tesi R. Valiante.pdf - EleA@UniSA

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2 2
On the other hand, if ω c < K F , then the wavenumber k is imaginary.
0
2 2
Defining k = −k
, the motion of the string is given by Eq.1.29.
y
± kx −iωt
= Ae e (1.29)
This corresponds to a spatially varying, but non-propagating disturbance.
Since the interest is on the conditions under which a harmonic wave can exist,
the results for imaginary wavenumbers are not considered in the study of
propagating waves, since they are non-propagating. Finally, the case
2 2
ω c = K F represents the transition from propagation to non-propagation.
0
Defining ω = c K F , the string motion is the one reported in Eq.1.30.
c 0
y
Ae
−iωct
= (1.30)
The frequency ω c is called cut-off frequency of the propagating mode.
There is no spatial variation in the motion, so the string is vibrating as a simple
spring-mass system. The basic factors governing propagation in a string on an
elastic foundation have been presented. Now, these results are displayed in
graphical form. Typically, two types of displays are used: the plot of frequency
versus wavenumber, which is called frequency spectrum of the system and the
plot of phase velocity versus wavenumber, which is called dispersion curve of
the system. To plot the frequency spectrum, the Eq.1.27 should be considered.
Assuming the frequency as real and positive, it is possible to get both real and
imaginary wavenumbers, if ω < ωc
and ω > ωc
, respectively. The results are
shown in Fig.1.8. The curves on the real plane are hyperbolas, while the
imaginary curves are ellipses. The line K = 0 is the non-dispersive result for the
taut string. It is possible to extract the phase velocity from the frequency
c p
spectrum by the relation ω = k .