# Question #8c553

Oct 26, 2016

C

#### Explanation:

We have 2 sound waves of same amplitude $A$, with frequencies ${f}_{1} \approx {f}_{2}$, so

• ${y}_{1} = A \sin 2 \pi {f}_{1} t$

• ${y}_{2} = A \sin 2 \pi {f}_{2} t$

The resultant wave is

$y = {y}_{1} + {y}_{2} = A \sin 2 \pi {f}_{1} t + A \sin 2 \pi {f}_{2} t$

And by the sum to product identity:

$y = 2 A \cdot \textcolor{g r e n}{\sin \pi \left({f}_{1} + {f}_{2}\right) t} \cdot \textcolor{red}{\cos \pi \left({f}_{1} - {f}_{2}\right) t}$

The red cosine term is the beat frequency we can perceive, ie ${f}_{b} = \left\mid {f}_{1} - {f}_{2} \right\mid$. It acts as an amplitude envelope for the green sine term.

The amplitude of the resultant beat can therefore be seen to be

$A \left(t\right) = 2 A \cdot \cos \pi \left({f}_{1} - {f}_{2}\right) t$

And that is what is perceived when, say, 2 musical notes that are very close in pitch are plucked on a guitar: in the model proposed above, a sinusoid pattern to its amplitude.

Because Intensity $I \propto {A}^{2}$, we can see that at max amplitude, ie when $\cos \pi \left({f}_{1} - {f}_{2}\right) t = \pm 1$, intensity is quadrupled as amplitude is doubled from A to 2A.