Dispersion of phase velocities in horizontally layered anisotropic slightly-contrasted periodic media
Keywords:periodical medium, dispersion, phase velocity, Floquet wave, perturbation theory
In this article, authors developed a method for estimating the Floquet waves velocity dispersion in a periodic horizontally layered medium with anisotropic layers. The method is based on the calculation of the effective system matrix — the logarithm of the period propagator. In the low-frequency range, the effective system matrix is approximated by the three first terms of the BCH (Baker-Campbell-Hausdorff) series. The eigenvalues of the effective system matrix are the vertical slowness of different wave modes of Floquet waves propagating up and down. To estimate the dispersion of the Floquet waves, the difference matrices are computed from the system matrices of the layers and the system matrix of the Backus averaged medium for the period — the increment matrices. By assuming that the increment matrices are small compared to the system matrix of the Backus medium, a second-order perturbation theory is applied. That allows to compute the eigenvalues of the effective system matrix. As a result, formulas for calculating the approximation of the dispersion of the vertical slowness and phase velocity of Floquet waves in a periodic horizontally layered medium with anisotropic layers are derived. These formulae are given by a weighted sum of the products of various pairs of increments of the system matrices of the layers and allow a very accurate approximation of the dispersion of phase velocities and vertical slowness in the low-frequency range. The accuracy of the calculations is demonstrated in a three-layer periodic medium with orthorhombic layers with different azimuths of symmetry planes. The obtained approximation of the dispersion of the squares of the vertical slowness and phase velocity of the Floquet waves is very accurate in the low-frequency range and gives satisfactory result in the first third of the corresponding pass band.
Brekhovskikh, L. M. (1973). Waves in layered media. Moscow: Nauka, 343 p. (in Russian).
Dynkin, E. B. (1947). Calculation of the coefficients in the Campbell-Hausdorff formula. Doklady Akademii Nauk SSSR, 57, 323—326 (in Russian).
Lankaster, P. (1978). Theory of matrices. Moscow: Nauka, 343 p. (in Russian).
Molotkov, L. A. (1979). Equivalence of periodically layered and transversally isotropic media. Matematicheskiye voprosy teorii rasprostraneniya voln. Zapiski nauchnogo seminara LOMI, 89, 219—233 (in Russian).
Molotkov, L. A., & Khilo, A. E. (1983). The effective media for periodic anisotropic systems. Matematicheskiye voprosy teorii rasprostraneniya voln. Zapiski nauchnogo seminara LOMI, 128, 130—138 (in Russian).
Rytov, S. M. (1956). Acoustical properties of a thinly laminated medium. Akusticheskiy zhurnal, 2(1), 71—83 (in Russian).
Roganov, Yu. V., Stovas, A., & Roganov, V. Yu. (2014). Analysis of pass and stop bands in a periodically layered medium. Tekhnologii seysmorazvedki, (2), 34—41 (in Russian).
Roganov, Yu. V., & Roganov, V. Yu. (2016). Wave propagation in periodic fluidsolid layered media. Geofizicheskiy zhurnal, 38(6), 111—117. doi.org/10.24028/gzh.0203-3100.v38i6.2016.91877 (in Russian).
Sibiryakov, B. P., Maksimov, L. A., & Tatarnikov, M. A. (1980). Anisotropy and dispersion of elastic waves in layered periodic structures. Novosibirsk: Nauka, 72 p. (in Russian).
Yakubovich, V. A., & Starzhinskiy, V. M. (1972). Linear differential equations with periodic coefficients and their applications. Moscow: Nauka, 720 p. (in Russian).
Backus, G. E. (1962). Long-wave elastic anisotropy produced by horizontal layering. Journal of Geophysical Research, 67(11), 4427—4440. https://doi.org/10.1029/JZ067i011p04427.
Braga, A. M., & Herrmann, G. (1992). Floquet waves in anisotropic periodically layered composites. Journal of the Acoustical Society of America, 91(3), 1211—1227. http://dx.doi.org/10.1121/1.402505.
Chicone, C. (2006). Ordinary Differential Equations with Applications. New York: Springer, 635 p. doi: 10.1007/0-387-35794-7.
Coppel, W. A., & Howe, A. (1965). On the stability of linear canonical systems with periodic coefficients. Journal of the Australian Mathematical Society, 5(2), 169—195. https://doi.org/10.1017/S1446788700026756.
Daley, P. F., & Hron, F. (1979). SH waves in layered transversely isotropic media — An asymptotic expansion approach. Bulletin of the Seismological Society of America, 69, 689—711.
Delph, T. J., Hermann, G., & Kaul, R. K. (1978). Harmonic wave propagation in a periodically layered, infinite elastic body: antiplane Strain. Journal of Applied Mechanics, 45(2), 343—349. https://doi.org/10.1115/1.3424299.
Floquet, G. (1883). Sur les equations differentielles lineaires a coefficients periodiques. Annales Scientifiques de l’Ecole Normale Superieure, 12, 47—88.
Gilbert, F., & Backus, G. E. (1966). Propagator matrices in elastic wave and vibration problems. Geophysics, 31, 326—332. https://doi.org/10.1190/1.1439771.
Haskell, N. A. (1953). The dispersion of surface waves on multilayered media. Bulletin of the Seismological Society of America, 43(1), 17—34.
Helbig, K. (1984). Anisotropy and dispersion in periodically layered media. Geophysics, 49(4), 364—373. https://doi.org/10.1190/1.1441672.
Madelung, E. (1964). Die mathematischen Hilfsmittel des Physikers. Berlin, Heidelberg: Springer-Verlag. https://doi.org/10.1002/zamm.19650450221.
Nayfeh, A. H. (1991). The general problem of elastic wave propagation in multilayered anisotropic media. Journal of the Acoustical Society of America, 89, 1521—1528. https://doi.org/10.1121/1.400988.
Nayfeh, A. H. (1989). The propagation of horizontally polarized shear waves in multilayered anisotropic media. Journal of the Acoustical Society of America, 86, 2007—2012. https://doi.org/10.1121/1.398580.
Nayfeh, A. H. (1995). Wave Propagation in Layered Anisotropic Media. Amsterdam: North-Holland, 347 p.
Norris, A. N. (1993). Waves in periodically layered media: A comparison of two theories. Journal of Applied Mathematics, 53, 1195—1209. https://doi.org/10.1137/0153058.
Norris, A. N., & Wang, Z. (1994). Low frequency bending waves in periodic plates. Journal of Sound and Vibration, 169, 485—502. https://doi.org/10.1006/jsvi.1994.1030.
Roganov, Yu., & Stovas, A. (2012). Low-frequency wave propagation in periodically layered media. Geophysical Prospecting, 60(5), 825—837. http://dx.doi.org/10.1111/j.1365-2478.2011.01028.x.
Roganov, Yu., & Stovas, A. (2014). Low-frequency normal wave propagation in a periodically layered medium with weak contrast in elastic properties. Geophysical Prospecting, 62(4), 1205—1210. https://doi.org/10.1111/1365-2478.12167.
Roganov, Yu., Stovas, A., & Roganov, V. (2019). Low-frequency layer-induced dispersion in a weak-contrast vertically heterogeneous orthorhombic. Geophysical Prospecting, 67, 2269—2279. https://doi.org/10.1111/1365-2478.12804.
Schoenberg, M. (1984). Wave propagation in alternating solid and fluid layers. Wave Motion, 6, 303—320. doi: 10.1016/0165-2125(84) 90033-7.
Schoenberg, M., & Muir, F. (1989). A calculus for finely layered anisotropic media. Geophysics, 54, 581—589. https://doi.org/10.1190/1.1442685.
Stovas, A., Roganov, Y., Duffaut, K., & Carter, A. J. (2013). Low-frequency layer-induced anisotropy. Geophysics, 78, WC3—WC14. https://doi.org/10.1190/geo2012-0301.1.
Stroh, A. N. (1962). Steady state problems in anisotropic elasticity. Journal of Mathematics and Physics, 41, 77—103.
Thomson, W. T. (1950). Transmission of elastic waves through a stratified solid medium. Journal of Applied Physics, 21(1), 89—93. https://doi.org/10.1063/1.1699629.
Wang, L., & Rokhlin, S. I. (2002). Floquet wave homogenization of periodic anisotropic media. Journal of the Acoustical Society of America, 112, 38—45. https://doi.org/10.1121/1.1488942.
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