The radiated acoustic pressure and time scales of a spherical bubble

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 ${\overline{R}}_{\text{max}}{\left(\right[{\overline{p}}_{\infty }-{\overline{p}}_v\left]/\rho c^2\right)}^{1/\left(3\kappa \right)}/c$, where c is the speed of sound in the liquid, ${\overline{R}}_{\text{max}}$ the maximum bubble radius, ρ the liquid density, ${\overline{p}}_{\infty }$ the hydrostatic pressure of the liquid, ${\overline{p}}_v$ 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 $\rho c^2{\overline{R}}_{\text{max}}{\left(\right[{\overline{p}}_{\infty }-{\overline{p}}_v\left]/\rho c^2\right)}^{1/\left(3\kappa \right)}/R$, where $R$ is the radial distance from the bubble centre to the point of measurement. The results are validated by comparison with experimental results. (paper)