Skip to main content
SpringerLink
Log in

Scattering and attenuation of elastic waves in random media

  • Published:
pure and applied geophysics Aims and scope Submit manuscript

Abstract

In seismic exploration, elastic waves are sent to investigate subsurface geology. However, the transmission and interpretation of the elastic wave propagation is complicated by various factors. One major reason is that the earth can be a very complex medium. Nevertheless, in this paper, we model some terrestrial material as an elastic medium consisting of randomly distributed inclusions with a considerable concentration. The waves incident on such an inhomogeneous medium undergo multiple scattering due to the presence of inclusions. Consequently, the wave energy is redistributed thereby reducing the amplitude of the coherent wave.

The coherent or average wave is assumed to be propagating in a homogeneous continuum characterized by a bulk complex wavenumber. This wavenumber depends on the frequency of the probing waves; and on the physical properties and the concentration of discrete scatterers, causing the effective medium to be dispersive. With the help of multiple scattering theory, we are able to analytically predict the attenuation of the transmitted wave intensity as well as the dispersion of the phase velocity. These two sets of data are valuable to the study of the inverse scattering problems in seismology. Some numerical results are presented and also compared, if possible, with experimental measurements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barker, A., andHenderson, D. (1971),Monte Carlo Values for the Radial Distribution Function of a System of Fluid Hard Spheres, Mol. Phys.21, 187.

    Google Scholar 

  • Batchelor, G. K., andGreen, J. T. (1972),The Determination of the Bulk Stress in a Suspension of Spherical Particles to Order c 2, J. Fluid Mech.56, 401.

    Google Scholar 

  • Bedeaux, D., andMazur, P. (1973),On the Critical Behavior of the Dielectric Constant for a Nonpolar Fluid, Physica67, 23.

    Google Scholar 

  • Biot, M. A. (1956a),Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid. I. Low Frequency Range, J. Acoust. Soc. Am.28, 168.

    Google Scholar 

  • Biot, M. A. (1956b),Theory of Propagation of Elastic Waves in a Fluid-saturated Porous Solid. II. High Frequency Range, J. Acoust. Soc. Am.28, 179.

    Google Scholar 

  • Bose, S. K., andMal, A. K. (1973),Longitudinal Shear Waves in a Fiber Reinforced Composite, Int. J. Solids Struct.9, 1975.

    Google Scholar 

  • Bose, S. K., andMal, A. K. (1976),Elastic Waves in Fiber Reinforced Composites, J. Mech. Phys. Solids22, 217.

    Google Scholar 

  • Bringi, V. N., Seliga, T. A., Varadan, V. K., andVaradan, V. V.,Bulk propagation characteristics of discrete random media, InMultiple Scattering and Waves in Random Media (eds. Chow, P. L., Kohler, W. E., and Papanicolaou, G. C.) (North-Holland, Amsterdam 1981).

    Google Scholar 

  • Bringi, V. N., Varadan, V. V., andVaradan, V. K. (1982a),Coherent Wave Attenuation by a Random Distribution of Particles, Radio Sci.17, 946.

    Google Scholar 

  • Bringi, V. N., Varadan, V. V., andVaradan, V. K. (1982b),The Effects of Pair Correlation Function on Coherent Wave Attenuation in Discrete Random Media, IEEE Antennas Propaga.AP-30, 805.

    Google Scholar 

  • Bringi, V. N., Varadan, V. K., andVaradan, V. V. (1983),Average Dielectric Properties of Discrete Random Media Using Multiple Scattering Theory, IEEE Trans. Antennas Propag.AP-31, 371.

    Google Scholar 

  • Bruggeman, D. A. G. (1935),Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen Mischkörpern aus Isotropen Substranzen, Ann. Phys.24, 636.

    Google Scholar 

  • Chaban, L. A. (1964).Self-consistent Field Approach to Calculation of the Effective Parameters of Microinhomogeneous Media, Sov. Phys. Acoust.10, 298.

    Google Scholar 

  • Chaban, L. A. (1965),Calculation of the Effective Parameters of Microinhomogeneous Media by the Self-consistent Field Method, Sov. Phys. Acoust.11, 81.

    Google Scholar 

  • Chatterjee, A. K., Mal, A. K., andKnopoff, L. (1978),Elastic Moduli of Two-component Systems, J. Geophys. Res.83, 1785.

    Google Scholar 

  • Christensen, R. M.,Mechanics of Composite Materials (Wiley, New York 1979).

    Google Scholar 

  • Corsaro, R. D., andKlunder, J. D.,A Filled Silicone Rubber Materials System with Selectable Acoustic Properties for Molding and Coating Applications at Ultrasonic Frequencies, Naval Res. Lab. Rep. 8301 (1979).

  • Cruzan, O. R. (1962),Translational Addition Theorems for Spherical Vector Wave Equations, Q. Appl. Math.20, 33.

    Google Scholar 

  • Datta, S. K.,Scattering by a random distribution of inclusions and effective elastic properties, InContinuum Model of Discrete Systems (ed. Provan, J. W.) (University of Waterloo, Waterloo, Illinois 1978).

    Google Scholar 

  • Datta, S. K., andLedbetter, H. M.,Anisotropic elastic constants of a fiber reinforced boron-aluminum composite, InMechanics of Nondestructive Testing (ed. Stinchcomb, W. W.) (Plenum, New York 1980).

    Google Scholar 

  • Datta, S. K., Ledbetter, H. M., andKinra, V. K.,Wave propagation and elastic constants in particulate and fibrous composites, InComposite Materials: Mechanical Properties and Fabrication, (eds. Kawatz, K. and Akasada, T.) Proceedings of the Japan-U.S. Conference, Tokyo, 30 (1981).

  • Domany, E., Gubernatis, J. E., andKrumhansl, J. A.,The Elasticity of Polycrystals and Rocks Materials Science Center Report (Cornell University, New York 1974).

    Google Scholar 

  • Edmond, A. R.,Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey 1957).

    Google Scholar 

  • Eshelby, J. D. (1957),The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems, Proc. R. Soc. LondonA, 376.

  • Fikioris, J. G., andWaterman, P. C. (1964),Multiple Scattering of Waves. II. Hole Corrections in the Scalar Case. J. Math. Phys.5, 1413.

    Google Scholar 

  • Fisher, I. Z.,Statistical Theory of Liquids (Univ. Chicago Pub., Chicago 1965).

    Google Scholar 

  • Foldy, L. L.,The Multiple Scattering of Waves, Phys. Rev.67, 107–119.

  • Garbin, H. D., andKnopoff, L. (1975),Elastic Moduli of a Medium with Liquid-filled Cracks, Quart. Appl. Math.33, 301.

    Google Scholar 

  • Gassmann, F. (1951),Elastic Waves Through a Packing of Spheres, Geophysics,16 673 and18, 269.

    Google Scholar 

  • Hudson, J. A. (1980),Overall Properties of a Cracked Solid, Math. Proc. Camb. Phil. Soc.88, 371.

    Google Scholar 

  • Hashin, Z. (1962),The Elastic Moduli of Heterogeneous Materials, J. Appl. Mech.29, 143.

    Google Scholar 

  • Ishimaru, A.,Wave Propagation and Scattering in Random Media (Academic, New York 1978).

    Google Scholar 

  • Jeng, J. H.,Numerical Techniques for Wave Propagation and Scattering Problems Associated with Anisotropic Composites, Ph.D. Dissertation (Pennsylvania State University, University Park, Pennsylvania 1988).

    Google Scholar 

  • Junger, M. C. (1981),Dilatational Waves in an Elastic Solid Containing Lined, Gas-filled Spherical Cavities, J. Acoust. Soc. Am.69, 1573.

    Google Scholar 

  • Keller, J. B. (1962),Wave Propagation in Random Media, Proc. Sympos. Appl. Math.13, 227.

    Google Scholar 

  • Keller, J. B. (1964),Stochastic Equations and Wave Propagation in Random Media, Proc. Sympos. Appl. Math.16, 145.

    Google Scholar 

  • Kinra, V. K., Petraitis, M. S., andDatta, S. K. (1980),Ultrasonic Wave Propagation in a Random Particulate Composite, Int. J. Solids Struct.16, 301.

    Google Scholar 

  • Kinra, V. K., andDatta, S. K. (1982a),Influence of Particle Resonance on Wave Propagation in a Random Particulate Composite, Mech. Res. Commun.9, 109.

    Google Scholar 

  • Kinra, V. K., andAnand, A. (1982b),Wave Propagation in a Random Particulate Composite at Long and Short Wavelengths, Int. J. Solids. Struct18, 367.

    Google Scholar 

  • Kligman, R. L., Madigosky, W. M., andBarlow, J. R. (1981),Effective Dynamic Properties of Composite Viscoelastic Materials, J. Acoust. Soc. Am.70, 1437.

    Google Scholar 

  • Korringa, J. (1973),Theory of Elastic Constants of Heterogeneous Media, J. Math. Phys.4, 509.

    Google Scholar 

  • Kröner, E. (1967),Elastic Moduli of Perfectly Disordered Composite Materials, J. Mech. Phys. Solids15, 319.

    Google Scholar 

  • Kuster, G., andToksöz, M. N. (1974),Velocity and Attenuation of Seismic Waves in Two-phase Media, Part I: Theoretical Formulations, Geophysics39, 587.

    Google Scholar 

  • Lax, M. (1951),Multiple Scattering of Waves, Rev. Mod. Phys.23, 287.

    Google Scholar 

  • Lax, M. (1952),Multiple Scattering of Waves. II. The Effective Field in Dense Systems, Phys. Rev.88, 621.

    Google Scholar 

  • Lloyd, P., andBerry, M. V. (1967),Wave Propagation Through an Assembly of Spheres IV. Relation Between Different Multiple Scattering Theories, Proc. Phys. Soc. London91, 678.

    Google Scholar 

  • Ma, Y., Varadan, V. K., andVaradan, V. V. (1984),Application of Twersky's Multiple Scattering Formalism to a Dense Suspension of Elastic Particles in Water, J. Acout. Soc. Am.75, 335.

    Google Scholar 

  • Ma, Y., Varadan, V. V., andVaradan, V. K. (1988),Scattered Intensity of a Wave Propagating in a Discrete Random Medium, Applied Optics27, 2469.

    Google Scholar 

  • Magnuson, A. H., Sundkvist, K., Ma, Y., andSmith, K. (1981),Acoustic Sounding for Manganese Nodules, Proc. 13th OTC Conf., 147.

  • Mal, A. K., andBose, S. K. (1974),Dynamic Elastic Moduli of a Suspension of Imperfectly Bonded Spheres, Proc. Camb. Philos. Soc.76, 587.

    Google Scholar 

  • McQuarrie, D. A.,Statistical Mechanics (Harper and Row, New York 1976).

    Google Scholar 

  • Merkulova, V. M. (1965),Acoustical Properties of Some Solid Heterogeneous Media at Ultrasonic Frequencies, Sov. Phys. Acoust.11, 55.

    Google Scholar 

  • Moon, F. C., andMow, C. C.,Wave Propagation in a Composite Material Containing Dispersed Rigid Spherical Inclusions, Rand Corp. Rep. RM-6134-PR (Rand, Santa Monica, California 1970).

  • O'Connell, R. J., andBudiansky, B. (1974),Seismic Velocities in Dry and Saturated Cracked Solids, J. Geophys. Res.79, 5412.

    Google Scholar 

  • O'Connell, R. J., andBudiansky, B. (1977),Viscoelastic Properties of Fluid-saturated Cracked Solids, J. Geophy. Res.82, 5719.

    Google Scholar 

  • Percus, J. K., andYevick, G. J. (1950),Analysis of Classical Statistical Mechanics by Means of Collective Coordinates, Phys. Rev.110, 1.

    Google Scholar 

  • Peterson, B., Varadan, V. K., andVaradan, V. V. (1983),Scattering of Elastic Waves by a Fluid Inclusion, J. Acoust. Soc. Am.73, 1487.

    Google Scholar 

  • Lord Rayleigh (1899),On the Transmission of Light Through an Atmosphere Containing Small Particles in Suspension, and on the Origin of the Blue Sky, Philos. Mag.47, 375.

    Google Scholar 

  • Reed, T. M., andGubbins, K. E.,Applied Statistical Mechanics (McGraw-Hill, New York 1973).

    Google Scholar 

  • Rowlinson, J. S. (1965),Self-consistent Approximation for Molecular Distribution Function, Mol. Phys.9, 217.

    Google Scholar 

  • Talbot, D. R. S., andWillis, J. R. (1980),The Effective Sink Strength of a Random Array of Voids in Irradiated Material, Proc. R. Soc. London,A370, 351.

    Google Scholar 

  • Talbot, D. R. S.,Variational estimates for attenuation and dispersion in elastic composites, InContinuum Models of Discrete Systems (eds. Burlin, O., and Hsich, R. K. T.) (North-Holland, Amsterdam 1981).

    Google Scholar 

  • Throop, G. J., andBearman, R. (1965),Numerical Solutions of the Percus-Yevick Equation for the Hard Sphere Potential, J. Chem. Phys.42, 2408.

    Google Scholar 

  • Twersky, V. (1962a),On Scattering of Waves by Random Distributions, I. Free-space Scatterer Formalism, J. Math. Phys.3, 700.

    Google Scholar 

  • Twersky, V. (1962b),On Scattering of Waves by Random Distributions, II. Two-space Scatterer Formalism, J. Math. Phys.3, 724.

    Google Scholar 

  • Twersky, V. (1962c),Multiple Scattering of Waves and Optical Phenomena, J. Opt. Soc. Am.52, 145.

    Google Scholar 

  • Twersky, V. (1964),On Propagation in Random Media of Discrete Scatterers, Proc. Sympos. Appl. Math.16, 84.

    Google Scholar 

  • Twersky, V. (1970),Absorption and Multiple Scattering by Biological Suspensions, J. Opt. Soc. Am.60, 1084.

    Google Scholar 

  • Twersky, V. (1976),Propagation Parameters in Random Distributions of Scatterers, J. Anal. Math.30, 498.

    Google Scholar 

  • Twersky, V. (1977),Coherent Scalar Field in Pair-correlated Random Distributions of Aligned Scatterers, J. Math. Phys.18, 2468.

    Google Scholar 

  • Twersky, V. (1978a),Coherent Electromagnetic Waves in Pair-correlated Random Distributions of Aligned Scatterers, J. Math. Phys.19, 215.

    Google Scholar 

  • Twersky, V. (1978b),Acoustic Bulk Parameters in Distributions of Pair-correlated Scatterers, J. Acous. Soc. Am.64, 1710.

    Google Scholar 

  • Twersky, V.,Scattering theory and diagnostic applications, InMultiple Scattering and Waves in Random Media (eds. Chow, P. L., Kohler, W. E., and Papanicolaou, G. C.) (North-Holland, Amsterdam 1981).

    Google Scholar 

  • Varadan, V. K., andVaradan, V. V.,Dynamic Elastic Properties of a Medium Containing a Random Distribution of Obstacles—Scattering Matrix Theory, Materials Science Center Report, 2740 (Cornell University, New York 1976a).

    Google Scholar 

  • Varatharajulu, V., andVezzetti, D. J. (1976b),Approach of the Statistical Theory of Light Scattering to the Phenomenological Theory, J. Math. Phys.17, 232.

    Google Scholar 

  • Varatharajulu V., andPao, Y. H. (1976c)Scattering Matrix for Elastic Waves. I. Theory, J. Acoust. Soc. Am.60, 556.

    Google Scholar 

  • Varadan, V. K., andVaradan, V. V.,Multiple Scattering of Elastic Waves by Cylinders of Arbitrary Cross-section. II. P and SV Waves, Materials Science Center Report 2937 (Cornell University, New York 1977).

    Google Scholar 

  • Varadan, V. K., Varadan, V. V., andPao, Y. H. (1978a),Multiple Scattering of Elastic Waves by Cylinders of Arbitrary Cross-sections. I. SH-Waves, J. Acoust. Soc. Am.63, 1310.

    Google Scholar 

  • Varadan, V. K., andVaradan, V. V.,Characterization of dynamic shear modulus in inhomogeneous media using ultrasonic waves, InFirst International Symposium on Ultrasonic Materials Characterization, NBS (Washington, D. C. 1978b).

  • Varadan, V. K., (1979a),Scattering of Elastic Waves by Randomly Distributed and Oriented Scatterers, J. Acoust. Soc. Am.65, 655.

    Google Scholar 

  • Varadan, V. K., andVaradan, V. V. (1979b),Frequency Dependence of Elastic (SH) Wave Velocity and Attenuation in Anisotropic Two Phase Media, Wave Motion1, 53.

    Google Scholar 

  • Varadan, V. K., Bringi, V. N., andVaradan, V. V. (1979c),Coherent Electromagnetic Wave Propagation Through Randomly Distributed Dielectric Scatterers Phys. Rev.19, 2480.

    Google Scholar 

  • Varadan, V. V., andVaradan, V. K. (1980a),Multiple Scattering of Electromagnetic Waves by Randomly Distributed and Oriented Dielectric Scatterers Phys. Rev.D21, 388.

    Google Scholar 

  • Varadan, V. K.,Multiple scattering of acoustic, electromagnetic and elastic waves InAcoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-matrix Approach (eds. Varadan, V. K., and Varadan, V. V.) (Pergamon, New York 1980b).

    Google Scholar 

  • Varadan, V. K., andVaradan, V. V. (Eds.),Acoustic, Electromagnetic and Elastic Wave Scattering —Focus on the T-matrix Approach (Pergamon, New York 1980c).

    Google Scholar 

  • Varadan, V. V., Bringi, V. N., andVaradan, V. K.,Frequency-dependent dielectric constants of discrete random media, InMacroscopic Properties of Disordered Media, Lecture Notes in Physics Vol. 154 (eds. Burridge, R., Childress, C., and Papanicolaou, G. C.) (Springer-Verlag, New York 1982).

    Google Scholar 

  • Varadan, V. V., Bringi, V. N., Varadan, V. V., andMa, Y. (1983a),Coherent Attenuation of Acoustic Waves by Pair-correlated Random Distribution of Scatterers with Uniform and Gaussian Size Distributions, J. Acoust. Soc. Am.73, 1941.

    Google Scholar 

  • Varadan, V. K., Bringi, V. N., Varadan V. V., andIshimaru, A. (1983b)Multiple Scattering in Discrete Random Media and Comparison with Experiments, Radio Sci.18, 321.

    Google Scholar 

  • Varadan, V. K., Varadan, V. V., andMa, Y. (1984a),Frequency-Dependent Elastic Properties of Rubberlike Materials with a Random Distribution of Voids, J. Acoust. Soc. Am.76, 296.

    Google Scholar 

  • Varadan, V. K., Varadan, V. V., andMa, Y.,Wave propagation in composite media, InAdvances in Aerospace Sciences and Engineering (eds. Yuceoglu, U., and Hesser, R.) (ASME, 1984b).

  • Varadan, V. K., Ma Y., andVaradan, V. V. (1985),A Multiple Scattering Theory for Elastic Wave Propagation in Discrete Random Media, J. Acoust. Soc. Am.77, 375.

    Google Scholar 

  • Varadan, V. K., Varadan, V. V., andMa, Y. (1986a),Multiple Scattering of Elastic (SH-) Waves by Piezolectric Cylinders in Rubberlike Materials, J. Wave-Material Interaction1, 291.

    Google Scholar 

  • Varadan, V. K., Ma, Y., andVaradan, V. V. (1986b),Multiple Scattering of Compressional and Shear Waves by Fiber Reinforced Composite Materials, J. Acoust. Soc. Am.80, 333.

    Google Scholar 

  • Varadan, V. K., Ma, Y., andVaradan, V. V. (1987a),Multiple Scattering of Elastic (P and SV-) Waves by Piezolectric Cylinders in Rubberlike Materials, J. Wave-Material Interaction2, 305.

    Google Scholar 

  • Varadan, V. V., Varadan, V. K., Ma, Y., andSteele, W. (1987b),Effects of Nonspherical Statistics on EM Wave Propagation in Discrete Random Media, Radio Sci.22, 491.

    Google Scholar 

  • Wang, S. W.,Nondestructive Evaluation of Fracture Strength of Graphites by Ultrasonic Methods M. S. Thesis (Ohio State University, Columbus, Ohio 1983).

    Google Scholar 

  • Waterman, P. C., andTruell, R. (1961),Multiple Scattering of Waves, J. Math. Phys.2, 512.

    Google Scholar 

  • Waterman, P. C. (1979),Matrix Theory of Elastic Wave Scattering. I, J. Acoust. Soc. Am.60, 567.

    Google Scholar 

  • Watts, R. O., andHenderson, D. (1969),Pair Distribution Function for Fluid Hard Spheres, Mol. Phys.16, 217.

    Google Scholar 

  • Wertheim, M. S. (1963),Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres, Phys. Rev. Lett.10, 321.

    Google Scholar 

  • Wood, A. B., andWeston, D. E. (1964),The Propagation of Sound in Mud, Acoustica14, 156.

    Google Scholar 

  • Wu, R.-S. (1982),Mean Field Attenuation and Amplitude Attenuation Due to Wave Scattering, Wave Motion,4, 305.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Center for the Engineering of Electronic and Acoustic Materials, Department of Engineering Science and Mechanics, The Pennsylvania State University, 16802, University Park, PA, USA

    V. K. Varadan, Y. Ma & V. V. Varadan

Authors
  1. V. K. Varadan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. V. Varadan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Varadan, V.K., Ma, Y. & Varadan, V.V. Scattering and attenuation of elastic waves in random media. PAGEOPH 131, 577–603 (1989). https://doi.org/10.1007/BF00876265

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00876265

Key words

Search

Navigation