= Pressure field of a pulsating sphere
{title2=$p/\rho=p_\infty/\rho+r^{-1}(R^2\dot R)_t-R^4\dot R^2/(2r^4)$}
In an unbounded <incompressible flow> with <spherical symmetry>, the <kinematic boundary condition> on a sphere of radius $R(t)$ gives $u_r=R^2\dot R/r^2$ and <velocity potential> $\phi=G(t)-R^2\dot R/r$. The <Unsteady Bernoulli equation> at infinity has $F(t)=\dot G+p_\infty/\rho$. Subtracting it from the finite-radius equation gives the displayed <pressure>, independently of $G$. Time differentiation is at fixed $r$. Evaluating at $r=R(t)$ yields the <Rayleigh equation for an inviscid spherical bubble> when the boundary <pressure> is prescribed.
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