On a well-tuned instrument, the middle-C note has fundamental frequency 264 Hz.

Question

On a well-tuned instrument, the middle-C
note has fundamental frequency 264 Hz. A
crazy musician tries to accompany a real or-
gan with an electronic synthesizer. On a
cold day, when the speed of sound in air
was 336 m/s he managed to get both instru-
ments in tune; i.e., at the same frequency
when the same key was pressed. Then it got
hot, the speed of sound in air increased to
348.6 m/s, and consequently all of the organ’s
pipes changed their fundamental frequencies.
(Same L, different v, so different f). Natu-
rally, the synthesizer’s frequencies remained
unchanged so the two instruments became badly out of tune.
Calculate the beat frequency of the ca-
cophony that resulted when the musician
pressed the middle C keys on both instru-
ments. Don’t round-off intermediate calcula-
tions.
Answer in units of Hz

Explanation / Answer

Since v = f, f = v/ and we see that frequency is directly proportional to speed. The problem gives us the fact that is constant, so the speed of sound is the only variable.

Our speed increases by the ratio 348.6/336. So the new frequency is:

(264 Hz)(348.6/336) = 273.9 Hz

Beat frequency is simply the difference between two frequencies, so the answer is:

273.9 Hz - 264 Hz = 9.9 Hz

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