Sound Speed Behind Shock Fronts

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-... 

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)... 

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 =... 

Figure 1. Schematic of a shock wave showing the conditions on sound speed, particle velocity, and shock speed required for mechanical stability in front of and behind the shock front.

During a detonation, a shock wave is produced outside this shock wave the pressure is ambient, but inside the shock wave the pressure is above ambient. This shock wave continues to travel outward until, as a result of the expansion of the gas behind the shock front, the pressure inside the wave falls to ambient pressure and the temperature is below the initial temperature of the gas. If the initial pressure is sufficiently high, the shock wave will travel at, but not exceed, the speed of sound. This is the strict definition of an explosion. If the shock wave travels at a slower speed, the event is referred to as a deflagration. It is the overpressurization or pressure difference resulting from the passage of a shock wave that causes most of the damage in an explosion. 

Behind the shock front various relaxation processes occur in the potential and the kinetic energies. The potential energy relaxation is associated both with thermal relaxation (see below) and with structural rearrangement within the system. Structural relaxation generally occurs at a speed considerably lower than that of the shock front (e.g., plastic flow), but it may be as fast as the shock front (e.g., martensitic transformation). It is accompanied by stress relaxation which occurs at the appropriate speed of first sound. 

The shock-compressed mixture pressure level directly behind the wave front for various initial pressures Pq is shown in Fig. 6.3a for stoichiometric (0=1) H2 + O2 mixtures [9]. On the basis of the same work, the initial pressure effect on the shock-compressed mixture temperature T1 (Fig. 6.3b) and the detonation wave Mach number M = D/c (Fig. 6.3c) can be neglected. Here D is the detonation velocity, and c is the sound speed. 

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