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 a frequency of 440 Hz and the note E should be

at 659 Hz.

a. What is the frequency difference between the third harmonic

of the A and the second harmonic of the E?

b. 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?

c. The 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?

Short answer

The solutions give a detailed description of the different phases of amplitude and

the frequency difference between the third harmonic

of the A and the second harmonic of the E

Step-by-step solution

Discussion on the path difference

The path difference also deals with the phase differences between them and the third speaker.

Explanation on phase difference

Provided sloution

$\mathcal{l}=1320-1318=2Hz$

Step 3:

The most clearly audible beat will be just the one found. Therefore, the beat frequency will be 2Hz. The frequency given by$$\begin{gathered} f_b=f_{3a}=f_{2e} \\ {f}_{3a}=1320 \end{gathered}$$