Biomedical Engineering Reference
In-Depth Information
maximum, that at
x
=
L
the amplitude of the oscillation is always zero, and
that there is no other position at which these situation occur within the tube
for this particular choice of
ω
(and hence
k
).
What about a signal generated at the second lowest natural frequency
of the tube, at which the tube is also capable of sustaining an important
excitation? Again, we want the contributions to the density fluctuations in
the vicinity of the membrane to be in phase. The wave will contribute (after
the perturbation has traveled for a time 2
τ
=2
L/c
) constructively to the
fluctuations generated by the membrane if the travel time equals a period
and a half of the excitation, that is, 3
T/
2. Let us recall that in the case of the
first resonance, the travel time was equal to half the period of the oscillation.
This is so because two signals out of phase by a period and a half will be
in counterphase (see Fig. 2.5). Taking into account the inversion that occurs
at the reflection, we have the result that the fluctuations induced by the
reflected wave are in phase with the ones generated by the membrane. The
second natural frequency is therefore
3
c
4
L
,
F
2
=
(2.19)
which is triple the frequency of the first mode.
Now, something very interesting happens in the tube at a special point. At
a distance from the membrane equal to one-third of the length of the tube, the
signal that arrives directly from the membrane will be always out of phase by
a quarter of an oscillation with respect to the signal at the exciting membrane.
The reason is the following: the period of the oscillation is
T
2
=4
L/
(3
c
),
so
L/
(3
c
) (the time it takes to travel a distance
L/
3 at speed
c
)is
T
2
/
4.
The signal that arrives at this point in space after undergoing a reflection
was emitted some time before: the time interval is the time it takes for the
sound to travel 5
/
3 of the length of the tube. This is so because the wave,
before affecting the pressure at our special position in space, had to travel the
length of the tube, plus the 2
/
3 of the tube length from the reflection point
to our point of observation. At the frequency of the oscillations considered,
the fluctuation induced by the reflected wave has been delayed in this time
interval by five-quarters of an oscillation with respect to the fluctuations at
the membrane. This is the same as saying that they are one quarter out of
phase. This is just the same as the delay of the fluctuation arriving at our
observation point directly from the excited membrane! But we still have to
consider the change in sign induced in the reflected wave. Therefore the two
contributions to the fluctuations, namely that from the wave coming directly
from the membrane and that arriving after a reflection, will be exactly out of
phase and their contributions will cancel each other. At this particular point,
at a distance one-third of the tube length from the membrane, there is no
oscillation. The air in this region is exposed to one wave trying to increase
the density, and to another one trying to decrease it. This is called a
node
:a
