Physics in Motion

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.


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.