= Solution
Write $V=\dot R$, $a=2/(5-\beta)$ and
$$
\xi=\frac rR,\qquad u(r,t)=V U(\xi),\qquad
\rho(r,t)=CR^{-\beta}D(\xi),\qquad
p(r,t)=CR^{-\beta}V^2P(\xi).
$$
Here $u$ is the outward radial <velocity>; $U,D,P$ are dimensionless. The relations
$$
\partial_t\xi=-\frac VR\xi,\qquad
\partial_r\xi=\frac1R,\qquad
q\equiv\frac{R\dot V}{V^2}=\frac{a-1}{a}=\frac{\beta-3}{2}
$$
are sufficient for the reduction. In smooth post-shock gas the spherical <continuity equation>, <Euler equations> and adiabatic <pressure> equation are
$$
\partial_t\rho+u\partial_r\rho=-\rho(\partial_ru+2u/r),\qquad
\partial_tu+u\partial_ru=-\frac1\rho\partial_rp,
$$
$$
\partial_tp+u\partial_rp=-\gamma p(\partial_ru+2u/r).
$$
For example, the <mass density> <material derivative> is $CR^{-\beta}(V/R)[(U-\xi)D'-\beta D]$, and the <velocity> <material derivative> is $(V^2/R)[(U-\xi)U'+qU]$. After cancellation of the dimensional factors the <spherical blast wave in a power-law ambient density> obeys
$$
\boxed{(U-\xi)D'+D\left(U'+\frac{2U}{\xi}-\beta\right)=0,}
$$
$$
\boxed{(U-\xi)U'+\frac{\beta-3}{2}U=-\frac{P'}D,}
$$
$$
\boxed{(U-\xi)P'+\left[\gamma\left(U'+\frac{2U}{\xi}\right)-3\right]P=0.}
$$
Primes here mean $d/d\xi$. In the <pressure> equation the scale derivative $2q-\beta=-3$ explains the constant term. Once the profiles are found, their finite <energy> integral fixes $A$ through
$$
E=4\pi CR^{3-\beta}V^2\int_0^1\left[\frac12DU^2+\frac{P}{\gamma-1}\right]\xi^2\,d\xi.
$$
No global constant in $p/\rho^\gamma$ is assumed: the <shock wave> generates different <entropies> in shells swept up at different times.
Back to article page
