Sine Waves and the Geometry of Sound
Real-world waveforms aren’t perfect, regular up and down oscillations, nor
do they have just one frequency: sound almost always has energy at multiple
frequencies simultaneously. This fact is central in the construction of digital
audio devices such as equalizers and filters (Chapter 7). However, audio
engineers have found it extremely useful to have a basic waveform that can
be used as a sort of benchmark or building block to describe more complex
waveforms. This basic waveform, the sine wave, oscillates repeatedly at a
single frequency, and has no sound energy at any other frequency.
Meet the sine wave
The sine wave is a basic, periodic wave (one that repeats at regular intervals).
It sweeps up and down in a motion that’s a bit like the shape of the humps
and dips of a roller coaster (Figure 1.5). Sounds that we’d describe as “pure,”
like a flute playing without vibrato, a person whistling, or a tuning fork, all
approximate the sine wave. It’s even the ideal wave shape for the electricity
in your home’s power outlets.
Figure 1.5 Computers are
capable of producing an
approximation of a pure sine
wave, as represented here in
the waveform display in
Apple Soundtrack Pro (top).
Shown in a graph of fre-
quency content over time
(bottom), this sine wave is a
single horizontal band; unlike
other sounds, the sine wave
carries energy at only a single
frequency.The haze around
the horizontal band is not due
to the presence of other fre-
quencies in the sine wave; its
caused by the limitations of
the analysis process.
1: U
The sine wave is the wave shape you’ll most often see illustrated, as in
Figures 1.3 and 1.5, but it’s not just useful in sound and physics textbooks:
any real-world sound can be analyzed mathematically as a combination of
sine waves, a property that underlies many digital audio techniques. As a
result, you can begin to understand how the complex, irregular world of
real sound can be understood as a combination of single frequencies.
Most of us would intuitively assume the sound of a single musical note has
one pitch, and thus one frequency. In fact, when you pluck the string on a
guitar, it produces a single pitch, but the string vibrates not just at one fre-
quency but at multiple frequencies simultaneously (Figure 1.6). The enor-
mous variety of sounds in the real world (and in digital audio applications)
is due to the complex blending of these various frequencies.
Figure 1.6 A string or other vibrating body doesn’t just vibrate at a single frequency
and wavelength; it vibrates at multiple frequencies at once.This string is vibrating not
only at the full length of the string, but also in halves, thirds, fourths, and so on.

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