Advertisement

pure and applied geophysics

, Volume 131, Issue 4, pp 577–603 | Cite as

Scattering and attenuation of elastic waves in random media

  • V. K. Varadan
  • Y. Ma
  • V. V. Varadan
Article

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.

Key words

Elastic wave random media multiple scattering phase velocity and attenuation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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
  2. 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
  3. Bedeaux, D., andMazur, P. (1973),On the Critical Behavior of the Dielectric Constant for a Nonpolar Fluid, Physica67, 23.Google Scholar
  4. 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
  5. 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
  6. Bose, S. K., andMal, A. K. (1973),Longitudinal Shear Waves in a Fiber Reinforced Composite, Int. J. Solids Struct.9, 1975.Google Scholar
  7. Bose, S. K., andMal, A. K. (1976),Elastic Waves in Fiber Reinforced Composites, J. Mech. Phys. Solids22, 217.Google Scholar
  8. 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
  9. 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
  10. 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
  11. 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
  12. Bruggeman, D. A. G. (1935),Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen Mischkörpern aus Isotropen Substranzen, Ann. Phys.24, 636.Google Scholar
  13. Chaban, L. A. (1964).Self-consistent Field Approach to Calculation of the Effective Parameters of Microinhomogeneous Media, Sov. Phys. Acoust.10, 298.Google Scholar
  14. 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
  15. Chatterjee, A. K., Mal, A. K., andKnopoff, L. (1978),Elastic Moduli of Two-component Systems, J. Geophys. Res.83, 1785.Google Scholar
  16. Christensen, R. M.,Mechanics of Composite Materials (Wiley, New York 1979).Google Scholar
  17. 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).Google Scholar
  18. Cruzan, O. R. (1962),Translational Addition Theorems for Spherical Vector Wave Equations, Q. Appl. Math.20, 33.Google Scholar
  19. 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
  20. 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
  21. 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).Google Scholar
  22. 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
  23. Edmond, A. R.,Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, New Jersey 1957).Google Scholar
  24. Eshelby, J. D. (1957),The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems, Proc. R. Soc. LondonA, 376.Google Scholar
  25. 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
  26. Fisher, I. Z.,Statistical Theory of Liquids (Univ. Chicago Pub., Chicago 1965).Google Scholar
  27. Foldy, L. L.,The Multiple Scattering of Waves, Phys. Rev.67, 107–119.Google Scholar
  28. Garbin, H. D., andKnopoff, L. (1975),Elastic Moduli of a Medium with Liquid-filled Cracks, Quart. Appl. Math.33, 301.Google Scholar
  29. Gassmann, F. (1951),Elastic Waves Through a Packing of Spheres, Geophysics,16 673 and18, 269.Google Scholar
  30. Hudson, J. A. (1980),Overall Properties of a Cracked Solid, Math. Proc. Camb. Phil. Soc.88, 371.Google Scholar
  31. Hashin, Z. (1962),The Elastic Moduli of Heterogeneous Materials, J. Appl. Mech.29, 143.Google Scholar
  32. Ishimaru, A.,Wave Propagation and Scattering in Random Media (Academic, New York 1978).Google Scholar
  33. 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
  34. Junger, M. C. (1981),Dilatational Waves in an Elastic Solid Containing Lined, Gas-filled Spherical Cavities, J. Acoust. Soc. Am.69, 1573.Google Scholar
  35. Keller, J. B. (1962),Wave Propagation in Random Media, Proc. Sympos. Appl. Math.13, 227.Google Scholar
  36. Keller, J. B. (1964),Stochastic Equations and Wave Propagation in Random Media, Proc. Sympos. Appl. Math.16, 145.Google Scholar
  37. 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
  38. 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
  39. 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
  40. 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
  41. Korringa, J. (1973),Theory of Elastic Constants of Heterogeneous Media, J. Math. Phys.4, 509.Google Scholar
  42. Kröner, E. (1967),Elastic Moduli of Perfectly Disordered Composite Materials, J. Mech. Phys. Solids15, 319.Google Scholar
  43. Kuster, G., andToksöz, M. N. (1974),Velocity and Attenuation of Seismic Waves in Two-phase Media, Part I: Theoretical Formulations, Geophysics39, 587.Google Scholar
  44. Lax, M. (1951),Multiple Scattering of Waves, Rev. Mod. Phys.23, 287.Google Scholar
  45. Lax, M. (1952),Multiple Scattering of Waves. II. The Effective Field in Dense Systems, Phys. Rev.88, 621.Google Scholar
  46. 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
  47. 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
  48. 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
  49. Magnuson, A. H., Sundkvist, K., Ma, Y., andSmith, K. (1981),Acoustic Sounding for Manganese Nodules, Proc. 13th OTC Conf., 147.Google Scholar
  50. 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
  51. McQuarrie, D. A.,Statistical Mechanics (Harper and Row, New York 1976).Google Scholar
  52. Merkulova, V. M. (1965),Acoustical Properties of Some Solid Heterogeneous Media at Ultrasonic Frequencies, Sov. Phys. Acoust.11, 55.Google Scholar
  53. 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).Google Scholar
  54. O'Connell, R. J., andBudiansky, B. (1974),Seismic Velocities in Dry and Saturated Cracked Solids, J. Geophys. Res.79, 5412.Google Scholar
  55. O'Connell, R. J., andBudiansky, B. (1977),Viscoelastic Properties of Fluid-saturated Cracked Solids, J. Geophy. Res.82, 5719.Google Scholar
  56. Percus, J. K., andYevick, G. J. (1950),Analysis of Classical Statistical Mechanics by Means of Collective Coordinates, Phys. Rev.110, 1.Google Scholar
  57. 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
  58. 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
  59. Reed, T. M., andGubbins, K. E.,Applied Statistical Mechanics (McGraw-Hill, New York 1973).Google Scholar
  60. Rowlinson, J. S. (1965),Self-consistent Approximation for Molecular Distribution Function, Mol. Phys.9, 217.Google Scholar
  61. 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
  62. 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
  63. 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
  64. Twersky, V. (1962a),On Scattering of Waves by Random Distributions, I. Free-space Scatterer Formalism, J. Math. Phys.3, 700.Google Scholar
  65. Twersky, V. (1962b),On Scattering of Waves by Random Distributions, II. Two-space Scatterer Formalism, J. Math. Phys.3, 724.Google Scholar
  66. Twersky, V. (1962c),Multiple Scattering of Waves and Optical Phenomena, J. Opt. Soc. Am.52, 145.Google Scholar
  67. Twersky, V. (1964),On Propagation in Random Media of Discrete Scatterers, Proc. Sympos. Appl. Math.16, 84.Google Scholar
  68. Twersky, V. (1970),Absorption and Multiple Scattering by Biological Suspensions, J. Opt. Soc. Am.60, 1084.Google Scholar
  69. Twersky, V. (1976),Propagation Parameters in Random Distributions of Scatterers, J. Anal. Math.30, 498.Google Scholar
  70. Twersky, V. (1977),Coherent Scalar Field in Pair-correlated Random Distributions of Aligned Scatterers, J. Math. Phys.18, 2468.Google Scholar
  71. Twersky, V. (1978a),Coherent Electromagnetic Waves in Pair-correlated Random Distributions of Aligned Scatterers, J. Math. Phys.19, 215.Google Scholar
  72. Twersky, V. (1978b),Acoustic Bulk Parameters in Distributions of Pair-correlated Scatterers, J. Acous. Soc. Am.64, 1710.Google Scholar
  73. 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
  74. 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
  75. 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
  76. Varatharajulu V., andPao, Y. H. (1976c)Scattering Matrix for Elastic Waves. I. Theory, J. Acoust. Soc. Am.60, 556.Google Scholar
  77. 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
  78. 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
  79. 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).Google Scholar
  80. Varadan, V. K., (1979a),Scattering of Elastic Waves by Randomly Distributed and Oriented Scatterers, J. Acoust. Soc. Am.65, 655.Google Scholar
  81. 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
  82. 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
  83. 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
  84. 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
  85. 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
  86. 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
  87. 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
  88. 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
  89. 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
  90. 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).Google Scholar
  91. 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
  92. 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
  93. 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
  94. 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
  95. 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
  96. Wang, S. W.,Nondestructive Evaluation of Fracture Strength of Graphites by Ultrasonic Methods M. S. Thesis (Ohio State University, Columbus, Ohio 1983).Google Scholar
  97. Waterman, P. C., andTruell, R. (1961),Multiple Scattering of Waves, J. Math. Phys.2, 512.Google Scholar
  98. Waterman, P. C. (1979),Matrix Theory of Elastic Wave Scattering. I, J. Acoust. Soc. Am.60, 567.Google Scholar
  99. Watts, R. O., andHenderson, D. (1969),Pair Distribution Function for Fluid Hard Spheres, Mol. Phys.16, 217.Google Scholar
  100. Wertheim, M. S. (1963),Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres, Phys. Rev. Lett.10, 321.Google Scholar
  101. Wood, A. B., andWeston, D. E. (1964),The Propagation of Sound in Mud, Acoustica14, 156.Google Scholar
  102. Wu, R.-S. (1982),Mean Field Attenuation and Amplitude Attenuation Due to Wave Scattering, Wave Motion,4, 305.Google Scholar

Copyright information

© Birkhäuser Verlag 1989

Authors and Affiliations

  • V. K. Varadan
    • 1
  • Y. Ma
    • 1
  • V. V. Varadan
    • 1
  1. 1.Research Center for the Engineering of Electronic and Acoustic Materials, Department of Engineering Science and MechanicsThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations