Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 106

3.5.3 Stability of irrotational flows.
Very often, the parallel flow velocity U may not depend on the cross-
coordinate y; instead, it may vary in the direction of the flow “x” only, and
additionally with time, so that U = U(x,t). For such problems, other stability
criteria can be defined, based on

E/ t + (

r k (k,x,t)E)/ x = E{2 i + r t (k,x,t)/ r } (2.104)

M/ t + (

r k (k,x,t)M)/ x = M{2 i - r x (k,x,t)/k} (2.105)

Landahl (1972), for example, studies shear flow stability over slowly
growing boundary layers , allowing dependencies of the horizontal velocity U on
x, t, and y. His classic paper uncovered a new wave instability, one in which
heterogeneities in the medium cause waves to focus and trap; energy “pile-up”
(as if in a traffic jam) and unstable amplification lead to catastrophic laminar
breakdown. This type of strong, sudden instability is studied in Chapter 4 in the
context of damaging lateral drilling vibrations that are found at the neutral point.
Consider the limit where i vanishes in Equations 2.104 and 2.105. The
remaining terms show that the existence of an energy source does not imply that
of a momentum source, and vice versa. If nonzero right sides are used to define
instability, one must specify whether E or M is under consideration. Even so,
subtleties still remain. For example, it is possible for the total energy (or
momentum) between two rays to be constant, while energy (or momentum)
density versus position within the wave group changes with time, in such a way
that integrable amplitude singularities appear. Interestingly, this is the case for
the drillstring bending instabilities considered next. Finally we note that, while
wavenumber, amplitude, energy, momentum and action propagate with the same
group velocity in linear theory, the propagation velocities will differ with each
other in nonlinear wave flows. These conceptual ideas on stability are explored
in Chin (1976).

3.6 References
Abramson, H.N., Plass, H.J. and Ripperger, E.A., “Stress Wave Propagation in
Rods and Beams,” in Advances in Applied Mechanics, Vol. V , edited by Hugh
Dryden and Theordore von Karman, Academic Press, New York, 1958.

Chin, W.C., Physics of Slowly Varying Wavetrains in Continuum Systems, Ph.D.
Th e s i s , Massachusetts Institute of Technology, Cambridge, Massachusetts,
1976.

Chin, W.C., “Stability of Inviscid Shear Flow Over Flexible Membranes,”
AIAA Journal, June 1979.