and isothermality which is not genuinely isothermal. As a result there exists an adiabatic flow through the discontinuity and the principle of energy conservation therefore holds, even in this [*[isothermal]{}*]{} case. The similarity variable yields $\eta=\left(r^5\rho t^{-2}E^{-1}\right)^{1/5}$ with $E$ the initial energy, $t$ the time, $r$ the radial distance and $\rho$ the initial density. This enables the immediate release of a specified amount of energy at even very small spatial extent. Hence, there are well defined solutions for point-like explosions e.g. [@Korobeinikov1956; @Solinger1975; @Lerche1976] with respect to the initial parameters energy and density. In this chapter we want to develop a concept of genuinely isothermal BWs without the aforementioned mixed adiabatic and isothermal assumption.

We already deduced genuinely isothermal equations from the system \[eq:odev,eq:odes\]. The only difference between the IPs treated earlier and the BWs is found in the initial conditions.

### Spherical isothermal BW

In the following, we discuss solutions in spherical coordinates ($\alpha=2$). Although the values for the RHCs \[eq:RHCV,eq:RHCS\] are computed with the same routine as described earlier, there is a little difference due to the BC $\left(u_\infty=0\right)$, and hence the simple relations depend only on the shock position $$u_\mathrm{d}=\xi_0-\xi_0^{-1},~~S_\mathrm{d}=S_\mathrm{u}\xi_0^{-2}.\label{eq:RHCBW}$$ In \[fig:pp\_blast\_k0\] we plot the phase diagram 