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.

  • $\begingroup$ 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. $\endgroup$
    – Kaster
    May 2, 2013 at 21:48
  • $\begingroup$ @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. $\endgroup$
    – abiyo
    May 2, 2013 at 21:56
  • $\begingroup$ Hi @abiyo Consider including a reference to the analytical weak shock solution. $\endgroup$
    – Qmechanic
    May 2, 2013 at 23:39
  • $\begingroup$ @Qmechanic, thanks for the reply.I have created an account.It will be hard to put all the equations here but the main idea behind weak shock approximation is that for a weak shock wave the entropy change across the shock can be neglected and simple wave approximations can be employed. Refer to archive.org/details/interactionofsho00cour for details. I haven't seen any satisfactory treatment for strong shocks. $\endgroup$
    – felasfa
    May 3, 2013 at 23:30
  • $\begingroup$ It is possible for a forward and reverse shock to interact. This happens with coronal mass ejections, or CMEs, when one CME launches from the sun and is overtaken by a second, faster CME. Granted, these types of CME-CME interactions are usually not weak shocks, but they can occur. The first CME can have a forward shock on its leading edge and a reverse shock on its trailing edge. Then the 2nd CME can run into the first, causing a forward-reverse shock interaction. $\endgroup$ Oct 18, 2014 at 14:19

1 Answer 1


I will give this a crack as it appears unanswered and I have dealt with this problem in the past. The OP is correct, it is in fact possible for a rarefaction wave to overtake a traveling shock wave from behind. The classical problem usually arises from the one-dimensional motion of a piston inside a channel. The instantaneous motion of the piston sends a shock wave through an initially stationary gas. At a later time, the piston abruptly comes to a stop. A simple rarefaction wave is sent into the post-shock gas and eventually catches up to the traveling shock. Below is a schematic of the problem in the space-time ($x-t$) diagram.

Decaying Shock Wave

Once the rarefaction overtakes the shock, a non-uniform region behind the shock begins to develop with entropy variations, which are carried along the particle paths in the region behind the shock wave. The entropy variations are what make the analytical approach very difficult. The governing equations for this system are, $$ \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho \frac{\partial u}{\partial x} = 0 $$ $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \frac{1}{\rho} \frac{\partial p}{\partial x} = 0 $$ $$ \frac{\partial s}{\partial t} + u\frac{\partial s}{\partial x} = 0 $$

The first attempt at a solution was carried out by Friedrichs 1948. His solution was for an infinitely weak shock where the entropy variations across the shock could be neglected. This assumed the flow behind the decaying shock was isentropic and the Riemann invariant is constant through the shock.

Other researchers provided ways to approximately account for the variation in entropy behind the decaying shock. Seek the works of Pillow 1949 and Lighthill 1950.

A solution for a moderately strong shock was presented by Ardavan-Rhad 1970. However, this solution only approximately satisfies the Rankine-Hugoniot shock jump conditions, and displays increasing error with high shock Mach numbers.

Finally, the solution you are seeking was provided by Sharma, Ram, and Sachdev 1987. The solution is for an arbitrary strength shock and describes the decay of the shock propagation from the interaction of an overtaking rarefaction wave. Moreover, the solution is uniformly valid for the entire flowfield behind the decaying shock wave. It is noted, the solution satisfies the governing equations exactly, but only approximately satisfies two of the Ranking-Hugoniot shock jump conditions. However, the error incurred is shown to be small for strong shocks. Also, in the limit of a weak shock, the solution converges to that proposed by Friedrichs.

I will be honest, the solution is mathematically heavy for myself. I have implemented the solution in Matlab for various calculations and have experienced some difficulty. Occasionally, I get solutions that are non-physical as their proposed solution method requires implementing a Newton-Raphson method very close to a singularity for a certain parameter. However, the more experienced may not have this issue. Anyways, I hope this answer/solution helps.


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