A 440.0-Hz tuning fork is sounded together with an out-of-tune guitar string, and a beat frequency of 3 Hz is heard. When the string is tightened, the frequency at which it vibrates increases, and the beat frequency is heard to decrease. What was the original frequency of the guitar string?

It is an interesting phenomenon that when two waves of slightly different frequencies (f1 and f2) interfere they produce beats, and the frequency (fb) of those beats is:

[tex] f_{b}=\mid f_{1}-f_{2}\mid [/tex] (1)

Note that because the equation takes the absolute value of the difference between f1 and f2 we have two possible results of [tex] f_{1}-f_{2} [/tex] that satisfy the equation, those are:

[tex] 440.0-f_{2}=3\Longrightarrow f_{2}=437.0\,Hz [/tex]

[tex] -(440.0-f_{2})=3\Longrightarrow f_{2}=443.0\,Hz [/tex]

Because the frequency of the beat is heard to decrease when the frequency of the guitar string is increased, that means the guitar string was at first under the pitch, so the original frequency of the guitar string is 437.0 Hz