Q.Explain how beats are formed when two sound waves of slightly different frequency are superposed, and derive the expression for the beat frequency in terms of the two individual frequencies.

When two sound waves of equal amplitude $A$ but slightly different frequencies $f_1$ and $f_2$ overlap at a point, taking $y_1=A\sin(2\pi f_1t)$ and $y_2=A\sin(2\pi f_2t)$, superposition gives:

This is a wave oscillating at the average (audible) frequency $\bar f=(f_1+f_2)/2$, whose amplitude is not constant but is modulated by the slowly varying envelope $2A\cos\!\left(2\pi\,\tfrac{f_1-f_2}{2}t\right)$, of frequency $(f_1-f_2)/2$. Loudness depends on the SIZE of the amplitude, not its sign, so within one full cycle of this envelope, the amplitude reaches its maximum SIZE ($2A$) twice -- once when the cosine equals $+1$, and again half an envelope-cycle later when it equals $-1$ (since $|-2A|=2A$ too). The listener therefore hears a loud beat twice per envelope cycle, so the number of …

Superpose the two equal-amplitude waves using the sum-to-product identity, identify the slow envelope term and the fast audible-tone term, and note that loudness (depending on amplitude magnitude) pe …

• Stating the beat frequency as $(f_1-f_2)/2$, forgetting that loudness maxima occur twice, not once, per cycle of the envelope term. …