Biomedical Engineering Reference
In-Depth Information
point in space where the amplitude of the fluctuation is zero (in other words,
there are no fluctuations).
We can also obtain this result by writing explicitly the spatiotemporal
pattern of pressure fluctuations along the tube as we did in (2.18). Adding the
forward-traveling wave generated at the vibrating membrane to the reflected,
backward-traveling wave, both oscillating at triple the frequency of the first
case (i.e., at the second resonance), we obtain
P
e
=
P
e
0
cos(3
kx
3
ωt
)+
P
e
0
cos(3
kx
+3
ωt
)
=2
P
e
0
cos(3
kx
)cos(3
ωt
)
,
−
(2.20)
which is a standing wave like that of the first mode, (2.18). Notice that now
there are
two
points at which the amplitude of the pattern is always zero.
One is the open end as before, and the other is, as we expect, at
x
=
L/
3.
Figure 2.7a shows the first two spatial configurations associated with the
excitation of a tube at the two lowest resonant frequencies. In this figure, the
lines describe the maximum amplitude of the oscillations that can occur at
each point of space. The points at which the lines touch are points with no
0
T
/2
T
0
T
/2
T
(a)
(b)
Fig. 2.7.
Standing sound waves in a tube with one end open and the other closed.
(
a
) Position-dependent amplitude of the oscillation, and (
b
) successive snapshots
of the spatial configuration of the air pressure within the tube when the tube
is excited at its lowest natural frequency (top) and at its second lowest natural
frequency (bottom), the first and second peaks in Fig. 2.6. A higher density of dots
means higher pressure
