# Superposition of Waves

An important property of the wave equation is known as the superposition principle. If \( y_1 (x,t) \) and \( y_2 (x,t) \) are two solution of the wave equation, the so is their sum, \( y_{sum} (x,t) = y_1 (x,t) + y_2 (x,t) \). This simple observation leads to a variety of interesting phenomena.

### Beats

The phenomenon of beats arises when two waves of similar, but not equal, frequencies overlap at some region of space. Let's consider then two sources of sound, a piano and a tuning fork, that produce sound waves of different frequencies, say \( f_1 \) and \( f_2 \), respectively. When the sound waves reach our ear they combine together,

$$ y (0,t) = y_{piano} (0,t) + y_{fork} (0,t) $$

$$ y (0,t) = \sin (2 \pi f_1 t) + \sin (2 \pi f_2 t) $$

Where I have assumed our ear or listening device is position at \( x = 0 \). Also, I have chosen the phases and amplitudes for convenience and simplicity, but the effect we are about to discuss does not depend on these choices. Using a trigonometric identity we find that,

$$ \sin (2 \pi f_1 t) + \sin (2 \pi f_2 t) = 2 \cos \left( 2 \pi \left( \frac{f_1 - f_2}{2} \right) t \right) \sin \left( 2 \pi \left( \frac{f_1 + f_2}{2} \right) t \right) $$

The second factor represents a note with frequency,

$$ f_{sum} = f_1 - f_2 $$

Typical notes are measured in hundred of \( Hz \), and the human ear cannot discern the rapid oscillations associated with the notes. However, \( f_{beat} \) can easily be only a few \( Hz \) or less as one tunes the piano to the fork and bring \( f_1 \rightarrow f_2 \). Such amplitude modulations are certainly detectable by our ears and one can perceive the note's intensity increasing and decreasing in a periodic fashion. The piano is tuned when \( f_{beat} = f_1 - f_2 \rightarrow 0 \). This can be detected by listening to the beat as its period grows larger and larger.