Question

Question #8c553

Answer 1

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 * color(gren)(sin pi ( f_1 + f_2) t) * color(red)( cos pi (f_1 - f_2) t )`

The red cosine term is the beat frequency we can perceive, ie `f_b = abs (f_1 - f_2)`. It acts as an amplitude envelope for the green sine term.

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

`A(t) = 2 A * cos pi (f_1 - f_2) 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 (f_1 - f_2) t = pm1`, intensity is quadrupled as amplitude is doubled from A to 2A.