Ed runs a tuning company and has asked his friend to observe a new employee who is having trouble determining the frequency of a tuning fork. The employee notes that the beat frequency between a 702 Hz tuning fork and the unknown fork is 20 Hz, and between a 708 Hz tuning fork and the unknown is 26 Hz. Using the principles of superposition and beat frequencies, the friend determines that the unknown fork's frequency must be 683 Hz to satisfy both conditions.

1. Physics LO 101
Beats
Q. Ed, a friend of yours, has a tuning company and is
receiving numerous complaints on a new
employee. He asks you to tag along during the
next tune up to find out what the problem is.
During the tune up the employee is unable to
determine the frequency of a tuning fork. He asks
for your help and tells you the following things:
 The beat frequency between a 702 Hz
tuning fork and an unknown tuning fork is
20 Hz
 With an 708 Hz tuning fork the beat
frequency is 26 Hz
Find the frequency of the unknown tuning fork.

2. Solution
After applying the principle of superposition and adding two waves the expression
given is:
stotal(0,t) = 2sm cos(-ϖt)cos(Δωt)
ϖ = (ω1 +ω2)/2 Mean angular frequency
Δω = (ω1 -ω2)/2 Angular frequency difference
Although the difference of frequency is defined as Δω = (ω1 -ω2)/2 for the wave.
What is heard is by our ears are the two individual frequencies that make up the
wave. However, since the wave is getting louder and softer at the frequency of (ω1 -
ω2)/2 it results in us hearing “beats”. The beat frequency is given by the absolute
value of the two slightly different frequencies.
Therefore,
If x = the unknown frequency, then
 20 Hz = | x-702| Hz (1)
 26 Hz = |x-708| Hz (2)
Both of these have two solutions,
In (1) the unknown frequency could be 722 Hz or 682 Hz
In (2) the unknown frequency could be 734 Hz or 682 Hz
We know that the beat frequency increased from 20 to 26 Hz as the known
frequency increased from 702 to 708 Hz. So, in order to distinguish between these
answers, the unknown frequency must be lower than 702 Hz and satisfy both
equations.
The frequency of the unknown tuning fork = 683 Hz