General behavior of the curves

 

ms_quadratic
Figure 9 Comparison between the points that solve the dispersion relation in the complex k_z²-plane, and a simple quadratic fit in water saturation S, for the Massillon sandstone measured by Murphy. The fit is nearly perfect for this case.

 

swg_quadratic
Figure 10 Comparison between the points that solve the dispersion relation in the complex k_z²-plane, and a simple quadratic fit in water saturation S, for the Sierra White granite measured by Murphy. The fit is also good for this case, but not as good as it was for Massillon sandstone.

Since the curves obtained in FIGS.6 and 8 are very well-behaved, it seemed like a good idea to check for simple dependencies on the saturation parameter S. Both curves look like they might be quadratic in S. This hypothesis is tested in FIGS.9 and 10. We find that the quadratic dependence is essentially exact to graphical accuracy for Massillon sandstone, and it is close but not exact for the Sierra White granite. We have not yet tried to analyze this behavior and will therefore not attempt an explanation of it at this time.