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For fully saturated porous cylinders, the factor that determines the
torsional modes of propagation is *m*_{11}(*r*) in (46). The
critical factor here is the Bessel function *J*_{2}(*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 *J*_{2}'(*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} - *J*_{2}(*j*_{2,n}^{old})/*J*_{2}'(*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

T

ABLE 1.The zeroes

*j*_{2,n} of

*J*_{2}(

*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 (which could then
be safely eliminated) if we first multiply the
bottom row by a factor of . Having made these
simplifications, we find

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

Some elementary consequences of this equation are: (a)
As so there is no liquid left in the system,
*J*_{1}(*k*_{sr}*R*_{1}) and like *R*_{1}, while
*Y*_{1}(*k*_{sr}^{*}*R*_{1}) and like 1/*R*_{1}.
So the dispersion relation is always satisfied in the limit when
*J*_{2}(*k*_{sr}^{*}*R*_{2}) = 0, which is exactly the condition for the fully
dry cylinder as expected. (b) If , then the first
determinant vanishes identically. The second determinant does not
vanish in general since it approaches the Wronskian
, so the condition
becomes *k*_{sr}*J*_{2}(*k*_{sr}*R*_{1}) = 0, again as expected.
(c) The special case of does
not affect these conclusions, as both *J*_{1}(*k*_{sr}*R*_{1}) and
in this limit, as they should. (d) The only case
that is missing from (detT3) is the one for . 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
.
We conclude that, with the one trivial exception just noted,
these simplifications have kept the basic nature of the
dispersion relation intact.

** Next:** Lower frequency results
** Up:** Berryman and Pride: Cylinder
** Previous:** ELEMENTARY TORSIONAL MODES
Stanford Exploration Project

11/11/2002