Wave Speed on a String

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.