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.
More to read
16.3 Wave Speed on a Stretched String
A wave on a string hurries when the string is tight and dawdles when the string is heavy. The exact rule is v equals the square root of tension over linear density. Tightening raises the speed, swapping in a thicker string lowers it. Notice what is missing: amplitude and frequency do not appear. Shouting louder into a guitar string changes nothing about the travel speed. The medium sets v, and the source then decides the frequency, with wavelength adjusting to fit.
16.2 Mathematics of Waves
Once the speed is known, the sine description locks everything else in place. Wavelength equals v over f, and the wave number k equals 2 pi over lambda. These turn a vague wiggle into numbers you can measure and predict. Guitarists use this daily. Pressing a fret shortens the string so the standing pattern fits a shorter length, and turning a peg raises tension and lifts the pitch.
17.2 Speed of Sound
Sound obeys the same moral with a different medium. Its speed depends on the stiffness and density of air, water, or steel, not on how loud the clap was. Warm air is slightly faster than cold air for the same reason a stiffer spring snaps back quicker. Comparing strings and sound cements the pattern. Change the medium and v changes; change the source and f changes; lambda always lands at v over f. Keep those roles straight and most wave confusion evaporates.
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2 · Watch
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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.