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Standing Waves on Vibrating Strings
(This exercise has been adapted from material in “Confined Waves”, a supplemental
chapter to Electric and Magnetic Interactions by Ruth Chabay and Bruce Sherwood.)
When transverse waves are travelling along a string that is fixed at both ends,
the wave motion must satisfy very specific “boundary conditions”: The total
displacement of the string at the fixed ends must be zero, since the ends can’t
move. Thus whatever kind of total wave motion is produced, the ends must be nodes.
To produce standing waves, we can cause the string to vibrate sinusoidally.
The waves undergo reflections from the fixed ends, creating sinusoidal waves
travelling back and forth along the string, so the net transverse displacement
of the string is the superposition of these waves going in opposite directions.
At most frequencies, the wavelengths don’s satisfy the condition that the ends
can’t move, so the net amplitude of the waves is small and the string just wiggles
in a complicated way. At special frequencies where the wavelengths happen to cause
nodes at the ends, we have resonance, and the maximum displacement of the string
can be very large at some places. For these resonant frequencies, there is always
a definite relationship between the wavelength and the length of the string. For
example, here are snapshots of two particular standing waves. For each, give the
wavelength in terms of the length L of the string:
 = _____
L
2 = _____
L
We see that halfway between nodes, the string occasionally reaches its greatest
possible maximum displacement away from equilibrium. These points are called
antinodes. Nodes and antinodes always alternate along the string, with a node
at each fixed end.
The two standing wave patterns shown above correspond to two different
frequencies. To discover what the pattern is for the allowed frequencies of
standing waves, we must first find out something important about the speed of the
waves on the string. In the box below, write the mathematical equation for the
speed of a transverse wave on a string:
Upon what does the speed depend? In particular, does the speed depend on the
frequency or wavelength of the wave?
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Write the mathematical relationship between the speed, wavelength, and frequency
of a wave:
Since the speed of the wave doesn’t depend on the wavelength (or the length of
the string), all waves travel along the string at the same rate, and the frequencies
of the waves can be calculated from the tension, linear density, and wavelength.
In the box below, derive the mathematical expressions for the frequencies of the
two waves on Page 1:
The longest wavelength standing wave is the one with wavelength 1 ( = 2L). The
frequency f1 of that wave is called the fundamental for the string. The next
highest allowed frequency is that for the wave with wavelength 2 ( = L). How is
f2 related to f1?
Now you’ll derive a few more wavelengths and frequencies.
Strings Classwork 2
Using the boundary conditions (Node at each end of the string) and the fact
that there is always an antinode between each pair of nodes, find the wavelengths
and frequencies for the cases given below. Give  in terms of L and f in terms
of f1.
3 = _____
L
f3 = _____
4 = _____
L
f4 = _____
5 = _____
L
f5 = _____
Do you feel you have found a general rule for predicting the allowed wavelengths
and frequencies of standing waves on a string that is fixed at both ends? Please
state the rule mathematically here:
Question
If you wanted to raise the frequency of the fundamental for a string, how could
you do that? You can’t change the string itself, so its density is constant.
Strings Classwork 3