Subwavelength localized modes for acoustic waves in bubbly crystals with a defect

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 interferences, these artificial media require structures with large dimensions. In [Ammari et al., J. Differential Equations, 263 (2017), pp. 5610-5629], it has been proved that a subwavelength bandgap opening occurs in bubble 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 increase 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.