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Let $D=u_1>0$ denote the front speed in the stationary upstream frame. In the shock frame, the upstream and downstream velocities are $-D$ and $-D/r$. Taking $M_1\to \infty$ in the Rankine-Hugoniot conditions for a perfect gas givesThe downstream laboratory velocity is $D-D/r=2D/(\gamma +1)$, which distinguishes the gas speed from the front speed.

For the planar blast-wave energy scaling of a self-similar blast wave, integration of the total energy density over the shocked interval $0<z<Z$ gives the energy per unit area on this side:The similarity solution makes $I$ time-independent; a finite positive explosion energy requires $0<I<\infty$. Conservation of energy therefore gives $\dot{Z}=(E_{+}/({\rho }_{1}I){)}^{1/2}{Z}^{-1/2}$ for the expanding front. Integrating from $Z(0)=0$,If $E$ denotes the one-sided energy, $C=(9/(4I){)}^{1/3}$. If the released energy $E$ feeds two symmetric fronts, $E_{+}=E/2$ and $C=(9/(8I){)}^{1/3}$. In either convention the requested scaling is $Z=C(E/{\rho }_1{)}^{1/3}{t}^{2/3}$. The constant depends on the similarity profiles and the energy convention; energy conservation determines the exponent without solving those profiles. The Strong-shock Rankine-Hugoniot conditions additionally fix $f(1)=(\gamma +1)/(\gamma -1)$ and $g(1)=h(1)=2/(\gamma +1)$.

For a planar adiabatic explosion with energy per unit area $E$ in a uniform medium of mass density ${\rho }_0$, the conserved energy has form $E=I{\rho }_{0}Z{\dot{Z}}^2$, with a finite positive integral $I$ of the similarity profiles. Hence $Z=C(E/{\rho }_0{)}^{1/3}{t}^{2/3}$. If $E$ is shared by two fronts, its fraction assigned to either side is absorbed into $C$.

A similarity profile is the time-independent function left after expressing a evolving field in its scaled coordinate, for example $\rho (z,t)={\rho }_{0}f(z/Z(t))$. Integrating such profiles converts a global conservation law into an algebraic relation among the time-dependent scales.