Name (printed) ______________________________
First Day Stamp
LAB
STANDING WAVES ON STRINGS
INTRODUCTION
Now that you understand the basic physics of
waves and sound you can begin to apply this
knowledge to the physics of musical instruments. The
next two labs will illustrate how physical systems can
be made to vibrate at frequencies (pitches) that
correspond to the notes of the musical scales. The
phenomenon of standing waves occurs in the
structures of all musical instruments. It is the means
by which all musical instruments “make their music.”
This first lab will explore the general physics of
standing waves as well as the physics of standing
waves on strings. Standing waves get their name
from the way that the energy of the wave seems to be
“standing” at a certain point or points on the medium.
The first three “harmonics” of a standing wave on a
n1
n2
n3
string are shown in the diagram above. The parts of
the medium where the energy appears to be standing
are known as antinodes and the motionless parts of
the medium are known as nodes. Note that in the
third harmonic of the diagram above there are three
antinodes and four nodes.
EQUIPMENT
wave generator
string vibrator
various strings
super pulley
mass set
clamps
meter stick
PURPOSE
In this experiment you will investigate standing waves
on strings. Specifically, you will learn how harmonic
frequencies relate to each other and what factors govern
the frequency and speed of a standing wave on a string.
PROCEDURE – PART 1
CONSTANT WAVE SPEED
For any string in resonance, the harmonic frequency is determined by the harmonic number (n), the wave speed (v),
𝒏𝒗
and the string length (L): 𝒇𝒏 =
𝟐𝑳
1.
Slip one end of the white elastic cord through the hole in the metal bar of the string vibrator, and tie a double
knot underneath so that the string will not slip back through. Slide the other end of the cord through the notch in
the mass hanger and check that it will not slip.
2.
Use the mass set to hang 145 grams of mass from the mass hanger (which itself is another 5 grams). Measure
the string length from the knot where the string goes through the hole in the string vibrator to the top of the
pulley. Record the length of the string, L, in meters. (Note that L is not the total length of the string, only the
part that is vibrating.)
3.
Turn the amplitude knob to one-third, then turn on the wave generator. Use the 1.0 Hz and 0.1 Hz frequency
knobs of the wave generator to adjust the vibrations so that the string vibrates with one antinode in the middle.
This is the fundamental frequency, or the first harmonic. Adjust the amplitude and frequency to obtain a largeamplitude wave, but also check the end of the vibrating blade; the point where the string attaches should be a
node. It is more important to have a good node at the blade than it is to have the largest amplitude possible. The
best data results from having a “quiet” node (little vibration) and a “loud” antinode (much vibration). Take time
to get it right.
4.
Record the frequency of the wave generator for the first harmonic.
1
5.
Repeat steps 3-4 for the next three harmonics (don’t do the 5th harmonic yet). The string always vibrates with a
node at each end, so the total number of antinodes is the harmonic number. (The drawing shows n = 1, 2, and 3
only.)
harmonic, n
frequency, fn (Hz)
string length, L (m)
wave speed, v (m/s)
1
2
3
4
5
average:
QUESTIONS AND CALCULATIONS
1.
What relationship do you see between the various harmonic frequencies on the string?
2.
a. Use your answer to the previous question to calculate what the 5th harmonic frequency should be. Show your
calculations below.
b. Repeat steps 3-4 in the procedure section to physically determine the frequency of the 5th harmonic. What is
the percentage difference between this measured frequency and the calculated frequency above?
3.
What effect does the number of the harmonic frequency have on the wave speed on the string? Explain why.
2
PROCEDURE – PART 2
VARYING WAVE SPEED
For a wave on a string, the speed is related to the
tension (FT) in the string, and the linear mass density
(µ) of the string, by the equation: 𝑣 =
𝑭𝑻
𝝁
Tension is measured in units of Newtons (N). The
linear mass density is the mass per unit length of the
!"## !" !"#$%&
𝒎
string, measured in kilograms/meter: 𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑎𝑠𝑠 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =
→ 𝝁=
!"#$%& !"#$%!
𝑳
Tension is applied by the hanging mass, and is equal to the weight of the hanging mass. Weight is mass multiplied
!
by the strength of gravity, g, which is a constant near the surface of the Earth. 𝑔 = 9.8 . So the tension in the
!"
string can be calculated by 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 = 𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑎𝑠𝑠×𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 → 𝑭𝑻 = 𝒎𝒈
1.
Switch from the white elastic cord to the yellow string. Measure and record the total string length, in meters.
2.
Using an electronic balance, measure and record the mass, in kilograms, of the string.
3.
Pull the string through the hole in the vibrating blade. Run the string over the pulley and attach the mass hanger
to the end of the string.
4.
Hang 50 grams total mass on the string (45 g plus the 5 g mass hanger). Record this hanging mass, in kilograms,
in the table.
5.
Measure from the knot at the vibrating blade to the top of the pulley. Record this distance L, in meters, in the
table. (Note: L is not the total string length, only the part that’s vibrating.)
6.
Adjust the frequency of the wave generator so that the string vibrates at the 4th harmonic. Just like Part A, adjust
the driving amplitude and frequency to obtain a large-amplitude standing wave, and clean nodes, including the
node at the end of the blade. Record the frequency in the table.
7.
Add 50 g to the hanging mass and repeat the last step. Record your data in a table.
8.
Repeat at intervals of 50 g up to 250 g.
DATA AND ANALYSIS
1.
Calculate the known linear density by dividing the total string mass by the total string length.
2.
Calculate the wave speed v in each trial using the equation for harmonic frequencies on a string, 𝒇𝒏 = .
𝟐𝑳
String length L is a constant and n = 4 for all trials.
3.
Calculate the tension in the cord for each trial using the equation 𝑭𝑻 = 𝒎𝒈. Record results in the data table.
4.
Calculate the experimental linear density for each trial using the equation 𝑣 =
𝒏𝒗
𝑭𝑻
𝝁
. You’ll have to rearrange the
wave speed equation to solve for experimental linear density. Record results in the data table.
5.
Calculate the average experimental linear density.
6.
Calculate the percent error between the known linear density and the experimental linear density of the string.
3
total string length (m) ________________
total string mass (kg) ________________
known linear density μknown (kg/m)___________
harmonic, n
frequency, fn
(Hz)
string length, L
(m)
speed, v
(m/s)
mass, m
(kg)
tension, FT
(N)
exp. linear density,
μ (kg/m)
4
4
4
4
4
experimental
linear density
Percent error calculation:
QUESTIONS AND CALCULATIONS
1.
Discuss how each of these variables affects the pitch of a stringed musical instrument:
a. String tension
b. String linear mass density
c. String length
2.
A violin string has a linear density of 0.030 grams/centimeter and is put under 150 N of tension. What is the
speed of a wave on this violin string?
3.
The speed of a wave on a string is 170 m/s when the string tension is 120 N. What must the tension be to raise
the wave speed to 180 m/s?
4
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LAB
STANDING WAVES IN A COLUMN OF AIR
INTRODUCTION
In the previous lab, you created standing waves
on strings. Standing waves can also be produced in
the air inside a tube. You’ll do that by vibrating the
air with an audio speaker connected to a wave
generator (see photo to the right). The impedance
change at the ends of the tube is large enough to
strongly reflect the sound waves back and forth
within the tube. In the previous lab, you were able to
clearly see the standing waves on the string, but
inside the tube of this lab, the sound waves will be
invisible. You’ll observe instead, by listening to the
standing wave. The clue that the standing wave is
present will come from the loudness of the sound –
peak loudest sound means that there is resonance,
which means there is a standing wave present.
A major difference between standing waves on
strings and standing waves in tubes is the location of
the nodes and antinodes. With strings, the ends are
always nodes because at the point where the string is
fixed, its movement is completely limited. However,
when you consider the freedom of movement of air
within a tube at an open end versus a closed end, it is
at the open end where you have greater freedom of
movement. Therefore, standing waves of sound in
tubes will have antinodes at the open ends and nodes
at the closed ends.
EQUIPMENT
wave generator
open speaker
resonance tube
patch cords
PURPOSE
In this experiment you will investigate standing waves in the air within open and closed tubes. Specifically, you will
learn how harmonic frequencies relate to each other and what the similarities and differences are between the
standing waves in strings, open tubes, and closed tubes. You will also investigate the relationship between the tube
length and frequency of sound waves resonating in closed and open tubes. You will also learn about the “end effect”
in both closed and open tubes.
PROCEDURE – PART A
HARMONICS OF OPEN TUBES
1.
Extend the white tube out of the blue tube (if necessary) so the total tube length is at least 10 cm different from
the other groups at your table. Do not extend it more than 60 cm.
2.
Place the speaker at a 45° angle near the end of the blue tube, as shown in the picture above.
3.
With the amplitude knob all the way down (counterclockwise), turn on the wave generator. Turn up the
amplitude to a very low level, no more than one-quarter of the dial. Starting at 40 Hz, slowly raise the wave
generator frequency until you reach resonance. When resonance occurs the loudness of the sound will increase
noticeably even though the amplitude of the generator is still low. This is the frequency of the first harmonic.
When you feel that you are at resonance, turn the amplitude knob down so that other groups can hear their own
resonance clearly. Record the frequency for your first harmonic in the data table on the next page.
4.
Starting from the first harmonic frequency, continue to raise the generator frequency to find the frequencies of
the next two harmonics. Record these next two harmonics in the data table. (There is a spot for a fourth
harmonic in the data table. Leave this blank for now. Also, leave the “harmonic #” column blank for now.)
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PROCEDURE – PART B
HARMONICS OF CLOSED TUBES
1.
Keep the same length used for the previous part of the experiment, but hold a textbook firmly against the end of
the white tube away from the speaker. This makes a closed tube of the same length as the open tube before.
2.
Keep the speaker at a 45° angle near the end of the blue tube.
3.
With the amplitude knob all the way down (counterclockwise), turn on the wave generator. Turn up the
amplitude to a very low level, no more than one-quarter of the dial. Starting at 40 Hz, slowly raise the wave
generator frequency until you reach resonance. This is the frequency of the first harmonic. When you feel that
you are at resonance, turn the amplitude knob down so that other groups can hear their own resonance clearly.
Record the frequency for your first harmonic in the data table below.
4.
Starting from the first harmonic frequency, continue to raise the generator frequency to find the frequencies of
the next two harmonics. Record these next two harmonics in the data table. (There is a spot for a fourth higher
harmonic in the data table. Leave this blank for now. Also, leave the “harmonic #” column blank for now.)
DATA AND ANALYSIS
Open Tubes
Harmonic
Frequency (Hz)
Closed Tubes
Harmonic #
Harmonic
First
First
Second higher
Second higher
Third higher
Third higher
Fourth higher
Fourth higher
Frequency (Hz)
Harmonic #
1.
Looking at the data for the open tube, what is the relationship between the higher harmonics and the first
harmonic frequency?
2.
Predict the frequency of the fourth higher harmonic and then use the wave generator to measure the actual
frequency (show calculations).
Predicted fourth
higher harmonic: __________
Actual fourth
higher harmonic: __________
6
% difference =
3.
Looking at the data for the closed tube, what is the relationship between the higher harmonics and the first
harmonic frequency?
4.
You should see one similarity and two differences between the harmonics of closed pipes and open pipes. State
these.
Similarity:
Difference #1:
Difference #2:
5.
Predict the frequency of the fourth higher harmonic and then use the wave generator to measure the actual
frequency (show calculations).
Predicted fourth
higher harmonic: __________
Actual fourth
higher harmonic: __________
% difference =
6.
In the numbering of harmonics, the harmonic numbers are integer multiples of the first harmonic. For example,
if the 1st harmonic were 100 Hz, 500 Hz would have to be the 5th harmonic (even if no other harmonics existed
between the 1st and 5th harmonic). Use this rule to fill in the harmonic numbers column in the data table for both
tubes. What does this imply about the possible harmonics for open versus closed tubes?
7.
If an open tube had a first harmonic of 150 Hz, what would be its next three harmonics as an open tube and
what would its first four harmonics be as a closed tube of the same length?
Open tube:
Closed tube:
8.
If a closed tube had a first harmonic of 150 Hz, what would be its next three harmonics as a closed tube and
what would its first four harmonics be as an open tube of the same length?
Open tube:
Closed tube:
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QUESTIONS AND PROBLEMS
1. Draw sketches of the first three harmonics of the
open tube below using the ruler for proper scale.
2. Draw sketches of the first three harmonics of the
closed tube below using the ruler for proper scale.
The following two questions pertain to the diagram below. It is a closed tube that you could blow into to produce a
pitch. Assume the temperature is 22°C.
3.
What is the lowest frequency that can be produced when blowing into the tube?
4.
If you overblew to get the next higher harmonic, what is the new note you would produce? Explain, or show
work to justify your answer. Draw what this standing wave looks like in the tube.
5.
If you were to cut off the bottom of the tube (but not change its length at all), what is the lowest note on the 12tone Equal Tempered scale that could be produced when blowing into this open tube?
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LAB
STANDING WAVES IN A VIBRATING BAR OR PIPE
INTRODUCTION
In a bar or pipe whose ends are free to vibrate, a
standing wave condition is created when it is struck
on its side, like in the case of the marimba or the
glockenspiel. The constraint for this type of vibration
is that both ends of the bar must be antinodes. The
simplest way a bar can vibrate with this constraint is
to have antinodes at both ends and another at its
center. The nodes occur at 0.224 L and 0.776 L. This
produces the fundamental frequency
If the bar is struck on its side, so that its vibration is
like that shown, the frequency of the nth mode of
vibration will be:
𝜋𝑣𝑘𝑚 !
𝑓! =
𝐿!
Where: v =
Material
Bar length, L
The mode of vibration, producing the next higher
frequency, is the one with four antinodes including
the ones at both ends. This second mode has a node
in the center and two other nodes at 0.132 L and
0.868 L
the speed of sound in the material of
the bar or pipe (some speeds for
common materials are shown below
Speed of sound, v (m/s)
Brass
3500
Copper
3650
Iron
4500
Aluminum
5100
Steel
5250
L = the length of the bar
m = 3.0112 when n = 1, 5 when n = 2,
7 when n = 3, … (2n + 1)
K=
thickness of bar
3.46
for rectangular bars
or
Bar length, L
K=
€
PROCEDURE
(inner radius)2 + (outer radius)2
2
for tubes
€
1. There is a copper pipe in your box that was once part of a copper-pipe xylophone. In the space below, make
measurements and calculations that lead to a prediction of the note on the Equal Tempered Scale it was
designed to play. Carefully show all measurements and calculations below.
Pipe #: _____
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2. There is an aluminum bar in your box that was once part of an aluminum-bar xylophone. In the space below,
make measurements and calculations that lead to a prediction of the note on the Equal Tempered Scale it
was designed to play. Carefully show all measurements and calculations below.
Bar #: _____
QUESTIONS AND PROBLEMS
1.
Let’s say you were going to make a set of wind chimes with four pipes. The lowest frequency will be C5
(523.25 Hz). The other frequencies will be an F5, a G5, and a C6. You will use aluminum pipe that has a
5.0 cm outside diameter and a 4.8 cm inside diameter. Assuming you are going for first mode vibrations,
how long would you make each of the pipes?
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In the space below and on the following page, do the following problems from the Giancoli book:
Pages 319 - 321, problems 52, 54, 57 -andPage 348, problems 25, 31, 33, 41
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