Modeling frequency shifts of collective bubble resonances with the boundary element method

Increasing the number of closely-packed air bubbles immersed in water changes the frequency of the Minnaert resonance. The collective interactions between bubbles in a small ensemble are primarily in the same phase, causing them to radiate an axially symmetric field that peaks at a frequency lower than the Minnaert resonance for a single bubble. Differently, large periodic arrays include bubbles that are further apart than half the wavelength so that collective resonances have bubbles oscillating in opposite phases, ultimately creating a fundamental resonance at a frequency higher than the single-bubble Minnaert resonance. This paper explains the transition in resonance behavior through a modal analysis of a mass-spring system and a boundary element method. We significantly reduce the computational complexity of the full-wave solver to a linear dependence on the number of bubbles in a rectangular array. The simulated acoustic fields confirm the initial downshift in resonance frequency and the strong influence of collective resonances when the array has hundreds of bubbles covering more than half the wavelength. These results are essential in understanding the low-frequency resonance characteristics of bubble ensembles, which have important applications in diverse fields such as underwater acoustics, quantum physics, and metamaterial design.