= Thin-shell approximation for a spherical blast wave
{title2=$\alpha=1/2$}
Let a strong <shock wave> of radius $R$ and speed $V$ sweep a uniform <mass density> $\rho_0$ into a shell of mass $4\pi\rho_0R^3/3$ and speed $2V/(\gamma+1)$. If the cavity <pressure> is $\alpha$ times the immediate post-shock <pressure>, shell <momentum> balance gives $R\dot V=3(\alpha-1)V^2$. Constant total <energy> gives $R^3V^2=\mathrm{constant}$ and hence $\alpha=1/2$. Including shell internal and kinetic <energy>, and treating the cavity volume as $4\pi R^3/3$ to thin-shell accuracy, yields $R=\xi(E/\rho_0)^{1/5}t^{2/5}$ with $\xi^5=75(\gamma-1)(\gamma+1)^2/[16\pi(5\gamma-3)]$. This coefficient is approximate; resolving the shell's finite volume changes it.
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