Spherical blast-wave similarity energy integrals ($E_{\mathrm{kin}}=2\pi E{\int }_0^{{\xi }_0}{\xi }^4{f}_{\rho }{f}_u^{2}d\xi$)

Use the fields of pressure similarity in a spherical blast wave, with front radius $R={\xi }_0\ell$. Since ${\rho }_0{\ell }^5/t^2=E$, changing the radial integration variable to $\xi$ givesBoth are time independent for fixed integrable similarity profiles and constant front coordinate ${\xi }_0$. With explosion energy $E$ and negligible external energy input or losses, their sum equals $E$. This normalization, together with shock matching and regularity, fixes the dimensionless blast coefficient rather than dimensional analysis alone.