Chapter 6.3, Problem 58E

Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. Beats occur when two tones vary in frequency by only a few hertz. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of this slight variation in the frequency. (Source: Pierce. J., The Science of Musical Sound. Scientific American Books.)

(a) Consider the two tones with frequencies of 220 Hz and 223 Hz and pressures P₁ = 0.005 sin 440πt and A = 0.005 sin 446πt, respectively. A graph of the pressure P = P₁ + P₂ felt by an eardrum over the 1-sec interval [0.15. 1.15] is shown here. How many beats are there in 1 sec?

(b) Repeat part (a) with frequencies of 220 and 216 Hz.

(c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.