Lab II: Resonance Tube
George Wong
Jocelyn Schulz
Instructor: Patrick Cooper
Experiment Date: 4 October 2012
Due Date: 22 October 2012
1
Objective
The objectives of this lab were the following: experimentally determine the speed of sound in
air; examine way different boundary conditions (open or closed end of tube) affect resonance
frequency versus tube length.
2
Theory
Gas (such as air) acts like a spring (and has mass) and thus allows promulgation of oscillations
and waves.
Air is approximately composed of 80% N2 and 20% O2 .
The standardly accepted relationship between wavelength λ, velocity of wave v, and frequency
f is: v = f λ.
Further, for a gas velocity can be expressed
r as a function of air pressure p, air mass density
γp
ρ, and a ratio γ given to be 1.4 as: v =
ρ
In a tube (open or closed), there are two sound waves–one that travels down the tube in one
direction, and another that is “reflected” back to the source in the other direction. Resonance
is achieved when the original wave and the reflected wave are in phase and additively interfere.
nv
Generally, it is said that for an open tube, resonant frequencies are f =
for n = 1, 2, 3, . . .
2L
nv
For closed tubes, resonant frequencies are f =
for n = 1, 3, 5, . . .
4L
The speed of sound is accepted to be approximately: v = 331.5 + .606 T m/s where T is
temperature in centigrade.
3
Set Up
The set up involved using a long plastic resonance tube (cylindrical tube open at both ends)
with a speaker mounted at one end. The speaker was driven by the DataStudio software and
a signal generator (capable of producing frequencies between 10 Hz and 1500 Hz). At the end
with the speaker, there was a place for a microphone to be positioned; it was possible to move
the microphone into the tube. The microphone was attached to the computerized DataStudio
software and a virtual oscilloscope (measuring the voltage outputted by the microphone). A
long piston, the radius of which matched the tube, was inserted into the tube and moved
back and forth (depending on which measurement was being taken). A bendable rubber
stopper was also used to close off the end of the tube and gradually “crack” open the end of
the tube during the last part of the experiment. The complete setup can be seen in diagram
form below.
1
Figure 1: Set up diagram showing location of resonance tube (long plastic tube), microphone (attached to voltage sensor at computer), and speaker (driven by signal generator at
computer).
4
Procedure
1. The resonance tube was set up on supports, the piston was inserted into the end of the
tube.
2. The microphone was attached to the speaker stand, which was placed at the end of the
tube without the piston.
3. Both the microphone and the speaker were connected to the computer interface (via
DataStudio).
4. The microphone was attached to the “output” of the digital signal generator, set at 2
volts.
5. The speaker was attached to the voltage sensor input, set to medium sensitivity.
6. Both the signal generator and the voltage sensor were set to display to a scope data
window, with the signal generator voltage being the trigger for the scope.
RESONANT FREQUENCIES / DISTANCES OF THE TUBE
7. The speaker was set to emit a 500 Hz sound wave.
8. The piston was moved completely into the tube, so that the effective length of the tube
(in which the sound was traveling) was 0.0 m.
9. The piston was pulled out of the tube until the registered voltage (from the microphone)
reached a relative maximum. The distance between the speaker and the piston was
recorded.
10. Step (9) was repeated for the length of the tube.
11. The speaker was set to emit a 1000 Hz sound wave.
12. Steps (8) through (10) were repeated.
SPEED OF SOUND MEASUREMENTS / OPEN VERSUS CLOSED TUBE
13. The microphone was moved 0.10 m into the tube from its initial position in line with
the speaker.
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14. The piston was moved to be 0.80 m away from the speaker (0.70 m away from the
microphone).
15. A 10 Hz square wave was produced and sent to the speaker.
16. The amount of time it took for the sound wave to pass the microphone, strike the piston,
and return to the microphone was found from the scope trace of the microphone.
17. The piston was removed completely from the tube, the signature of the sound wave (as
it changed) was noted.
18. As the rubber stopper was slowly bent towards (/and away) from the tube, the changes
in the nature of the scope trace was noted.
EXTRA ACTIVITIES
19. While the tube was closed, the oscilloscope trace was recorded with a normal 10 Hz
square wave.
20. One of the holes along the tube was opened; the traces before and after the opening of
the hole were compared.
21. The end of the tube was closed.
22. Step (20) was repeated.
23. With the tube open and a normal 10 Hz square wave being sounded, air was simulated
being blown into the tube / sucked out of the tube by moving the piston both into and
out of the tube at a somewhat quick pace. Changes in the trace were noted.
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Data
In below table, distances (d1 , d2 , . . .) are in meters, with uncertainty of 0.0005 m.
Closed tube resonant distances (meters) for given frequencies
frequency (Hz)
500
1000
1500
d1
0.123
0.065
0.003
d2
d3
d4
d5
d6
0.454
–
–
–
–
0.226 0.402 0.462 0.735
–
0.1375 0.251 0.378 0.489 0.597
10Hz square wave
distance traveled: 0.70 m × 2
uncertainty in distance ±0.04m
first peak: 0.0151 seconds
second peak: 0.0191 seconds
uncertainty in times recorded above ±0.010 seconds
3
For the first and second peak (as identified above), the first major pulse (first overall pulse)
and the second major pulse (forth overall pulse) were chosen. These two pulses were selected
as the two intermediate pulses (that clearly illustrate a dampening effect) were likely a result
of the air in the tube attempting to reestablish equilibrium after the initial sound wave
passed. The next pulse that was larger than these pulses (yet still smaller than the original
pulse) must then, obviously, be the reflected wave striking the mic again.
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Plots and Calculated Values
Closed tube calculated resonant distances for given frequencies
frequency (Hz)
500
1000
1500
f1
700
1323
28670
f2
f3
f4
f5
f6
568
–
–
–
–
1142 1070 1303 1035
–
1876 1713 1593 1583 1585
Above calcuations follow the formula f =
e.g.
f=
nv
for n = 1, 3, 5, . . .
4L
1 × 344 m/s
= 700 Hz
4 × 0.123 m
λ = 2 × avg(dn+1 − dn )
frequency (Hz)
500
1000
1500
λexp (m)
λcalc (m)
0.682 ± 0.008
0.688
0.342 ± 0.004
0.344
0.236 ± 0.004
0.230
Speed of sound in air
d
v is velocity, d is distance traveled, t is time elapsed
t
d = 1.4 m
t = 0.0191 − 0.0151 s
1.4
v=
m/s = 350 m/s
0.004
v=
4
Figure 2: Frequency number versus distnace plot illustrating what should be linear
relationships for each of the various frequencies. The slope from each point to the next (for
a certain curve) should be constant.
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Error Analysis
7.1
Numerical analysis
measured − reported × 100%
percent error = reported
Wavelength / frequency of resonance
By frequency calcuation:
frequency (Hz)
500
1000
1500
f1
f2 f3
f4
f5
228
5 –
–
–
81
20 11 113 7
∼ 30000 55 27 17 13
f6
–
–
11
By wavelength (averaged) calculation:
frequency (Hz)
500
1000
1500
λ (m) out-of-bounds percent error
0.008
0
0.004
0
0.004
1
Speed of sound
reported = 344 m/s
calculated = 350 m/s
5
percent error = 1.7 %
q
dv = (t−1 dd)2 + (dt−2 dt)2
dv = 10 m/s
Speed of sound within acceptable bounds as suggested by uncertainties.
7.2
Reasons / Causes
As the piston was moved back and forth, the distance between its end and the speaker
/ microphone needed to be measured; however, this was not at all an easy task. Mere
millimeters of difference in length could easily drastically affect the calculated frequency
(from experimental values) for resonance. This is especially true for those measurements
that tend to be on the order of magnitude of a few millimeters.
Variations in the air temperature / humidity / density (due to elevation) may also have
affected the speed of sound in air, which was taken to be 344 m/s for all calculations. Alterations of this number would necessarily affect calculated results.
Also, any air pressure changes in the room (such as someone walking by the experiment)
might drastically affect the way that the microphone picked up the sound. It was shown that
a quick movement of the piston (simulating air pressure change) had a drastic effect on the
amplitude and dampening rate of the sound wave.
Finally, the microphones (and potentially speakers) used in this experiment were not of the
best quality. While they were sensitive, they could have been much more sensitive to allow for
better (higher resolution / greater accuracy) representation of what was actually happening.
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Questions
§5.a In units of wavelength, how close is the speaker to a displacement minimum or maximum? Can you make a comparison to the driven string experiment?
The speaker produces the sound directly at its location, so when it “beats”, it is at a displacement maximum. In the driven spring experiment, at the fundamental frequency, the
location of the driving was very close to a displacement minimum (however, it was not perfectly at that location). Unfortunately, the exact displacement (in terms of wavelengths) was
not noted.
§5.b As you increase the frequency does the wavelength change as you would expect?
Qualitatively, yes the frequency does change the wavelength as would be expected. Higher
frequency led to lower wavelengths; though, some of the wavelengths recorded did not yield
perfect numbers for according frequency (after calculation).
§6.1.a Explain the scope trace seen in Fig. 4 and reproduced experimentally.
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When the sound wave first hits the mic, it causes a drastic change in pressure which is
recorded on the mic and on the scope (as a change in voltage). As the sound wave continued
to travel down the tube, the air by the mic attempted to reach pressure equilibrium and thus
“bounced” back and forth (with amplitude dampening). When the reflected pulse returned
and struck the mic again, it caused another spike in voltage (as recorded on the scope) which
was higher than the latest equalizing pulse.
§6.1.b Is the second (reflected) pulse inverted?
The reflected pulse was NOT inverted with respect to the original pulse.
§6.1.c As the piston is moved towards the speaker, what is seen?
As the piston moved closer to the speaker, the peaks (between original and reflected) became
closer and closer together. The second peak also appeared to be greater in magnitude; though
this is not necessarily the case.
§6.2.a Discuss what is seen with the rubber stopper closing the end of the tube versus not
closing it.
With an open tube, the scope trace seemed to drop off faster (quicker dampening) than with
the closed tube. This makes sense, as pressure was more able to “leak out” with the open
tube.
§6.2.b How much of a crack is necessary to change the reflected pulse?
While it became very difficult to tell exactly what was happening as the rubber stopper was
being cracked open, it appeared that the stopper needed to be removed about 21 of the way
in order for it to change the reflected pulse.
1 Let the length of the tubes in Fig. 1 be L. What are the wavelengths at the fundamental
frequencies of the two types of tubes?
According to the formulæ given above, the wavelength of the fundamental frequency of a
closed tube of length L would be = 4 L and the wavelength of the fundamental frequency of
an open tube of length L would be = 2 L.
nv
These values can be found with the knowledge of f =
(for closed pipe; otherwise 2
4L
substituted for 4) and λ = v/f .
2 What is the wavelength at the fundamental frequency of a tube of length L closed at both
ends?
If the tube is closed at both ends, the wavelength would be the same as a tube open at both
ends (simply a phase shift). Thus λ = 2 L.
3 Show that Eq.(2) follows from Eq.(1).
r
γp
Eq.(1) states v =
ρ
A known gas law states P V = nRT , which can be rewritten as P = nRT V −1 .
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Density ρ is given to be ρ = m/v.
s
Substituting the above P and ρ in Eq.(1) gives: v =
γnRT
=
v mv
r
γnRT
m
m
Molar mass is given as M = , and with that substitution, we arrive at Eq.(2): v =
n
9
r
γRT
M
Conclusion
The speed of sound was experimentally found to be 350 ±10 m/s, which is well within error
bounds with regard to the reported speed of sound, v = 344 m/s.
When the average wavelengths were used for resonance calculations, data agreed to within
2 % at all times; whereas, error was often on the order of 10 % by frequency calculation.
This is likely related to the fact that there were no perfect boundary conditions at the
end(s) of the tube; therefore, the only true measurement that could be relied upon was the
distance between one resonance distance and the next–which should be approximately 12 of
a wavelength.
Regardless of the fact that there was a bit of uncertainty and error that was not accounted for
for all calculated values, the close agreeance of the processed data suggests that the formulas
used to calculate wavelength / frequency / etc. are correct. Specifically, the formulas for
finding length L for a closed or for an open tube seem to hold.
As would be expected, the experimentally calculated values that involved relatively small
numbers had the greatest amount of error in them.
Somewhat interesting to note: every calculated value that was inaccurate (i.e. all of them)
turned out to be too high. None of the experimental (or calculated from experimental) values
were too low. This suggests that there may have been a systematic error in the experiment
(speed of sound incorrect, ruler consistently measuring the wrong length).
In the extra time at the end, several other mini-experiments were performed. One involved
opening the holes alongside the tube as the scope was running and comparing the traces.
This did not yield interesting results. Theoretically, I would expect that this change would
affect the overall amplitude of the waves on the trace; however, the microphone was not truly
sensitive enough to notice that much. In addition, the mere act of opening the hole likely
produced enough of a physical disturbance to affect air pressure waves within the tube.
Also, a qualitative analysis of the effect of moving the piston into and out of the tube rather
quickly as the speaker / mic set up was working was performed. It was found that moving the
piston drastically changed the profile of the wave and seemed to shift amplitude of measured
voltage in a way semi-directly related to the speed at which the piston was being moved.
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