Biomedical Engineering Reference
In-Depth Information
by the membrane. Since the frequency of the vibration is higher now, the
perturbations created by the membrane will be “advanced” with respect to
the returning wave. Neither the travel time nor the sign inversion at the end
changes for the reflected wave. Since the two signals in the neighborhood of
the membrane are no longer in phase, the total amplitude of the signal will
not be as large as before (when the membrane was oscillating at the natural
frequency of the tube).
If we increase the membrane frequency even more, the phase difference
between the perturbations created by the membrane and the ones induced
by the reflected wave will be larger. Eventually, there will be a frequency
for which the two contributions to the total fluctuation will be completely
out of phase. For example, the membrane might be compressing the air in
its surroundings at the same time as an expansion has traveled back after
reflection at the open end. The interesting aspect of this is that if we increase
the frequency even more, the situation will reverse and, eventually, the two
contributions to the total pressure fluctuation will be back in phase. The
oscillation of the membrane is in this case too fast to use the reflected wave
to increase the total fluctuation in its surroundings, but it is now su
ciently
fast to take advantage of its “second chance” to increase the total amount of
fluctuation. The argument can be repeated, and in principle we can see that
there are an “infinite” number of natural frequencies.
The concept of natural frequencies is illustrated in Fig. 2.6, where we
show the gain of a tube that is open at one end and closed at the other, as
a function of the frequency of excitation. The gain is a function that shows
us the response of a filter at every excitation frequency. In our case, this
frequency is the one at which we make the membrane oscillate. The peaks of
the function are the resonances of the tube: each frequency that corresponds
to a peak is a natural frequency. Any time we make the membrane vibrate at
any of these frequencies, the air column in the tube will vibrate with increased
amplitude. In contrast, if the frequency is anywhere between resonances then
the oscillation amplitude of the air column will be zero.
The natural frequencies of the tube are easy to identify if we look at the
gain curve, but we have to note that the peaks are not ideally narrow but
have a finite width. In any real tube there will be dissipation, i.e., inevitable
energy losses. In this case, a wave returning to the exciting membrane will
do so with its amplitude greatly reduced owing to energy losses. Therefore
neither will the peak at a resonance diverge, nor will the total amplitude of
the resulting wave be zero at frequencies different from a natural frequency.
In the case without losses, the zero amplitude of the wave at a nonresonant
frequency is due to the waves added after succesive bounces not being in
phase. The infinite summation of all the positive and negative perturbations
will give zero. This cannot happen if losses diminish the amplitudes of the
waves after a few bounces.
