Harmonic Analysis: Six-Segment Downhole Acoustic Waveguide - Measurement While Drilling Signal Analysis, Optimization and Design - page 63

His classic paper provides plots for both measured and calculated
reflection and transmission coefficients, versus the dimensionless quantity 2f a /c,
where f is the frequency in Hertz, a is the width of the conduit, and c is the
sound speed. Lippert shows that long, low frequency sound waves are
transmitted through such bends with very minimal reflection. If we take f = 50
Hz (which is high by present MWD standards), and assume a = 0.5 ft, with a
sound speed c of 5,000 ft/sec, we find that for 2f a /c = 0.01, more than 99% of
the incident power will pass through the bend. In other words , elbows are
effectively straight for acoustical purposes . It is interesting that an increase of f
to 500 Hz increases 2f a /c to 0.1; this increases the reflection coefficient to 0.05
and decreases the transmission coefficient to 0.99, which is still extremely close
to unity. In conclusion, for frequencies up to several hundred Hertz, a loss-free
wave assumption is perfectly reasonable - channels with bends, for the purpose
of acoustical analysis, can be assumed as straight without loss of generality, as
shown in Figure 2.5, thus significantly simplifying the mathematical formulation
and its solution. Further discussion and other essential acoustic ideas are found
in the classic topic of Morse and Ingard (1968).

Annulus-1 Annulus-2 Bit Motor Collar Drillpipe

MWD

x s

x = x s

x a x b

x m

x c

x = -x m -x b -x a x = -x m -x b x = -x m x = 0 x = x c

Figure 2.5. “Unwrapped” downhole MWD waveguide structure,
to be compared with illustration in Figure 2.4a.