Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 68

2.4.4 Generalized wave results.

We now substitute Equation 2.81 into Equation 2.80, and apply the
derivative operations defined by Equation 2.89 throughout. After some algebra,
we obtain a complicated equation, whose real and imaginary parts lead to

a

i nk,0 /n!) [nK n-1 a X + 1/2 n(n-1)K n-2 K X a]

(2.90)

r nk,0 /n!) [aK n - 2 { 1/2 n(n-1)K n-2 a XX
1/2 n(n-1)(n-2) K n-3 K X a X + 1/8 n(n-1)(n-2)(n-3)K n-4 K X 2 a
+ 1/6 n(n-1)(n-2)K n-3 K XX a}]

a T

i 0 a- r nk,0 /n!) [nK n-1 a X +1/2 n(n-1)K n-2 K X a]

(2.91)

i nk,0 /n!) [aK n - 2 { 1/2 n(n-1)K n-2 a XX
1/2 n(n-1)(n-2) K n-3 K X a X + 1/8 n(n-1)(n-2)(n-3)K n-4 K X 2 a
+ 1/6 n(n-1)(n-2)K n-3 K XX a}]

3 r nk,0 /n!) [1/12 n(n-1)(n-2)(n-3)(n-4)K n-5 K X K XX a
1/6 n(n-1)(n-2)(n-3)K n-4 K XX a X
+ 1/6 n(n-1)(n-2)K n-3 a XXX
+ 1/4 n(n-1)(n-2)(n-3)K n-4 K X a XX
+ 1/24 n(n-1)(n-2)(n-3)K n-4 K XXX a
+ 1/8 n(n-1)(n-2)(n-3)(n-4)K n-5 K X 2 a X
+ 1/48 n(n-1)(n-2)(n-3)(n-4)(n-5)K n-6 K X 3 a]
Here, the summations are taken from n=1 to infinity. Equation 2.90
describes the perturbation phase, while Equation 2.91 describes the amplitude of
the slowly varying wave. Both of these equations are cumbersome, though,
since they contain infinite numbers of Taylor series terms. But our use of
centered wavenumber expansions , while the end objective itself in many
stability analyses, in our study serves only an intermediate purpose. It functions
as a mathematical artifice, allowing us to obtain Equations 2.90 and 2.91, which
we will not retain as our final results.
Since these two equations must apply independently of any specific
centered values, we use binomial formulas to sum them analytically. We had
assumed that Taylor expansions in k were legitimate: having carried our
derivations using these expansions, we now to sum the series analytically to
obtain more compact results. This is analogous to our viewing the Taylor series
1 + x + x 2 + x 3 + ... , which is convergent only for | x | < 1, as implying the more
general function 1/(1-x), which is defined for all x, except x = 1.