Higher string modes vibrate faster because greater curvature raises restoring force

Evidence

Now a different type of vibration is seen, having a new shape and a higher frequency. Notice that the second mode is oscillating more rapidly than the first. If you look at how the filaments are oriented relative to the masses, you can see why the force is higher and therefore the acceleration is greater than in the first mode, leading to higher frequency and a larger force constant. — Eric J. Heller, Why You Hear What You Hear- An…, 783

Categorization

• 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