Piano Tuning Problem: Note A on the piano has a frequency of 220 cycles/sec. Inside the

Chapter 5, Problem 31

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Piano Tuning Problem: Note A on the piano has a frequency of 220 cycles/sec. Inside the piano there are two strings for this note. When the note is played, the hammer attached to the A key hits both strings and starts them vibrating. Suppose that one of the two A strings is tuned to 221 cycles/sec and the other is tuned to 219 cycles/sec. a. The combined sound of these two notes is the sum of the two sound waves. Write an equation for the combined sound wave, where the independent variable is t, in seconds. Use the (undisplaced) cosine function for each sound wave, and assume that each has an amplitude of 1. b. Use the sum in part a and transform it to a product of two sinusoids. c. Explain why the combined sound is equivalent to a sound with a frequency of 220 cycles/sec and an amplitude that varies. What is the frequency of the variable amplitude? Describe how the note sounds.