For fully saturated porous cylinders, the factor that determines the torsional modes of propagation is m11(r) in (46). The critical factor here is the Bessel function J2(js) and, specifically, the whereabouts of its zeroes. One source of this information, to five figure accuracy, is the reference of Abramowitz and Stegun (1965), which provides not only the location of the zeroes j2,n, but also the values of the corresponding derivatives J2'(j2,n). Having these derivatives is useful for improving the accuracy of the zeroes with a Newton-Raphson iterative method, based on j2,n = j2,nold - J2(j2,nold)/J2'(j2,nold). This approach gives a very rapid improvement to the values of the j2,n's within 2 to 3 iterations. The results to order n=3 are shown in TABLE 1.
Having already understood the zeroth order contributions to the dispersion relation (detT) due to zeroes of ksr and ksr*, we are now free to consider that neither of these factors vanishes for the higher order modes. This assumption permits us to factor these wavenumbers in or out of the determinant whenever it is convenient to do so. In particular, we note that the first two columns of (detT) would have a common factor of (which could then be safely eliminated) if we first multiply the bottom row by a factor of . Having made these simplifications, we find
Expanding the determinant along the third column, we have
We conclude that, with the one trivial exception just noted,
these simplifications have kept the basic nature of the
dispersion relation intact.