Linear-profile spherical blast wave ($\beta =(7-\gamma )/(\gamma +1)$)

The similarity profiles $U=2\xi /(\gamma +1)$, $D=(\gamma +1)\xi /(\gamma -1)$ and $P=2{\xi }^3/(\gamma +1)$ solve the spherical blast equations at the displayed ambient exponent. For $\gamma =5/3$, $\beta =2$ and $R^3=3E{t}^2/(2\pi C)$. The dimensional fields are $u=r/(2t)$, $\rho =8\pi C^{2}r/(3E{t}^2)$ and $p=2\pi C^2{r}^3/(9E{t}^4)$ for $0<r<R$. Their kinetic and internal energies are equal. Substituting these profiles into all three equations and the shock conditions verifies the solution.