Objects vibrate sinusoidally when restoring force is proportional to displacement

Evidence

The tines of the tuning fork are responsible for both its mass and its springiness. This is typical of most objects: the “massiness” and the “springiness” are both distributed throughout the object. For small displacements of the forks, the restoring force is proportional to the displacement. Therefore, we expect the vibration of the fork to be sinusoidal (figure 3.7), provided only one mode of oscillation is excited. — Eric J. Heller, Why You Hear What You Hear- An…, 285

Categorization

• DEFINE: Explains why sinusoidal motion is the natural result of linear restoring force — the bedrock mechanism of harmonic oscillation.
• XREF: connects to: Instrument sound depends on whether restoring force comes from material stiffness or from stretching — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1310