Numerical simulations of violent bubble dynamics are often associated with numerical instabilities at the end of collapse, when a shock wave is emitted. Based on the Keller–Miksis equation, we show that this is caused by two time scales associated with the phenomenon. Nonsingular equations are thus formed based on asymptotic expansion theory and the time derivatives of the bubble radius are shown to have algebraic singularities in the Mach number. The period of oscillation is shown to divide into two asymptotic layers: a long and short time scale. The short time scale, on which significant acoustic radiation is emitted from the bubble, has been determined to be , where c is the speed of sound in the liquid, the maximum bubble radius, ρ the liquid density, the hydrostatic pressure of the liquid, the vapour pressure of the liquid and κ the polytropic index of the bubble gas. Using the scalings for this short time scale, the radiated acoustic pressure scale has been deduced to be , where is the radial distance from the bubble centre to the point of measurement. The results are validated by comparison with experimental results.

当发出冲击波时，剧烈气泡动力学的数值模拟通常与坍塌结束时的数值不稳定性相关。基于Keller-Miksis方程，我们表明这是由与该现象相关的两个时间尺度引起的。因此，基于渐近展开理论形成了非奇异方程，并且气泡半径的时间导数在马赫数中具有代数奇异性。振荡周期显示为分为两个渐近层：长时标和短时标。已确定从气泡发出大量声辐射的短时间标度为，其中c是液体中的声速，最大气泡半径ρ，液体密度，液体的静水压力，液体的蒸气压和κ气泡气体的多折射率。使用此短时间标度的标度，已得出辐射声压标度为，其中是气泡中心到测量点的径向距离。通过与实验结果进行比较来验证结果。