Suppose a piano tuner hears 2 beats per second when listening to the combined sound from her...

Question:

Suppose a piano tuner hears 2 beats per second when listening to the combined sound from her tuning fork and the piano note being tuned. After slightly tightening the string, she hears 1 beat per second. Should she loosen or continue to further tighten the string? Why?

The Beat Phenomenon

When two wave sources with slightly differing frequencies {eq}\displaystyle {\nu_1} {/eq} and {eq}\displaystyle {\nu_2} {/eq} generate waves simultaneously and these waves are superposed then an interfernce effect in time will occur. The intensity is found to oscillate with time with a frequency {eq}\displaystyle {\nu} {/eq} called the beat frequency. It is given by {eq}\displaystyle {\nu = \pm (\nu_1-\nu_2)} {/eq}.

The frequency of a string clamped at both ends is given by,

{eq}\displaystyle {\nu=\frac{n}{2L}\sqrt{\frac{T}{\mu}}}-----(1) {/eq}.

Here T is the tension, L is the length of the string, {eq}\displaystyle {\mu} {/eq} is the mass per unit length and n is a natural number.

Clearly with increased tension the frequency increases.

Here it is given that the piano tuner initially hears 2 beats per second when she sounds the tuning fork and the piano string together. Next, she tightens the string and observes that the beat frequency has reduced to 1 per second. Thus the increased tension in the string has increased its frequency according to (1) and thereby reduced the number of beats. Since the beat frequency is the difference between the two sounded frequencies it follows that the process of tightening has caused a convergence of the two frequencies. Therefore she should further tighten the string until there are no beats.