Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 6

relevant to MWD signal enhancement and subject of several domestic patents).
Also, from Equation 1.5, an incident displacement pulse +f(x+ct) reflects from x
= 0 as -f(-x+ct): it reflects with the same shape and size, but “flips over” with a
sign change to enforce u = 0. Since both terms in u x (x,t) = f '(x+ct) + f '(-x+ct)
have like signs, an incident stress pulse reflects with the same sign.
1.1.2.2 Example 1-2. Stress-free end.
On the other hand, consider a stress-free boundary condition satisfying
u x (0,t) = 0. Obviously, the choice
u(x,t) = f(x+ct) + f(-x+ct) (1.6)
solves the problem, since u x (x,t) = f '(x+ct) - f '(-x+ct) vanishes at x = 0. Note
that u(0,t) = 2 f(ct); thus the displacement and the velocity u t (0,t) = 2 cf '(ct)
double (the latter relative to the incident cf '(ct) value) at a stress-free boundary.
Unlike Example 1-1, a free end causes an incident displacement wave f(x+ct) to
reflect with the same sign as f(-x+ct), again with its shape and size the same.
But since the terms in u x (x,t) = f '(x+ct) - f '(-x+ct) have unlike signs, an incident
stress pulse reflects with a sign change.
For both rigid and free ends, incident pulse shapes remain unchanged. But
shape invariance is not the norm, even for our nondispersive Equation 1.1;
distortions can arise, as we will show, from non-standard elastic boundary
conditions. How might our results be relevant to drillstring vibrations? Suppose
our drillbit, assumed rigidly anchored to the rock, were modeled as a zero
displacement condition at x = 0 (this crude model is not used later). Then, it is
clear that an approaching wave must double in stress at the bit as it reflects,
increasing its tendency to break rock. And to MWD? An incident pressure
signal from downhole will double at mud pump pistons, a phenomenon that has
been measured experimentally - a novel “one hundred feet hose”
implementation of this idea has been awarded two United States patents. The
same pulse might, however, depending on amplitude and frequency, reflect at a
desurger with significant signal distortion, causing surface detection problems.
1.1.2.3 Note on acoustics.
Equation 1.1 applies to acoustics problems, where axial or longitudinal
vibrations of fluid columns are of interest. The only change is the new
definition c 2 = B / , where B is the bulk modulus of the fluid. For air, the stress-
free boundary condition models long wave sound emission by an open pipe into
a large empty space such as a room. New acoustics students often find difficulty
envisioning waves reflecting at “nothing.” After all, since ping-pong balls pass
through and out, why shouldn' t sound waves? They don't: in fluid mechanics,
acoustic and hydraulic flow problems differ distinctly and satisfy different
equations. In this topic, by “acoustics,” we mean wave motions arising from the