Last updated: 2026-06-15

Wave Speed on a String

Pluck a guitar string and a wave races back and forth along it faster than a car on a motorway. How fast depends on just two things: how tightly the string is stretched and how heavy it is per unit length. This calculator turns those into the wave speed, and — for a string fixed at both ends — the fundamental frequency that sets its musical pitch.

Where the formula comes from

A taut string resists being displaced (tension pulls it back) and resists being accelerated (its mass). The wave speed balances these: more tension means a stronger restoring force and a faster wave, while more mass per length means more inertia and a slower wave. Working through the dynamics gives $v = \sqrt{T/\mu}$, with the tension $T$ on top and the linear mass density $\mu$ underneath, both inside a square root.

Formula

The speed depends only on the string, not on the wave — frequency and amplitude do not enter.

Worked example

A guitar string under $T = 80\ \text{N}$ with $\mu = 0.001\ \text{kg/m}$ and a vibrating length $L = 0.65\ \text{m}$:

That is close to the note A3 (220 Hz), which is exactly the kind of pitch such a string is meant to sound.

Tension, length and pitch

Because $f_1 = v/(2L) = \tfrac{1}{2L}\sqrt{T/\mu}$, a player has three handles on pitch. Tightening the string raises the speed and the pitch — but only with the square root of tension, so an octave up needs about four times the tension. Pressing a fret shortens $L$ and raises the pitch directly. And a thicker, denser string lowers it, which is why the low strings on a guitar or piano are the heavy, wound ones.

Limitations

The formula treats the string as perfectly flexible, uniform and thin, with the tension constant along its length. Real strings have a little stiffness, which raises the higher harmonics slightly (making them sharp), and a vibrating string's tension can change a touch with amplitude. The fundamental-frequency formula assumes the string is fixed at both ends; other boundary conditions give different overtone patterns.