


Standing Waves and Resonance
So far, our discussion has focused on traveling waves,
where a wave travels a certain distance through its medium. It’s
also possible for a wave not to travel anywhere, but simply to oscillate
in place. Such waves are called, appropriately, standing waves.
A great deal of the vocabulary and mathematics we’ve used to discuss
traveling waves applies equally to standing waves, but there are
a few peculiarities of which you should be aware.
Reflection
If a stretched string is tied to a pole at one end, waves
traveling down the string will reflect from the pole
and travel back toward their source. A reflected wave is the mirror
image of its original—a pulse in the upward direction will reflect
back in the downward direction—and it will interfere with any waves
it encounters on its way back to the source. In particular, if one
end of a stretched string is forced to oscillate—by tying it to
a mass on a spring, for example—while the other end is tied to a
pole, the waves traveling toward the pole will continuously interfere
with their reflected copies. If the length of the string is a multiple
of onehalf of the wavelength, , then the superposition
of the two waves will result in a standing wave that appears to
be still.
Nodes
The crests and troughs of a standing wave do not travel,
or propagate, down the string. Instead, a standing wave has certain
points, called nodes, that remain fixed at the equilibrium
position. These are points where the original wave undergoes complete
destructive interference with its reflection. In between the nodes,
the points that oscillate with the greatest amplitude—where the
interference is completely constructive—are called antinodes.
The distance between successive nodes or antinodes is onehalf of
the wavelength, .
Resonance and Harmonic Series
The strings on musical instruments vibrate as standing
waves. A string is tied down at both ends, so it can only support
standing waves that have nodes at both ends, and thus can only vibrate
at certain given frequencies. The longest such wave, called the fundamental,
or resonance, has two nodes at the ends and one antinode
at the center. Since the two nodes are separated by the length of
the string, L, we see that the fundamental
wavelength is . The string can also support standing waves
with one, two, three, or any integral number of nodes in between
the two ends. This series of standing waves is called the harmonic
series for the string, and the wavelengths in the series
satisfy the equation , or:
In the figure above, the fundamental is at the bottom,
the first member of the harmonic series, with n = 1.
Each successive member has one more node and a correspondingly shorter
wavelength.
Example

Sound comes out of bottles when you blow on them because
your breath creates a series of standing waves inside the bottle.
The pitch of the sound is inversely proportional to the wavelength,
according to the equation . We know that the wavelength
is directly proportional to the length of the standing wave: the
longer the standing wave, the greater the wavelength and the lower
the frequency. The tall bottle is twice as long as the short bottle,
so it vibrates at twice the wavelength and onehalf the frequency
of the shorter bottle. To make both bottles sound at the same pitch,
you would have to alter the wavelength inside the bottles to produce
the same frequency. If the tall bottle were halffilled with water,
the wavelength of the standing wave would decrease to the same as
the small bottle, producing the same pitch.
Pitch of Stringed Instruments
When violinists draw their bows across a string, they
do not force the string to oscillate at any particular frequency,
the way the mass on a spring does. The friction between the bow and
the string simply draws the string out of its equilibrium position,
and this causes standing waves at all the different wavelengths
in the harmonic series. To determine what pitches a violin string
of a given length can produce, we must find the frequencies corresponding
to these standing waves. Recalling the two equations we know for
the wave speed, and , we can solve for the frequency, , for any term, n,
in the harmonic series. A higher frequency means a higher pitch.
You won’t need to memorize this equation, but you should
understand the gist of it. This equation tells you that a higher
frequency is produced by (1) a taut string, (2) a string with low
mass density, and (3) a string with a short wavelength. Anyone who
plays a stringed instrument knows this instinctively. If you tighten
a string, the pitch goes up (1); the strings that play higher pitches
are much thinner than the fat strings for low notes (2); and by
placing your finger on a string somewhere along the neck of the
instrument, you shorten the wavelength and raise the pitch (3).
