Abstract
Solutions of P-SV equations of motion in a homogeneous transversely isotropic elastic layer contain a factor exp(±ν j z), where z is the vertical coordinate and j = 1, 2. For computing Rayleigh wave dispersion in a multi-layered half space, ν j is computed at each layer. For a given phase velocity (c), ν j becomes complex depending on the transversely isotropic parameters. When ν j is complex, classical Rayleigh waves do not exist and generalised Rayleigh waves propagate along a path inclined to the interface. We use transversely isotropic parameters as α H , β V , ξ, ϕ and η and find their limits beyond which ν j becomes complex. It is seen that ν j depends on ϕ and η, but does not depend on ξ. The complex ν j occurs when ϕ is small and η is large. For a given c/β V , the region of complex ν j in a ϕ -η plane increases with the increase of α H /β V . Further, for a given α H /β V , the complex region of ν j increases significantly with the decrease of c/β V . This study is useful to compute dispersion parameters of Rayleigh waves in a layered medium.
References
Anderson DL (1961) Elastic wave propagation in layered anisotropic media. J Geophys Res 66:2953–2963
Anderson DL (1966) Recent evidence concerning the structure and composition of the Earth’s mantle. Physics and Chemistry of the Earth, vol 6. Pergamon Press, Oxford, pp 1–131
Anderson DL (1989) Theory of the Earth. Blackwell Scientific Publications, Boston
Bhattacharya SN (1987) Reduction of deep layers in surface wave computation. Geophys J R Astron Soc 88:97–109
Dziewonski AM, Anderson DL (1981) Preliminary reference Earth model. Phys Earth Planet Inter 25:297–356
Harkrider DG, Anderson DL (1962) Computation of surface wave dispersion for multilayered anisotropic media. Bull Seismol Soc Am 52:321–332
Ikeda T, Matsuoka T (2013) Computation of Rayleigh waves on transversely isotropic media by reduced delta matrix method. Bull Seismol Soc Am 103:2083–2093
Ke G, Dong H, Kristensen A, Thompson M (2011) Modified Thomson-Haskell matrix methods for surface wave dispersion curve calculation and the accelerated root-searching schemes. Bull Seismol Soc Am 101:1692–1703
Love AEH (1927) A treatise on the mathematical theory of elasticity. Cambridge Univ. Press, Cambridge
Nishimura CE, Forsyth DW (1989) The anisotropic structure of the upper mantle in the Pacific. Geophys J R Astron Soc 96:203–229
Rabinowitz S (1993) How to find the square root of a complex number? Math Inform Q 3:54–56
Takeuchi H, Saito M (1972) Seismic surface waves. Methods in computational physics, vol 11. Academic, New York, pp 217–295
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Bhattacharya, S.N. Limits of transversely isotropic elastic parameters for the existence of classical Rayleigh waves. J Seismol 21, 237–241 (2017). https://doi.org/10.1007/s10950-016-9565-9
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DOI: https://doi.org/10.1007/s10950-016-9565-9