Physics Lab
Waves on a String
1
Purpose
To investigate standing waves in a string, and to use measured quantities (mass, length, etc.) to
predict the frequencies of “normal modes” of oscillation.
Theory
Equation 15.19 in your textbook tells you that the speed of waves traveling in a string is given by:
v
F

where F is the tension in the string, and µ is the mass per unit length of the string. In addition,
equation 15.1 tells us that the speed of a wave is related to the frequency and wavelength by:
v  f
From the figures on pages 571 and 576 of the text, we see that at the “fundamental frequency” (that
is, the lowest frequency at which there is a standing wave, also referred to as the “first harmonic”)
for a string that is fixed at both ends, λ/2 = L (where L is the total length of the vibrating string).
Likewise, the second harmonic (also known as the “first overtone”) has a wavelength that is equal to
the length of the string; the third harmonic has a wavelength that is 1.5 times the length of the string,
etc. Using these two equations, we can predict the frequency of any harmonics by measuring F, µ,
and L. In this lab, we will predict the frequency of the first two harmonics.
Procedure
Materials needed: wave driver, 2 patch cords (banana plug type), string, pulley, slotted mass set,
balance, meter stick, support rod.
1.
Set up the apparatus as shown by the instructor, before turning on any of the equipment or
launching DataStudio. (The apparatus will look somewhat similar to Figure 15.2, p. 550,
except that the right end of the string passes over a pulley, beyond which a mass will be hung.)
2.
Choose some amount of mass to place at the end of the string. The weight of this mass (i.e.
mg) will be the same as the tension in the string, as long as the equipment is set up properly
(since the acceleration of the mass should be zero).
3.
Measure the mass of a short length of the string using the balance.
4.
The string is elasticized, so that it is able to stretch by a very significant amount; how much it
stretches will depend on how much mass you hang from the end of the string. Estimate how
much the rope has stretched as a result of the hanging mass (no number required; just get a feel
for it). Based on that estimate, stretch the short length of string, and measure the length in its
stretched position. Because this is a rough estimate, record the shortest length that you think is
reasonably probable, what you actually expect the length to be, and the longest length you
Physics Lab
Waves on a String
2
think is reasonably probable. Also note that you should measure the length of the inner elastic,
not the length of the outer fabric, which has become permanently stretched.
5.
Based on the tension, mass, and lengths found in steps 2-4, calculate the velocity (using
Equation 15.19). You should have a low estimate (from your shortest probable length), an
expected estimate (from your expected length), and a high estimate (from your longest
probable length).
6.
Measure the length of the string from the wave driver to the pulley. This will serve as L in
further calculations.
7.
From Equation 15.1 and your calculation of velocity in step 5, calculate the frequency you
expect for the fundamental frequency (where λ = 2L). Again, you should have 3 numbers: a
low estimate (from your low velocity), an expected frequency, and a high estimate.
8.
Launch DataStudio, and create a new experiment. In the Setup window, double-click on the
“Signal Output” icon. Make sure the banana-plug cables are connected both to the Science
Workshop interface, and to the Mechanical Wave Driver. IMPORTANT: make sure the Wave
Driver is in the “unlocked” position before starting the Signal Output in step 10.
9.
Set the Amplitude of the signal generator to 4.0 V (down from 5.0 V), and set the frequency to
your expected frequency.
10.
Click the Start button. Observe the string, to see if it really is vibrating in its fundamental
mode (that is, similar to the first diagram in Figure 15.20, on page 571 of your text). NOTE:
you may need to press the string to ensure that it stays connected to the wave driver. Adjust
the frequency up and down through the whole range you found via your high and low estimate,
looking for the frequency that gives the greatest amplitude. This will be the actual
fundamental frequency. Record the actual fundamental frequency.
11.
Adjust the frequency of the signal generator, to make it double the fundamental frequency.
Observe the string, to ensure that it is vibrating in its second harmonic. Adjust the frequency
slightly higher and lower, to find the frequency that best matches the second harmonic.
Extra Credit:
Repeat the above procedure, for a different mass than the one you chose initially, and inspect
and describe several higher harmonics.
Notes regarding your lab report
The Data section of your report should include all of the measurements you actually made (that is,
the mass and length of the short piece of rope, the hanging mass, the length of the vibrating string,
and the observed frequencies).
The Analysis section of your report should include the calculations you made (that is, the velocities
and frequencies). In addition, you should calculate the percent difference between the expected
frequency and actual frequency, for the fundamental and second harmonic.
Physics Lab
Waves on a String
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The Results section of your report should be a table, giving the high, low, expected, and actual
(observed) frequencies for the fundamental and second harmonic, together with their percent
differences.
In addition to the general sorts of comments required in the Conclusion section of every report (as
described in the “Lab Overview” handout), you should speculate about what may have caused the
differences between the actual frequencies and your expected values. If you do the extra credit, be
sure to have comments in the conclusion that relate to both systems (the original mass, and the
second mass).