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Under proper conditions, bubbles driven by a sound field will pulsate periodically with a frequency equal to one-half the frequency of the sound field. This frequency component is the subharmonic of order one-half and is generated when the acoustic-pressure amplitude exceeds a threshold value. The threshold for subharmonic generation is calculated by means of a theory that relates the presence of the subharmonic to properties of Hill's equation. It is found that, for a given bubble, the threshold is a function of the driving frequency and is a minimum when the driving frequency is approximately twice the linear resonance frequency of the bubble. In addition, solutions of a set of nonlinear equations for the motion of the bubble wall, obtained on a computer, illustrate the growth of subharmonics and are used to determine the steady-state amplitude and phase of the subharmonic for a sequence of values of various parameters.
Eller et al. (1969) studied this question.