A piano tuner hears a beat every
1.00 seconds when listening to a 164-hertz frequency tuning fork and a single piano
string. What is the frequency of the string
if it is of a lower frequency than the tuning fork?
In this scenario, the tuner is
hearing a beat every 1.00 seconds. That means the beat has a frequency
of 1.00 hertz. We can recall that beat frequency,
𝑓 sub 𝐵, is equal to the difference between the frequency of the one tone being
mixed and the frequency of the other tone. And in case 𝑓 two is greater than
𝑓 one, we put absolute value bars around this difference. Since the tuner in our example
hears a beat every 1.00 seconds, that means there’s one complete wave cycle each
second, or that this beat frequency is 1.00 hertz.
By our beat frequency equation, we
can say that this is equal to 164 hertz, the frequency of the tuning fork, minus the
frequency of the piano string, which we’ve called 𝑓 sub 𝑠. Rearranging, we can say that 𝑓 sub
𝑠 is equal to 164 minus 1.00 hertz, or 163 hertz. That’s the frequency of this piano
string before it’s tuned to the tuning fork.