Solving the full complex dispersion relation is somewhat tedious, and we will not try to explain this in detail here. Instead we will show results for two cases: first the Massillon sandstone (at 560 Hz) and then the Sierra White granite (at about 200 kHz). We might expect based just on the frequencies that the sandstone behavior will be close to that predicted by Gassmann, while that of the granite may differ from Gassmann.

An important observation concerning how to proceed with the analysis
follows from the fact that we are seeking a curve in the complex
plane, points along the curve depending on the level of saturation *S*.
We know (at least in principle)
the locations of the end points of this curve since they are exactly
the points for full liquid saturation and full gas saturation. If we
assume that the attenuation is relatively small so the wavenumbers
*k*_{s} and *k*_{s}^{*} have small imaginary parts, then to a reasonable
approximation it is the case that the curve of interest lies close to
the real axis in the complex *k*_{z}^{2}-plane. If the imaginary parts
exactly vanish, the curve reduces to a straight line on the real axis
in this plane. These observations suggest that it might be helpful to
trace rays in the complex plane radiating out from the origin, and in
particular
a ray (*i.e.*, a straight line) passing through the origin and also through
the point corresponding to whichever point, *k*_{s}^{2} or (*k*_{s}^{*})^{2},
happens to lie closest to the origin should provide a good starting
point for the analysis. Another alternative is to consider the
straight line that connects these two points directly, even though it
would generally not also be a ray through the origin (except for the
special case when there is no attenuation). Both of these alternatives have
been tried.

The first alternative, a ray through the origin and then passing
through the closest point *k*_{s}^{2} or (*k*_{s}^{*})^{2}, has the very
important
characteristic that the values of the dispersion function become
purely imaginary in the shadow of the starting point of the curve. This
fact provides a great simplification because we need the dispersion
function to vanish identically - both in real and imaginary parts,
and this shadow region has the nice characteristic that the real part
is automatically zero. So the only remaining issue is to check where
the imaginary part vanishes. This procedure is much easier to
implement and to understand intuitively than trying to find the
complex zeroes using something like a Newton method, which could also
be implemented for this problem.

The second alternative is not as rigorous as the first, but for the
case of small attenuation gives very similar results and is especially
easy to implement. In this case we need only consider the line
connecting the two points *k*_{s}^{2} and (*k*_{s}^{*})^{2} in the complex plane.
It turns out that in the two cases considered here, the real part of the
dispersion function is again either zero or very small, so that it
makes sense to treat this approach as an approximation to the first
one in that we need only seek the points where the imaginary part
vanishes. This procedure is very intuitive and examples are shown
in FIGS. 5 through 8.

11/11/2002