Wave Speed on a String from Tension and Linear Density
The speed of a wave on a stretched string comes from the restoring tension and the inertia per unit length, with no reference to how the wave was made. Tuning an instrument is a change to one of those two quantities.
What a learner can do afterwards
- Calculates wave speed from tension and mass per unit length
- Predicts how the note of a string changes when it is tightened or thickened
- Explains why speed is set by the medium and frequency by the source
1 · Read
A stretched string carries waves at v equals the square root of T over mu, where T is tension and mu is mass per length. Tension restores, inertia resists, and amplitude and frequency never enter. Four times the tension gives twice the speed.
Tighten a string and v rises, so at fixed length the pitch climbs since f equals v over 2L. Swap in a thicker string at the same tension and mu grows, v falls, and the pitch drops.
Tune a guitar by turning the peg: more tension means a higher note on the same string. Fit a heavier gauge string instead and the same peg position sounds lower, because the thicker string carries waves more slowly.
Always split the roles: the medium sets v while the source sets f, and lambda follows as v over f. Warmer air changes the speed of sound the same way, by shifting the medium.
Tension up or mass down means faster waves and higher notes, and the medium always owns the speed.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.