The tuner begins by muting the right and center A4 strings. He then strikes the A4 key, and tunes the left A4 string to be exactly the same frequency as a 440 Hz tuning fork. Next, he unmutes the center string, and plays the A4 key. Now he hears both the left and center A4 strings vibrating at the same time. He can hear that the center string is slightly flat, and he can hear beats between the two strings with a beat frequency of 0.40 Hz. What is the frequency of the center string

In this exercise we have two waves with slightly different frequencies, the A4 chord with f = 440 Hz and the beating with the central chord fbeats = 0.40 Hz, let's use the relation

f_{beat} = | f - f₀ |

f = fo + - f _{beat}

f = 440 + 0.4 = 440.4 Hz

f = 440-0.4 = 439.6 Hz