  This paper is concerned with the high-frequency homogenization of bubbly
phononic crystals. It is a follow-up of the works [H. Ammari et al.,
Sub-wavelength phononic bandgap opening in bubbly media, J. Diff. Eq., 263
(2017), 5610--5629] which shows the existence of a sub-wavelength band gap.
This phenomena can be explained by the periodic inference of cell resonance
which is due to the high contrast in both the density and bulk modulus between
the bubbles and the surrounding medium. In this paper, we prove that the first
Bloch eigenvalue achieves its maximum at the corner of the Brillouin zone.
Moreover, 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. They rigorously justify, in the
nondilute case, the observed super-focusing of acoustic waves in bubbly
crystals near and below the maximum of the first Bloch eigenvalue and confirm
the band gap opening near and above this critical frequency.
