Solids velocity propagation

Ultrasonic methods have been considered for non-destructive evaluation of compositional changes and mixture uniformity in filled polymers. In principle, it is feasible to determine the size, shape and distribution of filler particles, to detect agglomeration and to assess the extent of filler-matrix interaction, through appropriate application of ultrasonic procedures [48]. The method involves determination of the elastic behaviour of solids by measurement of ultrasonic wave velocity. Propagation of a plane wave in a linear elastic material can be related to its elastic modulus (E) and density (p) according to ... 

The basics of the method are simple. Reflections occur at all layers in the subsurface where an appreciable change in acoustic impedance is seen by the propagating wave. This acoustic impedance is the product of the sonic velocity and density of the formation. There are actually different wave types that propagate in solid rock, but the first arrival (i.e. fastest ray path) is normally the compressional or P wave. The two attributes that are measured are... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

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. 

Material behavior have many classifications. Examples are (1) creep, and relaxation behavior with a primary load environment of high or moderate temperatures (2) fatigue, viscoelastic, and elastic range vibration or impact (3) fluidlike flow, as a solid to a gas, which is a very high velocity or hypervelocity impact and (4) crack propagation and environmental embrittlement, as well as ductile and brittle fractures. 

More recently, Rosen (R3), Spalding (S5), and Johnson and Nachbar (J4) have considered a simplified approach using the analysis of laminar-flame propagation velocities. According to these investigators, the principal exothermic reactions occur in the gas phase. Some of the heat liberated by these reactions is then transferred back to the solid surface to sustain the endothermic surface-gasification processes. Thus, the temperature profile within the reactive zone is quite similar to that of Rice and Crawford. However, gasification of the solid surface is assumed to be endothermic, while exothermic reactions were considered in the studies discussed previously. 

The Hydrodynamic Theory of fluidized bed stability was proposed by Foscolo and Gibilaro who adapted the stability principle of Wallis. They postulated that a fluidized bed is composed of two interpenetrating fluids. One fluid is the gas phase, and the solids phase is also considered as a continuous fluid phase. In this theory, voidage disturbances in the bed propagate as dynamic and kinetic waves. The stability of the fluidized bed depends upon the relative velocities of these two waves. The velocities of the kinetic wave (ue) and the dynamic wave (nj are ... 

Figure 8.8 Advective propagation of a chemical wave of tracer i moving with a velocity v in a wetted porous solid at times t= 1,2, 3 for different values of d2Cwl7(dCUq )2. Breaking takes place at t = 3 in cases b and c.

Refractive Index and Dispersion. The velocity of light varies depending upon the density of the medium in which it is propagating. In a vacuum, the speed of light is a constant, c, which has a value of 3.08 x 10 m/s. In any other medium, such as a gas, liquid, or solid, the velocity is given by the variable, v. The ratio between these two velocities determines the index of refraction, n, sometimes called the refractive index ... 

A simple geometrical analysis of this can be given in terms of third-order aberration theory for a single surface (Jenkins and White 1976 Hecht 2002). The situation is illustrated in Fig. 4.1. Rays propagating towards a virtual focus at a distance s below the surface of a solid are refracted so that a ray that passes through the surface at a distance h from the axis crosses the axis at a depth sa, with the paraxial focus at Sb- The refractive index n is the ratio of the velocity in the liquid to the velocity in the solid in acoustics this usually has a value... 

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