Velocity, of elastic waves

From Eq. (3.1) we see that the Debye frequency iw varies proportionally to the velocity of elastic waves, divided by the cube root of the volume, and from Eq. (3.9) wc see that the velocity of either longitudinal or transverse waves varies inversely as the square root of the compressibility times the density, if we assume that Poisson s ratio is independent of the volume. As we shall see later, this assumption can hardly be... 

In addition, the connection between the characteristic temperature and the elastic constants and the heat of atomization can be established directly by the Debye method. We note, for example, that the Debye characteristic temperature is directly proportional to the average velocity of elastic waves in a crystal (0= kc) and it is proportional to the square root of the atomization energy per unit atomic weight, at least in the first approximation (0= kVU/A). 

Measurement and assessment of ground vibrations Table 16 Propagation velocity of elastic waves in the subsoil... 

Given are measured input parameters (for example density, neutron porosity, velocity of elastic waves). 

In dense, non-fractured igneous rocks, the velocity of elastic waves is controlled by mineral composition. This fact is illustrated by the correlation between the longitudinal wave velocity and the Si02 content of igneous rocks. Quartz is characterized by a relatively low velocity. Thus, acidic rocks have lower velocities than basic rocks (Fig. 6.6). 

The effect of a changing pore fluid on the velocities of elastic waves results in... 

Geomechanical rock properties are a specific group of petrophysical parameters, directly measured in rock mechanics laboratories or by specific field tests. But they are also more or less strongly correlated to other petrophysical parameters (for example velocities of elastic waves) and therefore an indirect derivation from geophysical measurements is the subject of research and application. With respect to this application of geophysical methods, we can distinguish between ... 

The effects of mineral composition, porosity, and saturation are also different for sedimentary rocks with respect to the petrophysical properties density, velocity of elastic waves, and thermal conductivity. For a discussion of expected correlations, it is important to note ... 

Elastic constants of minerals are the key to understanding geophysical properties of the Earth s interior. Bulk modulus and rigidity parameters, for example, influence the velocities of seismic waves through the Earth. Numerous experi-... 

First, let us find the number of overtone vibrations in the range dv. We follow Chap. XIV in detail and for that reason can omit a great deal of calculation. In Sec. 2 of that chapter, we found the number of overtones in the range dv, in a problem of elastic vibration, in which the velocity of longitudinal waves was Vi, that of transverse waves vt. From Eqs. (2.20) and (2.21) of that chapter, the number of overtones of longitudinal vibration in the range dv, in a container of volume F, was... 

Theoretical relationship of the elastic parameters with the velocities of sound waves... 

Measurements of elastic wave velocities in olivine and wadsleyite at high pressures and... 

The vector form of the equations of motion (13.26) is called the Lame equation. The constants Cp and Cs have clear physical meaning. We will see below that equation (13.26) characterizes the propagation of two types of so called body waves in an elastic medium, compressional and shear waves, while the constants Cp and c are the velocities of those waves respectively. We will call them Lame velocities. 

The parameters measured in an ultrasonic experiment are the amplimde and phase of the signal. They are determined by attenuation and phase velocity of a wave. In turn, the attenuation and phase velocity are associated with material constants. In our case they are elastic coefficients (or elastic moduli). These constants can be calculated using quanmm-mechanical approach. Finally, we will obtain the expressions for the measured (phenomenological) parameters in terms of the microscopic ones. In the present section we will discuss the basics of the phenomenological elasticity theory and the microscopic description of the Jahn-Teller contribution to the elastic moduli will be discussed later. 

Mumaghan FD (1937) Finite deformations of an elastic solid. Am J Math 49 235-260 Niesler H, Jackson I (1989) Pressure derivatives of elastic wave velocities from ultrasonic interferometric measurements on jacketed poly crystals. J Acoust Soc America 86 1573-1585 Nomura M, Nishizaka T, Hirata Y, Nakagiri N, Fujiwara H (1982) Measurement of the resistance of Manganin under liquid pressure to 100 kbar and its application to the measurement of the transition pressures of Bi and Sn. Jap J Appl Phys 21 936-939 Nye JF (1957) Physical Properties of Crystals. Oxford University Press, Oxford... 

Li, B. and Zhang, J., 2005. Pressure and temperature dependence of elastic wave velocity of MgSi03 perovskite and the composition of the lower mantle. Phys. Earth Planet. Interior, 151, 143-54. 

Oxygen adsorption effect on the Van der Waals interaction contribution to the elastic moduli of ordered carbon nanotube arrays was estimated. Mechanical instability of square lattice of nanotubes with respect to the transition to triangular lattice was demonstrated. Variation of the elastic moduli due to the adsorption was shown to be of the same order of magnitude as the moduli themselves. This leads to variation of phase velocities of acoustic waves propagating across the array which proved to be dozens of times greater than mass loading by adsorbate for the acoustic wave frequency range of 0.1-1.0 GHz. 

The velocity of elastic ultrasonic waves in solution is strongly influenced by solute-solvent and solute-solute interactions which are determined by the chemical structure of the solute and solvent molecules. Still, acoustical methods have made only minor contributions to the detailed description of solute-solvent interactions. Ultrasonic velocity measurements are mostly limited to obtaining hydration numbers of molecules in aqueous solution [Br 75]. The successful application of acoustical methods to physico-chemical investigation of solutions became possible after development of adequate theoretical approaches and methods for precise ultrasonic velocity measurements in small volumes of liquids [Sa 77, Bu 79]. 

The velocity of the wave is related to the density (p) and the elastic constant (Cm) of the medium through which it is propagating (in the equation shown below). The elastic constant is unique to the mode of propagation and to the material. For example, in liquids Cm is the adiabatic bulk modulus (B) ... 

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