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boundary layer transition to turbulence, and hydraulic jump formation in water.
Landahl (1972) gave the first exposition of the ideas to be developed below, for
dynamically steady, and also for fully transient continuum systems. The
mathematical formalism for general continuous media is developed in Chin
(1976, 1980a). Chin (1979, 1980b, 1981) discusses applications to water waves,
whereas Chin (1988a,b) extends these ideas to drillstring failure.
We will develop the fundamental concepts from elementary notions. For
simplicity, consider the classical vibrating string, i.e., a violin or guitar string,
executing small transverse displacements at some fixed frequency. These
strings do not support bending as “drillstrings” do, but the qualitative ideas
introduced here can and will be generalized without difficulty. It is not even
necessary to solve differential equations in order to illustrate the physical ideas.
We only need to recognize the fact that disturbances propagate at wave speeds
of (T/ )
1/2
, where T represents tension and is the mass density (see Chapter 1).
Now suppose that (x) increases in the direction of propagation. If the rate
of increase is slow enough, it is clear that the foregoing expression for speed still
applies locally; more formal asymptotic
WK B
analysis can be used to find
explicit high-order corrections. Eventually, as becomes large, this (T/ )
1/2
expression shows that the disturbance slows down to the point where all wave
motion effectively ceases. Physically, the wave cannot overcome the inertia of
the system. Because it cannot propagate beyond the “focus,” and since the
system is essentially nondissipative, the wave must remain there and vibrate in
place. The number of waves per unit length, or “wavenumber” k, therefore
“piles up” and increases, while significant local accumulations of energy build,
as if in a traffic jam. This leads to violent “trapped” oscillations which cannot
be detected downstream. In this sense, the focus represents a “singularity” or
“black hole” that constrains the flow of information. Drillstring bending
vibrations similarly focus, with the focus located close to the neutral point,
defined as the transition point between positive and negative axial stress. This
will be demonstrated analytically and explained in physical terms.
4.3.3.4 Extension to general systems.
Boundary and initial conditions, typically used in solving differential
equations, were not used in our explanation. It was only necessary to consider
“wave speed.” When this vanishes due to “refractive effects” which effect slow
changes (as opposed to reflections that arise from sudden changes) to wave
speed in the propagation medium, energy can
pile up
the way cars pile up near
narrow necks of highways. This explanation is appealing because it is simple
and physically intuitive. And, it is easily generalized for more complicated
vibrations in arbitrary continuous media. Typically, a wave with a
frequency
and a
wavenumber
k will satisfy a
dispersion relationship
= (k) determined
by the host vibrations model. These ideas were covered in detail in Chapter 1.
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