Rheological properties and velocity dispersion of a medium with power-law dependence of Q on frequency

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G. Muller


The Kramers-Kronig relations for magnitude and phase of a linear causal filter are used to derive an exact general expression for the viscoelastic modulus M, corresponding to power laws for the quality factor, Qωγ. The exponent γ varies from -1 to + 1, such that the spectrum of rheologies extends from a Kelvin-Voigt to a Maxwell body. High- and low-frequency approximations for M(ω) are derived, and in the special cases γ = ±1, ±1/2, ±1/3, ±1/4,... closed-form solutions are given which apply for arbitrary frequencies. With M(ω) at hand, both high-frequency phenomena such as velocity dispersion and low-frequency phenomena such as creep and stress relaxation can be investigated. Results for phase-velocity dispersion are given as well as short- and long-time-scale approximations of the creep and relaxation functions. Simple dissipation operators are derived which can be convolved with theoretical seismograms in order to correct these for the influence of absorption. Some results on relaxation spectra for the case 0 ≤ γ ≤ 1 are summarized in an appendix. Taken together, the results of this paper suggest that media with 0 < γ < 1 should be considered as generalized Maxwell bodies and media with -1 < γ < 0 as generalized Kelvin-Voigt bodies. Application of the Kramers-Kronig relations to the viscoelastic modulus is better than the use of those relations in conjunction with the wavenumber of a plane wave, which is the procedure that has been employed so far.

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Muller, G. 1983. “Rheological Properties and Velocity Dispersion of a Medium With Power-Law Dependence of Q on Frequency”. Journal of Geophysics 54 (1), 20-29. https://journal.geophysicsjournal.com/JofG/article/view/82.
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