Biomedical Engineering Reference
In-Depth Information
(a)
(b)
0
T/2
T
3T/2
2T
0
T/2
T
3T/2
2T
Fig. 2.5.
(
a
) Harmonic oscillations
in phase
. Two harmonic oscillations are said
to be in phase when both oscillations reach a maximum (or a minimum) at the
same time. (
b
) In contrast, they are said to be
in counterphase
when one reaches a
maximum when the other reaches a minimum. Notice that two harmonic oscillations
in counterphase can be seen as two signals either delayed half a period relative to
each other or differing only by an overall factor of
−
1(or
inversion
)
a possible frequency of membrane vibrations that will allow the generated
wave to be well supported by the tube:
c
4
L
.
F
1
=
(2.16)
2.2.4 Modes and Natural Frequencies
The idea is not complex: in order to have a constructive effect, the harmonic
signal generated by the membrane must be in phase with the signal that re-
turns after being reflected at the open end of the tube. The theory of waves
expresses this idea in the following way: if the tube has a length
L
and the
sound propagates at a speed
c
, then the membrane must vibrate with a fre-
quency
F
1
=
c/
(4
L
) in order to contribute constructively and generate a
signal with augmented amplitude. This frequency is called the
natural fre-
quency
of the tube, and a pressure fluctuation oscillating at this frequency
is called a
mode
of the tube. Since frequency and wavelength are related
through
f
=
c/λ
, we have, for this mode, the result that the wavelength
λ
cannot take any value but
λ
1
=4
L
.
If it vibrates at a frequency different from the natural frequency, the mem-
brane can still induce pressure perturbations in the tube, but of a smaller
amplitude owing to the reflected wave not arriving exactly in phase with the
oscillation generated by the membrane. Is there only one natural frequency for
the tube? The answer is no, and this can be understood in the following way.
Let us suppose that the membrane is oscillating at just the natural frequency
of the tube. As we have discussed, the reflected wave returns in phase with
the perturbation established by the membrane. Let us now slightly increase
the vibration frequency of the membrane. Now, when the reflected perturba-
tion returns, it will no longer be in phase with the perturbation established
