Example from Physics, 14.4 Sound Interference and Resonance

Using Beat Frequency to Tune a Piano

Piano tuners use beats routinely in their work. When comparing a note with a tuning fork, they listen for beats and adjust the string until the beats go away (to zero frequency). If a piano tuner hears two beats per second, and the tuning fork has a frequency of 256 Hz, what are the possible frequencies of the piano?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Since we already know that the beat frequency f_B is 2, and one of the frequencies (let’s say f₂) is 256 Hz, we can use the equation $f_{B} = | f_{1} - f_{2} |$ to solve for the frequency of the piano f₁.
Solution
Since $f_{B} = | f_{1} - f_{2} |$ ,

we know that either $f_{B} = f_{1} - f_{2}$ or $- f_{B} = f_{1} - f_{2}$ .

Solving for f₁,

$f_{1} = f_{B} + f_{2} or f_{1} = - f_{B} + f_{2} .$
14.19

Substituting in values,

$f_{1} = 2 + 256 Hz or f_{1} = −2 + 256 Hz$
14.20

So,

$f_{1} = 258 Hz or 254 Hz.$
14.21

0 of 2 steps open

How did it go?

Strategy

Solution