Shock Waves versus Sound Waves

1 Shock Waves versus Sound WavesSIDEBAR 1 Explosions, projectiles whizzing by atsupersonic speeds, high-speed colli-sions of solids what do these phe-nomena have in common?They all create very large changes in localpressure over very short times, and theseviolent pressure changes self-steepen intoshock fronts, or Shock Waves . Unlike a soundwave, which is a small-amplitude com-pression wave that propagates at the localsound speed and leaves the state of the me-dium unchanged, a Shock front is a nonlinearwave that abruptly changes the state of thesupersonically approaching gas. The gas isgenerally at a higher temperature and pres-sure after it has passed through the Shock ( , after the Shock has passedthrough the gas).

2 Moreover, the Shock -heated gas moves subsonically with respectto the Shock . As we describe below, the nar-row region defined as the Shock front is aregion where thermodynamic processes cartoon from Courant and Friedrichs classic book on supersonic flow illustratesthe formation of a steep front in a discon-tinuous medium, namely, a train of skiers, barreling single file down thenarrow run. pile up in a heap as first one skiergets wrapped around a tree, and then, beforehe can warn the skier behind him to slowdown, the next skier crashes into him, and soon. The pileup of human wreckage creates asteep front moving up the slope away fromthe tree analogous to a receding Shock in a continuous medium.

3 Formation of thefront depends critically on the fact that theflow of skiers is supersonic in the sensethat it is faster than the speed with which themedium, in this case the skiers, can respondto new boundary conditions. The high pres-sure of the skiers behind this front isanalogous to the change of state experiencedby Shock -processed a hundred years ago Stokes,Earnshaw, Riemann, Rankine, Hugoniot,and Lord Rayleigh deduced the conditionsthat prevail at Shock fronts in gases. Theframework they used to analyze this com-pressible flow are the equations of ideal gas42An example of a receding Shock wave. From Supersonic Flow and Shock Waves byR. Courant and K. O. Friedrichs (New York:Interscience Publishers, Inc.)

4 , 1948),dynamics. These nonlinear equations de-within this context, infinitesimal pressurescribe conservation of mass, momentum,changes generate linear compression Waves ,and energy in an ideal fluid, a fluid in whichbetter known as Sound Waves , that travelall changes in kinetic and internal energy aredue to pressure forces. Heat conduction andBut what happens when the amplitude of aviscous stress are ignored and entropy iscompression wave is finite?constant, so all thermodynamic changes areFigure A illustrates what happens. At timeadiabatic and reversible. It was known that,tl we have a finite-amplitude Sound waveSpring/Summer 1985 LOS ALAMOS SCIENCES upersonic JetsP. Variations in pressure imply variations inThus each point on the waveform propagateswith its local Sound speed, which is greater atthe peaks than in the troughs.

5 With time thewaveform steepens (as shown at t2) andeventually breaks (as shown at t3) to producemultiple values for the state variables of thegas, P, p, and T. Of course this prediction iswrong. Instead nature inserts a Shock front(dashed line) just before the wave breaks,and the flow variables remain does this happen? On a microscopiclevel large gradients in temperature and ve-locity at the front of the steepening waveX ctFig. A. Self-steepening of a finite-amplitude Sound wave. In the regionwhere the state variables of the wave(here, pressure) would become multi-valued, irreversible processes dominateto create a steep, single-valued shockfront (vertical dashed line).LOS ALAMOS SCIENCE Spring/Summer 1985 Inverse DensityFig. B, Effects of the passage of a Sound wave and of a Shock wave.

6 As a Sound wavepasses through a gas, the pressure and density of the gas oscillates back and forthalong an adiabat (a line of constant entropy], which is a reversible path. Incontrast, the passage of a Shock front causes the state of the gas to jump along anirreversible path from point 1 to point 2, that is, to a higher pressure, density, andentropy. The curve connecting these two points is called a Hugoniot, for it wasHugoniot (and simultaneously Rankine) who derived, from the conservation laws,the jump conditions for the state variables across a Shock front, After passage of theshock, the gas relaxes back to point 3 along an adiabat, returning to its originalpressure but to a higher temperature and entropy and a lower density. The shockhas caused an irreversible change in the the irreversible processes of heat con-duction and viscous stress to dominate in aregion with a width equal to a few collisionmean free paths and to counteract the self-steepening process so that a single-valuedshock front forms.)

7 The net effect on a macro-scopic level is that mass, momentum, andenergy are conserved across the Shock front,but entropy is not; it increases as relativekinetic energy is dissipated into heat throughatomic or molecular 1864 Riemann was the first to analyzewave-steepening within the context of idealgas dynamics. He mistakenly assumed thatentropy was conserved (in other words, thatall processes were adiabatic) across shockfronts as it is for finite-amplitude soundwaves. Later, Rankine, Rayleigh, andHugoniot showed that an adiabatic shockfront would violate conservation of energy,and therefore Shock fronts must be non-adiabatic and irreversible. Figure B showsthe irreversible changes caused by the pas-sage of a Shock front in contrast to the re-versible changes produced by a Sound model these irreversible effects, the formof the Euler equations of gas dynamics wehave adopted must be modified as describedin Fig.

8 2 of the main dissipative nature of a Shock frontimplies that it can maintain itself only inthe presence of a driving force. A simpleexample of a driven Shock front is given in43