EXAMPLE
Modern 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 orchestra: Thus, there will
be two beats per second, and the period for each beat will be T = 1 /f = 0.5 s.
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 one-half of the wavelength, , then the superposition of the two waves will
result in a standing wave that appears to be still.