Chapter 6.3, Problem 59E

Hearing Difference Tones When a musical instrument creates a tone of 110 Hz, it also creates tones at 220. 330. 440. 550. 660, … Hz. A small speaker cannot reproduce the 110-Hz vibration but it can reproduce the higher frequencies, which are the upper harmonics. The low tones can still be heard because the speaker produces difference tones of the upper harmonics. The difference between consecutive frequencies is 110 Hz. and this difference tone will be heard by a listener. (Source: Benade. A., Fundamentals of Musical Acoustics. Dover Publications.)

(a) In the window [0.0.03] by [-1,1], graph the upper harmonics represented by the pressure

$P=\frac{1}{2}\sin [2\pi (220)t]+\frac{1}{3}\sin [2\pi (330)t]+\frac{1}{4}\sin [2\pi (440)t].$

(b) Estimate all t-coordinates where P is maximum.

(c) What does a person hear in addition to the frequencies of 220. 330. and 440 Hz?

(d) Graph the pressure produced by a speaker that can vibrate at 110 Hz and above.