# If a piano tuner hears 3 beats each second when listening to a 8 Hz tuning fork and a single...

## Question:

If a piano tuner hears 3 beats each second when listening to a 8 Hz tuning fork and a single string on the piano which is of higher frequency.. What is the frequency of the single string from the piano?

## Beat frequency

Beats occur when two sound waves of slightly different frequencies {eq}f_a {/eq} and {eq}f_b < f_a {/eq} interferes. The beat frequency, or the number of beats per unit of time, is

{eq}f_\text{beat} = f_a - f_b. {/eq}.

Since the string on the piano is the one with higher frequency, let it be denoted by {eq}f_a {/eq}. Given that {eq}f_b = 8 \ \text{Hz} {/eq} and {eq}f_\text{beat} = 3 \ \text{Hz} {/eq} (3 beats per second), then

{eq}f_\text{beat} = f_a - f_b \quad \Rightarrow \quad f_a = f_\text{beat}+f_b = 3 \ \text{Hz} + 8 \ \text{Hz} = 11 \ \text{Hz}. {/eq}

Therefore, the frequency of the piano string is 11 Hz.