Shock Stability

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. 

The shock wave is subject to other dissipative effects, however, such as viscosity and heat transport. It is these dissipative mechanisms that are responsible for preventing the shock from becoming a true, infinitesimally thin discontinuity. In reality, the velocity gradient can only increase until 

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 

Substituting (2.33) and (2.28) into the stability condition (2.32) yields the inequality 

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A number of papers are devoted to the theory of shock stability. The majority of them consider the problem of shock wave propagation in space without boundaries. In present paper we consider three-dimensional flow disturbances downstream of the shock propagating in a cylindrical tube. The possible types of transversal waves at the shock front are found. We study the properties of equations of state which guarantee the fulfilment of shock stability conditions and obtain the classification of equations of state. Stability of shock waves in two-phase flows is discussed. 

Inequalities (22), (25) classify the equations of state in connection with the shock stability conditions. Notice that (22), (25) may be used not only in the case of class (19) equations of state. 

Sufficient shock stability can be achieved by introducing polymer walls, running parallel to the direction of rubbing, within the FLC panel [30, 35, 36]. Figure 6.1.18 illustrates an example of such a device structure. Gass etal. have... 

Another reason for this is the complexity of molecular orientation in FLC devices. FLC devices have layered structures, three surfaces (that is, two substrate surfaces and a chevron interface), tilt angles, spontaneous polarization, etc. These features lead to more complex phenomena such as layer leaning, twisted orientations, low shock stability, layer rotation, etc. 

The above relations enable shock stability criteria to be expressed very simply. For upwards travelling shocks we apply the positive alternatives on the right of eqns (14.20) and (14.21). The stability criteria, eqn (14.18), then becomes ... 

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