FourierSeries-1

These vibrations are amplied and transmitted to the

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Unformatted text preview: instruments? The graphs in Figure 4 show these fluctuations (deviations from average air pressure) for a flute and a violin playing the same sustained note D (294 vibrations per second) as functions of time. Such graphs are called waveforms and we see that the variations in air pressure are quite different from each other. In particular, the violin waveform is more complex than that of the flute. t F IGURE 4 Waveforms t (a) Flute (b) Violin We gain insight into the differences between waveforms if we express them as sums of Fourier series: Pt a0 a1 cos t L t L b1 sin a2 cos 2t L b2 sin 2t L In doing so, we are expressing the sound as a sum of simple pure sounds. The difference in sounds between two instruments can be attributed to the relative sizes of the Fourier coefficients of the respective waveforms. The n th term of the Fourier series, that is, a n cos nt L nt L bn is called the nth harmonic of P. The amplitude of the n th harmonic is An and its square, A2 n a2 n sa 2 n b2 n b2 , is sometimes called energy of the n th harmonic. (Notice that n F OURIER SERIES ■ 9 bn and for a Fourier series with only sine terms, as in Example 1, the amplitude is A n the energy is A2 b 2.) The graph of the sequence A2 is called the energy spectrum of n n n P and shows at a glance the relative sizes of the harmonics. Figure 5 shows the energy spectra for the flute and violin waveforms in Figure 4. Notice that, for the flute, A2 tends to diminish rapidly as n increases whereas, for the violin, the n higher harmonics are fairly strong. This accounts for the relative simplicity of the flute waveform in Figure 4 and the fact that the flute produces relatively pure sounds when compared with the more complex violin tones. A@ n A@ n 0 FIGURE 5 Energy spectra 2 4 6 8 0 n 10 2 4 (a) Flute 6 8 n 10 (b) Violin In addition to analyzing the sounds of conventional musical instruments, Fourier series enable us to synthesize sounds. The idea behind music synthesizers is that we can combine various pure tones (harmonics) to cr...
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This note was uploaded on 12/05/2011 for the course MATH 41 taught by Professor Bray,c during the Spring '08 term at Duke.

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