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?

  • $\begingroup$ Shockwaves? Sounds like you are describing something like a tuning fork. $\endgroup$
    – user137289
    Jan 4, 2017 at 1:38
  • $\begingroup$ No, I am looking that the piston generated shock wave and then the piston pull back, one of the example is the N wave in the book "supersonic flow and shock waves". $\endgroup$
    – kevin Z
    Jan 4, 2017 at 2:05
  • $\begingroup$ Well, there is ultrasonic cleaning. Maybe that involves supersonic speeds of the driver? $\endgroup$
    – user137289
    Jan 4, 2017 at 2:48
  • $\begingroup$ Here is a paper that might help you get started (link.springer.com.ezproxy.uta.edu/article/…). The piston motion induces nonlinear wave motion in a semi-infinite gaseous medium. The nonisentropic solution is obtained using the method of characteristics. $\endgroup$
    – TRF
    Jan 6, 2017 at 7:06
  • $\begingroup$ Thank you so much, Also is there any numerical approach? $\endgroup$
    – kevin Z
    Jan 9, 2017 at 3:20

2 Answers 2



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


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


  1. 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.
  2. Whitham, G. B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.
  • $\begingroup$ Thank you for your answer, but from the book " Supersonic flow and shock waves" R. Courant and K.O. Friedrichs pp 166-167 said a decaying shock wave of stop the piston or retracted it, produced by a rarefaction wave overtaking it. $\endgroup$
    – kevin Z
    Jan 9, 2017 at 3:05
  • $\begingroup$ @kevinZ - I think you are saying the same thing that I did in the first part of my answer, namely that a rarefaction shock wave is unstable. $\endgroup$ Jan 9, 2017 at 14:27
  • $\begingroup$ @ honeste_vivere It looks like not the rarefaction shock, more like a shock front and followed by the rarefaction. they are two wave not the same one. $\endgroup$
    – kevin Z
    Jan 10, 2017 at 1:21
  • $\begingroup$ @kevinZ - No, that is precisely my point. Namely, that there will not be a rarefaction shock, just a rarefaction wave. This is the downward slope of the sawtooth behind the discontinuity. A regular shock with a unidirectional piston looks more like a step function (i.e., no rarefaction behind the shock, just compression). $\endgroup$ Jan 10, 2017 at 14:00
  • $\begingroup$ @ honeste_vivere is there any numerical approach about this problem? $\endgroup$
    – kevin Z
    Jan 12, 2017 at 0:52

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.


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