Question #8c553

Question

Question #8c553

1 Answer

Impact of this question

2071 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-26T13:09:31

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.

Science

Math

Humanities

... and beyond

Question #8c553

1 Answer

Explanation:

Related questions

Impact of this question