The Rheological Models of Becker, Scott Blair, Kolsky, Lomnitz and Jeffreys Revisited, and Implications for Wave Attenuation and Velocity Dispersion

Abstract

The rheological models of Lomnitz and Jeffreys have been widely used in earthquake seismology (to simulate a nearly constant Q medium) and to describe the creep and relaxation behavior of rocks as a function of time. Other similar models, such as those of Becker, Scott Blair and Kolsky, show similar properties, particularly the Scott Blair model describes a perfectly constant Q as a function of frequency. We first give a historical overview of the main scientists and the development and versions of the various models and priorities of discovery. Then, we clarify the relationship between the different versions of these models in terms of mathematical expressions of the complex modulus and calculate the phase velocity and quality factor Q as a function of frequency, illustrating the various special cases. In addition, we give useful hints for the numerical calculation of these moduli, which include special cases of the hypergeometric function.

Appendix: Computation of the Kummer Function

The Kummer function is a special case of the hypergeometric function, which is based on hypergeometric series, a term introduced by John Wallis in his 1655 book Arithmetica Infinitorum.

We have coded the hypergeometric function $_2 \textrm{F}_1 (a,b,c,z)$ in Fortran 77 (Abramowitz 1970; Press et al. 1997, p. 263), such that the Kummer function is

$$\begin{aligned} \textrm{U} (a,c,z) = \frac{\Gamma (1-c)}{\Gamma (a+1-c)} {{\mathcal {M}}} (a,c,z)+ \frac{\Gamma (c-1)}{\Gamma (a)} z^{1-c} {{\mathcal {M}}} (a+1-c,2-c,z), \ \ \ z \text{ complex }, \end{aligned}$$

(50)

where

$$\begin{aligned} {{\mathcal {M}}} (a,c,z) = \textrm{lim}_{b \rightarrow \infty } \ _2 \textrm{F}_1 \left( a,b,c,\frac{z}{b}\right) \end{aligned}$$

(51)

is the confluent hypergeometric function. Then,

$$\begin{aligned} \textrm{U} (1,2+r,\textrm{i}{\bar{f}}) = \frac{\Gamma (-1-r)}{\Gamma (-r)} {{\mathcal {M}}} (1,2+r,\textrm{i}{\bar{f}})+ \frac{\Gamma (1+r)}{\Gamma (1)} (\textrm{i}{\bar{f}})^{-1-r} {{\mathcal {M}}} (-r,-r,\textrm{i}{\bar{f}}) \end{aligned}$$

(52)

or

$$\begin{aligned} \textrm{U} (1,2+r,\textrm{i}{\bar{f}}) = \frac{\Gamma (-1-r)}{\Gamma (-r)} \ {_2 \textrm{F}_1} \left( 1,b,2+r,\frac{\textrm{i}{\bar{f}}}{b}\right) + \frac{\Gamma (1+r)}{\Gamma (1)} (\textrm{i}{\bar{f}})^{-1-r} \ {_2\textrm{F}_1} \left( -r,b,-r,\frac{\textrm{i}{\bar{f}}}{b}\right) , \end{aligned}$$

(53)

provided that b is very large.

The Kummer function can also be used to compute the sine and cosine integrals, namely Eq. (43), and

$$\begin{aligned} \textrm{Si}(z) = \frac{\pi }{2}+ \textrm{Im} [\textrm{E}_1 (z)], \ \ \ \textrm{and} \ \ \ \textrm{Ci}(z) = - \textrm{Re} [\textrm{E}_1 (z)], \end{aligned}$$

(54)

where $\textrm{E}_1 (z) = \exp (-z) \textrm{U}(1,1,z) = \Gamma (0,z)$ (Amos 1990).