Chapter 3
Experiment 1:
Sound
3.1
Introduction
Sound is classified under the topic of mechanical waves. A mechanical wave is a term which
refers to a displacement of elements in a medium from their equilibrium state; but to be
a wave this displacement must then propagate through the medium. The speed at which
the wave propagates is inversely related to the mass density of the propagating medium and
directly related to the forces attempting to restore the equilibrium condition.
A mechanical wave can propagate through any state of matter: solid, liquid, and gas.
Mechanical waves can be of two types: transverse or longitudinal. A transverse wave is
characterized by a displacement from equilibrium which takes place at right angles to the
direction the wave propagates; longitudinal waves have the displacement from equilibrium
along the axis of propagation.
Since two directions are perpendicular to the direction of propagation, transverse waves
have two independent polarization directions. The form of the equations describing these
two types of waves is very similar. However, transverse waves can only exist in solid
media, where intermolecular bonds prevent molecules from sliding past one another easily.
Such sliding motion is called shear. Solids support shear forces and will spring back rather
than continue to slide; this intermolecular connection will transmit the transverse wave from
molecule to molecule.
Longitudinal waves rely only on pressure and can exist in both solids and fluids. They
depend on the compressibility of the media. Solids and fluids all show a resistance to compression. Sound waves are longitudinal waves that are transmitted as a result of compression
displacement of molecules of the medium. We usually discuss sound in air, but sound travels
in everything except empty space. A sound wave can be generated in solids, liquids, or gasses
and can continue to propagate in a different medium.
The equation used to describe a simple sinusoidal function that propagates in space is
given by
h
i
Y(x, t) = A0 sin k(x − vt) p̂
(3.1)
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CHAPTER 3: EXPERIMENT 1
where Y is the time and position dependent displacement of the media from equilibrium,
A0 is the maximum displacement or amplitude of the medium’s motion, v is the velocity of
the wave which depends on the characteristics of the media. This particular wave travels
along the x-axis. . . x must increase at speed v to keep up with vt. p̂ is the polarization of
the wave. The case a longitudinal wave has p̂ = x̂ and a transverse wave has p̂ = py ŷ + pz ẑ
some combination of y and/or z polarization. k is a constant that is determined by both the
speed of the wave and the frequency of the wave. The constant k is usually expressed as
k=
2π
,
λ
(3.2)
where λ is the wavelength. The wavelength is related to the wave velocity v and the wave
frequency, f , by the expression
v = λf.
(3.3)
A periodic mechanical wave is characterized by a frequency of oscillation, f , which is
determined by the source of vibration motion that creates the disturbance. Thus, the
frequency and the speed of the wave in the media determines the wavelength. The source
can choose to oscillate at any frequency it chooses, but the medium decides the velocity of
propagation.
Checkpoint
What is the difference between a displacement wave and a pressure wave?
Checkpoint
Is sound a displacement wave, a pressure wave, or may it be considered as both?
Equation (3.1) describes the oscillations of particles with equilibrium position x. These
equations describe either longitudinal or transverse waves. The difference lies in the interpretation of the displacement which is described in the equation. For a transverse wave,
Equation (3.1) describes oscillations of the y and/or z coordinates of the particles at x.
For transverse waves the actual wave looks very similar to the plot of the displacement and
is easily visualized. For a longitudinal wave, Equation (3.1) describes oscillations of the x
coordinate of the particles at x in equilibrium. This results in a sinusoidal variation in the
density of media along the axis of propagation. This generally is much harder to visualize,
and there are few natural examples that can be easily observed. One such example would
be the pulse of compression which can be generated in a slinky spring.
A sound wave is a longitudinal wave and since the displacement of the wave causes a
variation in the density of air molecules along the direction of the wave, it can be viewed as
either a displacement wave or a pressure wave. The above equation may be used to describe
either picture. The displacement maximum is usually 90 degrees out of phase with the
26
CHAPTER 3: EXPERIMENT 1
Pressure Wave
(a)
(b)
(c)
Displacement Wave
Figure 3.1:
An illustration of the two representations of a longitudinal wave. The
displacement representation is the position of particles with respect to their average positions,
but the pressure increases as the particles move toward each other (compression) and
decreases as the particles move away from each other (rarefaction). The displacement wave
leads the pressure wave by 90◦ .
pressure maximum as shown in Figure 3.1. A sound wave is shown with both displacement
and pressure representations. The picture represents the density of the medium as the wave
passes through it.
Checkpoint
What determines the pitch of a sound wave? The source, the medium? Which
determines the speed of sound, the sound generator, the medium, or both? Which
determines the wavelength of sound, the sound generator, the medium, or both?
3.1.1
Superposition
When two sound waves happen to propagate into the same region of a medium, the instantaneous displacement of the molecules of the medium is normally the algebraic sum of the
displacements of the two waves as they overlap. If at one time and place each individual
wave would happen to be at a maximum amplitude, say Y1max and Y2max the net result would
be a displacement of the medium at a value equal to the sum of Y1max and Y2max . This is
27
(a)
Y1 + Y2
Y2
Y1
CHAPTER 3: EXPERIMENT 1
(c)
(b)
Figure 3.2: An illustration of constructive interference. Y1 and Y2 are in phase at all times
so that their sum has amplitude equal to the sum of Y1 ’s and Y2 ’s amplitudes.
(a)
Y1 + Y'2
Y'2
Y1
shown in Figure 3.2. If on the other hand, the second wave were at Y20max = −Y2max , the
net displacement equal to the sum of Y1max + Y20max , which in effect would be the difference
Y1max − Y2max or zero if the amplitudes are equal, as shown in Figure 3.3.
(c)
(b)
Figure 3.3: An illustration of destructive interference. Y1 and Y2 ’ are out of phase by 180◦
or half a wavelength. The sum of the two waves is zero if the two amplitudes are equal.
Checkpoint
What happens when two sound waves overlap in a region of space?
3.1.2
Reflection
We most often think of a reflection as occurring when a wave encounters the border of the
medium in which it is traveling. Anytime a wave encounters a sharp change in wave velocity,
due to a change in the nature of the medium, a reflection is generated and some or all of
the energy of the wave is redirected to the reflected wave. The amplitude and phase of the
reflected wave is determined by the boundary conditions at the point of reflection.
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CHAPTER 3: EXPERIMENT 1
In this lab, we will consider the effects of reflection from a solid boundary, such that the
boundary condition requires that the sum of the waves have a displacement of zero at the
point of reflection. Air molecules cannot be displaced from equilibrium at the wall. They
cannot move into the wall and atmospheric pressure presses them into the wall; they simply
have nowhere to go. This condition can only exist if we were to superpose a second wave
moving in the opposite direction with exactly the same amplitude, and 180 degrees out of
phase with the original wave. Hence, in order to satisfy the boundary condition, a reflection
wave is generated with exactly these properties.
Pressure
0.97
0.47
(a)
-0.03
AN
N
AN
N
-0.53
AN
N
AN
N
-1.03
(b)
(c)
Wall
The resulting superposition of incident and reflected
waves in the region in front of
the boundary also sets up a
second null area where the amplitudes cancel at a distance
of one half of a wavelength
from the boundary as shown
in Figure 3.4. The null areas are called nodes. If the
wave didn’t loose amplitude
as it traveled, a null would
be present at successive half
wavelength intervals over the
entire region. As it is, the
wave looses amplitude as it
propagates, and the cancellation is only partial.
0.97
0.47
(d)
-0.03
AN
-0.53
-1.03
N
AN
N
AN
N
AN
N
Displacement
Figure 3.4: Illustrations of pressure waves and displacement waves reflected at a wall. The incident wave and
reflected waves interfere to produce a series of nodes (N) and
anti-nodes (AN) spaced every half wavelength. (a), (b), and
(c) are pressure standing waves and (d) is the displacement.
The pressure in (b) becomes the pressure in (c) after the gas
moves like the arrows between indicate. The gas ‘sloshes’
back and forth between the dotted lines, but on average does
not cross them.
29
The same boundary does
not place such restrictions on
the pressure wave. The pressure at the boundary may
rise and fall, as is required.
The wall can easily support
whatever pressure results from
the superposition of any two
waves. A suitable pressure
wave which takes advantage
of the boundary restriction
(which is none) is directed in
the opposite direction with an
equal amplitude to conserve
energy, and is in phase with
the incident wave. The resulting superposed waves show a
maximum or anti-node at the
CHAPTER 3: EXPERIMENT 1
boundary (see Figure 3.4) and a null point or node at a distance of one quarter wavelength
away from the boundary. If the amplitude is sustained as the wave travels, a second null
appears a half wavelength from the first, or at a point three quarters of a wavelength from the
boundary. Between each node, the wave is seen to oscillate between maximum positive and
maximum negative amplitudes, where the maximum amplitude is the sum of the maximum
amplitudes of the waves considered separately. These areas are called anti-nodes. Such a
wave is referred to as a standing wave because it stands still. In either case, successive null
points or nodes occur at intervals of half of the wavelength of the traveling waves.
By measuring the distance between nodes of the standing wave, we can determine the
wavelength of the incident and reflected traveling waves. Since we also know the frequency
at which we excited the wave, we can find the speed of the wave.
3.2
Measuring the Speed of Sound in Air
From personal experience one can get a sense that the speed of sound in air is rapid. You
notice no delay in hearing a word that is spoken by a person nearby and the movement of the
speaker’s mouth. That would be quite a distraction, like watching a movie with the sound
track out of synch! And yet, when you sit in the outfield bleachers at a baseball game, you
can sense a noticeable delay between the arrival of the light showing the ball being hit and
the sound of the crack of the bat on the baseball.
The speed of sound is noticeably slower than the speed of light over distances the size
of a baseball field. In principle one could measure the speed of sound by timing how long
after ones sees the ball hit that the sound arrives if one knew how far away they were from
home plate. Instead, we will employ an oscilloscope simulation to observe the very short
time delay as sound travels a distance on the order of a meter.
3.2.1
The Speed of a Sound Pulse
In this experiment, a signal generator is used to produce a repeating electrical pulse to drive
a speaker. The pulse causes the speaker to emit a ‘click’ or pulse of sound whose speed
we will measure. A small microphone is used as a sensor. Its output is connected to one
input of the oscilloscope. The wave generator signal is also fed directly into the oscilloscope.
This signal will serve as a time reference against which to compare the microphone signal.
The delay introduced due to the distance in air that the sound travels from the speaker to
the microphone could be used to measure the speed of the click. Are the delays introduced
to the process of converting the electrical signal to mechanical sound and back, and in the
travel of electrical signals through the wire negligible? They probably are; however, we do
not have an exact location for where the sound is produced in the speaker or sensed in the
microphone. This could be a problem which we must deal with.
A sound wave will be sent down a tube and be reflected off a piston head back to a
microphone. We will measure the speed of the sound wave by observing the amount of time
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CHAPTER 3: EXPERIMENT 1
Figure 3.5: A sketch of the apparatus we will use to measure the speed of sound in air. A
wave generator and speaker will create a wave that travels down a tube and reflects from a
piston. A computer senses the travel time for the wave.
delay that is introduced to the arrival of the echo as the distance between the speaker (and
microphone) and the piston head is increased. This way we do not need to know exactly
where the sound originates or is detected. If the position of the speaker and microphone are
unchanged, the only contribution to time is piston position.
Helpful Tip
To avoid unnecessary interference with the measurements of other lab students, and to
spare the hearing and sanity of your Lab Instructor, leave your speaker on for ONLY
those times you are making measurements.
Set-up:
Familiarize yourself with the equipment as shown in Figure 3.5. Pasco’s 850 Interface will be
used to supply signals to the speakers from “Output 1”, to supply power for the microphone
from “Output 2”, to digitize the speaker’s signal, and to digitize the microphone’s output
in “Voltage D”. Check these connections. A suitable configuration for Pasco’s Capstone
program (“Sound 1.cap”) can be found on the lab’s website at
http://groups.physics.northwestern.edu/lab/sound.html
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CHAPTER 3: EXPERIMENT 1
Click the “Monitor” button at the bottom left to start taking data.
Observe the two signals from A and B inputs displayed at the same time. Without
disturbing the microphone, move the piston and watch the computer’s oscilloscope display.
Can you see the returned pulse(s) move as you move the piston? The oscilloscope graphs
microphone voltage on the vertical axis and time on the horizontal axis. Moving the
piston away from the speaker/microphone increases the distance the pulse must travel and
simultaneously increases the time needed.
Measure the Speed of Sound
To measure the speed of sound we want a square wave output and a frequency of about
5 - 20 Hz. To adjust the signal, click “Hardware” at the left and change only the settings for
“Output 1”. It is possible that the default signal needs no adjustments.
Figure 3.6: A sketch of the oscilloscope display showing the microphone’s response to
square wave ‘clicks’. The sound reflects off of the tube ends and travels back and forth down
the tube. The microphone measures each time the click passes by.
The pulse generated by the speaker travels down the tube and reflects off the moveable
piston back to the microphone. Drag the vertical numbers away from zero until the microphone signal occupies most of the Scope. Set the moveable piston to ∼80 cm from the
speaker. As you move the piston note that part of the signal moves to the right; these are
the clicks’ echoes passing the microphone as the sound bounces back and forth. Drag the
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CHAPTER 3: EXPERIMENT 1
time numbers to the right until the first echo observed by the microphone is near the right
side of the Scope and t = 0 is at the left side. You should see something similar to the
display shown in Figure 3.6. Move the piston closer to the mike.
Does the spike shift on the time scale? Sometimes it is hard initially to identify the
reflected pulse. It must move as the distance the clicks must travel changes and it must be
the first one to do so. The easiest way to identify the reflected click is to move the piston
around and to look for a pulse shifting around on the scope signal. As you move the piston
away from the mike you are introducing a delay in the time the microphone picks up the
sound. You might also see other peaks shifting as you move the piston. These may be second
and third echoes of the pulse bouncing off the speaker end of the tube.
Set the piston at some minimum distance for which you can readily observe the first echo
on the oscilloscope trace (∼20 cm is a good place). Note the piston position. How accurately
can you determine the piston’s position? Use the Smart Tool’s cross-hairs’ icon to locate
the leading edge of the pulse and note the time. Right-click the center of the SmartTool,
choose Properties, and increase the number of significant digits to 5-6. Be careful to write
the correct units and how accurately your time is known.
Now move the piston to a new position along the tube far from the speaker and note the
position again (∼70 cm is a nice choice. . . why?). Using the cross-hairs, determine the new
time of the shifted echo peak also. Remember that the extra distance you have introduced
to the sound travel is twice the change of position of the piston (going toward the piston
and coming back).
Calculate the speed of sound by dividing the extra distance added to the round trip of
the sound pulse by the corresponding increase in travel time. The pulse travels twice as far
as the piston moved but the oscilloscope measured the time (not double the time and not
half of the time),
2∆x
.
(3.4)
v=
∆t
The width of the echo’s leading edge can be an indicator of the uncertainty of the
measurement. If the echo moves around, this will increase your measurement error estimate.
Measure the latest time and subtract off the earliest time that might reasonably be assigned
to the time of the pulse’s echo. Let δ = latest - earliest and δt = 12 δ is a reasonable estimate
of the uncertainty in your time measurements. You need measure this uncertainty only once
since nothing substantial changes between measurements; each time measurement will have
this same uncertainty.
3.2.2
Measuring the Wavelength Observing Standing Waves
Sound incident on a barrier will interfere with its reflection, setting up a standing wave near
the reflector. The distance between nodes in the standing wave is a measure of half the
wavelength of the original sound wave when the wave travels at right angles to the reflector.
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CHAPTER 3: EXPERIMENT 1
Because of the inefficiency of the reflector and other losses, the nodes may be only partial
nodes. Additionally, the microphone has a finite size and will average the sound intensity over
a range of positions where only one position is at the intensity minimum. The wavelength of
sound should be expected to be on the order of meters for audible sounds. Diffraction effects
are commonly observed for sound waves passing through apertures like doors and windows
on the order of meters in size.
For this part of the experiment adjust the acrylic tube so there is about a centimeter gap
between the speaker and the end of the tube. This will release the pressure in the tube and
force this end of the tube to atmospheric pressure; this end will be a pressure node (and a
displacement anti-node). This part of the experiment will use “Sound 2.cap” that can be
downloaded from the lab’s website at
http://groups.physics.northwestern.edu/lab/sound.html
Set the frequency of the generator to 450 Hz. Click “Signal Generator” at the left to
access “Output 1” control panel. Keep the microphone near the piston and move the piston
and microphone to where the sound resonates in the tube (the microphone output goes to a
maximum).
Use the mike as a probe to measure the intensity of sound in the region between the
speaker and the piston by noting the amplitude of the signal on the scope as you move the
mike around back and forth inside the tube. This is the sound pressure level (SPL) as a
function of position, P (x), for the standing wave.
Place the mike near the piston and note whether the piston head is a node or an anti-node
by observing the variation in the intensity of the sound as you move the mike around near
the piston. Note your observation in your notebook. What would you expect for a pressure
wave or a displacement wave? Is the microphone a pressure sensor or an amplitude sensor?
Place the microphone near the speaker end of the tube, and note whether this is a maximum
or minimum. Explain this result in your notebook. You would think that near the speaker
you would get a large response from the microphone. Is that what you see?
Now, move the microphone to locate the first node away from the piston (remember that
right next to the piston is a displacement node) where the microphone’s output goes through
a maximum. Start with the mike right near the piston and move it away from the piston
and toward the speaker. Measure the first position of maximum response away from the
piston. Can you detect the next node? A third node? Record your observations. Record
the positions of the microphone where its output is minimum. Don’t forget your units and
error estimates; how accurately can you position the microphone at the maxima/minima and
how accurately can you read the centimeter scale? Move the microphone and remeasure one
node and one anti-node several times to check your error estimates. Are nodes and anti-node
measurements equally precise?
Calculate the wavelength of the sound from the distance between the first node of the
standing wave and the second node. If the nodes are close together, you can skip a node,
measure the distance between two nodes, and divide by two to get a better accuracy. After
all, your measurement errors will be divided as well since the denominator will be twice as
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CHAPTER 3: EXPERIMENT 1
large.
Calculate the speed of sound using the wavelength just determined and the frequency
from the signal generator’s LED display; use Equation (3.3). Note your results in your lab
book.
Checkpoint
The distance between successive nodes in a standing wave is a measure of what?
3.2.3
Sound Speed at Several Frequencies
Set the generator to something low around 450 - 500 Hz. Verify that the gap between the
speaker and tube end is still about 1 cm. Keep the microphone near the piston and vary the
position of the piston and microphone to maximize the microphone response.
Place the microphone at the end of the tube nearest the speaker. Put the piston all the
way into the tube so that it is close to the mike. Slowly withdraw the piston and observe
the position where the piston produces a maximum sound intensity. Move the microphone
to get a maximum response and see if the piston’s location can increase the microphone’s
output. Repeat until the piston’s position remains constant. This is where the tube is in
resonance with that particular frequency. Note this piston position, its units, and your error.
The position of the microphone will not need to change again until the frequency changes.
Withdraw the piston further and note a second resonance position. How precisely can
you position the piston and measure its position? If you cannot see a second resonance the
wavelength of the sound may be too long and you will need to increase the signal frequency.
These successive resonance positions are the positions of nodes for the standing wave at
this frequency. Since the distance between nodes is half a wavelength, merely doubling this
distance and multiplying it by the frequency of the sound as read off the signal generator
display will determine the speed of sound.
Repeat the measurements of successive resonance positions for four higher frequencies.
Make a table of wavelengths, frequencies, and calculated sound speeds for each frequency.
Plot your data using Vernier Software’s Graphical Analysis 3.4 (Ga3) program. A suitable
configuration for Ga3 can also be downloaded from the lab’s website. Does the speed vary
systematically with frequency? Use Ga3 to determine an average speed v̄ and a standard
deviation sv by drawing a box around the data points and by using the Analyze/Statistics
feature. Compute the deviation of the mean sv̄ using
sv
sv̄ = √
N
What frequency has a speed most similar to the speed measured in Experiment 1? Your
measurement’s best predictor is vs = v̄ ± sv̄
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CHAPTER 3: EXPERIMENT 1
Checkpoint
For a displacement wave reflecting off a solid wall is the boundary a node or an antinode in the resulting standing wave? For a pressure wave reflecting off a solid wall is
the boundary a node or an anti-node in the resulting standing wave?
3.2.4
Spectral Content
The Fourier Transform
Most of us are familiar with the link between musical pitch and sound frequency. A
soprano voice has high pitch and high frequency whereas a bass voice has low pitch and
low frequency. Probably you have already noticed while performing Section 3.2.3 above that
higher frequencies have higher pitch. You might also have noticed how ‘boring’ a pure sine
wave sounds. For contrast, select the square and triangle waveforms for “Output 1”; use a
frequency of a few hundred Hz and select the different wave shapes. Which shapes have the
most pleasant sound? Record your observations in your notebook.
We now want to investigate the frequency content of sounds more closely. The “Sound
FFT.cap” from the lab’s website provides a suitable configuration for Pasco’s Capstone
program. This will perform a ‘fast Fourier transform’ (FFT) on the microphone’s signal
and the speaker’s excitation. In graduate school you will learn more thoroughly that the
Fourier transform identifies the frequency or spectral content of functions of time. Study the
response vs. frequency for various waveform shapes and note your observations in your data.
The square wave should have responses at all integer multiples of the signal generator’s
frequency (the fundamental frequency). The triangle wave should have only odd multiples
of the fundamental. The amplitudes decrease for higher multiples or harmonics. Try singing
a pure note into the microphone and noting the spectral content of your voice.
3.3
Analysis
Calculate the Difference between the speed of sound found using the two different methods,
∆ = |v1 − v3 | .
Combine the uncertainties in these two measurements using
σ=
q
(δv1 )2 + s2v̄
Discuss the similarities and differences between this difference and the computed errors.
What other subtle sources of error can you think of that might have affected your measurements. Communicate with complete sentences. Which measurement method gives the best
accuracy and how do you know?
36
CHAPTER 3: EXPERIMENT 1
3.4
Conclusions
What physical relations have your measurements supported? Contradicted? Which were not
satisfactorily tested? Communicate with complete sentences and define all symbols. What
values have you measured that you might want to know in the future? Include your units
and errors.
37