  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 "Subwavelength focusing in bubbly
media using broadband time reversal", Physical Review, B (2015), by M. Lanoy et
al. 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.
