  The ability to control wave propagation is of fundamental interest in many
areas of physics. Photonic and phononic crystals have proved very useful for
this purpose but, because they are based on Bragg interference, these
artificial media require structures with large dimensions. In [Ammari et al.,
Subwavelength phononic bandgap opening in bubbly media, J. Diff. Eq., 263
(2017), 5610--5629], it has been proved that a subwavelength bandgap opening
occurs in bubbly phononic crystals. To demonstrate the opening of a
subwavelength phononic bandgap, a periodic arrangement of bubbles is considered
and their subwavelength Minnaert resonance is exploited. In this paper, this
subwavelength bandgap is used to demonstrate cavities, very similar to those
obtained in photonic and phononic crystals, albeit of deeply subwavelength
dimensions. The key idea is to perturb the size of a single bubble inside the
crystal, thus creating a defect. The goal is then to analytically and
numerically show that this crystal has a localized eigenmode close to the
defect bubble.
