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2
/c
4
)
1/4
cos{
½
arctan(- /
u
r
(x,t) = Aexp { (
4
/c
4
+
2
) + }}
2
/c
4
)
1/4
sin{
½
arctan(- /
X cos {
t + (
4
/c
4
+
2
) + }x}
(1.44)
Equation 1.44 clearly satisfies u(0,t) = A cos t. While this is reminiscent
of the u
r
(x,t) A cos (t - x/c) in Example 1-7, there are obvious differences:
an exponential damping with x, plus a complicated cosine argument showing
that dissipation affects the wave phase “ (t - x/c)” obtained in the undamped
case. Thus, in attenuative media, waves undergo shape distortions as well as
changes to propagation speed. From Equation 1.44, the exact speed is
dx/dt = - /{ (
4
/c
4
+
2 2
/c
4
)
1/4
sin{
½
arctan(- / ) + }} (1.45)
We emphasize that all of our results apply to Equation 1.35 only, and that this
damping model is applicable to a limited number of engineering problems.
1.4 Standing Versus Propagating Waves
In Examples 1-3 and 1-4 for the bounded domain 0 x L, we obtained
closed form solutions for problems with and without external excitation. In
Examples 1-7 and 1-8, we examined excitation sources located at x = 0, and
constructed right-going waves that propagated to infinity. More general kinds of
dissipation models will be studied using “kinematic wave theory” methods to be
discussed in Chapter 2.
1.4.1 Standing waves.
On bounded domains, superpositions of left and right-going waves may
combine to form “standing waves” that do not move either left or right, although
their amplitudes will locally vary with time. The simplest example is produced
by plucking a guitar string: the string executes back-and-forth motions which
appear geometrically similar, whose amplitudes increase and decrease with time.
Since the waves travel nowhere, and do not transport energy, they are called
standing waves. Standing waves contain systems of “nodes” or “minima,”
where amplitudes vanish, and “anti-nodes” or “maxima,” where amplitudes are
largest, which are of engineering interest. In machine vibrations, anti-nodes
indicate where shock absorbers should be placed. Nodes represent locations
where transducers should not be placed: they yield no information for “transfer
function” analysis.
1.4.2 Propagating waves.
On the other hand, the problems discussed in Examples 1-7 and 1-8
illustrate “propagating waves.” Here, the waves are not stationary, since a
reflective right boundary does not exist. “Wave energy” is propagated over a
distance, and the waves are said to be propagating waves. How are propagating
and standing waves related in bounded systems? All waves start in a highly
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