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At lower frequencies in the range *f* < 1 *kHz*, we may typically expect
that Gassmann's results hold for the poroelastic medium, where
. Also, to a very good approximation ,where the only deviations from equality are those due to the differences in the
densities of liquid and gas constituents. So deviations from this
approxiation are most substantial when the porosity is high.
From (detT2), we see that if the the products and
are equal, then these factors can be removed from the
third column of the determinant. Then, the resulting third column can
be subtracted from the first column, and the result can be expanded
along the first column to give:

| |
(46) |

having again used the fact that .So the important zeroes in this case are again those of *J*_{2}, some of
which are already displayed in TABLE 1.
Ignoring the imaginary part of *k*, which is usually quite small
in the limit, we have the analytical result that

| |
(47) |

Thus, at the higher frequencies, this velocity approaches that of the
shear wave as expected. When the lower frequencies are approached,
there is an obvious cutoff frequency, ,below which these torsional modes do not propagate for .Since this low frequency cutoff may often be in conflict with the
assumption under consideration here (*i.e.*,
frequencies low enough that Gassmann's equation is satisfied), we expect
generally that very few of the higher order modes can be excited in
this limit. The main result is therefore that *v*_{z} = *v*_{z}^{0} = *v*_{s}
is the velocity that will be observed in laboratory experiments in
this frequency domain, with only very few exceptions.
We will not consider this rather special case any further in this paper.

** Next:** Higher frequency results
** Up:** HIGHER ORDER TORSIONAL MODES
** Previous:** HIGHER ORDER TORSIONAL MODES
Stanford Exploration Project

11/11/2002