Pressure similarity in a spherical blast wave ($p=E^{2/5}{\rho }_0^{3/5}{t}^{-6/5}{f}_p(\xi )$)

For a spherical self-similar blast wave, put $\ell =(E{t}^2/{\rho }_0{)}^{1/5}$, $\xi =r/\ell$, $\rho ={\rho }_0{f}_{\rho }(\xi )$ and $u=(r/t)f_u(\xi )$. The radial Euler equation then requires $p={\rho }_0(\ell /t{)}^2{f}_p(\xi )$, whereThis follows from ${\partial }_t\xi =-2\xi /(5t)$ and ${\partial }_r\xi =1/\ell$. The momentum equation alone leaves a spatially uniform additive pressure undetermined. The scale-free strong-shock boundary data fix that freedom consistently with the displayed similarity scaling; an independently prescribed ambient pressure introduces another scale.