the change of the domains $\left(\mathrm{u}, \mathrm{d}\right)\mapsto\left(\mathrm{d}, \mathrm{u}\right)$ and the shock propagates into the quiescent downstream region $u\left(r\rightarrow\infty, t\right)=0$.

Mathematically, the up- and downstream regions are described with the 1D continuity and Euler equations. The jump itself must be determined via the Rankine-Hugoniot Conditions (see \[secRHC\]). We further use the isothermal condition $\left(T=\mathrm{const}\right)$ and assume an ideal gas equation of state. Hence, $P/\rho=\mathrm{const.}$, or $$P=\rho c_{\mathrm s}^2\label{eq:isocond}$$ where $P$ is the pressure, $\rho$ the density and $c_\mathrm{s}$ the constant speed of sound, leading to a violation of the principle of energy conservation.

The generalized equations e.g. [@Landau1974eng] together with equation \[eq:isocond\], the velocity normalized to the speed of sound and the density normalized to some characteristic density of the flow $\hat \rho$ can be written as a system of two nonlinear partial differential equations (PDE)

$$\begin{aligned}
   \partial_\tau\rho+\frac{1}{r^\alpha}\partial_r\left(r^\alpha\rho u\right)&=0\label{eq:origsys_c}\\
   \partial_\tau u+u\partial_r u +\frac{1}{\rho}\partial_r\rho &=0.\label{eq:origsys_i}
  \end{aligned}$$

The density $\rho=\rho\left(r,\tau\right)$ and the radial velocity $u=u\left(r,\tau\right)$ are functions of the spatial coordinate $r$ and the normalized time $\tau=c_\mathrm{s}t$.

The reduced differential equations {#redmeth}
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Self-similar solutions to a set of partial differential equations (PDEs) must satisfy two constraints. First, a group of transformations exists reducing the set of PDEs to a set of ordinary 