Question

Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have the frequency 440 Hz and the note E should be at 659 Hz. The tuner can determine this by listening to the beats between the third harmonic of the A and the second harmonic of the E. A tuner first tunes the A string very precisely by matching it to a 440 Hz tuning fork. She then strikes the A and E strings simultaneously and listens for beats between the harmonics. What beat frequency indicates that the E string is properly tuned

Answer 1

The beat frequency = 2.0 Hz

Step-by-step explanation:

From the given information

The 3rd harmonic frequency of the note is equivalent to thrice note A fundamental frequency.

i.e.

`f_(3A) = 3f_A`

where;

`f_A =` note A fundamental freq.

replacing 440 Hz for
`f_A`

`f_(#a) = 3 (440 \ Hz)`

`f_(3A) = 1320 Hz`

However, the 2nd harmonic of the E is equivalent to two times the fundamental frequency of the note E.

i.e.

`f_(2E) = 2f_E`

`f_E =` note E fundamental freq.

replacing 659 Hz for
`f_E`

`f_(2E) = 2(659)`

`f_(2E) = 1318 \ Hz`

Finally, the beat frequency when the E string is properly tuned is:

`\Delta f = f_(3A)-f_(2E)`

`\Delta f = 1320 \ Hz - 1318 \ Hz`

`\Delta f = 2.0 \ Hz`

Thus, the beat frequency = 2.0 Hz