Oscillating piston shock waves I am interesting to know whether there are analytical solutions for a piston gonging like a sine wave and generated shock wave and rarefaction. How the energy change during this process and how can we use mathematic equation to describe the process. More like the N-wave in the book "supersonic flow and shock waves". Also is there any numerical approach?
 A: Definitions
Let us first start with some definitions of parameters, in no particular order.


*

*Variables


*

*$T$ = the scalar temperature

*$U$ = the scalar bulk flow speed (in shock rest frame)

*$V$ = the scalar specific volume

*$\rho$ = the scalar mass density (or number density)

*$\epsilon$ = the specific internal energy

*$P$ = the scalar pressure

*$C$ = the scalar sound speed = $\left(\tfrac{\partial P}{\partial \rho}\right)_{S} = \tfrac{\gamma \ P}{\rho}$

*$S$ = the scalar thermodynamic entropy

*Subscripts


*

*$up$ = for upstream/ahead of shock

*$dn$ = for downstream/behind shock


*$C_{v}$ = specific heat at constant volume

*$\gamma$ = ratio of specific heats
Background
We know for a regular, compressive shock wave that the following are true:


*

*$P_{dn} > P_{up}$

*$T_{dn} > T_{up}$

*$\rho_{dn} > \rho_{up}$

*$S_{dn} > S_{up}$

*$U_{dn} < U_{up}$

*$U_{dn} < C_{s,dn}$

*$U_{up} > C_{s,up}$


The change in entropy across a compressive, hydrodynamic shock is given by:
$$
\Delta S = C_{v} \ \ln \lvert \frac{P_{dn} \ \rho_{dn}^{\gamma}}{P_{up} \ \rho_{up}^{\gamma}} \rvert \tag{1}
$$
The relationship between the bulk flow speed and speed of sound is given by:
$$
\begin{align}
  \left( \frac{U_{up}}{C_{s,up}} \right)^{2} & = \frac{\left( \gamma - 1 \right) + \left( \gamma + 1 \right) \ \tfrac{P_{dn}}{P_{up}}}{2 \ \gamma} \tag{2a} \\
  \left( \frac{U_{dn}}{C_{s,dn}} \right)^{2} & = \frac{\left( \gamma - 1 \right) + \left( \gamma + 1 \right) \ \tfrac{P_{up}}{P_{dn}}}{2 \ \gamma} \tag{2b}
\end{align}
$$
For a rarefaction wave, we know that:


*

*$P_{dn} < P_{up}$

*$T_{dn} < T_{up}$

*$\rho_{dn} < \rho_{up}$


Note that there is nothing in the Rankine-Hugoniot conservation relations that say anything about a rarefaction shock wave being disallowed.  However, from Equations 1, 2a, and 2b we can see that:


*

*$S_{dn} < S_{up}$

*$U_{dn} > C_{s,dn}$

*$U_{up} < C_{s,up}$


The following is from pages 61-62 of Zel'dovich and Raizer, [2002]:

According to the second law of thermodynamics the entropy of a substance cannot be decreased by internal processes alone, without the transfer of heat to an external medium.  This shows that it is impossible for a rarefaction wave to propagate in the form of a discontinuity...
The impossibility of the existence of a rarefactiion shock wave can be explained as follows.  Such a wave would propagate through the undisturbed gas with the subsonic velocity $U_{up} < C_{s,up}$... any disturbances induced by the density and pressure jumps will begin to travel to the right with the speed of sound $C_{s,up}$, and will outrun the "shock wave."  After a certain time the rarefaction region will include the gas in front of the "discontinuity", and the discontinuity will simply disappear.  In other words, a rarefaction shock wave is mechanically unstable...
Mechanical stability can be present only when the wave is propagated through the undisturbed fluid with supersonic speed, otherwise disturbances induced by the shock wave would penetrate the initial gas at the speed of sound, overtake the shock wave, and thus "wash out" the sharp wave front.

Questions and Answers

I am interesting to know whether there are analytical solutions for a piston gonging like a sine wave and generated shock wave and rarefaction. How the energy change during this process and how can we use mathematic equation to describe the process. More like the N-wave in the book "supersonic flow and shock waves"

If each cycle of the piston causes a compressive shock wave, then the retreat of the piston will not create a rarefaction shock wave as I discussed above.  The resultant wave profile is actually one like a sawtooth wave, which does have a well defined mathematical form given by:
$$
x\left(t\right) = \frac{A}{2} - \frac{A}{\pi} \ \sum_{n=1}^{\infty} \ \left( -1 \right)^{n} \ \frac{ \sin{\left( n \omega \ t \right)} }{n}
$$
where $A$ is the amplitude and $\omega$ is the angular frequency.  There are more examples in the link above but the general point is that yes, it can be described mathematically.
Additional Relevant Answers


*

*Notes on shock formation:  https://physics.stackexchange.com/a/139436/59023

*Notes on solutions to Burgers's equation:  https://physics.stackexchange.com/a/257904/59023
References


*

*Zel'dovich, Ya.B., and Yu.P. Raizer (2002) Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Ed. by W.D. Hayes and R.F. Probstein, Mineola, NY, Dover Publications, inc., The Dover Edition; ISBN-13: 978-0486420028.

*Whitham, G. B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.

A: This is not a simple problem to treat analytically for arbitrary amplitude of piston speed (see for example, Saenger, R. A., & Hudson, G. E. (1960). Periodic shock waves in resonating gas columns. The Journal of the Acoustical Society of America, 32(8), 961-970.). However, in the weak shock wave limit, this can be done as shown in the textbook  Unsteady motion of continuous media by Stanyukovich, K. P. See also Landau & Lifshitz $\S$101, problem 4.
