= Collapse time of a spherical cavity
{title2=$t_c=k a\sqrt{\rho/p_\infty}$}
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 $dt=dR/\dot R$ gives
$$
k=\sqrt{\frac32}\int_0^1\frac{s^{3/2}}{\sqrt{1-s^3}}\,ds.
$$
This <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.
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