Rarefaction waves

Expansion waves are the mechanism by which a material returns to ambient pressure. In the same spirit as Fig. 2.2, a rarefaction is depicted for intuitive appeal in Fig. 2.7. In this case, the bull has a finite mass, and is free to be accelerated by the collision, leading to a free surface. Any finite body containing material at high pressure also has free surfaces, or zero-stress boundaries, which through wave motion must eventually come into equilibrium with the interior. Expansion waves are also known as rarefaction waves, unloading waves, decompression waves, relief waves, and release waves. Material flow is in the same direction as the pressure gradient, which is opposite to the direction of wave propagation. 

Figure 2.7. When a freight train impacts a bull with finite mass, the resulting shock wave is quickly followed by a rarefaction wave, which overtakes and decays the original shock wave.

This constant is independent of position on the rarefaction wave. 

Riemann Invarient A constant defined by (2.61), which is independent of position on a rarefaction wave that propagates into a uniform state. 

Rarefaction wave A wave that reduces the normal stress (or pressure) inside a material as it propagates the mechanism by which a material returns to ambient pressure after being shocked (the state behind the wave is at lower stress than the state in front of it). Also known as unloading, expansion, release, relief, or decompression waves. 

Spall strength The dynamic tensile strength of a material associated with tension that results from the wave interaction of rarefaction waves. When the spall strength is exceeded, the material separates, or spalls. ... 

Figure 4.7. Internal reflection of a shock wave from a free surface, (a) Reflection of a shock wave from a free surface causes a reflected rarefaction wave. As indicated in (b), this increases the velocity of the shocked material from u, to Uf. The path upon shocking is Rayleigh line 0-1, whereas unloading occurs along release isentrope curve I -O, (c) Release isentrope path in P- V plane is indicated.

McQueen et al. (1982) demonstrated that by placing a series of high-impedance transparent fluids (called optical analyzers) over the sample at a series of thicknesses less than d in the target that the overtaking rarefaction (sound) velocity can be accurately obtained. Arrival of rarefaction waves rapidly reduce the shock pressure. These wave arrivals could be very readily detected by the change in light radiance caused by the onset of a decrease in shock amplitude when the rarefaction wave caught up to the shock front. The... 

McQueen, R.G., Hopson, J.W., and Fritz, J.N. (1982), Optical Technique for Determining Rarefaction Wave Velocities at Very High Pressures, Rev. Sci. Instrum. 53, 245-250. 

The peak overpressure developed immediately after a burst is an important parameter for evaluating pressure vessel explosions. At that instant, waves are generated at the edge of the sphere. The wave system consists of a shock, a contact surface, and rarefaction waves. As this wave system is established, pressure at the contact surface drops from the pressure within the sphere to a pressure within the shock wave. 

Initial shock-wave overpressure can be determined from a one-dimensional technique. It consists of using conservation equations for discontinuities through the shock and isentropic flow equations through the rarefaction waves, then matching pressure and flow velocity at the contact surface. This procedure is outlined in Liepmatm and Roshko (1967) for the case of a bursting membrane contained in a shock tube. From this analysis, the initial overpressure at the shock front can be calculated with Eq. (6.3.22). This pressure is not only coupled to the pressure in the sphere, but is also related to the speed of sound and the ratio of specific heats. 

However, for a rarefaction wave, the vapor becomes subcooled and the liquid becomes superheated. When the wave front passes, the liquid phase is assumed to adjust from the metastable state at an equilibrium rate. If isentropic processes are assumed, the mass transfer rate can be shown to be... 

The need for different expressions for compression and rarefaction waves is consistent with the experimental observation of Barclay et al. (1969) that the compression wave travels faster than the rarefaction wave. 

Rarefaction waves are generated circumferentially at the tube as the detonation leaves then they propagate toward the tube axis, cool the shock-heated gases, and, consequently, increase the reaction induction time. This induced delay decouples the reaction zone from the shock and a deflagration persists. The tube diameter must be large enough so that a core near the tube axis is not quenched and this core can support the development of a spherical detonation wave. 

The incident shock wave moves down the tube, heating and accelerating the test gas. In RST mode, the shock hits an end plate and reflects back to the test gas, further heating the gas and initiating stagnation conditions. Subsequently, a rarefaction wave travels down the tube and quenches all further reactions. 

The cutoff observed by Rozing Khariton (Ref 1) in detonation of explosives in tubes of small diameter and also discussed in the book of ZeTdovich Kompaneets (Ref 8, pp 213-16) is an abrupt extinction of the detonation by rarefaction waves from the sides of the charge reaching the axis before completion of the chemical reaction. It... 

Fig 2 Pressure in rarefaction wave behind Chapman-Jouguet point according to Taylor and to Langweilcr... 

Taylor (Ref 5) obtd a transient flow behind a C-J discontinuity using Riemann equations for polytropic gases. A plot of u/u2 vs x/Ut shown in Fig 12 of Ref 6 (See here Fig 1) is for u2 =U/3, c2 = 2U/3 and y=1.3, where u is material velocity in x direction, u2 is material velocity immediately behind the discontinuity at Ut (U = velocity of C-J wave t = time coordinate) C2 = sound velocity and y = Cj/cv (cp=specific heat at constant pressure and cv = sp heat at constant volume), Taylor calculated pressure in the rarefaction wave behind C-J point and plotted it in F ig given as Fig 12 of Ref 6 (Our Fig 2)... 

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