Collapse time of a spherical cavity ($t_c=ka\sqrt{\rho /p_{\infty }}$)

For a spherical vacuum cavity initially at rest with radius $a$, in an inviscid liquid of constant mass density $\rho$ and positive constant far-field pressure $p_{\infty }$, the Rayleigh equation for an inviscid spherical bubble has the first integral $R^3{\dot{R}}^2=2p_{\infty }(a^3-R^3)/(3\rho )$. To obtain it, multiply the radius equation by $2R^2\dot{R}$ and integrate the resulting exact derivative. Choosing the negative square root and integrating $dt=dR/\dot{R}$ givesThis improper integral converges at both endpoints, proving finite-time Rayleigh collapse of a spherical cavity in the ideal model. It has numerical value approximately $0.914681$. The prediction neglects viscosity, surface tension and liquid compressibility.