Translational motion of a bubble undergoing shape oscillations

This paper studies the nonlinear coupling between the volume pulsation, translational motion and shape modes of an oscillating bubble, especially in the context of translational instability, known as ‘dancing motion’, that is demonstrated by bubbles in acoustic standing waves. A set of coupled equations is derived that describes volume pulsations of a bubble, its translational motion and shape oscillations evolving on the bubble surface. The amplitudes of the surface modes and the translational velocity of the bubble are assumed to be small and allowed for in the equations of motion up to only second-order terms. The amplitude of the volume oscillation is not limited. Unlike earlier work on this subject, where only two adjacent shape modes with given natural frequencies are taken into account, we allow for all shape modes and do not impose any limitations on their natural frequencies. As a result, the present analysis reveals additional features, which have not been noted previously, inherent in the mutual interactions of the shape modes as well as in the interaction between the shape modes and the translational motion.