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Next: Conclusions Up: Berryman: Dynamic permeability Previous: Discussion of analytical examples

NUMERICAL EXAMPLES OF THE DYNAMIC PERMEABILITY FUNCTION

From (32) and the preceding discussion, we conclude that a reasonable choice of the functional form for dynamic permability is
   \begin{eqnarray}
{{\kappa(\omega)}\over{\kappa_0}} =
{{1}\over{(1-iP\omega/\omega_0)^{1/2} - i\omega/\omega_0}},
 \end{eqnarray} (43)
where $0 \le P \le 1$ and $\omega_0 = \eta\phi/\alpha\kappa_0$.We plot this function in the complex plane for five choices of P in Figure 1. The polar angle $\theta$ is displayed as a function of $\omega/\omega_0$ in Figure 2. Then, the real and imaginary parts are plotted in Figures 3 and 4 as a function of the quantity $\omega/\omega_0$.

Figure 1 shows that all these choices give very similar behavior in the complex plane. The function looks much like a semi-circle centered at the point (1/2,0) and having radius 1/2 in all cases. But this observation is only exactly true for the case P = 0. For that case it is also true that the relationship between $\omega$ and the polar angle $\theta$ in the complex plane is given exactly by $\omega/\omega_0 = \tan(\theta/2)$. For the other values of P, this relationship holds approximately true for $\omega/\omega_0 \le 1$and $P \le 1/2$, but deviations become substantial in all cases for higher values of $\omega$, as is observed in Figure 2.

 
fivemodels
Figure 1
lustration of the behavior in the complex plane of five model functions for $\kappa(\omega)/\kappa_0$of the form found in Eq. (43) for the different values of P = 0, 0.25, 0.4, 0.5, 1. Note that the case P=0 is exactly a semi-circle of radius 1/2 centered at the point (1/2,0) in the complex plane.
fivemodels
view

 
angledeg
Figure 2
Polar angle $\theta$ in degrees in the complex plane of points of five model functions of the form found in Eq. (43) for the different values of P = 0, 0.25, 0.4, 0.5, 1 (see Figure 1). Note that the case P=0 is exactly a semi-circle of radius 1/2 centered at the point (1/2,0) in the complex plane, and for this case $\omega/\omega_0 = \tan(\theta/2)$. All other cases are observed to deviate from this behavior.
angledeg
view

 
fivemodsre
Figure 3
Illustration of the behavior of the real part of five model functions for different values of P = 0, 0.25, 0.4, 0.5, 1 as a function of the argument $\omega/\omega_0$.Note that the real part of the dynamic permeability is essentially flat for a wide range of the smaller frequencies, but then falls off rapidly as the frequency approaches the resonance frequency $\omega_0$ from below.
fivemodsre
view

 
fivemodsim
Figure 4
Illustration of the behavior of the imaginary part of five model functions for different values of P = 0, 0.25, 0.4, 0.5, 1 as a function of the argument $\omega/\omega_0$.Note the main region of the deviation (half maximum and above) lies approximately in the range $0.2 \le \omega/\omega_0 \le 3.0$,which is just slightly over one decade in width.
fivemodsim
view

Figure 3 shows that the real part of the dynamic permeability is essentially flat for a wide range of the smaller frequencies, but then the function falls off rapidly as the frequency gets close to the resonance frequency $\omega_0$. Figure 4 shows that all the action occurs over six decades of frequency and the main region of the deviation (half maximum and above in the imaginary part) lies approximately in the range $0.2 \le \omega/\omega_0 \le 3.0$, which is just slightly over one decade in width.


next up previous print clean
Next: Conclusions Up: Berryman: Dynamic permeability Previous: Discussion of analytical examples
Stanford Exploration Project
7/8/2003