A gas-filled bubble in a liquid will generally dissolve because of diffusion of gas out of the bubble into the surrounding liquid. However, when set into motion by an acoustic field, a bubble may grow by a process called rectified diffusion. This process can counteract the effect of diffusion for values of the acoustic-pressure amplitude greater than some threshold value. This threshold has been determined by a theory that uses computed radius-time curves for bubbles pulsating nonlinearly rather than assumed infinitestimal, sinusoidal motions. For radius-time curves calculated by a digital computer, this threshold has been computed for a sequence of values of gas concentration, bubble radius, and acoustic frequency.

This paper describes two related instabilities of spherical bubbles that are set into pulsation by a sound field. One instability is the observed onset of erratic dancing by bubbles that are trapped in a standing wave. This instability occurs when the sound-pressure amplitude exceeds a threshold value, and we measured the threshold for bubbles driven below resonance in water and in isopropyl alcohol. The other instability, which also requires that the sound-pressure amplitude exceed a threshold value, is the theoretically predicted onset of oscillation of the bubble shape. This threshold was calculated for the conditions of the previous experiments by a theory of parametric excitation based on Hill's equation. All results refer to pressure amplitudes less than 0.7 bar and frequencies from 23.6 to 28.3 kHz. From the close agreement of the measured dancing thresholds and the calculated shape-oscillation thresholds, we conclude that the erratic dancing of pulsating bubbles in a sound field is caused by shape oscillations that are parametrically excited by the bubble pulsations.

LETTERS TO THE EDITOR 290 28O 270 260 250 silence 50 65 80 dB SPL FIG. 2. Average duration of the oral and whispered syllable (open and closed hexagons, respectively) for each of four auditory conditions. if and only if the stress vanishes3 My analysis, on the other hand, does not require any particular state of stress and thus refers to all conditions in an elastic body.In a material with positive longitudinal elasticity, Kolodner proves that the propagation condition for longitudinal waves is satisfied for at least two directions at an unstressed point, while I prove that it is satisfied for at least one direction at any point.

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.

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 12 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 Mathieu'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 close to twice the resonance frequency of the bubble. In addition, solutions of a nonlinear equation of motion for the bubble wall, obtained on a computer, illustrate the growth of subharmonics and are used to determine the steady-state amplitude of the subharmonic for a sequence of values of various parameters. [Work supported by Acoustics Programs, Office of Naval Research.]