LPC Physics 2
The Speed of Sound in Air
©
2003 Las Positas College, Physics Department Staff
The Speed of Sound in Air
Purpose:
In Part A, you will investigate the relationship between frequency, amplitude, and
character of sound waves using the Fourier Transform. In Part B, you will use the
principle of resonance to determine the speed of sound in a partially enclosed tube.
Equipment:
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Microphone
LabPro Kit
Resonance Tube Apparatus
Tuning Forks (3)
Rubber Mallet
Thermometer
Beaker
Rubber Bands
Dry Ice
Water
Theory:
Sound waves, like all waves, need a medium through which to travel, and are produced
by vibrations. The most common medium for sound is air, though liquids and solids will
work to some extent.
Sound is a longitudinal wave. This means that the direction of displacement of
the medium is the same as the direction of the wave’s velocity (remember that transverse
waves displace the medium in a direction perpendicular to the wave’s velocity). For this
reason, sound waves are sometimes referred to as compression waves.
So how is sound produced, and how does it travel from its source to your ear? Let’s look
at a simple speaker...the simplest speaker has simply a membrane that moves back and
forth under the influence of an alternating electrical current.
paper-y, movable
membrane
“cut-away”
of a simple
speaker
“box of air”
dots represent
molecules
Figure 1
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LPC Physics 2
The Speed of Sound in Air
©
2003 Las Positas College, Physics Department Staff
As the membrane of the speaker moves back and forth, it moves the molecules of air next
to it:
speaker
base movable membrane in
equilibrium position
membrane in
“compressed” position
equilibrium
position
“normal” air
a. Before the membrane
moves, the air molecules are
randomly scattered and
evenly distributed.
compressed
air
“normal”
air
b. As the membrane moves to
the right, it pushes the air
molecules nearest to it, creating
an area of higher density, or
“compressed” air.
Figure 2
membrane in
“rarefied” position
equilibrium
position
rarefied air
compressed
air
“normal”
air
c. When the membrane moves
to the left, the air molecules
must move to fill in the space
it’s created. At the same time,
the rightward- moving
molecules continue to push on
their neighbors. This creates an
area of lower density, or
“rarefied” air, next to an area
of compressed air.
As the speaker membrane continues to oscillate back and forth, it continues to compress
and then rarefy its neighboring air molecules, eventually creating a pattern that looks
something like this:
direction of wave motion
speaker
membrane
normal
normal
compressed
rarefied
normal
normal
compressed
rarefied
Figure 3
The compressed, or high density, areas have a local pressure that is greater than the
“normal” or equilibrium pressure. The rarefied, or low density, areas have a local
pressure that is less than the equilibrium pressure. For this reason, a sound wave is
sometimes referred to as a “pressure” or “compression” wave.
So, imagine you have a very large speaker. Say, one that takes up one entire wall of a
room. As the speaker membrane moves back and forth, it continually pushes on the air
molecules next to it, pushing them away. Do these molecules actually travel from one
side of the room to the other? Are we eventually left with a “pile-up” of air molecules on
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
one side of the room, and none on the other? No. Just like waves on a string, or other
solid medium, the molecules of a fluid medium, such as air or water, are merely
“wiggled” by the passing wave, not moved from one position to another. The air
molecules’ motion across the room is stopped when they run into the molecules next to
them, thus passing on their velocity.
If a source of sound waves is placed over a tube that is open at the top and closed at the
bottom, it will send these pressure disturbances made up of alternate compressions and
rarefactions (that is, a sound wave) down the tube. These disturbances will be reflected
at the tube’s closed end, thus creating a situation wherein identical waves are propagating
in opposite directions in the same region at the same time. As we remember from our
study of waves on string, this is the condition necessary to set up a standing wave.
However, whereas standing waves on a string require a fixed point, or node, at
each end, the open-closed tube must have a node at the closed end, and an anti-node (or
maximum amplitude) at the open end. These requirements will be met if the length of the
tube, L, is equal to one-quarter the wavelength, λ, of the sound wave.
L
λ
Figure 4
You’ll notice that the longitudinal sound wave has been illustrated by a transverse sine
wave. Not only does the transverse illustration make visualization easier, it is a
legitimate simplification: you can imagine the transverse wave as representative of the
pressure change – a positive amplitude means an increase in pressure, or a region of
compression; a negative amplitude means a decrease in pressure, or a region of
rarefaction.
Pressure
Pmax
equilibrium
distance
Pmin
distribution
of
molecules
equilibrium
compression
equilibrium
Figure 5
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rarefaction
equilibrium
LPC Physics 2
The Speed of Sound in Air
©
2003 Las Positas College, Physics Department Staff
To meet the condition of a node at the tube’s closed end and an anti-node at the open end,
standing waves can occur when the length of the tube and the wavelength follow the
equation:
L=
1
n
λ + λ where n = 0, 1, 2, 3, 4...
4
2
Eq. 1
L
L
L = ¼ λ, n = 0
L = ¾ λ, n = 1
L
L = 1 ¼ λ, n = 2
Figure 6 The first three standing wave configurations
Of course, sound waves are usually discussed not in terms of wavelength, but of
frequency. These variables are related by the velocity at which the waves travel through
the medium:
v = λf
v
λ=
f
Eqs. 2a and 2b
Substituting into Equation 1, we find that standing waves occur when:
f =
2n + 1 v
, where n = 0, 1, 2, ...
4 L
Eq. 3
These such frequencies are referred to as “resonance frequencies” or “driving
frequencies”.
As in the case of standing waves on a string, the lowest natural or resonant
frequency, f0, is called the fundamental frequency or first harmonic. Successively higher
frequencies are higher harmonics or overtones. For example, f2 is the second harmonic or
first overtone.
As can be seen from Eq. 3, the three experimental parameters involved in the resonance
condition of an air column are f, v, and L. To study resonance in this experiment, the
length L of an air column will be varied for a given driving frequency. The length of the
air column will be varied by raising and lowering the waver level in a tube.
As the length of the air column is increased, more wavelength segments will fit
into the tube, and will resonate when consistent with the node-antinode requirements at
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LPC Physics 2
The Speed of Sound in Air
©
2003 Las Positas College, Physics Department Staff
the ends (Fig. 6). The difference in the tube (air column) lengths between successive
points of resonance is equal to half a wavelength; for example:
λ
5
3
∆L = L2 − L1 = λ − L = ,
4
4
2
and
λ
3
1
∆L = L1 − L0 = λ − λ = .
4
4
2
Since the antinodes are the positions of maximum amplitude (particle displacement), the
antinodes correspond to maximum sound intensity. As a result, when an antinode is at
the open end of the tube, a loud resonance is heard. (If a node is at the open end – zero
amplitude – there is no resonance sound coming from the tube.) Hence, the tube lengths
for antinodes to be at the open end of the tube can be determined by lowering the water
level in the tube and “listening” for successive antinodes.
If the frequency, f, of the driving tuning fork is known and the wavelength is
determined by measuring the difference in tube length between successive antinodes
(resonance points), ∆L = λ 2 or λ = 2∆L , the speed of sound in air, vs, can be
determined from Eq. 2a.
The speed of sound in air is temperature-dependent and is given to a good
approximation by the relationship
v s = 331.4 + 0.6Tc m s
Eq. 4
where Tc is the air temperature in degrees Celsius. The equation shows that the speed of
sound at 0oC is 331.4 m/s and increases by 0.6 m/s for each degree of temperature
increase.
Fourier Analysis
When listening to musical sounds, it is generally agreed that a pure tone, or one that can
be mathematically described by a single sine wave, is often not very pleasant. Musical
sounds; a flute, a violin, a vocalist, even a pleasant speaking voice; are described by more
complicated periodic wave…something that may look more like this:
Figure 7
A possible representation of a sound wave produced by a musical instrument.
You can see that this wave has a general sine-like nature, and that it repeats itself. This
might represent a particular instrument playing (and holding) a certain note.
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LPC Physics 2
The Speed of Sound in Air
©
2003 Las Positas College, Physics Department Staff
The mathematician Jean Baptiste Joseph Fourier (Joe, to his friends) was an 18th
century mathematician and French Revolutionary who twice escaped the guillotine. A
contemporary (and sometime student) of Lagrange and Laplace, he was the first to
mathematically prove that any continuous function could be described by the addition of
an infinite number of harmonics (multiples of the fundamental frequency). (Note:
Although we say “infinite number of harmonics”, in many cases most of those harmonics
will have an amplitude (intensity) of zero, and thus not add to the overall sound. Most
sound waves will be able to be described by a finite number of harmonics.)
It is this addition of harmonics that allows us to distinguish between musical
instruments of different types. Each instrument’s tone includes different harmonics at
various relative intensities, resulting in a different timbre for each instrument. Fourier
analysis is the process of decomposing the sound wave into its constituent frequencies.1
Experiment:
Part A: Fourier Analysis
1. Connect the AC adapter to the LabPro by inserting the round plug on the 6-volt
power supply into the side of the interface. Shortly after plugging the power supply
into the outlet, the interface will run through a self-test. You will hear a series of
beeps and blinking lights (red, yellow, then green) indicating a successful startup.
2. Attach the LabPro to the computer using the USB cable that is Velcro-ed to the side
of the computer box (do not unplug the USB cable from the computer!). The LabPro
computer connection is located on the right side of the interface. Slide the door on
the computer connection to the right and plug the square end of the USB cable into
the LabPro USB connection.
3. Connect a motion detector to analog Channel 1 on the LabPro. The analog ports,
which accept British Telecom-style plugs with a right-hand connector, are located on
the same side of the LabPro as the AC Adapter Port. If you are using an older
microphone, you will need to use the DIN-BTA adapter from the LabPro Kit.
4. Open the file speed_of_sound.xmbl (or .cmbl) in the Experiments folder on the
desktop. This will start the Logger Pro program, and bring up the appropriate
experiment file. You should see a screen displaying graphs of Sound Pressure vs.
time and an FFT graph of amplitude vs. frequency. If you do not have an auto-ID
sensor (which is the likely case), a dialog box will pop up asking you to confirm the
sensors being used. If you have the suggested sensors attached to the LabPro in the
suggested ports, click “OK”. If the “OK” button is not active, ask your instructor for
help.
5. Try talking or singing into the microphone. Use Auto Scale to adjust the units on the
axis. You should also adjust the sampling rate at which data is collected (1000
1
The theory for this lab was written by Jennifer LK Whalen
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
pts/sec is a good starting place) until the variations in amplitude as a function of time
become apparent.
6. Reset, and Sing as pure a tone as possible into the microphone. Observe and sketch the
wave pattern. Select an area of your graph by dragging over it with the mouse. Select
View > Graph Options > Axis Options. This will provide a Fourier analysis of your
sound wave, i.e. the amplitude of each harmonic is displayed in graphical form. It
should be possible to increase the amplitudes displayed. Sketch this graph, and have
your lab partner(s) repeat the procedure. Describe in your own words what these
patterns represent, and how it accounts for the difference in character of your voices.
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
Part B: Determining the speed of sound in air
1. Set up resonance tube as shown below. The microphone will be added later.
Tuning Fork
Resonance Tube
Illustration by Jay Ayers
2. Fill the reservoir with water, with the reservoir at the bottom. Next, raise the reservoir
to bring the water to the top of the resonance tube.
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
3. Slowly lower the water level while a struck (humming) tuning fork is held over the
opening. Be careful not to hit the fork into the glass tube! To direct the maximum
sound energy into the resonance tube, hold the tuning fork so that the two tines are in
a vertical plane (as shown).
4. Continue to lower the water in the tube until resonance is heard. Raise and lower the
water slightly with the humming tuning fork in place to better establish the level of
water for resonance. Estimate the distance h from the tuning fork to the top of the
tube and record. Try to maintain this distance each time.
5. Have one partner hold the Microphone just above the opening of the tube (below and
slightly to the side of the tuning fork). Reset the Sound Program before you raise or
lower the water level near the resonance point. When the resonance occurs, the
microphone will help you see the exact spot. Use WINDOW > BIG NUMBERS to
make use of the digital data.
6. The first resonance occurs when the water level is roughly 1/4λ from the top of the
tube. Record water level y1. You may wish to mark the resonant point with a rubber
band.
7. Using 1/4λ = y1+ h, estimate 3/4λ = y2+ h and find the second resonance point.
8. Repeat Step 5 for this resonance and record water level, y2.
9. Record the temperature of the air in the lab.
10. Repeat Steps 4-8 for two more tuning forks of different frequencies (three forks total).
11. Lower the reservoir to its lowest position - in order to lower the water level as much
as possible. Add a small piece of dry ice to the water and wait for it to sublime.
12. When bubbling stops, see if the resonance tube is full of vapor; if not, add another
small piece of dry ice and repeat the procedure. (Although CO2 gas is invisible, the
cold gas evaporating from the dry ice causes water vapor in the air to condense into a
fog, which makes the CO2 column visible. You may neglect the small amount of
water in the CO2 column in your analysis of this experiment.) Make sure that the
white CO2 column fills the resonance tube to the top and that bubbling has stopped
before you go on to the next step.
13. While raising the reservoir slowly, use the tuning fork to locate resonance points as
before. Work slowly enough to avoid overshooting a resonance by more than a few
millimeters. Avoid backing up (lowering the water level), so air doesn’t enter the
resonance tube; if you must back up, use a little more dry ice to expel the air.
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
14. Find two resonance points ym and yn in the CO2 column. Also record (m-n). (In our
notation, y1 is the water level nearest the top of the tube at which resonance occurs;
y2 is the second highest water level, and so on. Thus m-n = 1 if y and yn are
consecutive resonance points. You will not need the values of m and n individually.)
15. Repeat steps 10-12 with a second tuning fork of a different frequency.
Analysis:
1. Question: Show that the boundary conditions applied to a standing wave in an open
tube requires that:
L = [(2m - 1)/4] λ
where m is an integer
2. Since ym+ h = [(2m - 1)/4] λ the wavelength λ (for a given gas and tuning fork)
can be found from:
( y m + h ) − ( y n + h ) = 2m − 1 λ + 2n − 1 λ = (m − n ) λ
4
4
2
In practice, this means that the first and second resonance's occur for m = 2, n = 1, so
l
that y2 - y1 = 2 λ. In other words, the distance between the resonance points
equals one-half wavelength.
(NOTE: As long as h is constant, it drops out of the calculation.)
Find λ and vo = λf for each tuning fork/gas combination. For each gas, find the best
value of v by averaging the measurements made at different frequencies.
3. Calculate the uncertainty in each of your velocities.
4. Compare your velocity, vo, with the standard equation:
vc= 331.4 m/s + 0.60T(o C)
Is vc within the estimated uncertainty of vo?
5. Compare your experimental velocity, vo, and the velocity determined in Step 4, vc,
with the value determined from the kinetic theory relation.
vkin. th =
γP
ρ
≅
γRT
M
where M is molecular weight and γ = 1.40 for air and 1.30 for CO2.
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
Results:
Write at least one paragraph describing the following:
• what you expected to learn about the lab (i.e. what was the reason for conducting
the experiment?)
• your results, and what you learned from them
• Think of at least one other experiment might you perform to verify these results
• Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in
this laboratory.
2
22
The Theory section was made possible in part by the following references:
Jerry D. Wilson. Physics Laboratory Experiments, 2nd Edition. Lexington MA: D.C. Heath and
Company, 1986.
Dean S. Edmonds, Jr. Cioffari’s Experiments in College Physics. 8th Edition. Lexington, MA: D.
C. Heath and Company, 1988.
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
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LPC Physics 2
The Speed of Sound in Air
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Clean-Up:
Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. If you do not turn in this page with your instructor’s signature with
your lab report, you will receive a 5% point reduction on your lab grade. How you divide
clean-up duties between lab members is up to you.
Clean-up involves:
• Completely dismantling the experimental setup
• Removing tape from anything you put tape on
• Drying-off any wet equipment
• Putting away equipment in proper boxes (if applicable)
• Returning equipment to proper cabinets, or to the cart at the front of the room
• Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
• Shutting down the computer
• Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
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LPC Physics 2
The Speed of Sound in Air
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2003 Las Positas College, Physics Department Staff
Data Tables
Resonance in Air
Tuning Fork
Frequency
h : distance
from fork to
tube
First
Resonance: y1
Estimate of
y2
Second
Resonance: y2
Temperature of
Air in Lab
Resonance in CO2
Tuning Fork
Frequency
h : distance from
fork to tube
Resonance ym
Resonance yn
(m-n)
Analysis Tables
Tuning Fork
Frequency
Wavelength λ
Velocity v
Average
Velocity
δv
vc
vkin. th
Does your experimental value of velocity match the two theoretical values within the
experimental uncertainty?
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