctions in the periodic structure near that critical frequency, we demonstrate that these eigenfunctions can be decomposed into two parts: one part is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is varying. This is very different from the usual homogenization where the second part is constant. This homogenization theory is termed high frequency homogenization in the sense that it is concerned with the asymptotic of wave fields near the critical frequency where a subwavelength band gap opens rather than the zero frequency. Our results shed light into the wave propagation theory in metamaterials. In particular, they rigorously justify, in the nondilute case, the observed superfocusing of acoustic waves in bubbly crystals near and below the maximum of the first Bloch eigenvalue.

Keywords: bubble, phononic crystal, band gap, Bloch theory, homogenization, metamaterial

@Techreport{ALZ17_736,
  author = {H. Ammari and H. Lee and H. Zhang},
  title = {High frequency homogenization of bubbly crystals},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-40.pdf },
  year = {2017}
}

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