Summary
The problem of propagation of plane harmonic waves in an infinite elastic solid containing a random configuration ofN identical scatterers of finite size is considered. The paper presents the general formalism of the analysis of mean (or coherent) elastic waves for an arbitrary shape of the scatterers and the solutions of some specific problems in the case of spherical scatterers. The equations expressing the average total field in terms of the probability density function and scattering properties of individual scatterers are presented. These equations concern an arbitrary shape of the scatterers and include the effects of single and multiple scattering. In the case of dilute and uniform distribution of spherical scatterers the more detailed analysis of average total field leads to the determination of the equivalent homogeneous medium in which the mean wave propagates. For the limit of the low frequences the effective parameters of the medium are given and they turn out to be the same as those obtained within effective modulus theory in static case.
Zusammenfassung
Das Problem der Ausbreitung von ebenen harmonischen Wellen in einem unendlichen, elastischen Körper, derN identische Streukörper endlicher Größe enthält, wird betrachtet. Der allgemeine Formalismus der Untersuchung der mittleren (oder kohärenten) Wellen für beliebige Gestalt der Streukörper und Lösungen einiger spezieller Probleme für kugelförmige Streukörper wird angegeben. Die angegebenen Gleichungen für das mittlere totale Feld in Abhängigkeit von der Dichtefunktion und den Streueigenschaften der Streukörper schließen einfache und mehrfache Streuung ein. Im Fall schwacher, gleichförmig verteilter, kugelförmiger Streukörper führt eine detailliertere Untersuchung des mittleren totalen Feldes zur Bestimmung eines äquivalenten homogenen Werkstoffes, in dem sich die mittlere Welle ausbreitet. Für den Grenzfall niedriger Frequenzen werden die effektiven Parameter angegeben. Es zeigt sich, daß es dieselben wie nach der Theorie der effektiven Moduli im statischen Fall sind.
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References
Sobczyk, K.: Methods of statistical dynamics (in polish). Warsaw: Polish Scientific Publishers. 1973.
Sobczyk, K.: Wave propagation in random media (in Russian). Miechanika, Izd. “Mir”, Nr. 6*148 (1974).
Twersky, Y.: On multiple scattering of waves. J. Res. Nat. Bur. StandardsD/64. 715–730 (1960).
Burke, J. E., and Y. Twersky: On scattering of waves by many bodies. Radio Sci., Journ. of Res. Nat. Bur. Standards68Nr. 4 (1964).
Foldy, L. L.: The multiple scattering of waves. Phys. Rev.67Nr. 3-4 (1945).
Lax, M.: Multiple scattering of waves. Rev. Modern Phys.23, 287–310 (1951).
Twersky, Y.: On scattering of waves by random distributions. Part I: J. Math. Phys.3, 700 (1962); Part II: J. Math. Phys.3, 724 (1962).
Mathur, N. C., andK. C. Yeh: Multiple scattering of electromagnetic waves by random scatterers. Report of Dept. Electr. Eng. Univ. of Illinois, Urbana 1963.
Barrat, P. J.: Scattering of plane harmonic waves in an infinite elastic solic by an arbitrary configuration of two-dimensional obstacles. Journ. Inst. Math. and App.4, 233 (1968).
Barrat, P. J.: Multiple scattering of plane harmonic elastic waves in an infinite solid by an arbitrary configuration of obstacles. Proc. Cambr. Phil. Soc., Part II,66, 469 (1969).
Guz, A. N., andD. T. Golowczan: Diffraction of elastic waves in multiconnected media (in Russian). Kiev. 1972.
Batchelor, G. K.: Sedimentation in a dilute dispersion of spheres. J. Fluid Mech.52, Part 2, march (1972).
Kupradze, V. D.: Dynamical problems in elasticity (Progress in solid mechanics, Vol. 3).Sneddon, I. N., andR. Hill, eds.) North-Holland. 1963.
Sobczyk, K.: Elastic wave propagation in a discrete random medium (in Polish). IFTR Reports 5/1974.
Einspruch, N. G., E. J. Witterholt, andR. Truell: Scattering of plane transverse wave by a spherical obstacle in an elastic medium. J. Appl. Phys.31Nr. 5 (1960).
Morse, P. M., andH. Feshbach: Methods of mathematical physics. McGraw-Hill. 1953.
Mal, A. K., andI. Herrera: Scattering of Love waves by a construction in the crust. J. Geoph. Res.70Nr. 4 (1965).
Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inculusion and related problems. Proc. Roy. Soc. (London) Ser.A 241, 376 (1957).
Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids13Nr. 4 (1965).
McCoy, J. J.: On the dynamic response of discordered composities. J. Appl. Mech. June (1973).
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Sobczyk, K. Elastic wave propagation in a discrete random medium. Acta Mechanica 25, 13–28 (1976). https://doi.org/10.1007/BF01176926
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DOI: https://doi.org/10.1007/BF01176926