Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 284

Mud

X = X

s

Figure 5.5. Negative pressure pulser.
It is clear that Equation 5.61 does not apply: as shown in Chapter 1, the
material velocity u/ t is continuous through the source point, so that it is never
antisymmetric . We do know that some sort of delta function is required, the
only problem being, the choice of dependent variable that correctly yields the
velocity jumps characteristic of negative pulsers. It turns out the velocity
potential formulation discussed in earlier in this chapter works elegantly (see
Morse and Feshbach (1953) for detailed developments and discussion).
The main velocity potential results of Chapter 9 can conveniently
summarized by Equations 5.49a, 5.49b, and 5.50, which we re-order as,
2 (x,t)/ t 2 - c 2 2 (x,t)/ x 2 = 0 (5.62)
(x,t)/ x = u t (x,t) (5.63)
(x,t)/ t = c 2 u x (x,t) (5.64)
That is, (x,t) satisfies the classical wave equation; the first spatial derivative
physically equals the material velocity, while the first time derivative is
proportional to the strain as shown in Equation 5.64. The velocity potential,
therefore, provides an alternative but equivalent vehicle for solving mud pulse
acoustic problems.