 Solution

In the energy-conserving blast regime, neglect ambient pressure, radiative losses, ejecta mass and gravity compared with the explosion. The cold ambient ideal gas has no sound speed scale, and its power-law mass density has no preferred radius. The point explosion introduces only its energy $E$ and elapsed time $t$. This scale-free setting supports a self-similar blast wave; extra physics would limit the time interval over which that description applies.

The mass swept up to radius $R$ isIt is finite because $\beta <3$. With no energy loss, the kinetic energy and internal energy both scale as $C{R}^{3-\beta }{\dot{R}}^2$. Therefore $R^{3-\beta }{\dot{R}}^2$ is constant and integration givesThe dimensionless coefficient $A$ depends on the adiabatic exponent and ambient exponent through the similarity profiles. Indeed, $[E/C]=L^{5-\beta }{T}^{-2}$, which independently verifies the scaling. The shock wave decelerates for $0<\beta <3$, since its time exponent lies between $2/5$ and $1$.