tesi R. Valiante.pdf - EleA@UniSA

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the Re ( k) > 0 and Im ( ) > 0
k axes. Phase velocity is often presented
independently by dispersion curves. Although it is possible to consider c p as
positive, negative, real and imaginary, depending on k, the most physical
c and
meaningful information is contained in a plot which has as axes Re ( ) > 0
Re ( k ) > 0 , as shown in Fig.1.10. The horizontal line is the result of the non-
dispersive string, where all the wavelengths propagate at the same velocities c 0 .
Usually, in structural health monitoring applications the dispersion curves
present the phase velocity as function of frequency ω .
Fig. 1.10 - Dispersion curve for a string on an elastic foundation
1.7.1. GROUP VELOCITY
Group velocity is associated with the propagation velocity of a group of
waves of similar frequency. In reference books this concept is always introduced
by means of the pool example. A stone dropped in a pool of still water creates an
intense local disturbance which does not remain localized, but spreads outward
over the pool as a train of ripples. In this phenomenon, it can be observed that,
when a group of waves advance into still water, the velocity of the group is less
than the velocity of individual waves of which it is composed. The waves appear
to originate at the rear of the group, propagate to the front and disappear. A
simple analytical explanation is to consider two propagating harmonic waves of
equal amplitude, but slightly different frequency, ω 1 and ω 2 . Such harmonic
waves will have the expression of Eq.1.31,
( k x − ω t)
+ Acos(
k x − t)
y = Acos
1 1
2 ω2
(1.31)
p