Equations elastic wave velocity

Walhs (1969) also derived an equation for the elastic wave velocity ... 

Ed and Foo are the values of drag force on a particle in a fluidized bed and on a particle in an infinite medium, respectively. is the dimensionless distance (x/dp) between the particles. The following equations were derived for the elastic wave velocity ... 

While estimating dFIde, the value of the superficial velocity was assumed constant. Substitution of Eqs. (48) and (49) in (47) results in the following equation for the elastic wave velocity ... 

The elastic-wave velocity in a polycrystalline material is given by the following equations. For the longitudinal... 

For the relationship between thermal conductivity and elastic wave velocity, the simple equation results ... 

For any other pore fluid, the Clausius-Mossotti equation can also be used for thermal conductivity. For elastic wave velocity a transformation from dry to any other pore fluid is possible using Gassmann s equation. 

Summary of the calculations is listed in table 2. This table displays the shock wave parameters calculated in the code and the corresponding theoretical values. The values of the axial stress (O33) obtained from our calculations are in a very good agreement with the corresponding theoretical values given by equation 16. However, the speeds of the longitudinal elastic wave are underestimated when compared to their theoretical values. This is attributed to the effect of FE mesh size. In fact when a fine mesh is used, the velocity approaches the theoretical wave speed value as shown in Fig. 19. 

The lattice dynamical method, which was originally proposed by Born and Huang [59], to calculate the velocity of sound waves in elastic solids and hence their elastic constants. The equation of motion of an atom is given by... 

We are now in a position to write down the closed formulation of the conservation equations for the particle and fluid phases that defines the one-dimensional particle bed model. The particle phase momentum equation, derived in Chapter 7, adopts the net primary force term F of eqn (8.4) and is augmented by the elasticity term Taz, which, in terms of the dynamic-wave velocity, becomes ppU dejdz. 

Necessary conditions for the existence of a shock are that mass and momentum are conserved across it. In order to quantify these conditions, we will apply the simplified particle bed model formulation introduced in Chapter 8, in which the condition pp > pf enabled the particle-phase equations to be treated independently eqns (8.21) and (8.25). These are reproduced below, eqns (14.1) and (14.2), with the expression for the dynamic-wave velocity, eqn (8.19), inserted in the elasticity term of... 

The ultrasonic relaxation loss may involve a thermally activated stmctural relaxation associated with a shifting of bridging oxygen atoms between two equihbrium positions (169). The velocity, O, of ultrasonic waves in an infinite medium is given by the following equation, where M is the appropriate elastic modulus, and density, d, is 2.20 g/cm. ... 

Equation 5.2 indicates that i is much lower than the corresponding value for the elastic-dielectric, while Eq. (5.3) indicates that if is significantly greater than the corresponding value for the elastic-dielectric. Since both of these current values depend upon the velocity of the plastic wave, measurements of these currents can, in principle, be used to evaluate the velocity of the plastic wave. Observation of a waveform described by Eq. (5.1) would confirm the presence of a second wave behind which the material is conductive. 

The passage of a sound wave along a tube, so that no energy is dissipated by friction, is an example of a compressional wave of permanent type, and Newton applied his equation (1) to determine the velocity of sound in air. For this purpose he took e as the isothermal elasticity of air, which is equivalent to assuming that the temperature is the same in all parts of the wave as that in the unstrained medium. Since air is heated by compression and cooled by expansion, the assumption implies that these temperature differences are automatically annulled by conduction. Taking the isothermal elasticity, we have ... 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

Fig. 4 General solution for the dispersion equation on water at 25 °C. The damping coefficient a vs. the real capillary wave frequency o> , for isopleths of constant dynamic dilation elasticity ed (solid radial curves), and dilational viscosity k (dashed circular curves). The plot was generated for a reference subphase at k = 32431 m 1, ad = 71.97 mN m-1, /i = 0mNsm 1, p = 997.0kgm 3, jj = 0.894mPas and g = 9.80ms 2. The limits correspond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit, V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coefficient for a Perfectly Viscous Surface Film...

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. 

Let us analyze the space and time structure of the elastic displacement field in detail. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988) ... 

As we can see here, both the potential and solenoidal components of the elastic displacement field satisfy wave equations and therefore represent waves traveling in space at velocities Cp and Cg respectively. Let us examine them in detail. 

Any disturbance W [e,g, the displacement of an elastic string or the height of a water wave) which is propagated along the r-direction with velocity u satisfies a simple partial differential equation, the wave equation ... 

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