Lower frequency results

At lower frequencies in the range f < 1 kHz, we may typically expect that Gassmann's results hold for the poroelastic medium, where $\mu^* \equiv \mu$. Also, to a very good approximation $k_s^* \simeq k_s$,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 $\mu k_{sr}$ and $\mu^* k_{sr}^*$ 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:  

$${{2J_2(k_{sr}R_2)}\over{\pi k_{sr}R_1}} = 0,$$ (230)

having again used the fact that $J_2(z)Y_1(z)-J_1(z)Y_2(z) = 2/\pi z$.So the important zeroes in this case are again those of J₂, 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  

$$v_z^{(n)} = {{v_s}\over{[1 - (j_{2,n}v_s/\omega R_2)^2}]^{1/2}}.$$ (231)

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, $f_c^n = j_{2,n}v_s/2\pi R_2$,below which these torsional modes do not propagate for $n \ge 1$.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_z⁰ = 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.