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Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency

by H. Ammari and H. Zhang

(Report number 2016-26)

Abstract
We derive an effective medium theory for acoustic wave propagation in bubbly fluid near Minnaert resonant frequency. We start with a multiple scattering formulation of the scattering problem of an incident wave by a large number of identical small bubbles in a homogeneous fluid. Under certain conditions on the configuration of the bubbles, we justify the point interaction approximation and establish an effective medium theory for the bubbly fluid as the number of bubbles tends to infinity. The convergence rate is also derived. As a consequence, we show that near and below the Minnaert resonant frequency, the obtained effective media can have a high refractive index, which is the reason for the super-focusing experiment observed in [M. Lanoy, R. Pierrat, F. Lemoult, M. Fink, V Leroy, A Tourin, Subwavelength focusing in bubbly media using broadband time reversal, Physical Review, B 91.22 (2015)]. Moreover, our results indicate that the obtained effective medium can be dissipative above the Minnaert resonant frequency, while at that frequency, effective medium theory does not hold. Our theory sheds light on the mechanism of the extraordinary wave properties of metamaterials, which include bubbly fluid as an example, near resonant frequencies.

Keywords: Minnaert resonance, bubbly media, point interaction approximation, effective medium theory, super-focusing, super-resolution.

@Techreport{AZ16_663,
  author = {H. Ammari and H. Zhang},
  title = {Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-26},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-26.pdf },
  year = {2016}
}

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