Attenuation of elastic waves in a cracked solid

SUMMARY The spectral ratios technique is used to measure the attenuation and phase dispersion of the compressional wave and two shear waves polarized parallel and perpendicular to the cleavage of a slate, before and after cracks had been induced in the cleavage plane. The experimental results show that the quality factor Q of the rock sample is affected significantly by the presence of cracks, and that Q is more sensitive to crack parameters than the corresponding wave velocity. The frequency dependence of the attenuation coefficient α is generally assumed to be proportional to the n th power of frequency, ω n , where it is suggested that n has a value lying between 0.5 and 4, depending on the mechanism of the attenuation. The measured frequency dependence of α in this study, where α is influenced strongly by presence of aligned cracks, is demonstrably non‐linear for both P ‐ and S 2 ‐ (polarized parallel to the cleavage) waves, implying that Q is frequency dependent. The convex shape of the attenuation coefficient curves for P ‐waves indicates a value of n of rather less than unity (0.5 ± 0.2), and the concave shape of the curves for S 2 ‐waves suggests a value of n of more than unity (1.8 ± 0.2), indicating that the frequency dependence of the attenuation also depends on wave mode. The attenuation coefficient for S 1 ‐waves (polarized perpendicular to the cleavage of the slate) has also been observed to behave in a non‐linear manner ( n = 3.5 ± 0.3) at low confining pressures. This is probably due to attenuation by scattering, where the frequency dependence in the long‐wavelength limit is predicted to be to the fourth power of frequency. At high confining pressures, the scattering factor is found to be negligible and a linear relationship (constant Q ) is observed, probably as a result of friction between crack surfaces. The crack‐induced phase dispersion for P ‐, S 1 ‐ and S 2 ‐waves has been observed to maintain causality. Although the P ‐ and S 2 ‐wave velocities are not found to be sensitive to cracking (as predicted by Hudson's theory), the corresponding Q values have been found to vary significantly with confining pressure. This behaviour can be explained as due to the closure of low aspect ratio cracks at high confining pressure. The S 2 ‐wave phase dispersion is the only one which can be demonstrated to obey the Kramers‐Kronig relation. This is made possible because the S 2 ‐waves can be recorded over a sufficiently wide frequency band with high enough signal‐to‐noise ratios.