This blog provides commentary on interesting geological events occurring around the world in the context of my own work. This work is, broadly, geological fluid dynamics. The events that I highlight here are those that resonate with my professional life and ideas, and my goal is to interpret them in the context of ideas I've developed in my research. The blog does not represent any particular research agenda. It is written on a personal basis and does not seek to represent the University of Illinois, where I am a professor of geology and physics. Enjoy Geology in Motion! I would be glad to be alerted to geologic events of interest to post here! I hope that this blog can provide current event materials that will make geology come alive.

Banner image is by Ludie Cochrane..

Susan Kieffer can be contacted at s1kieffer at gmail.com

Tuesday, November 9, 2021

Shock and rarefaction waves as a dynamic pair

In the previous post, I discussed the many meanings of the word shock.  In this post, I focus on the specific properties of shock waves as compression waves in air, and the expansion (rarefaction) waves that follow the shocks in order to bring the pressure back to ambient. In particular, I emphasize that there is one parameter, the sound speed, that determines the behavior of waves in a gaseous system.


Shock waves in air can be generated in a number of ways, e.g., by a supersonic aircraft, a speeding bullet, or an erupting volcano.  The general physics of such waves can be envisioned by considering a pipe that has a barrier (referred to as a diaphragm) separating two regions with different initial pressures. This is a so-called "shock tube" used in many laboratories to measure material properties.

 The two color images of shock tubes used here are from here Shengkai Wang. A typical shock tube is closed at both ends and contains a piece of material, the diaphragm, that separates a high-pressure gas (the driver gas) from a lower gas (the driven gas, typically the gas whose properties are being investigated). An experiment begins when the diaphragm bursts and waves propagate into the driver and driven gas. The closed ends generate reflected waves that complicate the discussion, but here we only consider a short time after the diaphragm bursts before the waves are reflected off the closed ends. Diaphragms are selected on the basis of material strength (e.g., aluminum diaphragms are common) and material thickness.  For example, by using aluminum diaphragms of different thicknesses, the pressure at which the diaphragm ruptures can be varied. Gas is pumped into the driver section (left, blue in this figure) until the diaphragm ruptures (sometimes expedited by knife blades placed in the path of the bulge that develops in the diaphragm when the driver gas is pressurized). Two waves are initially generated when the diaphragm bursts:

A shock wave (indicated by the inclined arrow) propagates toward the right into the low pressure "driven gas" (green). This wave increases the pressure in the driven gas (orange region).  The (now shredded diaphragm) is pushed downstream as shown by the boundary between the orange "and the green driven gas.

At the same time, an expansion wave (technically called a rarefaction wave) propagates back into the driver gas to reduce its pressure (shaded light blue region). Reflected waves are generated when the shock wave hits the wall of the shock tube on the right, and when the expansion wave hits the wall of the shock tube on the left. The discussion above only applies until the time one or the other of the reflected waves is affect the gas. The reason that the shock waves is shown as "very thin" and the expansion wave as "thicker" as indicated by the gradient blue shading is discussed below.

The parameter that determines the properties of waves in a medium (be it a gas, liquid or solid) is the speed of sound in that medium. The speed of sound depends on the temperature of the medium (strictly, on the square root of the temperature for gases). The sound speed is the speed at which a pressure wave with a very small amplitude is passed from particle to particle through the medium in which it is propagating. The speed of sound in air at 20 C (68 F) is 343 meters/second (about 1,125 ft/s, or 767 mph). Materials moving faster than the sound speed are referred to as supersonic.


To understand why shock waves exist and how they form, consider two compression waves traveling close together through air.  The first wave propagates at a speed that depends on the temperature of the air. This wave compresses the air as it passes through and therefore it also heats that air.  A second wave following close behind is thus traveling into air that has been warmed by the first wave and any second (or later) wave will overtake the first wave.  The two waves thus merge into one wave,  a shock wave. Such shocks are only around 200 nm (10E-5 inch) thick, about the mean free path of the air molecules.   


The situation is reversed in an expansion wave: the head of an expansion wave will travel at a speed appropriate to the temperature of the medium into which it is propagating.  But expansion cools a gas and hence any trailing wave will travel at a slower speed. Trailing expansion waves therefore cannot overtake the head of the expansion wave.  In contrast to shock waves of infinitesimal thickness, expansion waves have a finite thickness. 

Fluid dynamicists use so-called "x-t diagrams" to illustrate these wave properties (x=distance, t=time), this particular diagram from Cheng et al, 2010.  These diagrams aren't as difficult as they appear and, in fact, appear in many places in analyses of traffic flow***. The diaphragm initially separates two regions with different properties (pressure, density,..) and the plane of the diaphragm is referred to as the "contact surface" between these two different regions. When the diaphragm bursts, the contact surface travels toward the right, i.e., the interface between the two regions moves to the right as illustrated by the dashed line. A wave that "tells the low pressure fluid that the diaphragm has burst" propagates as a shock wave ahead of the contact surface.  This wave raises the pressure of undisturbed fluid in "region 1" to a new pressure, "2" in the diagram.  It has negligible thickness as discussed above.

Meanwhile, the expansion wave travels from the contact surface back into the high-pressure region, "telling it that the diaphragm has burst." This wave reduces the pressure in the high pressure region 4 to a new pressure 3. Because the material can't separate at the contact surface, the pressures in regions 3 and 2 must be equal. The rarefaction wave has a finite thickness as indicated by the "rarefaction fan" (shown, but labeled as "rarefaction wave" in this diagram). 


***Imagine cars traveling on a one-lane road during rush hour. Suddenly a deer leaps from the side and causes the leading car to crash.  This disturbance to the traffic flow propagates as a shock wave back into the trailing cars.  When the deer recovers and leaps off the road again, traffic starts to flow, but it takes longer for the cars to clear out than it took for the jam to form.  There is a semi-quantitative analysis of this "fluid dynamic behavior" of traffic flow in many text books.