HIGHER ORDER TORSIONAL MODES

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(j_s) 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 j_2,n, but also the values of the corresponding derivatives J₂'(j_2,n). Having these derivatives is useful for improving the accuracy of the zeroes with a Newton-Raphson iterative method, based on j_2,n = j_2,n^old - J₂(j_2,n^old)/J₂'(j_2,n^old). This approach gives a very rapid improvement to the values of the j_2,n's within 2 to 3 iterations. The results to order n=3 are shown in TABLE 1.

1.5

TABLE 1.The zeroes j_2,n of J₂(z) as a function of the order n of appearance along the real axis.

1.0

Having already understood the zeroth order contributions to the dispersion relation (detT) due to zeroes of k_sr and k_sr^*, 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 $\mu^* (k_{sr}^*)^2$ (which could then be safely eliminated) if we first multiply the bottom row by a factor of $\mu^* k_{sr}^*$. Having made these simplifications, we find

$$\begin{eqnarray} \left\vert\begin{array} {ccc} J_2(k_{sr}^*R_2) & Y_2(k_{sr}^*R_... ...\mu^*k_{sr}^*J_1(k_{sr}R_1)\end{array}\right\vert = 0,\nonumber\ \end{eqnarray}$$

after also eliminating a common factor of -1 from the top row, and -k_sr from the third column.

Expanding the determinant along the third column, we have

$$\begin{eqnarray} 0 = \mu^*k_{sr}^*J_1(k_{sr}R_1)\left\vert\begin{array} {cc} J_2... ...1(k_{sr}^*R_1) & Y_1(k_{sr}^*R_1) \ end{array}\right\vert.\quad \end{eqnarray}$$

Some elementary consequences of this equation are: (a) As $R_1 \to 0$ so there is no liquid left in the system, J₁(k_srR₁) and $J_2(k_{sr}R_1) \to 0$ like R₁, while Y₁(k_sr^*R₁) and $Y_2(k_{sr}^*R_1) \to \infty$ like 1/R₁. So the dispersion relation is always satisfied in the limit when J₂(k_sr^*R₂) = 0, which is exactly the condition for the fully dry cylinder as expected. (b) If $R_1 \to R_2$, then the first determinant vanishes identically. The second determinant does not vanish in general since it approaches the Wronskian $J_2Y_1-J_1Y_2 = 2/\pi k_{sr}^*R_2$, so the condition becomes k_srJ₂(k_srR₁) = 0, again as expected. (c) The special case of $k_{sr} \to 0$ does not affect these conclusions, as both J₁(k_srR₁) and $J_2(k_{sr}R_1) \to 0$ in this limit, as they should. (d) The only case that is missing from (detT3) is the one for $k_{sr}^* \to 0$. But this multiple zero of the original dispersion relation (detT) was eliminated when we removed two factors of (k_sr^*)² from the first and second column in the first step of our simplification of the dispersion relation - a step which is always legitimate except when $k_{sr}^* \equiv 0$.

We conclude that, with the one trivial exception just noted, these simplifications have kept the basic nature of the dispersion relation intact.