Real World Digital Audio

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; it’s

caused by the limitations of

the analysis process.

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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.

Overtones

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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