EFFECTIVE MEDIUM THEORY FOR ACOUSTIC WAVES IN BUBBLY FLUIDS NEAR MINNAERT RESONANT FREQUENCY

We derive an effective medium theory for acoustic wave propagation in a bubbly fluid near the Minnaert resonant frequency. We start with a multiple scattering formulation of the scattering problem in which an incident wave impinges on a large number of identical and small bubbles in a homogeneous fluid. Under certain conditions on the configuration of the bubbles distribution, 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 may be highly refractive, which can be used to explain the superfocusing experiment observed in [M. Lanoy et al., Phys. Rev. B, 91 (2015), 224202]. Moreover, above the Minnaert resonant frequency, the effective medium is dissipative, 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.