Name: _________________________________ Lab Section: _________ Group Number: __________
Lab Partners: _______________________________________________ Grade: ____________
Physics 202
Experiment #2
Vibrating String
Introduction We will be studying standing waves in a string. In standing waves, the nodes (points of zero
motion) and antinodes (points of maximum motion) are fixed. There are many possible modes of standing waves
in a string, each with a different number of antinodes. The mode with one antinode is called the first harmonic
(also called the fundamental), the mode with two antinodes is called the second harmonic, etc. The term harmonic
means that the frequencies of these modes are integer multiples of the fundamental, as we will see.
The frequency, f, and wavelength, , of a wave in the string are related to the speed of the wave, v, according to
v=f
eqn 1
and the speed of the wave, depends on the string properties: the linear density, , (mass per unit length), and the
tension in the string, T.
T
v
eqn 2

1) Write an equation for the frequency as a function of wavelength, linear density, and tension. Write this equation
on you in-class lab form, so you can use it for calculations.
If you observe a stringed instrument being played, you can hear how the pitch (or frequency) is affected by these
physical quantities.
First, if you look at the instrument itself, you will notice that the lower strings are “heavier,” i.e., if you had equal
length of low string and high string, the low string would have more mass.
2) According the equation you derived for question #1, should a larger linear density produce a higher or lower
pitch?
Vibrating String
1
Next, if you watch the musician tuning the instrument, you can see that they tighten or loosena string to adjust its
pitch.
3) According to the equation you derived for question #1, should you tighten or loosen a string that is playing a
little too low?
Usually when a string is played, the pitch corresponds to the fundamental mode (or first harmonic). This mode had
a node at each end of the string and an antinode in the center.
4)
a) Sketch this mode below. On your sketch, clearly label the length of the vibrating string.
Note: Make all sketches neat and symmetric.
b) If there is one or more wavelengths in your sketch, clearly label the distance corresponding to one
wavelength. If not, indicate what fraction of a wavelength is present in your sketch.
5) From your sketch, you should be able to determine the relationship between L and  for the first harmonic.
Write this equation in the first cell of the table on your in-class lab form, then use this equation to write the
frequency of string vibration as a function of string length, linear density, and tension for the first harmonic. Write
this latter equation on your in-class lab form also, so you can use it for calculations.
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Vibrating String
Now if you watch the musician play the instrument, you will notice that they sometimes place a finger on a string
to effectively shorten the length of the vibrating string.
6) According to the equation you derived in question #5, will this shortening of the string result in a higher or
lower pitch?
A good string player can also play harmonics other than the fundamental by forcing additional nodes in the string.
The second harmonic will not only have a node at each end, it will also have a node in the center.
7)
a) Sketch this mode below. On your sketch, clearly label the length of the vibrating string.
b) If there are one or more wavelengths in your sketch, clearly label the distance corresponding to one
wavelength. If not, indicate what fraction of a wavelength is present in your sketch.
8) From your sketch, you should be able to determine the relationship between L and  for the second harmonic.
Write this equation in the appropriate cell of the table on your in-class lab form, and then use this equation to write
the frequency of string vibration as a function of string length, linear density, and tension for the 2nd harmonic.
9) The frequency of the second harmonic should be twice the frequency of the first harmonic. Is that what you
find, if you compare your answers to questions 5 and 8?
Vibrating String
3
We have guided you through the first and second harmonics; let’s see if you can guess the pattern:
10) The second harmonic had twice the frequency of the first harmonic.
frequency to be ____ times the fundamental frequency.
You expect the third harmonic
11) The first harmonic had 2 nodes, the second harmonic had 3. How many nodes do you think the third
harmonic has?
12)
a) Sketch this mode below. On your sketch, clearly label the length of the vibrating string.
b) If there is one or more wavelengths in your sketch, clearly label the distance corresponding to one
wavelength. If not, indicate what fraction of a wavelength is present in your sketch.
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Vibrating String
13) From your sketch, you should be able to determine the relationship between L and  for the 3rd harmonic.
Write this equation in the appropriate cell of the table on your in-class lab form, and then use this equation to write
the frequency of string vibration as a function of string length, linear density, and tension for the 3rd harmonic.
14) Does your answer to question 13 agree with your guess in question 10?
15) Do you see a pattern to the relationship between wavelength, string length, and harmonic number? Write the
general relationship (true for any harmonic number, n) here and in the appropriate cell of the table on your in-class
lab form. Then evaluate it for the fourth harmonic and write that in the appropriate cell of the table on your inclass lab form.
Vibrating String
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Name: _________________________________ Lab Section: _________ Group Number: __________
Lab Partners: _______________________________________________ Grade: ____________
Physics 202
Experiment #2
Vibrating String
Introduction We will be studying standing waves in a string. In standing waves, the nodes (points of zero
motion) and antinodes (points of maximum motion) are fixed. There are many modes of standing waves in a
string, each with a different number of antinodes. The mode with one antinode is called the first harmonic (also
called the fundamental), the mode with two antinodes is called the second harmonic, etc. The term harmonic
means that the frequencies of these modes are integer multiples of the fundamental, as we will see.
The frequency, f, and wavelength, , of a wave in the string are related to the speed of the wave, v, according to
v=f
eqn 1
and the speed of the wave, depends on the string properties: the linear density, , (mass per unit length), and the
tension in the string, T.
T
v
eqn 2

Answer to question#1 in prelab:
Answer to question #5 in prelab:
1st harmonic
2nd harmonic
3rd harmonic
4th harmonic
Relationship
between
L and 
6
Vibrating String
generalized to nth harmonic
Part I: Varying Tension:
Attach the string to the vibrator and plug the vibrator into the function generator (which will provide oscillations for
your string). Hang the string over the pulley, and place weights on it, so that it is under tension.
Determine the tension in the string.
Turn on the function generator and vary the frequency until you have a single loop (single antinode), and thus
the first harmonic. Measure the vibrating string length, which equals the distance from the node closest to the
vibrator to the node at the pulley.
1. Use this length and the table above to determine the wavelength of the first harmonic.
If we vary the tension in the string, the speed of the wave will vary according to eqn 2. If the speed of the wave
varies, for a fixed string length, the fundamental frequency will vary according to eqn 1.
2.
From pre-lab question #1, we have an expression for the frequency as a function of wavelength, tension,
and linear density. Do you expect the fundamental frequency to increase or decrease as you increase the
tension? Are the two proportional?
Measure the fundamental frequency for four slightly different tension values.
3.
To make a straight-line graph that shows the relationship between tension and frequency,
a) what should you graph on the x-axis?
b) what should you graph on the y-axis?
c) what will your slope be (in terms of linear density and wavelength)?
Vibrating String
7
Make this plot and determine the slope.
4.
From the slope, calculate the linear density of the stretched string, and the error in this value.
We would like to compare the linear density determined from the graph to a direct measurement of the linear
density. It is easy to measure the linear density of the unstretched string, but when it is stretched, its linear
density changes, so we need to find this value.
5.
Describe a method for directly determining the linear density of the string when stretched.
Following your method, determine the linear density of the stretched string, and the error in this value.
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Vibrating String
6.
Do your two values of the linear density agree within the stated error? Explain.
Part II: Length Variation
Now we would like to determine how the fundamental frequency varies with the length of the string.
7.
From pre-lab question #5, we have an expression for the fundamental frequency as a function of string
length, tension, and linear density. Will a longer string resonate at a higher or lower frequency? Are
frequency and length proportional?
You have determined the fundamental frequency for one length from Part I. Measure the fundamental
frequency for three more lengths.
Include in your spreadsheet the harmonic number, frequency, string length, and wavelength for all of these
cases as well as the case from Part I.
8.
According to eqn 1, what quantities should you graph on the x- and y-axes in order to determine the
speed of the wave from the slope of the graph?
Make this graph, and determine its slope. Find the speed of the wave in the string, and the error in this speed.
Vibrating String
9
9.
Determine the speed of the wave from the tension and the linear density, as well as the error in this
quantity.
10.
Do your two values of the speed agree within the stated error? Explain.
Part III: Harmonics
Choose a length and tension for which you have already determined the fundamental frequency. Write the
length, tension, and fundamental frequency here, and record in spreadsheet.
Using these values of length and tension, vary the frequency of the function generator to produce three more
harmonics. For each of these, record the harmonic number, the frequency and the length of the vibrating string.
11.
In each case, use the table above and the measured length of vibrating sting to determine the wavelength of
the wave. Then use equation 1 to determine the frequency for each harmonic.
a) 2nd harmonic
b) 3rd harmonic
c) 4th harmonic
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Vibrating String
12.
Are the measured frequencies of these harmonics integer multiples of the fundamental frequency?
13.
Are any integers missing (for example, do you find a first and third harmonic, but not a second)?
Some instruments do not play all harmonics; do stringed instruments play all harmonics?
Vibrating String
11
Post-lab Questions:
1) In this lab, we did not investigate what would happen if we changed the linear density. Describe an
experiment that would determine the relationship between frequency and linear density.
2) If you did the experiment you described in #1, what quantities would you graph in the x- and y-axes to
obtain a linear graph?
3) What was the objective of this lab? Do you feel the objective was appropriately achieved? Why or why
not?
4) Name the two most significant sources of error in this experiment (Be specific – do NOT say, for
example, “human error” or “equipment limitations”). Are these errors likely to be random or systematic?
Explain.
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Vibrating String
5) Describe some ways that the error in this lab could be reduced, including the possible purchase of
additional equipment.
6) What were the results of this experiment? Was there a physical law you were trying to verify, if so, state
that law. Were you trying to determine the value of some quantity or quantities? If so, what was (were)
the value(s)? Do the error(s) in the value(s) make sense? If not, please explain why.
Vibrating String
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