Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 195

Thus, we in effect adopt the phase velocity convention c p =

0 /k > 0 for upgoing

waves. Since A

r 2 = EI k 4 - Nk 2 , it follows that A 0 2 = EI k 4 - Nk 2 > 0 , and

hence, that (EI k 3 - Nk) > 0. Therefore,
C g = r / k = (2EI k 3 - Nk)/ A 0
= ( EI k 3 - Nk)/ A 0 +EI k 3 / A 0 > 0 (4.3.11)
since each of the constituent terms on the right side is greater than zero. In other
words, a wave that is moving up toward the surface in the positive direction will
never reflect. Thus, a wave encountering a focus along its direction of travel
must remain there, since it cannot reverse direction.
Now, let us combine Equations 4.3.6 and 4.3.10, and solve for the
wavenumber distribution k(x). We have, using the quadratic formula,
k 2 = { N + (N 2 + 4 EI A

0 2 )}/(2EI)

(4.3.12)

and hence,
k = [{N + (N 2 + 4 EI A 0 2 )}/(2EI)] (4.3.13)
Our results can be expressed in a more concise and physically revealing form if
we introduce the nondimensional (starred) quantities,
N * = N / { 2 0 (EI A)} (4.3.14)
k * = { EI / ( A 0 2 )} 1/4 k (4.3.15)
C g * = 1/2 { A/(EI 0 2 )} 1/4 C g (4.3.16)
Then, straightforward algebraic manipulations lead to the polynomial formulas
C g * = {N * + (N *2 +1 ) } 1/2 (N *2 +1)

(4.3.17)

k * = { N * + (N *2 +1 ) } 1/2 (4.3.18)

which can be readily programmed and evaluated.