# Attenuation and dispersion of*SH* waves due to scattering by randomly distributed cracks

- 62 Downloads
- 19 Citations

## Abstract

The effect of randomly distributed cracks on the attenuation and dispersion of*SH* waves is theoretically studied. If earthquake ruptures are caused by sudden coalescence of preexisting cracks, it will be crucial for earthquake prediction to monitor the temporal variation of the crack distribution. Our aim is to investigate how the property of crack distribution is reflected in the attenuation and dispersion of elastic waves.

We introduce the stochastic property, in the mathematical analysis, for the distributions of crack location, crack size and crack orientation. The crack size distribution is assumed to be described by a power law probability density (*p(a) ∞ a*^{−γ} for*a*_{min}≤*a*≤*a*_{max} according to recent seismological and experimental knowledge, where*a* is a half crack length and the range 1≤γ≤3 is assumed. The distribution of crack location is assumed to be homogeneous for the sake of mathematical simplicity, and a low crack density is assumed. The stochastic property of each crack is assumed to be independent of that of the other cracks. We assume two models, that is, the aligned crack model and the randomly oriented crack model, for the distribution of crack orientation. All cracks are assumed to be aligned in the former model. The orientation of each crack is assumed to be random in the latter model, and the homogeneous distribution is assumed for the crack orientation. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed.

We observe the following features common to both the aligned crack model and the randomly oriented crack model. The attenuation coefficient*Q*^{−1} decays in proportion to*k*^{−1} in the high frequency range and its growth is proportional to*k*^{2} in the low frequency range, where*k* is the intrinsic wave number. This asymptotic behavior is parameter-independent, too. The attenuation coefficient*Q*^{−1} has a broader peak as γ increases and/or*a*_{min}/*a*_{max} decreases. The nondimensional peak wave number*k*_{p}*a*_{max} at which*Q*^{−1} takes the peak value is almost independent of*a*_{min}/*a*_{max} for γ=1 and 2 while it considerably depends on*a*_{min}/*a*_{max} for γ=3. The phase velocity is almost independent of*k* in the range*ka*_{max}<1 and increases monotonically as*k* increases in the range*ka*_{max}>1. While the magnitude of*Q*^{−1} and the phase velocity considerably depend on the orientation of the crack in the aligned crack model, the above feature does not depend on the crack orientation.

The accumulation of seismological measurements suggests that*Q*^{−1} of*S* waves has a peak at around 0.5 Hz. If this observation is combined with our theoretical results on*k*_{p}*a*_{max}, the probable range of*a*_{max} of the crack distribution in the earth can be estimated for γ=1 or 2. If we assume 4 km/sec as the*S* wave velocity of the matrix medium,*a*_{max} is estimated to range from 2 to 5 km. We cannot estimate*a*_{max} in a narrow range for γ=3.

### Key words

Cracks scattering attenuation phase velocity*SH*waves

## Preview

Unable to display preview. Download preview PDF.

### References

- Achenbach, J. D., andLi, Z. L. (1986),
*Propagation of Horizontally Polarized Transverse Waves in a Solid with a Periodic Distribution of Cracks*, Wave Motion*8*, 371–379.Google Scholar - Aki, K. (1980),
*Scattering and Attenuation of Shear Waves in the Lithosphere*, J. Geophys. Res.*85*, 6496–6504.Google Scholar - Aki, K., andRichards, P. G.,
*Quantitative Seismology*(W. H. Freeman and Company, San Francisco 1980).Google Scholar - Aki, K. (1981),
*Scattering and Attenuation of High-frequency Body Waves (1–25 Hz) in the Lithosphere*, Phys. Earth Planet. Inter.*26*, 241–243.Google Scholar - Ang, D. D., andKnopoff, L. (1964),
*Diffraction of Elastic Waves by a Finite Crack*, Proc. Nat. Acad. Sci. U.S.A.*51*, 593–598.Google Scholar - Angel, Y. C., andAchenbach, J. D. (1985a),
*Reflection and Transmission of Elastic Waves by a Periodic Array of Cracks*, J. Appl. Mech.*52*, 31–41.Google Scholar - Angel, Y. C., andAchenbach, J. D. (1985b),
*Reflection and Transmission of Elastic Waves by a Periodic Array of Cracks: Oblique Incidence*, Wave Motion*7*, 375–397.Google Scholar - Crampin, S. (1987),
*Geological and Industrial Implications of Extensive-dilatancy Anisotropy*, Nature*328*, 491–496.Google Scholar - Foldy, L. L. (1945),
*The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers*, Phys. Rev.*67*, 107–119.Google Scholar - Ishimaru, A.,
*Wave Propagation and Scattering in Random Media*(Academic Press, London 1978).Google Scholar - Kikuchi, M. (1981),
*Dispersion and Attenuation of Elastic Waves Due to Multiple Scattering from Inclusions*, Phys. Earth Plant, Inter.*25*, 159–162.Google Scholar - Loeber, J. F., andSih, G. G. (1968),
*Diffraction of Antiplane Shear Waves by a Finite Crack*, J. Acoust. Soc. Am.*44*, 90–98.Google Scholar - Mal, A. K. (1970),
*Interaction of Elastic Waves with a Griffith Crack*, Int. J. Engng. Sci.*8*, 763–776.Google Scholar - Matsunami, K. (1988),
*Laboratory Measurements of Elastic Wave Attenuation by Scattering Due to Random Heterogeneities*, Bull. Disas. Prev. Res. Inst., Kyoto Univ.*38*, 1–16.Google Scholar - Sato, H. (1979),
*Wave Propagation in One-Dimensional Inhomogeneous Elastic Media*, J. Phys. Earth*27*, 455–466.Google Scholar - Sato, H. (1984),
*Scattering and Attenuation of Seismic Waves in the Lithosphere — Single Scattering Theory in a Randomly Inhomogeneous Medium*, Rep. Nat. Res. Cent. Disas. Prev.*33*, 101–186 (in Japanese).Google Scholar - Segall, P., andPollard, D. D. (1983),
*Joint Formation in Granite Rock of the Sierra Nevada*, Geol. Soc. Am. Bull.*94*, 563–575.Google Scholar - Sih, G. C., andLoeber, J. H. (1969),
*Wave Propagation in an Elastic Solid with a Line of Discontinuity or Finite Crack*, Quart. Appl. Math.*27*, 193–213.Google Scholar - Tan, T. H. (1977),
*Scattering of Plane, Elastic Waves by a Plane Crack of Finite Width*, Appl. Sci. Res.*33*, 75–88.Google Scholar - van der Hijden, V. H. M. T., andNeerhoff, F. L. (1984),
*Scattering of Elastic Waves by a Plane Crack of Finite Width*, J. Appl. Mech.*51*, 646–651.Google Scholar - Yamashita, T. (1986),
*Preparation Process of Large Earthquake as the Growth and Coalescence Process of Multiply Interactive Cracks*, Programme and abstracts, The Seismological Society of Japan, No. 2, 69 (in Japanese).Google Scholar - Yamashita, T., andKnopoff, L. (1987),
*Models of Aftershocks Occurrence*, Geophys. J. R. Astr. Soc.*91*, 13–26.Google Scholar - Yamashita, T., andKnopoff, L. (1989),
*A Model of Foreshock Occurrence*, Geophys. J.,*96*, 389–399.Google Scholar