Answer to Question #163233 in Physics for Earl

Question

Two speakers are set simultaneously in the auditorium. The two speakers are set with different frequency. If the first speaker is 267Hz and the other is 273 Hz. Determine the beat frequency and the observed frequency heard by crowd.

Answer 1

(a) We can find the beat frequency as follows:

$$f_{beat}=|f_2-f_1|=|273\ Hz-267\ Hz|=6\ Hz.$$

(b) If two sounds differ in frequencies, the sound waves can be modeled as:

$$y_1=Acos(k_1x-2\pi f_1t), y_2=Acos(k_2x-2\pi f_2t).$$

Using the trigonometric identity

$$cosu+cosv=2cos(\dfrac{u+v}{2})cos(\dfrac{u-v}{2})$$

and considering the point in space as $x=0\ m$ we find the resulting sound at a point in space, from the superposition of the two sound waves:

$$y(t)=2Acos(2\pi f_{avg}t)cos(2\pi(\dfrac{|f_2-f_1|}{2})t),$$

here, $f_{avg}=\dfrac{f_1+f_2}{2}$ is the observed frequency heard by crowd.

Then, we get:

$$f_{obs}=f_{avg}=\dfrac{273\ Hz+267\ Hz}{2}=270\ Hz.$$