On resonant nonlinear bubble oscillations

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Abstract
If a bubble were produced with an initial surface distortion, the energy carried by surface modes could be converted to other modes by nonlinear interaction, a conversion that provides a possible mechanism of second generation by bubbles. Longuet-Higgins (1989a,b) has argued that volume pulsation would be excited at twice the frequency of the distortion mode and that the response to such excitation is ‘surprisingly large’ when its frequency is close to the natural resonance frequency of the volumetrical mode. It is shown in this paper that this is feasible only if the driving system is sufficiently energetic to supply the energy involved in those volume pulsations, and that this is not generally the case. In the absence of external sources, the sum of energies in the interacting modes cannot exceed the initial bubble energy; an increase in one mode is always accompanied by a decrease in another. In contrast to any expectation of significant pulsations near resonance, we find that, once modal coupling is admitted, the volumetrical pulsation has very small amplitude in comparison with that of the initial surface distortion. This is because of the constraint of energy, a constraint that becomes more severe once damping is admitted. Our conclusion therefore is that the distortion modes of a bubble are unlikely to be the origin of an acoustically significant bubble response.

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