Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 110

4.1 Typical Downhole Vibration Environment
4.1.1 What is wave motion?
Rigorous definitions for wave propagation can be found in physics and
mathematics topics. Here, we will introduce fundamental ideas and terminology
using intuitive arguments and examples, as we wish only to provide beginning
students with a physical feeling for the subject. Imagine a metal bar without
sudden changes (or “discontinuities”) in cross-sectional area or material
properties. If this bar (or better off, a stretched toy Sl i n ky ) is struck at one end
with a hammer parallel to the axis, energy propagates along axis through “wave
motions” to be felt at the opposite end; in general, multiple reflections may
occur, depending on material attenuation and the range of excitation frequencies.
It is important to observe that the material at the struck end is not
physically transported to the opposite end: there is no permanent material
transport anywhere, although material elements will be displaced temporarily as
the wave propagates through. This transient displacement is associated with
local changes to material velocity, acceleration and stress. In fluid mechanics,
this is called “sound” or “acoustic wave motion,” as opposed to “wind” or
“hydraulic flow.” Wave motions arise from the “compressibility” of the
medium. If the opposite end of our struck bar responded immediately, there
would be no wave motion, since the “incompressible” medium will have
transmitted the excitation instantaneously at infinite speed. Such materials act
like infinitely rigid billiard balls: the displacement of any one immediately
affects the others. Compressibility gives rise to wave propagation having a
distinct speed called the “sound speed.” In the case of one-dimensional motions,
where the wavelength is large compared to a typical cross-sectional length scale,
the sound speed is c = (E/ ), where E is “Young' s modulus” and is the mass
density per unit volume (for waves in fluids, the “bulk modulus” B replaces E).
Many subtleties arise in wave propagation. For one, the sound speed may
not be the only speed with which disturbances propagate in three-dimensional
systems (see Chapter 3). The incorrect conclusion may have been reached by
those whose only exposure to waves is the classical equation 2 u/ t 2 - c 2 2 u/ x 2
= 0. And in physical systems that engineers may perceive as one-dimensional,
the propagation may in fact be three-dimensional if the wavelengths in question
are small compared to typical cross-dimensions - one needs, literally, to “zoom
out” in order to see the big picture. A vigorous “handshake” will excite one-
dimensional long waves on a stretched jump rope; however, merely “pinching”
the rope will cause little in the way of large-scale wave propagation.
At sudden changes in geometry or material properties, an “incident wave”
arriving at the discontinuity may partially reflect and partially transmit; if so, we
refer to the change as an “impedance mismatch” (such reflections and
transmissions need not occur if impedances remain unchanged through the