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. The beat frequency that indicates that the E string is properly tuned is 2.0 HzThe tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

Respuesta :

Answer:

658 Hz

Explanation:

Third harmonic is given by

[tex]f_A=3\times 440\\\Rightarrow f_A=1320\ Hz[/tex]

Second harmonic is given by

[tex]f_E=2\times 659\\\Rightarrow f_E=1318\ Hz[/tex]

The difference between the two frequencies is 1320-1318 = 2 Hz

Now, [tex]\Delta f=4\ Hz[/tex]

The beat frequency is given by

[tex]\Delta f=3f_A-2f_E'\\\Rightarrow f_E'=\dfrac{3f_A-\Delta f}{2}\\\Rightarrow f_E'=\dfrac{1320-4}{2}\\\Rightarrow f_E'=658\ Hz[/tex]

The frequency of the E string when she hears four beats per second is 658 Hz