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Physics 102/112--Lab 6: The Vibrating String
© James J. DeHaven, Ph.D., 1994-2009
When a string is fastened at both ends, and then set into vibration, standing waves are set up on the string.
The frequency of these waves is determined by the physical parameters of the string: its mass, the tension under
which it is held, and the length of the string between the points at which it is fastened. The frequencies of the
standing waves which can exist on the string will determine the frequencies of sound waves emitted by the string.
When we understand the physics of the vibrating string, we understand the basis for musical instruments
such as the guitar, violin, and other stringed instruments. In addition, we gain an insight into the production and
control of sound from all instruments, as well as an understanding of the measurable physical quantities which
determine our human, subjective perception of sound.
In this lab you will investigate the production of standing waves on a guitar string. You will measure the
value for the fundamental frequency of such a string, and predict and verify the frequencies of its overtones. You
will then systematically investigate the influence of
A) length
B) linear density
C) tension
on the frequencies of the fundamentals and their overtones, and use the data thereby obtained to infer the
nature of the dependence of the frequencies of vibration of the string on these variables.
Introduction
When a string fastened at both ends is subjected to a periodic applied force, only certain frequencies seem
to be effective in setting the string into vibration. We observed this behavior in the classroom, when we applied a
periodic (oscillatory) force to a piece of elastic cord held at each end by a student. As you will recall, only certain
vibrational frequencies were effective in getting the cord to vibrate, and these manifested themselves in the curious
phenomena of “standing waves” that we could observe on the cord. Only waves with a wavelength that went to
zero at each end were observed.
Observed
Not Observed
Figure 1: Only those standing waves which obey the boundary conditions are observed on a vibrating string
-2 The frequencies of these standing waves are determined by the length of the sting that is vibrating (L), and
the phase velocity of waves on the string (v):
[1]
f = nv
2L
n = 1, 2, 3 , . . .
Only certain distinct frequencies (sometimes called the eigenvalues of the string) can be observed. This is
because “n” is limited to the integers--”n” can never have a fractional value. In principal, there are an unlimited
number of possible frequencies that we could observe, because “n” can be as large as we want to imagine, but
there is still a strict limit that no frequencies in between these eigenvalues can be observed, unless we change L or
v, whereupon a different set of distinct frequencies is observed.
The phase velocity for a wave on a string is given by:
[2]
v =
Ft
µ
Ft
= tension
µ
= linear density
This means that the frequencies of the standing waves allowed on a string can be varied by varying the
density of the string or by varying the tension under which the string is held.
The lowest allowed frequency for a vibrating string (n =1) is referred to as the FUNDAMENTAL for that
string. When we listen to a stringed instrument, the PITCH that we hear is determined by the frequency of the
fundamental. From equations [1] and [2], it is clear that there are three ways to vary the fundamental frequency:
1) vary the length of the active region--the shorter this length the higher the pitch of the note we will hear;
2) vary the tension on the string--tighter strings will vibrate at a higher frequency--we do this when we tune
(for example) a guitar;
3) vary the linear density of the string--by playing strings with different densities, we can generate different
notes--bass strings on guitars are generally much heavier than the thin strings used to play the higher notes.
The higher permitted frequencies are called the OVERTONES. They are integral multiples of the
FUNDAMENTAL frequency as predicted by equation [1]. These frequencies determine our perception of the
timbre or quality of sound of a musical instrument. The relative intensity of these overtones will vary from
instrument to instrument, and this is how we tell, for example, whether a musical “A” is coming from a guitar or
from a clarinet. The fundamental frequency is the same for both instruments (for middle A this is 440 Hz) and so
we say they have the same pitch. The relative intensities of the overtones (also known as the Fourier Spectrum) of
the two instruments is very different, and this is how we tell one from another.
-3 As noted above, the allowed frequencies for a standing
wave on a string, are dictated by the boundary conditions at the
ends of the string. Simply stated, the amplitude of the wave
must always be zero at each end.
Nodes
Antinodes
Figure 2: Nodes and Antinodes
The allowed waves may also exhibit points other than at each
end at which the amplitude is zero all the time; these are called
nodes. The points at which the wave has a maximum amplitude
are called the antinodes:
The fundamental and various overtones for a musical “A” (440 Hz) being played on a mythical violin are
shown below graphically (figure 3), along with the applicable equations and the numerical values. Keep this figure
in mind, when observing the vibrations on the guitar strings later in the experiment.
Fundamental
f =
v
2L
440 Hz
1st Overtone
f =
v
L
880 Hz
2nd Overtone
v
f = 3
2L
1320 Hz
3rd Overtone
f =
2v
L
1760 Hz
4th Overtone
f =
5v
2L
2200 Hz
•
•
•
•
•
•
•
•
•
•
•
•
9th Overtone
f = 5v
L
4400 Hz
Figure 3: Fundamental and Overtones for a stringed instrument playing a musical “A”
-4 Experimental
The apparatus you will use is known as a “sonometer”. It consists of a string fastened at both ends, two
“bridges” which describe the active region for wave propagation on the string, a tensioning lever used to regulate
the tension in the string, a driver to vibrate the string, and a detector to detect the vibrations induced in the string by
the driver.
String
Adjustment
Knob
Bridge
Driver
Detector
Bridge
Tensioning
Lever
Figure 4: Schematic Diagram of a Sonometer
In order to operate the sonometer, an energy source, capable of sinusoidally exciting the string at a
frequency chosen by the user, is necessary. This is accomplished as follows: A “driver” coil. essentially a lowamplitude speaker is placed directly below the wire (see figure 4). The sound waves emitted by the driver, impart a
sinusoidal force to the wire. The driver itself is powered by a signal generator, which is a device which supplies
electrical energy with a sinusoidally varying voltage at a frequency which can be selected by the user.
The operation of the signal generator is straightforward. The user selects the wave form (sinusoidal,
sawtooth, square wave), and a frequency range (for example 100HZ) and then varies the frequency using the rotary
frequency knob in the lower left hand corner of the front panel.
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Frequency
Display
On/Off
Frequency
Control
Frequency
Range
Selection
Amplitude
Control
Figure 5: Signal Generator
The output from the generator is sent both to the sonometer driver and to channel 1 of the oscilloscope (for
detection).
Driver
Detector
The output from the sonometer detector is sent to
channel 2 of the oscilloscope, and in this way, you can
detect both the driving signal as well as the signal
induced by it in the string at the same time.
Frequency
Generator
Oscilloscope
Figure 6: Block diagram of electrical connections
in sonometer experiment
-6 It is extremely important to note that the string will not usually vibrate at exactly the same frequency
as the the driver coil. The most frequent situation encountered is one in which the string frequency is exactly twice
that of the frequency supplied by the signal generator.
Driving Waveform
(from Signal
Generator)
Vibrating Waveform
(from vibrating
string detector)
Figure 7: Typical scope display illustrating the relationship between the signal generator frequency and the
frequency detected in the string. Usually, the string vibrates at twice the frequency as the driver frequency
If you think about it, this makes sense. Think about a swing, and think about you swaying forwards and
backwards providing energy to the swing each time it reaches you. The first time it gets to you, you push it,
because you are swaying forward. The next time you pull it, because you are swaying backwards. The frequency at
which you are swaying is exactly 1/2 that of the swing. This is essentially what is happening in this experiment,
with the signal generator taking your place, and the string acting like the swing.
It may occur to you to ask whether it can happen that the string could vibrate at 3 times the frequency of the
signal generator, or perhaps 4 or 5 or more. The answer is yes, but you are unlikely to observe this in this
experiment--if you are careful, however, you may notice some evidence of this behavior.
What follows now is an extremely detailed experimental procedure for observing a fundamental vibration
and its overtones on a guitar string, with special attention paid to recommended oscilloscope and signal generator
settings, as well as a description of what you are likely to see.
Determination of the Fundamental Frequency of a Guitar String
The first step is to find out the value of the fundamental frequency of your guitar string.
First, prepare the sonometer. Your instructor will have demonstrated to you how to mount a string in the
sonometer. Use a 0.020 inch diameter string. Be careful because you can stick yourself with the sharps ends of the
wire.
-7 Apply tension to the string using a weight and weight hanger. Use a 500 gram weight and place it in the
third (or middle) slot of the tensioning lever on the sonometer. Make sure this end of the sonometer is close to the
end of the table so that the weight hangs freely. Make sure as well that there is some foam rubber under the weight
in case it drops off or in case the string breaks.
Make sure the tensioning lever is level. This is essential, and you should check this every time you alter the
string or the tension on the string in any way.
Set the bridges at the 10 centimeter mark and the 70 centimeter mark. (What is L?) Place the detector at the
40 cm mark, and place the driver at 15 cm (5 cm away from the bridge).
For this first experiment, you will leave the signal generator turned off.
Turn on the oscilloscope and make sure the following settings are being utilized:
Time Base--10ms/division
Trigger Source--Line
Trigger Mode--Normal
Channel 1 Volts--2.00V/division
Channel 2 Volts--5.00mV/division
Use the position knob to position the trace from channel 1 in the upper half of the scope screen, and that
from channel 2 in the lower half. You might consider making your life easier by allowing the scope to measure the
frequency for you. To do this, press the TIME button on the scope, and use the softkeys to tell the scope to give
you channel 1 frequency measurements and channel 2 frequency measurements.
When the scope is ready, gently pluck the guitar string. You should see a very large signal on the screen
which gradually decreases in size as the vibrations in the string die out. The signal may be temporarily off scale
(bigger than the space allotted for it on the oscilloscope screen). If you see no signal or your signal is very weak,
check your settings carefully. Is your time base 10 milliseconds (not Microseconds)? Is your channel 2 voltage
setting 5 millivolts per division (and not 5 volts)? Are your triggering settings correct? Is the probe attenuation in
the channel menus set to 1x (as opposed to 10x)?
When you are able to get a signal try to capture it. To do this, press the SINGLE SEQ button. If you want
to revert to the continuous running operation, press the Run/Stop button. Any time you want a single trace you
should follow this procedure.
Figure 8 (next page) shows a typical trace taken under these conditions. Note that the frequency is given at
91.24 Hz. Yours may be different. Remember!! Frequency is data--so note it down in your lab notebooks. Note
also that it may be necessary or it may give you a better display if you change the settings for time base or voltage
slightly from the recommended ones. The recommendations should work, but if they don’t, try to vary the settings
a little bit.
Print out your oscilloscope screen. Select the print utility. Press Print Screen. Print Screen will change to
Cancel Print during printing. When printing is done Printing done will flash briefly on the screen. If you have a
54102A oscilloscope, you must press the ONLINE button, followed by FORM FEED, followed by ONLINE
again, for the printer to print. If you have a 54102B scope, you don’t need to do this. If you want to have a record
of all your settings, go to the Hardcopy Menu and the the Printer Menu before you print and turn the factors
softkey top ON. If you do this your output will be similar to that in figure 9. If FACTORS are ON then you
SHOULD NOT do the online, form feed, online trick even if you have a 54102A scope.
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Figure 8: Scope display for Fundamental Frequency of plucked guitar string
The waveform is not precisely sinusoidal, because there are overtones present in all likelihood. Note also
that your waveform may look different than mine, and that the waveform may change in appearance as the
amplitude of the strings oscillations changes.
Note that the scope can find no frequency for channel 1--this is because there is no oscillating signal
connected to channel 1 at this time.
Observing Standing Waves
The fundamental frequency you obtained from plucking the string was from a waveform whose irregular
shape implies that it contains all the Fourier components that are observable on the string, within the constraints of
the experimental apparatus you are using. You will now observe individual standing waves corresponding to the
fundamental and each of the first three overtones. To do this you will add energy to the string using a signal
generator.
As noted above, the oscillator frequency will be exactly 1/2 the frequency of the standing wave that it
excites on the guitar string. Since we saw above that the natural frequency of the guitar string corresponds to about
9 wave crests on the scope display, then we expect that we can excite the fundamental using a frequency of just half
that (about 4.5 crests). YOUR SETTINGS AND RESULTS MAY DIFFER SOMEWHAT FROM MINE.
USE MY RESULTS AS A GUIDE, NOT AS A BIBLE!!
First, it is important to restore the scope to the proper settings. Note the change in the trigger source!
Time Base--10ms/division
Trigger source--Channel 1
Trigger Mode--Normal
-9 Channel 1 Volts--2.00V/division
Channel 2 Volts--5.00mV/division
Second: make sure the sonometer is still set up properly
Tensioning Lever--level by adjusting string adjustment knob
Hanging Mass--500 grams +weight hangar
Location of Hangar--Slot 3 (middle slot) of tensioning lever
Bridge 1--10cm mark
Bridge 2--70cm mark
Driver Coil--15cm mark
Detector Coil--40cm mark
Guitar String--0.020 inches in diameter
Third: Prepare the frequency generator
Turn it on
Make sure the following pushbuttons are depressed:
RANGE--100
FUNCTION--sine wave
ALL PUSHBUTTONS SHOULD BE OUT
Make sure that the AMPLITUDE control is set to MAX
ALL OTHER SMALL DIALS SHOULD BE SET TO THEIR MINIMUM VALUES
(CCW AS FAR AS THEY WILL GO) AND PUSHED IN
The oscilloscope should now show a sine wave signal in the upper trace (corresponding to channel 1, if you
have set the horizontal positioning knobs on the scope correctly) and, depending on the frequency being put out by
the signal generator, may show a flat or a sinusoidal trace coming from the detector (channel 2).
Search for the fundamental by carefully rotating the FREQUENCY control on the signal generator until the
detector trace (channel 2) shows a pronounced change in amplitude. The effect will be very dramatic: when you are
at the correct frequency, the standing wave will have an amplitude so large that the sine wave may well go off scale.
A trace of the experiment I performed is shown below (fig. 9). Note that you know roughly where you should
look--you already have the results for the fundamental frequency from the first part of this experiment.
Depending on how good our hearing is, you may hear the string vibrating. If you are pumping it at the right
frequency, you will definitely see it vibrating. You should see a single pronounced anti-node halfway between the
two bridges as expected. You may have to temporarily change the Volts/Div setting for channel 2 in order to allow
the signal to fit on the screen. If the signal doesn’t fit on the screen, the scope will not be able to measure the
frequency for you.
If the signal seems to drift in and out, don’t worry. This is because the signal generator drifts slightly. If at
any time you want to capture what is on screen, just press the SINGLE SEQ button. To return to continuous
operation, press the RUN/STOP button. Figure 10 shows what the signals looked like after I adjusted the volts/Div
in channel 2. Note the frequency of 89.35 Hz is not too far from that obtained in part 1 of this experiment (see fig.
8).
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Figure 9--Fundamental: when you have hit it right, the trace goes off scale at 5mv/Div
Figure 10--Frequency measurement of fundamental on vibrating guitar string
-11 Now return the scope to its original settings. Look for the 1st overtone by turning the frequency knob to a
higher value. You ought to be able to
predict where this should be.
IMPORTANT!!! To observe the
first overtone you should move the
detector to the 55cm mark. If you leave it
at 40cm, it is likely that you wont observe
anything at the first overtone setting.
Why is this? You should also be sure
you have returned the channel 2 Volts
setting to 2mV per division.
The results of the experiment that
I performed looking for the first overtone
are shown in figure 11. You should see
both anti nodes very clearly on the string
as well as the (motionless) node at the
center. You will probably be able to hear
the string vibrating when you are “on
frequency”.
Now perform the same
experiment for the 2nd and 3rd overtones
(Figs. 12 & 13). You should still be able
to see the anti nodes, and you will
definitely be able to hear them. If you
have set up the experiment correctly, all
this should be observable while you are
on the 100 Hz scale of the signal
generator (whose range is from 10 to 190
Hz). If you have trouble detecting those
overtones, try moving the detector coil to
a place where it is right under an anti
node. This is where the signal will be at
its highest.
Figure 11--1st Overtone
For the second and third
overtones, map the standing wave.
Carefully measure the amplitude of the
signal for several different detector
positions, and show that the proper
number of nodes and anti nodes are
present. (Note--when the detector is
brought too close to the driver, it picks up
the driver signal directly--don’t panic if
you suddenly see a “strange” signal
appear in channel 2, and grow as the
detector is brought nearer to the driver.
Figure 12--2nd Overtone
-12 Note that it is possible to “expand”
the wave horizontally by alerting the
Time/Div setting as we have done in
Figure 14. The smaller number of peaks,
the more precise the scope is in
measuring frequency. However, if the
time/Div is set so low that less than one
peak appears on the screen, the scope
will not be able to measure a frequency.
The frequencies for the standing
waves I observed can be read from the
data for the frequency of the channel 2
signal taken directly from the
oscilloscope screen. They are as
follows:
Fundamental
1st Overtone
2nd Overtone
3rd Overtone
Figure 13--3rd Overtone
89.35Hz
177.9Hz
271.7Hz
364.4Hz
Are these values reasonable given
reasonable expectations of experimental
error? How are yours in comparison?
Figure 14--3rd Overtone at Time/Div setting of
500 microseconds
-13 Observation of Higher Overtones
Return the scope to its original settings with one exception: set the Time/Div to 1 msec/Div. Push in the 1K
RANGE button on the signal generator. Starting with the FREQUENCY control knob in its ccw position, tune the
frequencies of the driver while watching the scope screen and listening to the string. Can you hear the distinct
overtones that the string can support? It really sounds quite spooky and musical. Try to capture and print out a
higher overtone. I observed the 9th overtone (note the scope settings) and my results are displayed in Figure 15. Is
the frequency shown for the signal in channel 2 consistent (within reasonable expectations for experimental error)
with what you would expect for the ninth overtone of a string whose fundamental frequency has been measured at
89.35Hz?
Figure 15--Ninth Overtone
Dependence of Fundamental Frequency on Length
You can alter the active length of the string by moving the bridges. Using the methods employed in part 1
of this lab, where you measured the fundamental frequency of the string by plucking it and observing the waveform
on the oscilloscope, find out how the frequency of the fundamental depends on the active length of the string.
Make sure you return to your original scope settings. Since you will not need the signal generator for this
part of the experiment, you can turn it off. Measure the fundamental frequency for active lengths of 60, 50, 40, 30,
and 20 cm. Make sure that everything else in your experiment is held constant. Vary only the length.
-14 Dependence of Fundamental Frequency on Tension
5 mg
4 mg
3 mg
2 mg
1mg
As shown on the left (Fig. 16), you can vary the tension on the
string from mg to 5 mg, in steps of mg, where m = the mass of the hangar
and the hanging weight and g is the acceleration due to gravity.
This time set the active length of the string to 60cm, and keep
everything constant while varying the tension by moving the hanging mass
from one notch to another in the tensioning lever. Again, use the oscilloscope
to measure the fundamental frequency for each value of tension.
You may have to change the Time/Div settings to accommodate the
wide range of frequencies you may encounter. Be sure to be careful that you
level the tensioning rod each time you change notches.
Figure 16--Tension provided by
tensioning lever
Dependence of Fundamental Frequency of Linear Density of the String
You can vary the linear density of the string simply by changing strings. Use the five strings provided-PLEASE DON’T MIX THEM UP!!!--and keep all other experimental parameters constant. Be careful--it is easy
to prick your fingers with the thinner strings. The linear densities of the strings are given below:
Diameter in Inches
0.010
0.014
0.017
0.020
0.022
Linear density in grams per meter
0.39
0.78
1.12
1.50
1.84
Data Reduction
Use the graphical analysis program to deduce the nature of the functional dependence of frequency on the
following three parameters:
1) Active Length
2) Tension
3) Linear Density
From your classroom work, you will probably already have a good idea of what the results should be and
so this will assist you in choosing the relationships to investigate with Graphical Analysis.
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Report:
Introduction: Write a brief introduction stating the objectives of the experiment, and a concise summary of the
methods that will be used.
Experimental: Describe the experimental apparatus and precisely what variables will be measured and how they will
be measured.
Results: Summarize the results of the experiment. Show sample calculations. If you are attaching computer
generated tables or graphs, briefly explain them here.
Discussion: Explain the significance of your results and their connection with more general physical principles.
Where it is possible, compare your numbers with accepted values. Explain any sources of error.