How Fast Does a Wave Travel on a String?

By Lucian — builder & engineer, LK Forge

Flick one end of a rope and a pulse races to the other end at a definite speed. Flick it harder or faster and the pulse looks different — but it still travels at the same speed. That is the surprising part: how quickly a wave moves along a string has nothing to do with the wave itself, only with the string. Two properties set it, and here is exactly how, with an animation of a pulse.

A single pulse in the wave-on-a-string simulator. Raise the tension and the pulse sprints; add mass to the string and it crawls.

Compare rows 1–3: tension ×4 (10→40) doubles the speed, ×9 (10→90) triples it. Compare rows 1 and 4: quadrupling the density halves the speed. Same square-root law, both ways.

Where frequency comes in: v = fλ

If the string fixes the speed, what does the frequency do? It sets the wavelength, through v = f·λ. Drive the string faster (higher f) and, since v cannot change, the wavelength shrinks to keep the product equal; drive it slowly and the waves stretch out. So at v = 31.6 m/s, a 5 Hz shake makes 6.3-metre waves and a 20 Hz shake makes 1.6-metre ones — same speed down the string, different sizes. This is also why a guitar string plays a higher note when you tighten it (raising v raises every resonant frequency) or fret it shorter, and a lower note when it is thicker (larger μ).

Watch the pulse in the animation hit the fixed end and flip upside down as it reflects — a fixed end reverses the pulse, a free end returns it upright. Standing waves are just these reflections adding up. Try the tension and density sliders, then the oscillate mode, in the wave-on-a-string simulator.