The resonance frequencies of two bubbles in compressible liquid

The nonlinear pulsation equations of bubbles are derived using energy-based analysis, and the resonance frequencies of a double-bubble system are obtained by linearizing these equations in a compressible liquid. The results indicate that the resonance frequency is lower than that of a single bubble when the two bubbles pulsate in phase but higher when they pulsate out of phase. Additionally, the resonance frequency increases linearly with the distance between in-phase oscillating bubbles but decreases inversely with distance for out-of-phase oscillations, while the consideration of compressibility leads to a further reduction in resonance frequency. When the initial perturbation ratio of two bubbles with different ambient radii equals 1, larger bubbles exert a stronger influence on smaller ones, potentially disrupting oscillation regularity. For bubbles of the same radius, an initial perturbation ratio below 1 leads to compressed and irregular oscillation waveforms, whereas a ratio above 1 causes excited and irregular waveforms that deteriorate more rapidly than those with equal initial perturbations.