= Spherical blast-wave similarity energy integrals
{title2=$E_{\rm kin}=2\pi E\int_0^{\xi_0}\xi^4f_\rho f_u^2d\xi$}
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 $\xi$ gives
$$
E_{\rm kin}=2\pi E\int_0^{\xi_0}\xi^4f_\rho f_u^2\,d\xi,\qquad
E_{\rm int}=\frac{4\pi E}{\gamma-1}\int_0^{\xi_0}\xi^2f_p\,d\xi.
$$
Both 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.
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