Dispersion of phase velocities in horizontally layered anisotropic slightly-contrasted periodic media
DOI:
https://doi.org/10.24028/gzh.0203-3100.v42i3.2020.204704Keywords:
periodical medium, dispersion, phase velocity, Floquet wave, perturbation theoryAbstract
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.
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