# Shock Rarefaction Interaction

I am interested to see/know if there exist analytical solutions for shock/rarefaction interaction. A rarefaction wave can catch up to a shock wave from behind. The shock will decay and the motion will no longer be uniform. I know analytical solutions exist assuming weak shocks and ignoring the entropy dependence. Is there a 1D solution that describes the interaction of a shock wave with rarefaction wave for arbitrary shock strength? \begin{align} \rho_{t}+u\rho_{x}+\rho u_{x}=0\\ \rho(u_{t}+uu_{x})+p_{x}=0\\ S_{t}+uS_{x}=0 \end{align}

where $\rho$ is density,$u$ is velocity, $p$ is pressure and $S$ is entropy.

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## migrated from math.stackexchange.comMay 2 '13 at 23:31

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What would the PDE/IVP be exactly? – Daniel Robert-Nicoud May 2 '13 at 21:39
I think this is better suited to physics. – Ross Millikan May 2 '13 at 21:39
Rarefaction wave doesn't catch up, since rarefaction wave is subsonic, and shock is obviously supersonic. Initial discontinuity will decay in time, but rarefaction will go backwards (if initially the state with higher values is on the left), whereas shock will go right. – Kaster May 2 '13 at 21:48
@Daniel, I have edited my question and added the equations. @Kaster; it is actually possible for a rarefaction wave to catch up with shock wave. It seems counterintuitive but if you consider the riemann problem with left side closed, the reflected expansion wave can actually attain speeds larger than the shock wave and catch up with it. – abiyo May 2 '13 at 21:56
Hi @abiyo Consider including a reference to the analytical weak shock solution. – Qmechanic May 2 '13 at 23:39