Suppose that two experimenters, holding opposite ends
of a stretched string, each shake their end of the string, sending
wave crests toward each other. What will happen in the middle of
the string, where the two waves meet? Mathematically, you can calculate
the displacement in the center by simply adding up the displacements
from each of the two waves. This is called the principle of
superposition: two or more waves in the same place are superimposed
upon one another, meaning that they are all added together. Because
of superposition, the two experimenters can each send traveling
waves down the string, and each wave will arrive at the opposite
end of the string undistorted by the other. The principle of superposition
tells us that waves cannot affect one another: one wave cannot alter
the direction, frequency, wavelength, or amplitude of another wave.
Suppose one of the experimenters yanks the string downward,
while the other pulls up by exactly the same amount. In this case,
the total displacement when the pulses meet will be zero: this is
called destructive interference. Don’t be fooled by
the name, though: neither wave is destroyed by this interference.
After they pass by one another, they will continue just as they
did before they met.
On the other hand, if both experimenters send upward pulses
down the string, the total displacement when they meet will be a
pulse that’s twice as big. This is called constructive interference.
You may have noticed the phenomenon of interference when
hearing two musical notes of slightly different pitch played simultaneously.
You will hear a sort of “wa-wa-wa” sound, which results from repeated
cycles of constructive interference, followed by destructive interference
between the two waves. Each “wa” sound is called a beat,
and the number of beats per second is given by the difference in
frequency between the two interfering sound waves:
orchestras generally tune their instruments so that the note “A”
sounds at 440 Hz. If one violinist is slightly out of tune, so that
his “A” sounds at 438 Hz, what will be the time between the beats
perceived by someone sitting in the audience?
The frequency of the beats is given by the difference
in frequency between the out-of-tune violinist and the rest of the
Thus, there will be two beats per second,
and the period for each beat will be T