747 7.6 Trigonometric Equations A piano string can vibrate at more than one frequency when it is struck. It produces a complex wave that can mathematically be modeled by a sum of several pure tones. When a piano key with a frequency of ƒ1 is played, the corresponding string vibrates not only at ƒ1 but also at the higher frequencies of 2ƒ1 , 3ƒ1 , 4ƒ1 , . . . , nƒ1. ƒ1 is the fundamental frequency of the string, and higher frequencies are the upper harmonics. The human ear will hear the sum of these frequencies as one complex tone. (Data from Roederer, J., Introduction to the Physics and Psychophysics of Music, Second Edition, Springer-Verlag.) EXAMPLE 10 Analyzing Pressures of Upper Harmonics Suppose that the A key above middle C is played on a piano. Its fundamental frequency is ƒ1 = 440 Hz, and its associated pressure is expressed as P1 = 0.002 sin 880pt. The string will also vibrate at ƒ2 = 880, ƒ3 = 1320, ƒ4 = 1760, ƒ5 = 2200, . . . Hz. The corresponding pressures of these upper harmonics are as follows. P2 = 0.002 2 sin 1760pt, P3 = 0.002 3 sin 2640pt, P4 = 0.002 4 sin 3520pt, and P5 = 0.002 5 sin 4400pt The graph of P = P1 + P2 + P3 + P4 + P5 can be found by entering each Pi as a separate function yi and graphing their sum. The graph, shown in Figure 36, is “saw-toothed.” −0.005 0.005 0 0.01 Figure 36 (a) Approximate the maximum value of P. (b) At what values of t = x does this maximum occur over 30, 0.014? SOLUTION (a) A graphing calculator shows that the maximum value of P is approximately 0.00317. See Figure 37. (b) The maximum occurs at t = x ≈0.000191, 0.00246, 0.00474, 0.00701, and 0.00928. Figure 37 shows how the second value is found. The other values are found similarly. S Now Try Exercise 109. −0.005 0.005 0 0.01 Figure 37
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