Any complex waveform can be built from independent orthogonal sine waves

Evidence

Fourier analysis is a way of building up a waveform from much simpler bits. There are many ways of building up complex waveforms from simple basic shapes, like rectangles, but Fourier analysis was developed (by a French mathematician called Jean Baptiste Joseph Fourier in 1807) to solve the problem of heat diffusion in a metal bar! His idea was very simple. He built up the more complicated shapes of the functions, or waveforms, that he was studying by using the humble sine wave. He found that by adding together sine waves with different frequencies, phases and amplitudes, he could very accurately approximate the functions he was interested in. He found that using sine waves as a basis for approximation was a very useful thing to do. For a start, different sine wave frequencies were independent from or, more strictly, orthogonal to each other and so could be manipulated, or used in equations, independently of each other. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440 (#corroborates, via linked claim)

So far we have mainly considered a sound wave to be a sinusoidal wave at a particular frequency. This is useful as it allows us to consider aspects of sound propagation in terms of the wavelength. However, most musical sounds have a waveform that is more complex than a simple sine wave; a variety of wave­ forms are shown in Figure 1.49. So how can we analyze real sound wave­ forms, and make sense of them in acoustical terms? The answer is based on the concept of superposition and a technique called Fourier analysis. — David Howard, Jamie Angus, coustics and Psychoacoustics, 436 (#corroborates, via linked claim)

Categorization

• corroborated: 2 independent passages support this claim
• XREF: connects to: All complex sounds, including noise, are superpositions of pure sinusoids; Only a sinusoid has a single frequency; every other shape contains multiple frequencies; Any complex waveform can be decomposed into simple sine waves — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 439 (+12 more)

Broader Connections

• Any waveform can be approximated by summing orthogonal sine waves #corroborates (1.00) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440
• A waveform’s sharpness dictates how many high-frequency sine waves it contains #expands (0.90)
• Accurate square waves require infinitely many harmonics #qualifies (0.85)
• Complex real-world sounds can be analyzed via Fourier analysis #corroborates (0.85) — David Howard, Jamie Angus, coustics and Psychoacoustics, 436
• A hypothetical compound tuning fork illustrates how complex vibrations decompose into sine waves #exemplifies (0.85)
• An open-open pipe’s modes form a complete harmonic series at integer multiples #exemplifies (0.80)
• Any sound wavefront can be synthesized from elemental spherical waves #expands (0.75) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 2235
• Complex numbers elegantly encode phase differences in acoustic signals #expands (0.75)