Description
This criterion is linked to a Learning Outcome
Section I – Report Format (5%)
You can use Owl Purdue website for help with format
•
https://owl.purdue.edu/owl/research_and_citation/apa_style/apa_style_introduction.html
Links to an external site.
• Title Page with Experiment Title, physics course no/section, report date & your name
• Use APA format for this report (also use APA for graphs and tables).
• References if needed
5
pts
Full Marks
0
pts
No Marks
5
pts
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Section II – Abstract (10%)
• In paragraph form, state the experiment objective(s) and how it was tested.
• Include a brief description of the experiment
• State the results and error results. Are the error % high or low and why important?
• Why is this experiment important and what are possible applications of this experiment
10
pts
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0
pts
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pts
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Section III – Introduction (15%)
• Write a brief paragraph stating significance and objectives of the experiment.
• Narrative should prove your understanding of the physics of the experiment.
• Include explanation/derivation of equations used. All symbols must be defined.
15
pts
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0
pts
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15
pts
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Section IV – Apparatus (5%)
• You may include drawings of the apparatus, if possible/applicable, and a listing of equipment if necessary. State what equipment was connected to what other piece of equipment. You may summarize the equipment used and for what purpose.
• Do not state numerical results.
• If you include photo, go to “paste, paste special, picture metafileâ€Â
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pts
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pts
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pts
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Section V – Experimental Procedure (5%)
• Write a brief narrative of the procedures followed to obtain data (summary of procedure). This may be 1-2 paragraphs in length.
• Do not copy all the detailed procedures from the manual.
• Include any problems you may have had and how you overcame them.
• Write in complete sentences and as if you are telling the reader about the process you used.
5
pts
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0
pts
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pts
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Section VI – Data (15%)
• You may include original data sheets initialed by instructor at completion of experiment. Ask your TA if this is required
• You must transfer the data to an excel sheet for easier analysis.
• Example: (note set up of this table, and check APA guidelines for table formatting)
 Example(Refer to your syllabus Lab report section)
15
pts
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pts
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15
pts
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Section VII – Calculations and Graphs (20%)
• You should show each type of calculation with appropriate tables, graphs, numerical results and errors.
• All tables/graphs must be referenced and labeled properly.
• All symbols must be defined. Units must be included.
• Discuss the graph and the results that the graph represents in terms of your overall goal of a physical constant.
• Graph Example: note set up of graph, and check APA guideline for graph formatting
 Example:(Refer to your syllabus)
20
pts
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pts
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pts
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Section VIII – Discussion of Results and Error Analysis (20%)
• Summarize any unusual problem or concerns with the experiment, including statements of how the experiment could be improved.
• When discussing error, make sure to draw from the following calculations to give quantitative results: Use appropriate error calculations for your experiment.
• The questions from your lab manual may help you with a better analysis and to support your discussion of the results.
20
pts
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pts
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20
pts
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Section IXâ€â€Conclusion (5%)
• Complete discussion of how the results of the experiment support the theory.
• How can errors be reduced?
• Is the method sufficiently precise and accurate?
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Experiment 7:
Pipes and Air Columns
Pre-lab (10 Points)
1. A pipe that is shaped like a cone acts like a/an: (circle one)
open pipe
closed pipe
2. If a 1 m long pipe open at both ends has a diameter of 2.54 cm and the velocity of
sound is 344 m/s, what is the fundamental frequency of the pipe?
3. Sketch the third mode of a pipe which is closed at one end.
133
4. What is the wavelength of the resonant frequency sketched in the following diagram:
5. Give the name of a real instrument that exhibits the properties of a cone.
134
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PRE-Lab Summary (15 points)
Read the experiment before coming to lab.
1) Summarize the procedure for this experiment on the page below.
2) Include the purpose, procedure and calculations that you will need.
3) This summary should be in your own words in bullet format. You may
use the back of this page as needed.
Last/First Name (print):
Experiment 7:
Pipes and Air Columns
1
1.1
Background
Definitions
1.1.1 A pipe or air column is defined by a tube. A tube is hollow with the length much
greater than the width. The cross–section of the tube may be circular, square, or any other
shape. The tube itself is rigid and to a good approximation does not take part in the movement
of the air inside of it.
1.1.2 In musical acoustics, an open pipe or tube is defined as open at both ends. This
occurs for flutes, piccolos, some organ pipes and a few other wind instruments. This is not
the case for instruments that use reeds or for brass instruments.
an open pipe
1.1.3 In musical acoustics, a closed pipe or tube is closed at one end and open at the
other. This occurs for clarinets. Brass instruments are closed at one end and open at the
other, but the mouthpiece and the bell are shaped such that these instruments are not tubes of
uniform cross–section.
a closed pipe
1.1.4 A cone in musical acoustics is a pipe with different diameters of the two ends. The
two ends are connected by a tube with straight sides. It is possible that one of the ends actually
comes to a point, so that it is closed. This occurs for oboes, English horns, bassoons and
saxophones.
135
a cone — d1 < d2
1.2
another code with a closed end
Pressure Nodes and Antinodes
The open ends of any pipe are pressure nodes. The closed ends are pressure antinodes.
Consider what happens to a sound wave when it reaches the closed end of a pipe. If all of the
air molecules inside a pipe were moving towards the closed end, the pressure at that end
would increase. There is no way for the air to escape. The air would reach a pressure
maximum at the closed end and the pressure would gradually decrease as the distance from
the closed end increased. This is obviously a pressure maximum or antinode.
The air molecules near the closed end cannot move because they are physically restricted by
the end of the tube. The closed end defines a displacement node. The air molecules farthest
from the closed end also move the farthest to increase the air pressure. Thus, the point farthest
away from the closed end is a displacement antinode. For an acoustically closed pipe, this is
the open end.
At the open end of a pipe, the air is free to move back and forth with no restrictions on its
movements. It is impossible to change the air pressure at that point since the air molecules do
not move appreciably closer together or farther apart. The open ends of a pipe are locations of
pressure nodes.
To understand the natural resonance modes of pipes or air columns, one must remember that:
•
•
1.3
open ends of pipes are pressure nodes and displacement antinodes
closed ends of pipes are pressure antinodes and displacement nodes
The Closed End of a Cone
Imagine changing the diameter of an open pipe so that one end is just slightly smaller than the
other. Both ends would still act as pressure nodes, since the air can freely move in and out of
the pipe at both ends. Even as the diameter of the smaller end decreases, it still acts as a
pressure node. In fact, we can make the smaller end so small that it is closed and it still will
define a pressure node! This is because it is not possible to build up the air pressure in the
point. The amount of air available to increase the air pressure is decreasing at the same rate
that the volume is decreasing, so it is not possible to form a high pressure or a low pressure
region. The closed, pointed end of a cone defines a pressure node.
136
1.4
The Natural Modes of an Open Pipe
Both ends of an open pipe are pressure nodes. This means that the simplest wave that occurs
inside a pipe has a node at each end and an antinode in the middle. The wavelength of this
vibration mode is twice the length of the pipe. The frequency of this vibration can be
calculated using:
v = ï¬n  fn
(1)
where v is the speed of sound, ï¬n is the wavelength of harmonic number n, and fn is the
frequency of harmonic number n. In an air column, the air itself vibrates hence the speed of
sound in this equation is the speed of sound in the air inside of the pipe. The frequency
of the first harmonic, or fundamental, is:
f1 = ð‘£ = v
ï¬1
(2)
2L
As was the case for a string, the higher vibration modes have additional nodes added
symmetrically between the open ends of the pipe. The first few vibration modes are shown in
Figure 1a and 1b with respect to the displacement and pressure of the wave.
The second vibration mode has λ2
f2 =
v
ï¬2
The third vibration mode has λ3
=
f3 = v =
ï¬3
v
2
L
3
= L, hence
v
=
=2·
L
2
3
v
2L
= 2 · f1
(3)
= 3 · f1
(4)
L and:
=
3v
2L
=3·
v
2L
Figure 1a. The First Few Vibration Modes of a Pipe Open at Both Ends (Displacement of Air)
137
Figure 1b. The First Few Vibration Modes of a Pipe Open at Both Ends (Pressure of Air)
In general, the wavelength of harmonic n, ï¬n= 2Lâ„n, so the frequency = nvâ„2L. Each vibration
mode has (n + 1) pressure nodes. The wavelengths of the first 10 vibration modes of an open
pipe are given in Table 1. This is for an “ideal pipe,†that is, the small corrections related to
the pipe diameter have been ignored.
Table 1. The Frequencies of Open Pipes
Mode
No.
Number of
HalfWavelengths
Wavelength
Frequency
Frequency
(re-written)
vs. 2L
Frequency
in terms of
f1
Number
of Nodes
1
1
2L
v/2L
1v/2L
1 f1
2
2
2
L
v/L
2v/2L
2 f1
3
3
3
2/3 L
3v/2L
3v/2L
3 f1
4
4
4
1/2 L
5
5
2/5 L
4v/2L
5v/2L
4 f1
5
2v/L
5v/2L
5 f1
6
6
6
1/3 L
3v/L
6v/2L
6 f1
7
7
7
2/7 L
7v/2L
7v/2L
7 f1
8
8
8
1/4 L
4v/L
8v/2L
8 f1
9
9
9
2/9 L
9v/2L
9v/2L
9 f1
10
10
10
1/5 L
5v/L
10v/2L
10 f1
11
The vibration modes of an open pipe all have frequencies that are members of a harmonic
series given by:
fnopen = nf 1open = n
v
2L
Therefore, a cylindrical pipe open at both ends will always have a discernible pitch.
138
(5)
1.5
The Natural Modes of a Cone
Since the closed, pointed end of a cone acts as an open end, a cone–shaped pipe has the same
vibration modes and harmonic structure as a pipe open at both ends. The frequencies of the
vibration modes are all the members of a harmonic series based on the wavelength of the
fundamental being twice the length of the pipe. So:
fncone = nf 1cone = n
ð‘£
(6)
2ð¿
A cone-shaped pipe will always have a discernible pitch. Its fundamental frequency is the
same as for a cylindrical pipe open at both ends. The cone, however, will have a different
vibration recipe. This is the primary reason that an oboe has a different timbre than a flute.
1.6
The Natural Modes of a Closed Pipe
A closed pipe has one open end and one closed end. The open end is a pressure node and the
closed end is a pressure antinode. This situation cannot occur for strings, since each end of a
string is physically restricted from moving and must be a node.
The first few vibration modes of a closed pipe are shown in Figure 2a and 2b with respect to
displacement and pressure of the wave.
Figure 2 a. The first few Vibration Modes of a Closed (Displacement of Air)
Figure 2b. The first few Vibration Modes of a Closed Pipe (Pressure of Air)
139
The fundamental vibration mode contains only one–quarter of the fundamental wavelength in
the pipe. This leads to a vibration frequency of:
v
f1 =
ï¬1
v
=
(7)
4L
Compare Equation (7) with Equation (2). The fundamental frequency of a closed pipe is
1â„ that of an open pipe. A closed pipe plays a note that is an octave lower than that played
2
by an open pipe or a cone of the same length! A clarinet plays an octave lower than an
oboe, even though they are about the same size.
The second vibration mode of a closed pipe has a pressure node at the open end, a second
node in the pipe, and an antinode at the closed end. From Figure 2b, it can be seen that
¾ of a wavelength fits inside of the closed pipe. Therefore, the wavelength of the second
node is 4 L, and
3
f2 =
4
v
ï¬2
=
v
4
L
3
=
3v
4L
=3·
=
5v
=5·
v
4L
= 3f1
(8)
= 5f1
(9)
Likewise, ï¬3 = L and
5
f3 =
v
ï¬3
=
v
4
L
5
4L
v
4L
Mode n of a closed pipe has n pressure nodes. The frequencies are the odd–numbered
members of a harmonic series. The even–numbered members of the harmonic series have zero
amplitude. The general equation for the frequencies of a closed pipe is:
f closed = (2n – 1)
n
v
4L
= (2n – 1)fclosed
1
(10)
Notice that in Equation (10), n is the number of the vibration mode, and is not the number of
the member of the harmonic series generated by the closed pipe.
The vibration recipe of a closed pipe contains only the odd–numbered harmonics of a
harmonic series. A square wave can be generated using only the odd–numbered harmonics of
a harmonic series. It should not be surprising that the waveshape of the sound produced by a
closed pipe is similar to a square wave. This is the primary reason that the timbre of a clarinet
is different than that of an oboe or a flute.
In summary, (1) a closed pipe plays an octave lower than an open pipe of the same length,
and (2) the vibration recipe of the sound produced by a closed pipe contains only the oddnumbered harmonics of the fundamental.
In general, for a closed pipe, the wavelength of harmonic n = 4Lâ„(2n–1), so the frequency =
(2n–1)v
â„4L. Each vibration mode has n pressure nodes. The wavelengths of the first 10
vibration modes of a closed pipe are given in Table 2. Again, the pipe diameter corrections
have been ignored.
140
Table 2. The Frequencies of Closed Pipes
1.7
Mode
No.
Number of
QuarterWavelengths
Wavelength
Frequency
Frequency
in terms of
f1
Number
of Nodes
1
2
3
4
5
6
7
8
9
10
1
3
5
7
9
11
13
15
17
19
4L
4/3 L
4/5 L
4/7 L
4/9 L
4/11 L
4/13 L
4/15 L
4/17 L
4/19 L
v/4L
3v/4L
5v/4L
7v/4L
9v/4L
11v/4L
13v/4L
15v/4L
17v/4L
19v/4L
1 f1
3 f1
5 f1
7 f1
9 f1
11 f1
13 f1
15 f1
17 f1
19 f1
1
2
3
4
5
6
7
8
9
10
The Natural Modes of Brass Instruments
Brass instruments are essentially cylindrical in shape with the mouthpiece and bell added on
to the pipe. The size and shape of the mouthpiece primarily affects the frequencies of the
upper harmonics causing these harmonics to decrease in frequency. The size and shape of the
bell primarily affect the frequencies of the lower harmonics causing these harmonics to
increase in frequency. By adjusting the size and shape of the mouthpiece and the bell, the
frequencies of the vibration modes can be adjusted to nearly match the harmonic structure of
a single harmonic series. All of the harmonics are present in the vibration recipe. The fundamental frequency of this new series is not directly related to the original length of the pipe.
1.8
The Natural Modes of Other Pipe Shapes
There are only two shapes other than cylinders and cones that give rise to vibration modes
belonging to one harmonic series. These shapes are described by mathematical functions
called Bessel functions. Benade has called the instruments based on these shapes Bessel
horns. The construction of metal instruments with precisely these shapes is very difficult, so
these horns are a curiosity known only to those of you who have studied musical acoustics.
The Bessel horn shapes are shown in Figure 3.
Figure 3. A Family of Bessel Horns Given by the Formula a = b(x + x 0)-y
The parameter is the flare constant y.
141
1.9
Some Vibration Recipes
The figures below present examples of the vibration recipes of some instruments based on
pipes and air columns. The vibration recipe can vary substantially depending on the note that
is played and its sound intensity level.
Frequency (in Hz)
Figure 4. The Vibration Recipe of a Flute
Frequency (in Hz)
Figure 5. The Vibration Recipe of an Oboe
142
Frequency (in Hz)
Figure 6. The Vibration Recipe of a Clarinet
Frequency (in Hz)
Figure 7. The Vibration Recipe of a Trumpet
Frequency (in Hz)
Figure 8. The Vibration Recipe of a Tuba
143
Frequency (in Hz)
Figure 9. The Vibration Recipe of a Tuba
2
Purpose of Laboratory Seven
In Laboratory 07: Pipes and Air Columns, we will measure the frequencies of the
vibration modes of open and closed cylindrical pipes. We will measure 10 resonant
frequencies for an open pipe of length L. We will then close one end of the pipe and measure
10 resonant frequencies for a closed pipe of length L. We will then set up vibration modes in
open and closed tubes, and will use a miniature microphone to examine the locations of the
pressure nodes and antinodes in the tube.
The goal of Laboratory 07 is to give you first-hand experience with the resonances which
occur in open and closed pipes.
3 Equipment
•
•
•
•
•
•
•
•
•
•
PASCO resonance tube, Model WA–9612, including:
90 cm clear plastic tube with moveable piston
tube mounting stands, one with speaker assembly
miniature microphone and probe rod
clamp-on hole covers
dual-trace oscilloscope
1/8†phone jack (female) to BNC adapter
wave function generator and frequency meter
thermometer (°C)
metric ruler
144
4 Experimental Procedures
4.1 Setting up the Resonance Tube
4.1.1 Measure the temperature in the room using the thermometer. Enter this value in
Worksheet Table 1. Calculate the speed of sound in the room using:
y= 344 + 0.6 (T-20) m/sec
(11)
and enter this value in Worksheet Table 1.
4.1.2 Measure the length and the diameter of the tube. Enter these values in Worksheet
Table 1.
4.1.3 Set up the resonance tube, oscilloscope, and computer–controlled oscillator/function
generator as shown in Figure 10.
Figure 10. Equipment Setup
4.1.4 Use the output of the oscillator as the input to channel 1 of the oscilloscope. Use the
output of the microphone as the input to channel 2 of the oscilloscope. Connect a cable from
the output of the oscillator/function generator to the trigger of the oscilloscope.
4.2 Setting up the Wave Function Generator and Frequency Meter
Make sure that the amplitude of the wave function generator is set at its minimum value.
Using BNC “T†connectors, connect the output of the wave function generator to the speaker,
to the trigger input of the oscilloscope and to the frequency meter.
145
4.3 The Resonant Frequencies of an Open Tube
4.3.1
Set the frequency of either wave function generator to 250 Hz. Turn up the amplitude
until it can be clearly heard. The starting frequency should be approximately 250 Hz because
the small diameter speakers used in this experiment cannot reproduce low frequency sounds.
Increasing the voltage to try to hear the sounds under 250 Hz may damage the speaker!
Slowly increase the frequency and listen carefully. In general, the sound will become louder
as you increase the frequency because the oscillator, speaker, and your ears are more efficient
at higher frequencies. Listen for a relative maximum in the sound intensity level, that is, a
frequency where there is a slight decrease in the sound intensity level as you increase or
decrease the frequency slightly. There should be a large number of resonances between 100
and 2000 Hz. You can also find the maxima by watching the trace on the oscilloscope.
Mount the small microphone near the speaker. Use the output of the microphone as the input
to channel 2 of the oscilloscope. Resonance is marked by a maximum in the microphone
signal. When the signal peaks are at a maximum height, you have found a resonance
frequency. Record these values in Worksheet Table 2.
Note: It can be difficult to find the resonant frequencies at low frequencies (0– 300 Hz). If
you cannot find the low frequency resonances, try finding some of the higher frequencies
first, and then using your knowledge of resonance modes in a tube, determine the
approximate resonant frequencies for the lower modes. Check to make sure that these modes
actually exist!
4.3.2 Raise the frequency slowly until you find a new resonant frequency. Measure and
record the frequency in Worksheet Table 2.
4.3.3. Find at least 10 consecutive resonant frequencies for the open tube, and record these
values in Worksheet Table 2.
4.3.4 Calculate the difference between each 2 consecutive frequencies and enter your
calculations in Worksheet Table 2. Calculate the average difference frequency.
4.3.5 Calculate the fundamental resonant frequency that you would expect to find for this
particular tube length, tube diameter and temperature using the equation:
fn =
nv
2(L+0.8d)
(12)
where fn is the frequency of harmonic number n, v is the speed of sound, L is the tube length
and d is the tube diameter. (The effective length of the tube is the actual length + 0.4 d for
each open end.) Enter these values in Worksheet Tables 1 and 2. Since each resonance differs
from the next lowest or next highest resonance by f1, the average experimental difference in
resonance frequencies should be equal to the fundamental frequency of the tube. Calculate
the harmonic numbers of the experimentally observed resonances. Explain any discrepancies
that you may observe.
146
4.4 The Resonant Frequencies of a Closed Tube
4.4.1 Close the end of the tube farthest from the speaker using the piston. Support the rod
on some convenient object like the storage box for the resonance tube.
4.4.2 Repeat the procedure used in Section 4.3 for the closed tube. Enter your data into
Worksheet Table 3. The equation for the frequencies of a closed tube is:
f=
(2n−1)v
4(L+0.4d)
(13)
Remember each resonance is separated by 2f1 from its nearest neighbors. Therefore, half of
the average difference in frequency = f1.
4.5 Standing Waves of an Open Tube
4.5.1 Remove the piston from the tube. Adjust the frequency of the speaker to excite a
strong harmonic of the open tube, with n closer to 5 (closer to 1000Hz).
4.5.2 Mount the microphone on the end of the probe arm and insert it into the tube through
the hole in the speaker/microphone stand. Move the microphone down the length of the tube
and note the positions where the oscilloscope signal is a maximum and where it is a
minimum. You will not be able to move the microphone completely down the tube from one
side because the cable is too short. However, you can move the microphone around to the
opposite side of the tube to complete your measurements. Enter the values in Worksheet
Table 4.
4.5.3 The microphone is sensitive to pressure. Therefore, the measured maxima are
pressure antinodes and the measured minima are pressure nodes. Sketch the wave activity
along the length of the tube in the space provided. By referring to Figure 1, you should be
able to see that one wavelength is defined by twice the distance between nodes or twice the
distance between antinodes. This is caused by the cooperative action of the wave traveling
downthe tube and its reflection from the end acting together. The result is a standing wave of
air pressure. To calculate the wavelength, multiply the distance between consecutive nodes or
antinodes by two.
4.6 Standing Waves of a Closed Tube
4.6.1 Insert the piston into the tube and slide it until it reaches the maximum point that the
microphone can reach coming in from the speaker end.
4.6.2 Adjust the speaker frequency to excite a strong harmonic of the closed tube,
with n closer to 5 (closer to 1000Hz).
147
4.6.3 Mount the microphone on the end of the probe arm and insert it into the tube through
the hole in the speaker/microphone stand. Move the microphone down the length of the tube
and note the positions where the oscilloscope signal is a maximum and where it is a
minimum. Enter the values in Worksheet Table 5
4.6.4 Sketch the wave activity along the length of the tube in the space provided. Remember
that the wavelength is twice the distance between nodes orantinodes.
4.7 Determination of Wavelength and Calculation of the Speed of Sound
4.7.1 Using the data in Worksheet Tables 4 and 5, determine the wavelengths for the waves
in the open and closed tubes used in the standing wave experiments. Enter these values in
Worksheet Table 6.
4.7.2 Given the frequency of the sound waves used in these experiments, calculate the
speed of sound in the tube. Compare these values with the speed of sound calculated for
Worksheet Table 1 and explain any discrepancies. Complete Worksheet Table 6
148
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Worksheets: Pipes and Air Columns
Worksheet Table 1. Preliminary Data
Variable
Value
Temperature (°C)
Speed of Sound (m/sec)
Tube Length (m)
Tube Diameter (m)
Open Tube Fundamental f1 (Hz)
Closed Tube Fundamental f1 (Hz)
v = 344 + [0.6 x (T – 20)]
fn =
nv
2(L+0.8d)
fn =
(2n−1)v
m/sec
open tube
closed tube
4(L+0.4d)
149
Worksheet Table 2. Resonance Frequencies of an Open Tube
Resonance
Number
Experimental
Frequency
(Hz)
Frequency
Difference
(Hz)
Calculated
Harmonic
Number
1
2
3
4
5
6
7
8
9
10
Average
Calculated f1
Difference
Explain any major differences between the calculated values and the experimentally
determined values.
150
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Worksheet Table 3. Resonance Frequencies of a Closed Tube
Resonance
Number
Experimental
Frequency
(Hz)
Frequency
Difference
(Hz)
Calculated
Harmonic
Number
1
2
3
4
5
6
7
8
9
10
Average
Average/2
Experimental f1
Calculated f1
Difference
Explain any major differences between the calculated values and the experimentally
determined values.
151
Worksheet Table 4. Standing Wave in an Open Tube
Microphone
Positions
Positions of
Maxima (m)
1
2
3
4
5
6
7
8
9
10
Resonant Frequency:
Sketch of the measured pressures for a mode of an open tube.
152
Positions of
Minima (m)
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Worksheet Table 5. Standing Wave in a Closed Tube
Microphone
Positions
Positions of
Maxima (m)
Positions of
Minima (m)
1
2
3
4
5
6
7
8
9
10
Resonant Frequency:
Sketch of the measured pressures for a monde of an open tube.
153
Worksheet Table 6. Determination of the Wavelengths and the Speed of Sound
Variable
Open Tube
Closed Tube
wavelength (m)
frequency (Hz)
measured v (m/sec)
calculated v (m/sec) from Table 1
difference (m/sec)
Explain any major difference between the calculated speed of sound and the experimentally
determined values.
154
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