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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
Pt a0 a1 cos 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 bn is called the nth harmonic of P. The amplitude of the n th harmonic is
and its square, A2
n sa 2
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
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
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.
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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