Speed of Transverse Wave on a Stretched String: v = √(T/μ)

Pluck a stretched string and the kink races along it at a speed given by the beautifully simple formula v = √(T/μ): the square root of the tension T (in newtons) divided by the linear mass density μ (kilograms per metre). Tighter strings snap disturbances along faster; heavier strings drag them slower.

This one formula is the entire physics of tuning: guitarists tighten strings to raise the pitch, and bass guitars use thick heavy strings to go low. It also governs power-line galloping in wind, crane cables, and the sonometer in your school lab.

Key Concepts

• Tension T (N): the pulling force stretching the string. More tension = stronger restoring pull on any displaced bit = faster wave.
• Linear mass density μ (kg/m): mass per unit length. More mass per metre = more inertia resisting the pull = slower wave.
• Non-dispersive: on an ideal string, v is the same for all frequencies — a bass note and a treble note travel together, which is why a chord arrives intact.
• Dimensional check: T/μ has units (kg m/s²)/(kg/m) = m²/s², and its square root is m/s — a speed, as required.

Formulas

• v = √(T/μ), with T in newtons and μ in kg/m, giving v in m/s.
• Combined with the string's allowed wavelengths, this gives the note frequencies: f_n = nv/(2L) for a string of length L fixed at both ends.
• Assumptions and limits: a perfectly flexible, uniform string; small-amplitude vibrations; constant tension. Real strings have slight stiffness, which makes high harmonics travel a touch faster (the "inharmonicity" piano tuners correct for).

Worked Example

A steel wire has tension T = 100 N and mass 10 g per metre (μ = 0.010 kg/m). It is 0.65 m long, fixed at both ends.

1. Wave speed: v = √(T/μ) = √(100/0.010) = √10000 = 100 m/s.
2. Fundamental frequency: f_1 = v/(2L) = 100/(2 x 0.65) = 100/1.3 = 76.9 Hz.
3. To raise the pitch by one octave (double f to ~154 Hz), you need v to double — and since v ∛ √T, tension must quadruple to 400 N. (Guitarists instead shorten the string with a fret — much easier!)
4. Swap in a string twice as heavy (μ = 0.020 kg/m) at the same tension: v = √(100/0.020) = √5000 = 70.7 m/s, and f_1 drops to 70.7/1.3 = 54.4 Hz — heavier means lower.

Common Misconceptions

• "A thicker string carries waves faster." Usually the opposite — thicker means larger μ, so v = √(T/μ) drops (unless you also crank up the tension a lot).
• "Plucking harder makes the wave travel faster." No — amplitude is absent from the formula. Harder plucks sound louder, not quicker; only the medium (T and μ) sets the speed.
• "High notes travel faster along the string." On an ideal string every frequency travels at the same v — that is exactly why the formula has no f in it.

Real-World Applications

• Tuning guitars, violins, and pianos: tuning pegs adjust T until v gives the right note.
• Sonometer experiments verify v = √(T/μ) by measuring how frequency changes with tension and length.
• Overhead power lines: engineers compute v to predict wind-driven "galloping" oscillations and space the dampers.
• Textile looms and cable cars: moving-string dynamics depend on this speed for stability.
• Crane and elevator cables: knowing v helps avoid resonant swaying when the load moves.

In short: a disturbance on a stretched string travels at v = √(T/μ) — faster with more tension, slower with more mass per metre, and independent of how hard you pluck or which note plays. Tighten for pitch, thicken for bass: the whole string family is this formula in action.