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

Answer:

The beat frequency = 2.0 Hz

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