Sound speed

The radius and velocity of the bubble waU are given by R and U respectively. The values for H, the enthalpy at the bubble waU, and C, the local sound speed, maybe expressed as foUows, using the Tait equation of state for the Hquid. 

In most solids, the sound speed is an increasing function of pressure, and it is that property that causes a compression wave to steepen into a shock. The situation is similar to a shallow water wave, whose velocity increases with depth. As the wave approaches shore, a small wavelet on the trailing, deeper part of the wave moves faster, and eventually overtakes similar disturbances on the front part of the wave. Eventually, the water wave becomes gravitationally unstable and overturns. 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

Since a compressional disturbance moves at the speed a + u, the sum of the sound speed and the particle velocity at the point through which the... 

It should be noted that not all materials satisfy these stability criteria. For example, over a range of low pressures, the sound speed of fused silica decreases with pressure, so shock waves cannot be supported. As pointed... 

In materials that support shock waves, the sound speed increases with pressure. It is this same property that causes rarefactions to spread out as they progress. In Fig 2.6(b), an unloading wave is shown propagating into a stationary material with some initial pressure Pq. This time, we consider the evolution of two small decompressional disturbances. The first disturbance moves at the local sound speed of a, into its surroundings, which have begun... 

Write the Eulerian sound speed, a, in terms of the Lagrangian sound speed, c. 

Shock velocity The velocity of the shock wave as it passes through the material. In the limit of an infinitesimally small shock wave it is equal to the bulk sound speed of the material. 

Shock-wave data have seen most applications in the measurement of density at high pressure. Other properties of compressed condensed materials whose measurements are discussed in this chapter include sound speed and temperature. Review articles by Grady (1977), Yakushev (1978), Davison and Graham (1979), Murri et al. (1974), Al tshuler (1965), and Miller and Ahrens (1991) summarize experimental techniques for measuring dynamic yielding. 

Another important method of determining the Gruneisen ratio in the shock state is the measurement of sound speed behind the shock front. The techniques employing optical analyzers (McQueen et al., 1982) piezoresistive (Chap-... 

Figure 4.21. Pressure-volume paths used to relate the slope of Hugoniot (dP/dV) to isentropic sound speed G (4.57).

Upon unloading from a high-pressure state the sound speed... 

C° = bulk sound speed in the compressed (shocked state) driver. 

When the elastic shock-front speed U departs significantly from longitudinal elastic sound speed c, immediately behind the elastic shock front, the decaying elastic wave amplitude is governed by (Appendix)... 

The Lagrangian sound speed is obtained in the following heuristic way. We consider small departures from the shock-compressed state, where the bulk and shear moduli are K and G. The Eulerian sound speed c is then given by... 

The motion of disloeations under eonditions of shoek-wave eompression takes plaee at sueh high veloeities (approaehing the elastie sound speed) that many vaeaneies and interstitials are left behind. However, these point defeets ean anneal out at room temperature and are thus diflieult to study by shoek-reeovery teehniques. The presenee of point defeets has little effeet on the material eompressibility and other properties related to equation of state. While they also have little direet influenee on the relief of shear stresses, point defeets do influenee the mobility and multiplieation of disloeations. This, in turn, affeets most of what happens under shoek-wave loading eonditions. 

Hugoniot. At low-stress amplitudes pot /o C S c,o, where c,o is the adiabatic longitudinal elastic sound speed at p = pg. 

This is expressed in terms of the particle acceleration immediately behind the shock front. Equation (A. 15) can be expressed in terms of the Lagrangian stress gradient (dff/dX), and the Lagrangian longitudinal sound speed Q =... 

If we accept the assumption that the elastic wave can be treated to good aproximation as a mathematical discontinuity, then the stress decay at the elastic wave front is given by (A. 15) and (A. 16) in terms of the material-dependent and amplitude-dependent wave speeds c, (the isentropic longitudinal elastic sound speed), U (the finite-amplitude elastic shock velocity), and Cfi [(A.9)]. In general, all three wave velocities are different. We know, for example, that... 

The isentropic sound speed c, differs in principle from the Hugoniot sound speed Cfj because of the entropy increase on the Hugoniot... 

There are few analytic solutions to the governing equations for interesting problems. The conservation equations are typically solved approximately on digital computers. It is assumed that the sound speeds are real and the system... 

Other cell variables such as sound speed and heat capacities can be calculated using similar techniques. Some codes allow a variety of multimaterial element thermodynamic treatments. For example, CTH allows all materials in an element to have the same or different pressures or temperatures [44], Material interfaces in multimaterial elements do not coincide with element boundaries, as shown in Fig. 9.14 [45]-[49]. The interfaces must be constructed using pattern matching or some other technique. 

The sound speed c, m s , is the velocity of propagation of the pressure variations. This depends on the physical properties of the medium and increases with the density of the medium. In air, for example, it is. 344 m s, while in water, 1410 m s and in concrete, 3000 m s . The elapsed time between successive compressions is called the period time T. 

Pq = atmospheric pressure Cq = atmospheric sound speed E = amount of combustion energy Rq = charge radius... 

The sound speed Oq of the contained gas has to be calculated for the temperature at failure ... 

It is not possible to obtain exactly identical flow conditions for the configurations explored. The level of velocity fluctuation at the burner outlet also differs in the various cases. This level was adjusted to get an acceptable signal-to-noise ratio. In the results presented here, the specific heat ratio was taken as equal to y= 1.4, the sound speed Cq = 343 m/s corresponds to a room temperature T = 293 K. The air density is taken equal to = 1.205 kg/m. Laminar burning velocities are... 

Subsonic low-velocity flame—flame propagates at a speed much lower than sound speed in the combustion products... 

Quasi-detonation—flame propagates with the velocity between the sound speed in the combustion products and CJ value... 

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