Compressional and shear waves

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)  

We can apply the same decomposition to the strength of an external volume force F  

Now let las substitute relations (13.36) and (13.39) into the equation of motion of a homogeneous isotropic medium (13.26)  

We will apply the divergence operation to both sides of this equation. Since 

The curl of the expression enclosed in parentheses is also zero. Thus the field (cpV Up — + Fp/p) has neither sources nor vortices in the whole space. 

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The static Poisson ratio is determined in a triaxial cell. The dynamic Poisson ratio is calculated with the sonic compressional and shear wave velocities. They usually are different. 

Example 2.4 Calculate the compressional and shear wave velocities in aluminum and polyethylene. 

In order to satisfy the stress-free boundary condition, coupled compressional and shear waves propagate together in a SAW such that surface traction forces are zero (i.e., T y = 0, where y is normal to the device surface). The generalized surface acoustic wave, propagating in the z-direction, has a displacement profile u(y) that varies with depth y into the crystal as... 

Such behaviour of fracture propagation can clearly be of extreme importance for practical purposes. For example, the fraction r, from the measurement of the sound velocity ratio of compressional and shear waves, can indicate the proximity of the imminent macroscopic failure or fracture of... 

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. 

A similar result holds true for any general dynamic equation of elasticity theory and for equations of compressional and shear waves as well. 

We found in the previous section that this equation describes separately the propagation of the compressional and shear waves in a homogeneous medium. 

We have learned already that any elastic oscillation can be represented as a superposition of the compressional and shear waves, which correspond to the potential and solenoidal parts of the elastic displacement field. Therefore, it is clear that the elastic tensor G can also be represented as the sum of the potential and solenoidal components, described by tensor functions G W and G respectively ... 

Since this field may be represented as a superposition of two types of waves, compressional and shear waves, we are faced with the problem of formulating a certain analytical criterion (similar to the Sommerfeld radiation conditions) that provides for the exclusion from the solution of the elastic field equations of compressional and shear waves that are convergent at infinity. It should also be pointed out that the radiation conditions are not included as some kind of heuristic principle in the initial mathematical formulation of the problem. 

Formulae (15.260) and (15.261) can be further simplified, if we assume that we have constant background velocities Cp (r) = const and c b (r) = const. In this case the incident field can be separated into the compressional and shear waves... 

Substituting representation (15.262) into (15.260) and (15.261), and taking into account the properties of the compressional and shear waves (15.263) and (15.264), we arrive at a result that calculation of the anomalous compressional wavespeed requires only the compressional incident wavefield, while the anomalous shear wavespeed is determined solely by the shear incident wavefield ... 

Manghnani, M. H. Ramananantoandro, R. 1974. Compressional and shear wave velocities in granulite facies rocks and eclogites to 10 Kb. Journal of Geophysical Research, 79, 5427-5446. 

Attenuation measurements are relatively easily obtained in the laboratory using various techniques (standing wave, pulse, torsional resonance, and torsional cyclic loading techniques— e.g., Hampton, 1967 Badiey et al., 1988 Bennell and Taylor-Smith, 1991 and other papers in Hovem et al., 1991). In-situ measurements are relatively rare compared with measurements of other soil properties (e.g., Hamilton, 1972, 1976 Taylor-Smith, 1974 Dunlop and Whichello, 1980 Dunlop, 1992). Attenuation, a, of both compressional and shear waves in surface sedi-menfs, is sfrongly dependenf on the frequency as presented in Equation 7.9... 

Nafe, J.E., and Drake, C.L. 1957. Variation with depth in shallow and deep water marine sediments of porosity, density, and the velodties of compressional and shear waves. Geophysics, 22(3) 523-552. 

Bouchon M (1979) Discrete wave number representation of elastic wave fields in three-space dimensions. J Geophys Res 84(B7) 3609-3614 Brocher TM (2008) Compressional and shear-wave velocity versus depth relations for common rock types in... 

Pure compressional and shear waves exist only for the propagation in the main axes. 

Increasing porosity decreases both compressional and shear wave velocity. 

Figure 6.4 shows the range of the compressional and shear wave velocity for... 

Gebrande et al. (1982) analysed compressional and shear wave velocities for three rock groups ... 

Figure 6.10 shows results of experimental investigations on sandstone samples with different porosity and different clay content. Both—porosity and clay content— result in a decrease of velocity for compressional and shear wave velocity. 

FIGURE 6.10 Compressional and shear wave velocities versus praosity (fraction) for 75 sandstone samples (water saturated) at a confining pressure 40 MPa and a pOTe pressure 1.0 MPa. Data after Han et al. (1986). 

The various pore fluids (water, oil, gas, and mixtures of them) influence elastic wave velocities as a result of effects that are different for compressional and shear waves ... 

Figure 6.14 shows the effect of three different pore fluids (air, water, and kerosene, but no mixture of them) upon the compressional and shear wave velocity. 

FIGURE 6.14 Compressional and shear wave velocities of Boise sandstone as a function of effective pressure at different pore fluids. Data after King (1966), converted units. 

FIGURE 6.15 Effect of porosity and pore fluid on velocities of unfractured chalk samples (North Sea, Cretaceous) (a) compressional and shear wave velocity for dry and water-saturated rock (b) ratio of wave velocities for water saturated and dry rock (compressional and shear wave). Data from Rogen et al. (2005). 

FIGURE 6.16 Influence of gas and water saturation on compressional and shear wave velocity. Experimental data for Ottawa sand from Domenico (1976). 

Lebedev et al. (2009) published a study of velocity (compressional and shear wave) measurement under different saturation conditions. Fluid distribution during the experiment was observed by X-ray computer tomography. 

TABLE 6.11 Compressional and Shear Wave Velocity in m s as a Function of Depth in m... 

Katahara (1996) published velocity data for kaolinite, illite, and chlorite. Results show a distinct anisotropy for compressional and shear wave. The clay can be described as a transverse isotropic material. 

White et al. (1983) published the results of an analysis of Pierre shale elastic anisotropy. A transverse isotropy was assumed and the vertical compressional and shear wave velocity and the anisotropy coefficients have been calculated for depth sections. 

Correlations Between Compressional and Shear Wave Anisotropy... 

Implementing Eqs. (6.93) and (6.94) into Eqs. (6.5) and (6.6) results in compressional and shear wave velocities. For sphere pack models, both wave velocities show identical dependence on porosity and presstue. Thus, the ratio Fp/Fs is independent of porosity and pressure and only controlled by Poisson s ratio of the solid material v. ... 

The fourth factor is an element of the structure tensor and depends on the internal structure (expressed by the angle a) and the parameter/, which is controlled by the bonding properties of the contact. Compressional and shear waves with propagation in vertical and horizontal directions differ only in this last term. It follows immediately that for a dry rock the velocity ratios (e.g. VpIV, elastic anisotropy parameters) depend only on structural and bonding properties. 

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