Mud Acoustics in Modern Drilling - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 298

Figure 5.15. Example MWD collar used for siren frequency
and source placement optimization analysis in six-section waveguide model.
In an infinite uniform pipe, “½ p” travels upstream and “-½ p” travels
downstream. In the diagram shown, what ultimately propagates up the drillpipe
is less certain. We have solved the six-section waveguide problem, together
with appropriate matching conditions at all impedance discontinuities, and using
algebraic manipulation software, obtained a closed form analytical solution.
The question addressed is simple: What is the signal propagated uphole
(expressed as a multiple of p) as a function of siren frequency and location
within the MWD drill collar? Figures 5.16a,b,c show interesting results.
Different bands of strong (red) and blue (weak) amplitudes are possible,
providing a range of frequencies for FSK usage. Importantly, the pulser need
not stop completely during signaling, e.g., “50-60-50 Hz” is acceptable. This
minimizes mechanical inertia effects that drive the need for strong motors.