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Transport equations for elastic and other waves in random media. (English) Zbl 0954.74533
Summary: We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to accoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.

MSC:
74J99 Waves in solid mechanics
74J10 Bulk waves in solid mechanics
74E35 Random structure in solid mechanics
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[1] Chandrasekhar, S., Radiative transfer, (1960), Dover New York · Zbl 0037.43201
[2] van de Hulst, H.C., ()
[3] Stott, P., A transport equation for the multiple scattering of electromagnetic waves by a turbulent plasma, J. phys., A 1, 675-689, (1968) · Zbl 0167.26701
[4] Watson, K.; Peacher, J.L., Doppler shift in frequency in the transport of electromagnetic waves in an underdense plasma, J. math. phys., 11, 1496-1504, (1970)
[5] Watson, K., Multiple scattering of electromagnetic waves in an underdense plasma, J. math. phys., 10, 688-702, (1969)
[6] Law, C.W.; Watson, K., Radiation transport along curved ray paths, J. math. phys., 11, 3125-3137, (1970)
[7] Watson, K., Electromagnetic wave scattering within a plasma in the transport approximation, Phys. of fluids, 13, 2514-2523, (1970) · Zbl 0199.60502
[8] Barabanenkov, Yu.; Vinogradov, A.; Kravtsov, Yu.; Tatarskii, V., Application of the theory of multiple scattering of waves to the derivation of the radiative transfer equation for a statistically inhomogeneous medium, Radiofizika, 15, 1852-1860, (1972), Engl. transl. pp. 1420-1425
[9] Besieris, I.M.; Tappert, F.D., Propagation of frequency modulated pulses in a randomly stratified plasma, J. math. phys., 14, 704-707, (1973)
[10] Howe, M.S., On the kinetic theory of wave propagation in random media, Phil. trans. roy. soc. London, 274, 523-549, (1973) · Zbl 0336.73012
[11] Ishimaru, A., ()
[12] Besieris, I.M.; Kohler, W.; Freese, H., A transport-theoretic analysis of pulse propagation through Ocean sediments, J. acoust. soc. am., 72, 937-946, (1982) · Zbl 0535.76084
[13] Barabanenkov, Yu.; Kravtsov, Yu.; Ozrin, V.; Saichev, A., Enhanced backscattering in optics, Progress in optics, 29, 67-190, (1991)
[14] Asch, M.; Kohler, W.; Papanicolaou, G.; Postel, M.; Sheng, P., Frequency content of randomly scattered signals, SIAM review, 33, 519-625, (1991) · Zbl 0736.60055
[15] Froelich, J.; Spencer, T., Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. math. phys., 88, 151-184, (1983) · Zbl 0519.60066
[16] Sheng, P., Introduction to wave scattering, localization, and mesoscopic phenomena, (1995), Academic Press San Diego
[17] K. Case and P. Zweifel Linear transport theory, Addison-Wesley Reading MA.
[18] Larsen, E.; Keller, J.B., Asymptotic solution of neutron transport problems for small Mean free paths, J. math. phys., 15, 75-81, (1974)
[19] Bensoussan, A.; Lions, J.L.; Papanicolaou, G., Boundary layers and homogenization of transport processes, Publ. RIMS, 15, 53-157, (1979) · Zbl 0408.60100
[20] Born, M.; Wolf, E., Principles of optics, (1986), Pergamon Press Oxford
[21] Burridge, R.; Papanicolaou, G., Transport equations for Stokes’ parameters from Maxwell’s equations in a random medium, J. math. phys., 16, 2074-2085, (1975) · Zbl 0307.35076
[22] Lewis, R., Geometrical optics and polarization, I.E.E.E. trans. on antennas and propagation, AP-14, 100-101, (1966)
[23] Wesley, J.P., Diffusion of seismic energy in the near range, J. of geophy. re., 70, 5099-5106, (1965)
[24] Nakamura, Y., Seismic energy transmission in an intensively scattering environment, J. of geophy., 43, 389-399, (1977)
[25] Dainty, A.M.; Toksöz, M.N., Elastic wave propagation in a highly scattering medium, J. of geophy., 43, 375-388, (1977)
[26] Wu, R.S., Multiple scattering and energy transfer of seismic waves-separation of scattering effect from intrinsic attentuation — I. theoretical modelling, Geophys. J. roy. astr. soc., 82, 57-80, (1985)
[27] Wu, R.S.; Aki, K., Multiple scattering and energy transfer of seismic waves — separation of scattering effect from intrinsic attentuation-II, (), 49-80
[28] Toksöz, M.N.; Dainty, A.; Reiter, E.; Wu, R.S., A model for attentuation and scattering in Earth’s crust, Pageoph, 128, 81-100, (1988)
[29] Shang, T.L.; Gao, L.S., Transportation theory of multiple scattering and its application to seismic coda waves of impulse source, Scientia sinica, ser. B., 31, 1503-1514, (1988)
[30] Mayeda, K.; Su, F.; Aki, K., Seismic albedo from the total energy dependence on hypocentral distance in southern California, Phys. Earth planet. int., 67, 104-114, (1991)
[31] McSweeney, T.; Biswas, N.; Mayeda, K.; Aki, K., Scattering and anelastic attentuation of seismic energy in central and southcentral alaska, Phys. Earth planet. int., 67, 115-122, (1991)
[32] Fehler, M.; Hoshiba, M.; Sato, H.; Obara, K., Separation of scattering and intrinsic attentuation for the kanto-tokai region, Japan, Geophys. J. int., 108, 787-800, (1992)
[33] Zeng, Y.; Su, F.; Aki, K., Scattering wave energy propagation in a medium with randomly distributed isotropic scatterers, J. geophys. roy., 96, 607-619, (1991)
[34] Zeng, Y., Compact solutions of multiple scattering wave energy in the time domain, Bull. seism. soc. amer., 81, 1022-1029, (1991)
[35] Hoshiba, M., Simulation of multiple scattered coda wave excitation adopting energy conservation law, Phys. Earth planet inter., 67, 123-126, (1991)
[36] Chen, X.; Aki, K., Energy transfer theory of seismic surface waves in a random scattering and absorption in half space-medium, (), 58-64
[37] Sato, H., Multiple isotropic scattering model including P-S conversions for the seismogram envelope formation, Geophy. J. int., 117, 487-494, (1994)
[38] Zeng, Y., Theory of scattered P-wave and S-wave energy in a random isotropic scattering medium, Bull. of seism. soc. amer., 83, 1264-1276, (1993)
[39] Hansen, R.A.; Ringdal, F.; Richards, P., The stability of RMS lg measurements and their potential for accurate estimation of the yields of soviet underground nuclear explosions, Bull. seism. soc. amer., 80, 2106-2126, (1990)
[40] Keller, J.B.; Lewis, R., Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations, () · Zbl 0848.35068
[41] Wigner, E., On the quantum correction for thermodynamic equilibrium, Physical rev., 40, 749-759, (1932) · JFM 58.0948.07
[42] Courant, R.; Hilbert, D., ()
[43] Burridge, R., Some mathematical topics in seismology, (1976), Courant Inst. of Math. Sciences New York · Zbl 0354.73078
[44] Gihman, I.; Skorohod, A., ()
[45] Karlin, S., Total positivity, (1968), Stanford University Press Stanford · Zbl 0219.47030
[46] Akkermans, E.; Wolf, P.E.; Maynard, R.; Maret, G., Theoretical study of the coherent backscattering of light by disordered media, J. phys. France, 49, 77, (1988)
[47] van Tiggelen, B.A.; Langendijk, A., Rigorous treatment of the speed of diffusing classical waves, Europhys. letters, 23, 311, (1993)
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