Wave Speed on a String
Worked example: 100 N, mu = 0.01 kg/m → v = 100 m/s — press Try an example to run it live, then adjust anything.
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Grade 11Grade 11 Physics
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Wave Speed on a String explained
Tension provides the restoring force that snaps a displaced string back; mass per unit length provides the inertia that resists. Their tug-of-war sets the wave speed. A guitar's high E string, with μ ≈ 0.4 g/m under about 70 N of tension, carries waves at √(70/0.0004) ≈ 418 m/s — faster than sound in air. Turning a tuning peg raises F, speeds up the waves, and (since f = v/2L) sharpens the pitch.
The square root explains a luthier's dilemma: to double a string's fundamental you must quadruple its tension, which is why pianos don't just tighten one string type but vary μ instead. Bass strings are wrapped in heavy copper windings to raise μ, slowing the waves so low notes fit on a playable length — a piano's lowest string can be under more than 1000 N of tension yet still vibrate at a leisurely 27.5 Hz.
Wave Speed on a String formula
- = Wave speed (m/s)
- = String tension (N)
- = Linear mass density (kg/m)
Missing one of these? Work it out first, then come back
- Wave speed — Wave Speed (v = fλ), Fundamental Frequency of a String
- String tension — Rope Tension When Lifting a Mass, Newton's Second Law
- Linear mass density — Direct Count from Mass and Length — Tex, Decitex and Denier, Yarn Diameter from Linear Density