Any plucked string shape decomposes into sinusoidal standing-wave modes

Evidence

Fourier’s theorem describes the decomposition of any function into a sum of sinusoids. Initially, what concerned us was a time signal arriving, for example, at a microphone. Now, we have seen that Fourier’s theorem is also describing the initial shape of the string, that is, a function of position along the string. Fourier’s theorem tells us that any shape of the string can be represented as a linear combination of some or, if needed, all of the modes. Each mode is a standing wave, sinusoidal in shape, oscillating sinusoidally in time. — Eric J. Heller, Why You Hear What You Hear- An…, 846

A vibrating string fixed at both ends, for example by being stretched across two bridge-like supports as illustrated in Figure 4.4, has a unique set of standing waves (see Chapter 1). Any observed instantaneous shape adopted by the string can be analyzed (and synthesized) as a combination of some or all of these standing wave modes. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1076

A vibrating string fixed at both ends, for example by being stretched across two bridge-like supports as illustrated in Figure 4.4, has a unique set of standing waves (see Chapter 1). Any observed instantaneous shape adopted by the string can be analyzed (and synthesized) as a combination of some or all of these standing wave modes. The first 10 modes of a string fixed at both ends are shown in Figure 4.6. In each case the mode is illustrated in terms of the extreme positions of the string between which it oscillates. Every mode of a string fixed at both ends is constrained not to move; there­ fore there cannot be any velocity, or displacement, at the ends themselves Figure 4. — David Howard, Jamie Angus, coustics and Psychoacoustics, 1101

Categorization

• XREF: Fourier decomposition also anchors the soundscape framing of signal vs. backdrop in prior listening-focused reading. — connects to: Plucked-string spectra reveal the plucking point through nulled harmonics — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1127
• DEFINE: Clarifies how Fourier’s theorem applies equally to time signals and to spatial shapes of a vibrating string.
• corroborated: 3 independent passages support this claim — Eric J. Heller, Why You Hear What You Hear- An…, 846; David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1076; David Howard, Jamie Angus, coustics and Psychoacoustics, 1101