Number of Transition Frequencies of a System Containing an Arbitrary Number of Gas Bubbles

Abstract

``Transition frequencies'' of a system containing an arbitrary number of bubbles levitated in a liquid are discussed. Using a linear coupled-oscillator model, it is shown theoretically that when the system contains $N$ bubbles of different sizes, each bubble has $2N - 1$ (or less) transition frequencies which make the phase difference between an external sound and a bubble's pulsation $\pi / 2$. Furthermore, we discuss a discrepancy appearing between the present result regarding the transition frequencies and existing ones for the resonance frequencies in a two-bubble case, and show that the transition frequency, defined as above, and the resonance frequency have a different physical meaning when $N \ge 2$, while they are consistent for $N = 1$.