Spectral decompositions were important for characterizing light sources and reflections in Section 4.1. In the case of sound, they are even more important. A sinusoidal wave, as shown in Figure 11.3(a), corresponds to a *pure tone*, which has a single associated frequency; this is analogous to a color from the light spectrum. A more complex waveform, such the sound of a piano note, can be constructed from a combination of various pure tones. Figures 11.3(b) to 11.3(d) provide a simple example. This principle is derived from *Fourier analysis*, which enables any periodic function to be decomposed into sinusoids (pure tones in our case) by simply adding them up. Each pure tone has a particular *frequency*, *amplitude* or scaling factor, and a possible timing for its peak, which is called its *phase*. By simply adding up a finite number of pure tones, virtually any useful waveform can be closely approximated. The higher-frequency, lower-amplitude sinusoids are often called *higher-order harmonics*; the largest amplitude wave is called the *fundamental frequency*. The plot of amplitude and phase as a function of frequency is obtained by applying the *Fourier transform*, which will be briefly covered in Section 11.4.

Steven M LaValle 2020-01-06