Geology Reference
In-Depth Information
As a brief review, the Fourier assumption v(x,t) = sin(kx- t) when applied to the
transverse string equation v tt - Tv xx = 0 , l e ads to = ( T/ ) 1/2 k. Thus, every
wave of wavenumber k has a well-defined frequency that depends on the
properties of the medium. The dispersion relationship can be obtained from
differential equation analysis or empirically in the laboratory.
Two wave speeds, a “phase velocity” C p = /k, and a “group velocity” C g
= / k, are usually defined. These ideas originated with the classical
physicists of the nineteenth century, who were disturbed with the fact that the
“obvious” /k speed inferred from sin(kx- t) or sin k{x - ( /k)t} exceeded the
speed of light ( /k, it turns out, is not the speed with which energy propagates -
instead, it is the group velocity demonstrated in Chapter 2). The applications of
these speeds are now familiar to communications and electrical engineers, and to
acoustical designers. The velocity used in wave trapping analyses should,
naturally, be the one with which energy physically propagates, that is, we need
to consider the manner in which group velocity behaves in waveguides . For
vibrating strings, of course, both velocities happen to be identical (this is not so,
however, for lateral bending waves).
The ideas developed at the Massachusetts Institute of Technology can be
summarized in several succinct statements:
(1) Wave trapping represents a strong instability mechanism with a sound
physical basis; it may or may not coexist along with the resonant modal ones
well known in eigenfunction analysis.
(2) In dynamically steady systems, trapping is implied by zeros or
minimums in the expression for group velocity.
(3) In more general transient systems, with higher-order spatial
dimensions, the critical group velocity turns out to be a complicated (but well
defined) function of the local phase velocity (e.g., see Landahl (1972)).
4.3.4 Why drillstrings fail at the neutral point.
For the transversely vibrating strings described above, the density
nonuniformities responsible for vanishing group velocities are easily visualized.
In the case of uniform vertical drillstrings, with the exception of unimportant
area changes at collar/pipe junctions responsible for sudden reflections only, the
required inhomogeneities needed for refraction are less apparent. One might
reasonably ask, “What nonuniformity or heterogeneity can possibly induce
violent bending motions far downhole, together with the mysterious
disappearance of lateral vibrations at the surface? ” The relevant nonuniformities
turn out to be those “seen” by the linear equation governing bending motions
(for example, refer to Timoshenko and Goodier (1934), Love (1944), Den
Hartog (1952) or Abramson, Plass, and Ripperger (1958), Clough and Penzien
(1975)). That is, terms representing the varying axial load due to weight appear
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