Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 91

product eigenfunctions such as the “cos l x/L x cos m y/L y ” encountered above.
In principle, the lumpy geometry and its axial motion can be resolved into
components that are associated with the mode shapes cos l x/L x cos m y/L y .
The amplitudes P l,m can be determined uniquely in terms of integrals involving
surface features, just as we had indicated in Chapter 1 (Hildebrand, 1948;
Tychonov and Samarski, 1964, 1967). The complete solution is obtained by
superposing all modal contributions.
Each of the ( l,m ) modes so constructed either propagates with a speed of
its own (see Equations 3.55 and 3.56 below) or fails to propagate. In the above
example where the input frequency is below cutoff, the solution in Equation
3.51 is a standing wave that decays exponentially with z. The eigenfunction is
termed an “evanescent mode,” and no energy is propagated down the
waveguide. If the waveguide is excited with a frequency just below the cutoff
frequency of a particular mode, then this and higher modes are evanescent, and
not important at appreciable distances from the source. All modes having cutoff
frequencies below the driving frequency may propagate energy and be
detectable at large distances.

3.2.8 MWD notes.
In mud pulse telemetry, positive pressure pulsers and sirens create three-
dimensional wave modes by mechanically blocking mud flow in the drillpipe.
Bypass flows, vortexes and other local inefficiencies are created, but these
small-scale disturbances do not travel to the surface; only the plane wave
associated with area blockage propagates uphole. This low-order mode will, of
course, be affected by dissipation. Negative pressure pulsers introduce local
three-dimensional waves, creating pressure signals by porting drillpipe mud into
the lower pressure annular flow. Again, only the plane wave component
propagates to the surface. Downhole MWD turbines supply energy to power
sensor-telemetry electronics and hydraulic components. While thin turbine
blades superficially resemble thick mud siren lobes, they do not significantly
block the oncoming mud; thus, turbine noise does not propagate uphole and
cannot be used for MWD transmission. Again, only downhole events associated
with pressures uniform across the area impact MWD surface signal processing,
since this lowest-order mode propagates most efficiently. Such events include
shocksub and PDC drilling noise, and bit bounce. Details are developed later.

3.2.9 Phase and group velocity.
Let us consider “propagating waves” in more detail. From Equation 3.46,
we have 2 /c 2 = k x 2 + k y 2 + k z 2 or = c (k x 2 + k y 2 + k z 2 ). Now, from the
pressure solution in Equation 3.45, a phase velocity can be defined by c p =

/k z

or,