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 a spherically-symmetric field that peaks at a frequency lower than the Minnaert resonance for a single bubble. In contrast, 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 work investigates the transition in resonance behavior using 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.

翻译：在水中增加紧密排列的气泡数量会改变Minnaert共振频率。小规模气泡群内的集体相互作用主要处于同相位，导致其辐射出球对称场，该场在低于单个气泡Minnaert共振频率处达到峰值。相比之下，大型周期阵列中包含间距超过半波长的气泡，因此集体共振中气泡以反相位振荡，最终形成频率高于单个气泡Minnaert共振的基本共振。本研究利用质量-弹簧系统的模态分析与边界元方法探究共振行为的转变过程。我们将全波求解器的计算复杂度显著降低为与矩形阵列中气泡数量呈线性关系。模拟的声场证实了共振频率的初始下移，并揭示了当阵列包含数百个气泡且覆盖超过半波长范围时集体共振的强烈影响。这些结果对于理解气泡群的低频共振特性至关重要，在水声学、量子物理和超材料设计等多个领域具有重要应用价值。