Harmonic Analysis: Elementary Pipe and Collar Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 88

u 1 = C 1 e i Z(x/c + t) + C 2 e i Z(-x/c + t) , x < 0

(3.A.6)

u 2 = C 3 e i Z(x/c + t) + C 4 e i Z(-x/c + t) , x > 0

(3.A.7)

That u 1 and u 2 represent left and right-going waves, respectively, requires C 2 = 0
and C 3 = 0. This illustrates the implementation of so-called “radiation” or
“outgoing wave” conditions in classical physics. Continuity of fluid
displacement requires

u 1 (x,t) = u 2 (x,t) at x = 0 (3.A.8)

If the dipole source produces a 'p in the form p 2 - p 1 = p s e i Zt , then
B wu 1 /wx - B wu 2 /wx = p s e i Zt at x = 0 (3.A.9)

These conditions lead to the pressure solutions
p 1 (x,t) = - B wu 1 /wx = - ½ p s e i Z(x/c + t) , x < 0

(3.A.10)

p 2 (x,t) = - B wu 2 /wx = + ½ p s e i Z(-x/c + t) , x > 0

(3.A.11)

These equations state the obvious fact that a 'p pulse of strength p s splits into
two waves that travel in opposite directions having equal and opposite strengths
- ½ p s and + ½ p s . For this “infinite-infinite” system, nothing else happens that
is of physical interest. It is interesting to note that |p 1 | /'p = |p 2 | /'p = 0.5
identically for this ideal infinite-infinite system, a property explained previously.

3.1.2 Case (b), drillbit as a solid reflector.
Here we consider the wave motion in Figure 3.A.1b. This applies when
the drillbit nozzles are very small and when the drillbit is firmly pressing against
hard rock. The problem is defined on x > 0 and the dipole source is now located
at x = x s . For 0 < x < x s , we take u 1 with linear combinations of sin Zx/c and
cos Zx/c to model standing waves, but for x > x s , our u 2 is proportional to e i Z(-x/c
+ t) to represent a non-reflecting propagating wave traveling to the right. Since a
solid reflector satisfies u = 0 at x = 0, and Equations 3.A.8 and 3.A.9 now apply
at x = x s , we find the solution
p 2 (x,t) = + i {2 sin (Zx s /c)} ½ p s e i Z(-x/c + t) , x > x s (3.A.12)

3.1.3 Case (c), drillbit as open-ended reflector.
Here we consider the wave motion in Figure 3.A.1c. Although we often
think of a drillbit as a solid reflector because the nozzle area is small compared
to the cross-sectional area, this is not true most of the time. In Case (a), we
found that |p 1 | /'p = |p 2 | /'p = 0.5 in the absence of reflections. In Chapter 2 for
our six-segment waveguide, we found that at very low frequencies, source
position was unimportant in the drill collar and p pipe /'p | 0.95, approximately,
or almost 1.0. This result, as explained in that write-up, is consistent with
modeling the drillbit as an open-ended reflector. The problem here is defined on