STANDING WAVES IN AIR COLUMN – 11MAC
The purpose of the experiment is:
•
To show that standing waves can be set up in an air column.
•
To determine the velocity of sound in air
Apparatus: computer, FFTScope program, speaker, long cylindrical air column with
removable end cap, microphone probe, meter stick
INTRODUCTION
A traveling wave of sinusoidal shape moves a distance (the wavelength) in a time T
(the period). The wave velocity c is therefore equal to / T . Because f (the frequency )
is defined as 1 / T , we can also write c = f  . Light and sound are examples of traveling
waves, albeit very different kinds of waves.
A standing wave is caused by
Length, L
superposing two similar (same
frequency and wavelength)
Wavelength, 
traveling waves, moving in
opposite directions. The figure
shows a vibrating string. At the
Fixed
End
fixed ends the traveling waves
are reflected. If the wave length
is just right we see nodes where
there is no motion or variation
at all. The "just right" condition
occurs when the string length L
is an integral number of half-waves /2: L = n(/2), n = 1,2,… .
Fixed
End
Nodes
A vibrating string is an example of a transverse wave: its oscillation is perpendicular to
the direction of its velocity. On the other hand, a sound wave is a longitudinal wave: its
oscillation is in the same direction as its velocity. These waves have similar properties,
but they are quite different in detail, as you will see.
EXPERIMENT
PART I
Part A: Standing wave in an air column with one end closed and the other end open
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The sound waves in this experiment are made by a computer speaker placed near the open
end of a tube that is closed at the other end. The speaker diaphragm vibrates, which sets
the air molecules vibrating and produces a sound wave propagating down the tube. A
sound wave is a longitudinal vibration of air molecules which causes alternating areas of
high density and low density of air (density waves). Sometimes this is also referred to as
alternating areas of high pressure and low pressure (pressure waves, highs and lows are
with respect to the average air pressure). The wavelength of this kind of wave is twice the
distance between two adjacent points of highest density at a given time. In this case,
longitudinal sound waves (vibrating in the direction of wave propagation) travel from the
loudspeaker, strike the closed end of the tube, and reflect back to the speaker.
The superposition of incoming wave and reflected wave results in a standing pressure
wave. Standing pressure waves have nodes (no fluctuation in pressure, or zero sound) and
anti-nodes (large fluctuations of pressure about the average pressure, or loud sound). The
rule is that at open ends of the pipe the standing pressure wave has a node (the pressure is
constant and equals the outside pressure). At closed ends of the pipe the pressure wave
has an anti-node.
NOTE: Do not get confused with the corresponding “displacement‟ wave which is 90
degrees out of phase with the pressure wave (meaning that a “displacement” node
corresponds with a pressure anti-node).
The purpose of this experiment is to measure the positions of the pressure nodes within
the tube. In between two nodes is a point of large pressure fluctuations, i.e. loud sound.
The sound is measured with a small movable microphone. From the experiment you can
determine the wavelength of the sound waves because the nodes are separated by half a
wavelength /2 (or adjacent areas of loud sound are separated by /2 ). First, you have
to create a standing wave.
to jack on speaker
From Mac
sound
speaker
microphone, attached to movable probe
(Aa) Determination of a resonant frequency.
Preliminary diagnostics of microphone by measuring frequency of whistling and
„Ahh‟ sound
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0) Open the “Fftscope” program under the “Lab Apps” folder of the
computer‟s Desktop. The FFTScope program is able to show you the
frequency compositions of a time-dependent signal.
1) Select “Oscilloscope left channel” on the bottom by clicking the circle
next to the option. Click on the Autoscale button at the top, then click on
the “3x zoom out” button twice. Do you see the vibration-amplitude from
the background noise as function of time, updated in the window every 0.1
second (Sampling window = 100 millisecond)? As a small test, whistle
near the small microphone probe. See what the vibration-amplitude of
your whistle looks like. Make the "Ahh…" sound to see what the „shape‟
of this sound is in the oscilloscope mode. You can click and drag to
zoom-in a small portion of the signal. Click the Y-Autoscale button if the
signal is too large for the display (called “out-of range” or “clipped”) and
bring your graph within the proper vertical range. Do you see a SINE-like
wave? Now double-click anywhere in the window to reset the display (or
choose “View”/”Reset x and auto scale y”).
2) To see the frequency spectrum of the same whistle or “Ahh …” sound,
select “fft on left channel” on the bottom by clicking the circle next to that
option. Now you are looking at the „shape‟ of your “Ahh…” sound in the
FFT mode. The graph shows the amplitudes as a function of frequency. Do
you see a collection of small peaks on the left side of the window? Ignore
the seemingly very large peak at 0 Hz but zoom-in („3x zoom in‟ in top
bar) in the region above 50 Hz.
3) (IMPORTANT: YOU HAVE TO CHANGE THE FREQUENCY RANGE
ON THE HORIZONTAL AXIS from 0-22000 Hz to 0-1200 Hz by high
lighting, etc.)
4) You will notice the low frequency background due to the air-ventilation in
the ceiling. This unwanted background cannot be avoided but you should
not confuse it with your actual spectrum. Identify the fundamental
frequency and higher harmonics of the “Ahh…” sound made by your
voice. The fundamental is not always the highest peak. You may need to
Y0-Autoscale. Find out the frequency pattern of your “Ahh… “ sound.
When you move your mouse pointer over the peaks, the various
frequencies are shown in the left-of-middle-lower square. What is the
accuracy of this frequency reading? (check it by what increment the
frequency reading changes each time you move the cursor just a little).
5) Change Sampling window (ms). First stop data collection by selecting
“Stop taking data”. Then click on the box to the right of “Sampling
window (ms)” and and type “1000”. This sets the time collection time to 1
second, rather than the standard setting of 0.1 second. Find the
frequencies of your “Ahh…” sound again. Notice that the resolution of
the frequencies has now changed.
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Determination of resonant frequencies in a tube
6) Now you can turn on the „white noise‟ (which is made up of all
frequencies) generating by the speaker by clicking on the option “White
noise” in the left pane of the program (Signal generator). Turn on the
speaker and check that the volume is roughly in the mid-position; also
check that your computer‟s sound level (see the tiny speaker icon in the
right of the menu bar) is also roughly in the mid-position. Do you hear the
noise similar to what you hear if you tune a TV into an empty channel?
What frequencies are there when you bring the microphone near the
speaker? Are the waves traveling or standing?
7) Now place one of the speakers about 5 cm from the open-end of the tube.
Insert the microphone probe into the tube through the small hole in the cap
and place the microphone tip inside near the closed end of the tube (the
end with a cap). You should now see a discrete frequency spectrum, as
altered by the confinement of the tube, updated every second. Only
specific frequencies survive inside the tube as standing waves.
8) Explore the “FFT averaging On” option on the lower right of the program.
Do you see the change of the frequency spectrum? What did averaging do
to the spectrum? This option is sometimes usefull but do not use this
option when you want instantaneous data for example when moving the
little microphone around when you search for a node of the standing wave.
9) You can stop the data collection by hitting the Space bar, or selecting the
“Stop taking data” option. Stop the white noise by choosing the “Off”
option in the Signal generator. Is the frequency spectrum the same as the
case without the tube? Do you see peaks? Find the frequencies of the first
5 peaks. These are the resonant frequencies. They can be described as a
series of multiples of the fundamental (the lowest frequency) such as
1:2:3:4:.. or 1:3:5:7:… . To see this sequence it is crucial to always include
the lowest frequency.
Move the position of the microphone and restart the white noise and restart data
collection by again selecting “fft on left channel” and compare the resonant frequencies
you get. Do you see some frequencies disappear as you move the probe microphone
position inside the tube? Why does this happen?
(Ab) Determination of /2. Now have the signal generator drive the speaker at one of the
resonant frequencies you identified – at the bottom left of the program window, you
should see a box where you can enter the frequency in Hz; enter the value of the resonant
frequency, and then click on the Sine button above. You should hear the sine-wave at that
fixed frequency. You can now move the microphone around and watch the amplitude of
the peak at the frequency that you just typed in. Locate the nodes (where you find
minimum amplitudes). Write down the position of the microphone and determine the
distance between the nodes. From this you can infer the wavelength and with the known
frequency you can determine the velocity of sound.
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If there is time choose another resonant frequency and repeat above procedure.
PART II
Part B. standing wave in air column with both ends open
Remove the end cap from the air column. Follow the procedure in Part A (a) to measure
the resonant frequencies in this open ended air column.
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Standing Waves in Air Column
preliminary questions
Name: ____________________
Sec. ______
Date: ___________________
1. What is speed of sound in air at temperature 17 °C? See text book.
2. Consider a tube filled with air (at 17 °C) and open at both ends. Let the length of
the tube be 70 cm.
(a)What is the fundamental (lowest) resonance frequency in this tube?
Draw the allowed pattern of nodes and antinodes. Show your work.
(b) Draw the node-antinode pattern of the next 2 higher resonant frequencies. What
is the value of those frequencies? Do the frequencies follow a pattern of 1,2,3,4,….
Or is the pattern 1,3,5,7,… ?
(c) If one end is now closed (the other end is still open), what will be the
fundamental resonance frequency? What is the pattern of the higher harmonics?
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Lab Report (PART I)
Name: ______________________, Sec. _____
Partners: _____________________________________________________________
Diagnostic Phase:
Place the microphone in front of your mouth and make the “Ahh …” sound.
Show in a graph what this sound looks like in the oscilloscope mode. No need to zoom
in.
How does it compare to a pure SINE wave for a single frequency?
Show in a second graph what this sound looks like in the FFT mode. For this FFT mode
you should zoom in to a range of 0-1200 Hz (ALWAYS INCLUDE 0 Hz, otherwise it is
hard to recognize the lowest frequency. However some very low frequencies are due to
background noise!) On this reduced scale the fundamental frequency and the higher
harmonics become clearly visible. Before you print, make your screen image more stable
by clicking “FFT averaging On” (keep the data taking going so you can still measure the
frequencies and add them by hand to your just printed peaks).
Record the frequencies of the fundamental and higher harmonics. (Notice and ignore the
very low frequency background! The fundamental is not always the highest peak.)
Do the harmonics in the frequency spectrum follow the sequence 1:2:3:4:5…., or
1:3:5:7:9…? How can you infer from this that your voice box is an open-open tube or an
open-closed tube?
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Record the fundamental and harmonics generated by your partner for the same “Ahh…”
sound.
Part A: Standing wave in an air column with one end closed and the other end open
(Aa) Determination of resonant frequencies
(1) Turn on the Signal generator and select “white noise”. In open space, white noise
contains a continuous spectrum of frequencies. However inside the tube the traveling
waves are reflected and become standing waves. Due to the space restrictions only certain
wavelengths and their corresponding frequencies are possible and the frequency spectrum
becomes discrete.
Stick the small microphone inside the tube and keep it as close as possible to the closed
end of the tube. In this location it will detect all allowed frequencies.
List of resonant frequencies at microphone position near the closed end:
________, __________,____________,___________,_______________,___________
What is the accuracy of the above frequency readings? (You estimate this by slowly
moving the mouse-pointer horizontally over each peak and observe its reading jumping.)
Make a graph showing the frequency spectrum. Remember to switch to “FFT averaging
On” before printing. Include the graph in the lab report.
(2) The displacement-amplitude of the standing wave inside the tube varies with the
location x along the tube length. The text book discusses this. The amplitudes near the
closed end are significantly different from the amplitudes near the open end.
Sketch the expected amplitude as a function of location x for the three following cases:
a. For the lowest frequency (this would be the simplest pattern).
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b. For the second lowest frequency.
c. For the third lowest frequency.
(3) Based on the three patterns you sketched in (2), what is the expected ratio-pattern of
the frequencies.
How do those ratios compare with your observed spectrum in (1)? Try especially the ratio
f_2 / f_3. (The ratio f_1 / f_2 is less accurate for this purpose).
(4) Move the microphone inside the tube to a position x where at least one of the
resonant frequency peaks disappears.
In order to see a frequency disappear it is imperative to turn off “FFT averaging On” ,
because you need instantaneous data taking.
Record the new position x of the microphone. How far is it from the closed end?
x=
List the resonant frequencies at the new microphone position and place a cross at the
place of the disappeared frequency:
________, __________,____________,___________,_______________,___________
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Include a graph, showing the new frequency spectrum
(Ab) Determination of /2.
(0) Turn off the white noise. Set the function generator to SINE-wave. Set the frequency
of the generator to one of the resonant frequencies you found before in part (Aa). The
purpose of this part of the experiment is to identify the exact nodes of the pressureamplitude of one specific harmonic frequency, without the presence of the other
harmonics.
(1) Choice of specific resonant frequency: ________ (prefer the third lowest frequency) .
The frequency spectrum now should show only one peak. If not, alert your TA.
(2) Measure the distance between microphone locations for adjacent nodes (disappearing
peaks) : _________ Nodes are separated by /2, so infered wavelength: _________
Verify that the distance between adjacent anti-nodes (maximum height of peak) leads to
the same numbers.
(3) Calculate velocity of sound (show work): ______________
(4) If there is time: Choice of another resonant frequency: ________ (prefer the fourth
lowest frequency)
(5) Measure distance between nodes: _________
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(6) Calculate velocity of sound (show how you find it): _____________
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Lab Report (PART II)
Name: ______________________, Sec. _____
Partners: _____________________________________________________________
Part B. standing wave in air column with both ends open
(Ba) Resonant frequencies in open ended air column:
Make a graph showing the frequency spectrum and include it in your report.
(Bb) Expected resonant frequencies in open air column as a function of length of tube(L),
speed of sound(c) and integer (n). Consult your textbook.
(Bc) Do the resonant frequencies found in (Ba) follow the formula in (Bb)? Explain
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