§31 Summary

In the previous analysis, the problem of a collision of a plane parallel shock wave with a high density cloud bounded by two plane parallel tangential discontinuities has been discussed. Radiation losses, magnetic fields and self gravity of the cloud were neglected. General analytic solutions were found for the simple case in which the ratio of the environment's density to that of the cloud's density is a quantity of the first order.

When the shock collides with the boundary of the cloud, a discontinuity in the initial conditions is produced. This splits the incoming shock into two shock waves: one which penetrates the cloud and one which is reflected back to the external medium. When the transmitted shock into the cloud reaches the opposite boundary, another discontinuity in the initial conditions is produced, causing the transmission of a shock wave to the external medium and the reflection of a rarefaction wave from the point of collision.

Figure V.5: Variation of the pressure $p$ and density $\rho$ (with respect to the initial pressure $p_1$ and density $\rho_1$ of the environment) as a function of position $x$ (normalised to the initial width of the cloud $\Delta$) and dimensionless time $t$ (in units of the time $\Delta/c_1$ -where $c_1$ is the speed of sound in the external medium). Dashed lines represent shock waves (S), dot-dashed lines are tangential discontinuities (T), which are boundaries of the cloud, and short-long dashed lines represent weak discontinuities (W), which bound a rarefaction wave. The system of reference was chosen so that gas far away to the right of the diagram is at rest. The diagram shows the case for which $\rho_c/\rho_1\!=\!10^4$, and the polytropic indices correspond to a monoatomic gas.

Figure V.6: Variation of the width of the cloud in units of its original size $\Delta$ as a function of the dimensionless quantity $c_1t/\Delta$. Where $c_1$ represents the sound speed of the gas for the external environment and $t$ the time. The curve was produced under the assumption that $\rho_c/\rho_1\!=\!10^4$. The gas was considered to be monoatomic.