Acoustically excited microbubbles (MBs) are known to be nonlinear oscillators with complex dynamics. This has enabled their use in a wide range of applications from medicine to industry and underwater acoustics. To better utilize their potential in applications and possibly invent new ones a comprehensive understanding of their dynamics is required. In this work, we explore the effect of bubble-bubble interactions on the resonance frequency of MB suspensions. MBs oscillate in response to an external acoustic wave and since bubbles in a cluster are at different locations compared to the excitation source, they are excited at different times. In this work we refer to these delays as primary delays. Interactions between the scattered pressure fields from adjacent bubbles have also been shown to alter the dynamics of MBs that exist within clusters. These secondary waves generated by MBs reach MBs in their proximity at different times that depend on their spatial location in the cluster. Here we refer to these delays as secondary delays. Inclusion of the secondary delays modifies the class of the differential equations governing the oscillations of interacting MBs in a cluster from ordinary differential equations to neutral delay differential equations. Previous work has not considered the all the delays associated with the bubble distances when modeling the interactions between bubbles. In this work we investigate the effect of both the primary and secondary delays on the effective resonance frequency of MB clusters. It is shown that primary delays cause spreading the resonance frequency of identical MBs within a range where the closest MB to the acoustic source exhibits the lowest resonance frequency and the furthest MB resonates at the highest frequency. This range has been shown to be up to 0.12 MHz for the examples investigated in this work. The effect of secondary delays is shown to be very significant. In the absence of secondary delays, the ordinary differential equation model predicts a decrease of up to 26% in the resonance frequency of 4 identical interacting MBs as the inter-bubble distances are decreased. However, we show that inclusion of the secondary delays result in the increase of the resonance frequency of MBs if they are situated close to each other. This increase is shown to be significant and for the case of 4 identical interacting MBs we show an increase of 58% in the resonance frequency.

声激发微泡 (MBs) 是具有复杂动力学的非线性振荡器。这使得它们能够在从医学到工业和水下声学的广泛应用中使用。为了更好地利用它们在应用中的潜力并可能发明新的，需要全面了解它们的动态。在这项工作中，我们探讨了气泡-气泡相互作用对 MB 悬浮液共振频率的影响。MB 响应外部声波而振荡，并且由于簇中的气泡与激发源相比位于不同的位置，因此它们在不同的时间被激发。在这项工作中，我们将这些延迟称为主要延迟。来自相邻气泡的分散压力场之间的相互作用也被证明可以改变存在于簇内的 MB 的动力学。这些由 MB 产生的二次波在不同时间到达附近的 MB，这取决于它们在集群中的空间位置。在这里，我们将这些延迟称为次要延迟。包含次级延迟将控制集群中相互作用 MB 的振荡的微分方程的类别从常微分方程修改为中性延迟微分方程。以前的工作在对气泡之间的相互作用进行建模时没有考虑与气泡距离相关的所有延迟。在这项工作中，我们研究了初级和次级延迟对 MB 簇的有效共振频率的影响。结果表明，主延迟导致相同 MB 的共振频率在距离声源最近的 MB 表现出最低共振频率而最远的 MB 以最高频率共振的范围内扩展。对于这项工作中研究的示例，该范围已显示为高达 0.12 MHz。二次延迟的影响被证明是非常显着的。在没有二次延迟的情况下，常微分方程模型预测，随着气泡间距离的减小，4 个相同的相互作用 MB 的共振频率将降低多达 26%。然而，我们表明，如果它们彼此靠近，则包含次级延迟会导致 MB 的共振频率增加。