×

Multiple waves propagate in random particulate materials. (English) Zbl 1427.74073

Summary: For over 70 years it has been assumed that scalar wave propagation in (ensemble-averaged) random particulate materials can be characterized by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the best of the authors’ knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions.

MSC:

74J20 Wave scattering in solid mechanics
45B05 Fredholm integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82D15 Statistical mechanics of liquids
78A48 Composite media; random media in optics and electromagnetic theory
74A40 Random materials and composite materials
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965.
[2] G. Adomian, The closure approximation in the hierarchy equations, J. Stat. Phys., 3 (1971), pp. 127-133, https://doi.org/10.1007/BF01019846.
[3] G. Adomian and K. Malakian, Closure approximation error in the mean solution of stochastic differential equations by the hierarchy method, J. Stat. Phys., 21 (1979), pp. 181-189, https://doi.org/10.1007/BF01008697.
[4] T. Arens, S. N. Chandler-Wilde, and K. O. Haseloh, Solvability and spectral properties of integral equations on the real line: II. \(L^p\)-spaces and applications, J. Integral Equations Appl., 15 (2003), pp. 1-35. · Zbl 1043.45001
[5] C. Aristégui and Y. C. Angel, Effective material properties for shear-horizontal acoustic waves in fiber composites, Phys. Rev. E, 75 (2007), 056607, https://doi.org/10.1103/PhysRevE.75.056607.
[6] L. G. Bennetts and M. A. Peter, Spectral analysis of wave propagation through rows of scatterers via random sampling and a coherent potential approximation, SIAM J. Appl. Math., 73 (2013), pp. 1613-1633, https://doi.org/10.1137/120903439. · Zbl 1277.31001
[7] L. G. Bennetts, M. A. Peter, and H. Chung, Absence of localisation in ocean wave interactions with a rough seabed in intermediate water depth, Quart. J. Mech. Appl. Math., 68 (2015), pp. 97-113, https://doi.org/10.1093/qjmam/hbu024. · Zbl 1310.76017
[8] S. K. Bose and A. K. Mal, Longitudinal shear waves in a fiber-reinforced composite, Internat. J. Solids Structures, 9 (1973), pp. 1075-1085. · Zbl 0263.73018
[9] M. Chekroun, L. Le Marrec, B. Lombard, and J. Piraux, Time-domain numerical simulations of multiple scattering to extract elastic effective wavenumbers, Waves Random Complex Media, 22 (2012), pp. 398-422, https://doi.org/10.1080/17455030.2012.704432. · Zbl 1291.74114
[10] J.-M. Conoir and A. N. Norris, Effective wavenumbers and reflection coefficients for an elastic medium containing random configurations of cylindrical scatterers, Wave Motion, 47 (2010), pp. 183-197, https://doi.org/10.1016/j.wavemoti.2009.09.004. · Zbl 1231.74223
[11] F. de Hoog and I. H. Sloan, The finite-section approximation for integral equations on the half-line, J. Austral. Math. Soc. Ser. B, 28 (1987), pp. 415-434, https://doi.org/10.1017/S0334270000005506. · Zbl 0645.65093
[12] J. Dubois, C. Aristégui, O. Poncelet, and A. L. Shuvalov, Coherent acoustic response of a screen containing a random distribution of scatterers: Comparison between different approaches, J. Phys. Conf. Ser., 269 (2011), 012004, https://doi.org/10.1088/1742-6596/269/1/012004.
[13] J. G. Fikioris and P. C. Waterman, Multiple scattering of waves. \textupII. “Hole corrections” in the scalar case, J. Math. Phys., 5 (1964), pp. 1413-1420, https://doi.org/10.1063/1.1704077. · Zbl 0138.46503
[14] L. L. Foldy, The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev. (2), 67 (1945), pp. 107-119. · Zbl 0061.47304
[15] M. Ganesh and S. C. Hawkins, A far-field based T-matrix method for two dimensional obstacle scattering, ANZIAM J., 51 (2009), pp. C215-C230. · Zbl 1386.76146
[16] M. Ganesh and S. C. Hawkins, Algorithm 975: TMATROM-A T-matrix reduced order model software, ACM Trans. Math. Software, 44 (2017), 9, https://doi.org/10.1145/3054945. · Zbl 1484.78006
[17] A. L. Gower, EffectiveWaves.jl: A package to calculate ensemble averaged waves in heterogeneous materials, https://github.com/arturgower/EffectiveWaves.jl/tree/v0.2.0 (accessed 2018-24-10).
[18] A. L. Gower, I. D. Abrahams, and W. J. Parnell, A Proof that Multiple Waves Propagate in Ensemble-Averaged Particulate Materials, preprint, https://arxiv.org/abs/1905.06996, 2019. · Zbl 1472.74106
[19] A. L. Gower and J. Deakin, MultipleScattering.jl: A Julia library for simulating, processing, and plotting multiple scattering of waves, https://github.com/jondea/MultipleScattering.jl (accessed 2017-12-29).
[20] A. L. Gower, R. M. Gower, J. Deakin, W. J. Parnell, and I. D. Abrahams, Characterising particulate random media from near-surface backscattering: A machine learning approach to predict particle size and concentration, EPL, 122 (2018), 54001.
[21] A. L. Gower, M. J. A. Smith, W. J. Parnell, and I. D. Abrahams, Reflection from a multi-species material and its transmitted effective wavenumber, Proc. A, 474 (2018), 20170864, https://doi.org/10.1098/rspa.2017.0864. · Zbl 1402.74049
[22] M. Gustavsson, G. Kristensson, and N. Wellander, Multiple scattering by a collection of randomly located obstacles-Numerical implementation of the coherent fields, J. Quant. Spectrosc. Radiat. Transfer, 185 (2016), pp. 95-100, https://doi.org/10.1016/j.jqsrt.2016.08.018.
[23] G. Kristensson, Coherent scattering by a collection of randomly located obstacles-An alternative integral equation formulation, J. Quant. Spectrosc. Radiat. Transfer, 164 (2015), pp. 97-108, https://doi.org/10.1016/j.jqsrt.2015.06.004.
[24] G. Kristensson, Evaluation of some integrals relevant to multiple scattering by randomly distributed obstacles, J. Math. Anal. Appl., 432 (2015), pp. 324-337, https://doi.org/10.1016/j.jmaa.2015.06.047. · Zbl 1332.35352
[25] M. Lax, Multiple scattering of waves, Rev. Modern Phys., 23 (1951), pp. 287-310, https://doi.org/10.1103/RevModPhys.23.287. · Zbl 0045.13406
[26] M. Lax, Multiple scattering of waves. II. The effective field in dense systems, Phys. Rev., 85 (1952), pp. 621-629, https://doi.org/10.1103/PhysRev.85.621. · Zbl 0047.23501
[27] C. Layman, N. S. Murthy, R.-B. Yang, and J. Wu, The interaction of ultrasound with particulate composites, J. Acoust. Soc. Am., 119 (2006), pp. 1449-1456, https://doi.org/10.1121/1.2161450.
[28] C. M. Linton and P. A. Martin, Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber, J. Acoust. Soc. Am., 117 (2005), pp. 3413-3423, https://doi.org/10.1121/1.1904270.
[29] C. M. Linton and P. A. Martin, Multiple scattering by multiple spheres: A new proof of the Lloyd-Berry formula for the effective wavenumber, SIAM J. Appl. Math., 66 (2006), pp. 1649-1668, https://doi.org/10.1137/050636401. · Zbl 1111.76051
[30] P. Lloyd and M. V. Berry, Wave propagation through an assembly of spheres: IV. Relations between different multiple scattering theories, Proc. Phys. Soc., 91 (1967), pp. 678-688, https://doi.org/10.1088/0370-1328/91/3/321.
[31] P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles, Cambridge University Press, Cambridge, 2006, https://doi.org/10.1017/CBO9780511735110. · Zbl 1210.35002
[32] P. A. Martin, Multiple scattering by random configurations of circular cylinders: Reflection, transmission, and effective interface conditions, J. Acoust. Soc. Am., 129 (2011), pp. 1685-1695, https://doi.org/10.1121/1.3546098.
[33] P. A. Martin and A. Maurel, Multiple scattering by random configurations of circular cylinders: Weak scattering without closure assumptions, Wave Motion, 45 (2008), pp. 865-880, https://doi.org/10.1016/j.wavemoti.2008.03.004. · Zbl 1231.74228
[34] P. A. Martin, A. Maurel, and W. J. Parnell, Estimating the dynamic effective mass density of random composites, J. Acoust. Soc. Am., 128 (2010), pp. 571-577, https://doi.org/10.1121/1.3458849.
[35] M. I. Mishchenko, Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: A microphysical derivation from statistical electromagnetics, Appl. Optics, 41 (2002), pp. 7114-7134, https://doi.org/10.1364/AO.41.007114.
[36] M. I. Mishchenko, Multiple scattering, radiative transfer, and weak localization in discrete random media: Unified microphysical approach, Rev. Geophys., 46 (2008), RG2003, https://doi.org/10.1029/2007RG000230.
[37] M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media, Phys. Rep., 632 (2016), pp. 1-75, https://doi.org/10.1016/j.physrep.2016.04.002.
[38] M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering, Cambridge University Press, 2006.
[39] M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, T-matrix computations of light scattering by nonspherical particles: A review, J. Quant. Spectrosc. Radiat. Transfer, 55 (1996), pp. 535-575, https://doi.org/10.1016/0022-4073(96)00002-7.
[40] F. Montiel, V. A. Squire, and L. G. Bennetts, Evolution of directional wave spectra through finite regular and randomly perturbed arrays of scatterers, SIAM J. Appl. Math., 75 (2015), pp. 630-651, https://doi.org/10.1137/140973906. · Zbl 1317.31005
[41] K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Coherent backscattering verified numerically for a finite volume of spherical particles, Astrophys. J., 760 (2012), 118, https://doi.org/10.1088/0004-637X/760/2/118.
[42] A. N. Norris, Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites, Internat. J. Engrg. Sci., 24 (1986), pp. 1271-1282, https://doi.org/10.1016/0020-7225(86)90056-X. · Zbl 0594.73032
[43] A. N. Norris, F. Lupp, and J.-M. Conoir, Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers, J. Acoust. Soc. Am., 131 (2012), pp. 1113-1120.
[44] W. J. Parnell and I. D. Abrahams, Multiple point scattering to determine the effective wavenumber and effective material properties of an inhomogeneous slab, Waves Random Complex Media, 20 (2010), pp. 678-701, https://doi.org/10.1080/17455030.2010.510858. · Zbl 1267.74068
[45] W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, Effective properties of a composite half-space: Exploring the relationship between homogenization and multiple-scattering theories, Quart. J. Mech. Appl. Math., 63 (2010), pp. 145-175, https://doi.org/10.1093/qjmam/hbq002. · Zbl 1273.74415
[46] V. J. Pinfield, Thermo-elastic multiple scattering in random dispersions of spherical scatterers, J. Acoust. Soc. Am., 136 (2014), pp. 3008-3017.
[47] J. Przybilla, M. Korn, and U. Wegler, Radiative transfer of elastic waves versus finite difference simulations in two-dimensional random media, J. Geophys. Res., 111 (2006), B04305, https://doi.org/10.1029/2005JB003952.
[48] R. Roncen, Z. E. A. Fellah, F. Simon, E. Piot, M. Fellah, E. Ogam, and C. Depollier, Bayesian inference for the ultrasonic characterization of rigid porous materials using reflected waves by the first interface, J. Acoust. Soc. Am., 144 (2018), pp. 210-221, https://doi.org/10.1121/1.5044423.
[49] S. Rupprecht, L. G. Bennetts, and M. A. Peter, On the calculation of wave attenuation along rough strings using individual and effective fields, Wave Motion, 85 (2019), pp. 57-66, https://doi.org/10.1016/j.wavemoti.2018.10.007. · Zbl 07215317
[50] G. Sha, Correlation of elastic wave attenuation and scattering with volumetric grain size distribution for polycrystals of statistically equiaxed grains, Wave Motion, 83 (2018), pp. 102-110, https://doi.org/10.1016/j.wavemoti.2018.08.012. · Zbl 1469.74071
[51] P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, Springer Ser. Materials Sci. 88, Springer-Verlag, Berlin, Heidelberg, 2006.
[52] V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, Scattering of electromagnetic waves by ensembles of particles and discrete random media, J. Quant. Spectrosc. Radiat. Transfer, 112 (2011), pp. 2095-2127, https://doi.org/10.1016/j.jqsrt.2011.04.010.
[53] L. Tsang, C. T. Chen, A. T. C. Chang, J. Guo, and K. H. Ding, Dense media radiative transfer theory based on quasicrystalline approximation with applications to passive microwave remote sensing of snow, Radio Sci., 35 (2000), pp. 731-749, https://doi.org/10.1029/1999RS002270.
[54] L. Tsang and A. Ishimaru, Radiative wave equations for vector electromagnetic propagation in dense nontenuous media, J. Electromagn. Waves Appl., 1 (1987), pp. 59-72, https://doi.org/10.1163/156939387X00090.
[55] L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, John Wiley & Sons, 2004.
[56] V. Twersky, On scattering of waves by random distributions. \textupI. FreeSpace scatterer formalism, J. Math. Phys., 3 (1962), pp. 700-715, https://doi.org/10.1063/1.1724272. · Zbl 0111.41603
[57] V. K. Varadan, Scattering of elastic waves by randomly distributed and oriented scatterers, J. Acoust. Soc. Am., 65 (1979), pp. 655-657, https://doi.org/10.1121/1.382419.
[58] V. K. Varadan, V. N. Bringi, and V. V. Varadan, Coherent electromagnetic wave propagation through randomly distributed dielectric scatterers, Phys. Rev. D, 19 (1979), pp. 2480-2489, https://doi.org/10.1103/PhysRevD.19.2480.
[59] V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru, Multiple scattering theory for waves in discrete random media and comparison with experiments, Radio Sci., 18 (1983), pp. 321-327, https://doi.org/10.1029/RS018i003p00321.
[60] V. K. Varadan, Y. Ma, and V. V. Varadan, A multiple scattering theory for elastic wave propagation in discrete random media, J. Acoust. Soc. Am., 77 (1985), pp. 375-385, https://doi.org/10.1121/1.391910. · Zbl 0574.73040
[61] V. K. Varadan, V. V. Varadan, and Y.-H. Pao, Multiple scattering of elastic waves by cylinders of arbitrary cross section. I. SH waves, J. Acoust. Soc. Am., 63 (1978), pp. 1310-1319. · Zbl 0381.73028
[62] P. C. Waterman, Symmetry, unitarity, and geometry in electromagnetic scattering, Phys. Rev. D, 3 (1971), pp. 825-839, https://doi.org/10.1103/PhysRevD.3.825.
[63] P. C. Waterman and R. Truell, Multiple scattering of waves, J. Math. Phys., 2 (1961), pp. 512-537, https://doi.org/10.1063/1.1703737. · Zbl 0108.21403
[64] R. L. Weaver, Diffusivity of ultrasound in polycrystals, J. Mech. Phys. Solids, 38 (1990), pp. 55-86, https://doi.org/10.1016/0022-5096(90)90021-U. · Zbl 0706.73028
[65] U. Wegler, M. Korn, and J. Przybilla, Modeling full seismogram envelopes using radiative transfer theory with Born scattering coefficients, Pure Appl. Geophys., 163 (2006), pp. 503-531, https://doi.org/10.1007/s00024-005-0027-5.
[66] R. Weser, S. Wöckel, B. Wessely, and U. Hempel, Particle characterisation in highly concentrated dispersions using ultrasonic backscattering method, Ultrasonics, 53 (2013), pp. 706-716, https://doi.org/10.1016/j.ultras.2012.10.013.
[67] R. West, D. Gibbs, L. Tsang, and A. K. Fung, Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media, J. Opt. Soc. Amer. A, 11 (1994), pp. 1854-1858, https://doi.org/10.1364/JOSAA.11.001854.
[68] R.-B. Yang, A dynamic generalized self-consistent model for wave propagation in particulate composites, J. Appl. Mech., 70 (2003), pp. 575-582, https://doi.org/10.1115/1.1576806. · Zbl 1110.74773
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.