Beats phenomenon — how two waves of slightly different frequency create beats

When two sound waves of slightly different frequencies $f_1$ and $f_2$ superpose, the resultant intensity fluctuates — rising and falling periodically. Each cycle of rising and falling is called a beat. The number of beats per second equals the difference in frequencies.

Let the two waves be: $y_1 = A\sin(2\pi f_1 t)$ and $y_2 = A\sin(2\pi f_2 t)$.

By superposition: $y = y_1 + y_2$.

Using $\sin C + \sin D = 2\cos\left(\frac{C-D}{2}\right)\sin\left(\frac{C+D}{2}\right)$:

The resultant is a wave of average frequency $\frac{f_1 + f_2}{2}$ whose amplitude varies at frequency $\frac{|f_1 - f_2|}{2}$.

The amplitude term $2A\cos\left(\pi(f_1 - f_2)t\right)$ completes one full cycle in time $\frac{1}{|f_1 - f_2|}$ seconds.

But the intensity (proportional to amplitude squared) has maxima twice per cycle of the amplitude variation. So:

For our problem: $f_{beat} = |260 - 256| = \mathbf{4 \text{ beats per second}}$.