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Schematic
f(t - x/c)
g(t + x/c)
x a
x b
x
Figure 4.3a. General bi-directional waves.
4.3.2 Theory.
The pressure wave taken here is the sum of two waves traveling in
opposite directions. Let “f” denote the incident wave traveling from downhole
(again, this upgoing wave is not the “clean” 'p signal, but the intended signal
plus ghost reflections associated with reflections at the drillbit and collar-pipe
junction). Then, “g” will denote reflections of any type at the mudpump
(positive displacement pistons and centrifugal pumps both allowed), plus
reflections at the desurger with any type of shape distortion permitted, plus the
mudpump noise itself. In general, we can write
p(x,t) = f(t - x/c) + g(t + x/c)
(4.3a)
where c is the measured speed of sound at the surface. Two transducers are
assumed to be placed along the standpipe. Note that the impedance mismatch
between standpipe and rubber rotary hose does introduce some noise, however,
since this effect propagates downward, it is part of the “g” which will be filtered
out in its entirety. The rotary hose does not introduce any problem with our
approach. Now let x a and x b denote any two transducer locations on the
standpipe. At location “b” we have
p(x b ,t) = f(t - x b /c) + g(t + x b /c)
(4.3b)
If we define W = (x b - x a )/c > 0, it follows that
p(x b ,t - W) = f{t + (x a - 2x b )/c} + g(t + x a /c)
(4.3c)
But at location “a” we have
p(x a ,t) = f(t - x a /c) + g(t + x a /c)
(4.3d)
Subtraction yields
 
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