Shock front, speed

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)... 

Here, we will review basic properties of low-order wave equations that admit shocks, demonstrate that correct entropy conditions follow as direct consequences of high-order derivative terms, and show how artificial viscosity and upstream differencing can lead to errors in modeling important physical quantities and also in describing shock front speed. 

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-... 

The shock-change equation is the relationship between derivatives of quantities in terms of x and t (or X and t) and derivatives of variables following the shock front, which moves with speed U into undisturbed material at rest. The planar shock front is assumed to be propagating in the x (Eulerian spatial coordinate) or X (Lagrangian spatial coordinate) direction, p dx = dX. 

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 =... 

Conservation relations are used to derive mechanical stress-volume states from observed wave profiles. Once these states have been characterized through experiment or theory they may, in turn, predict wave profiles for the material in question. For the case of a well-defined shock front propagating at constant speed L/ to a constant pressure P and particle velocity level u, into a medium at rest at atmospheric pressure, with initial density, p, the conservation of momentum, mass, and energy leads to the following relations ... 

The elastic-shock region is characterized by a single, narrow shock front that carries the material from an initial state to a stress less than the elastic limit. After a quiescent period controlled by the loading and material properties, the unloading wave smoothly reduces the stress to atmospheric pressure over a time controlled by the speeds of release waves at the finite strains of the loading. Even though experiments in shock-compression science are typically... 

Detonation Propagation of a flame-driven shock wave at a velocity at or above the speed of sonnd in the nnreacted medinm as measnred at the flame front. The wave is snstained by chemical energy released by shock compression and ignition of the nnreacted medinm. The flame front is conpled in time and space with the shock front, and there is no pressnre increase significantly ahead of the shock-flame front. Propagation velocities in the range 1000-3500 m/s may be observed depending on the gas mixtnre, initial temperatnre and pressnre, and type of detonation. 

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. 

Figure 6-13 shows the physical differences between a detonation and a deflagration for a combustion reaction that occurs in the gas phase in the open. For a detonation the reaction front moves at a speed greater than the speed of sound. A shock front is found a short distance in front of the reaction front. The reaction front provides the energy for the shock front and continues to drive it at sonic or greater speeds. 

For a detonation, the reaction front moves faster than the speed of sound, pushing the pressure wave or shock front immediately ahead of it. For a deflagration, the reaction front moves at a speed less than the speed of sound, resulting in a pressure wave that moves at the speed of sound, moving away from the reaction front. A noticeable difference is found in the resulting pressure-time or pressure-distance plots. 

In the free field, the blast wave from an explosion travels at or above the acoustic speed for the propagating medium. TM 5-1300 provides plots of shock front velocity vs. scaled distance for high energy TNT explosives. There are no similar plots available for pressure wave propagation. However, for design purposes it can be conservatively assumed that a pressure wave travels at the same velocity as a shock wave. In the low pressure range, and for normal atmospheric conditions, the... 

Mathematical formulation of Prandtl-Meyer flow is given in Ref 66, p.p 162—64, equations 5.3 1 to 5.3.25 inclusive. In Fig 34 is shown the Prandtl-Meyer flow within a steady- deton zone characteristics are solid lines and stream lines are dashed line AB is the shock front, r = ratio of radius of axial stream tube to its initial radius and c = sound speed... 

Various means have been employed to measure shock and free surface velocities. We quote from Ref 7 The earliest work employed a pin technique. Pointed metal pins were spaced at graded distances from the free surface. When the surface was impelled forward by the impacting shock front, it made contact with each of the pins in turn. The pins were wired to separate pulseforming circuits, and the pulses produced on contact were displayed on a high-speed oscilloscope sweep. In this way the free surface velocity... 

The elapsed time for particles to pass through the shock front may be approximated by dv/U. Since U is of the same order of magnitude as the speed of sound in the gas, the ratio of the flying time to the Stokes relaxation time of a particle can be expressed by... 

Both 2m and s may be measured quite accurately by a variety of techniques such as precisely spaced pins that close electrical circuits and high-speed cameras. Then, from Eqs. (16) to (19) and the initial conditions, one can And P, E, and V for the compressed material behind the shock front and the equation of state E(P, V) of the material near the Hugoniot curve. Various other reasonable assumptions ultimately permit fairly accurate determinations of E(P, V) for pressures and densities further removed from the Hugoniot curve. For each value of P and V, a separate experiment producing particular values of x and m is needed. 

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