Thin-shell approximation for a spherical blast wave ($\alpha =1/2$)

Let a strong shock wave of radius $R$ and speed $V$ sweep a uniform mass density ${\rho }_0$ into a shell of mass $4\pi {\rho }_0{R}^3/3$ 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 $R^3{V}^2=\mathrm{constant}$ 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.