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. 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?

Answer 1

658 Hz

Step-by-step explanation:

Third harmonic is given by

`f_A=3* 440\\\Rightarrow f_A=1320\ Hz`

Second harmonic is given by

`f_E=2* 659\\\Rightarrow f_E=1318\ Hz`

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

Now,
`\Delta f=4\ Hz`

The beat frequency is given by

`\Delta f=3f_A-2f_E'\\\Rightarrow f_E'=(3f_A-\Delta f)/(2)\\\Rightarrow f_E'=(1320-4)/(2)\\\Rightarrow f_E'=658\ Hz`

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