If you have ever played around with a guitar, you've probably noticed that if two strings are fingered to be the ``same note'' but are really slightly out of tune and are struck together, the resulting sound ``beats'' - it modulates up and down in intensity at a low frequency often in the ballpark of a few cycles per second.
Beats occur because of the superposition principle. We can add any two
(or more) solutions to the wave equation and still get a solution to the
wave equation, even if the solutions have different frequencies. Recall
the identity:
(192) |
If one adds two waves with different wave numbers/frequencies and uses
this rule, one gets
(193) | |||
(194) |
This describes a wave that has twice the maximum amplitude, the average frequency (the first term), and a second term that (at any point ) oscillates like .
The ``frequency'' of this second modulating term is
, but the ear cannot hear the inversion of phase that
occurs when it is negative and the difference is small. It just hears
maximum amplitude in the rapidly oscillating average frequency
part, which goes to zero when the slowing varying cosine does,
twice per cycle. The ear then hears two beats per cycle, making
the ``beat frequency'':
(195) |