Wave Speed on a String PhysicsTension, Linear Density & √(T/μ) Formula
Introduction
The speed of a transverse pulse on a stretched string is set by exactly two quantities: the tension pulling the string taut and the linear density resisting the acceleration of each small segment. Combining Newton's second law with the geometry of a curved string element yields v = sqrt(T/μ), where T is tension in newtons and μ is mass per unit length in kilograms per metre. Every other property of the string, its total length, its material, its colour, is irrelevant once T and μ are known.
The formula appears throughout acoustics and mechanical engineering. Guitar and piano tuning, the design of overhead power cables, and the calibration of seismic sensors all require an accurate model of how fast a disturbance travels along a one-dimensional medium under tension. In each case the engineer reaches for the same square-root expression. The simulator makes both variables independently adjustable, so the nonlinear relationship between T and v is directly observable rather than inferred from algebra alone.
Most people expect that doubling tension doubles wave speed, the same linear proportionality that governs so many other physical quantities. The Wave Speed readout contradicts this: at μ = 0.020 kg/m, raising T from 20 N to 80 N, a factor of four, brings the readout from 31.6 m/s to 63.2 m/s, a factor of only two. The square root, not a linear function, connects tension to speed.
The Physics Explained
Deriving v = sqrt(T/μ) starts by isolating a short curved segment of string of arc length ds. The tension T acts tangentially at both ends, and for small displacements the net transverse component of the two tension forces equals T · (ds/R), where R is the local radius of curvature. That net force must equal the segment's mass μ · ds times its transverse acceleration. Setting force equal to mass times acceleration and recognising the acceleration as the second time derivative of displacement leads directly to a wave equation whose propagation speed is sqrt(T/μ). The derivation requires no elasticity, no material constants beyond μ, and no assumption about wave shape.
With the default configuration of T = 20 N and μ = 0.020 kg/m, the Wave Speed readout shows 31.6 m/s, matching sqrt(20/0.020) = sqrt(1000) ≈ 31.623 m/s. The pulse crosses the 8 m string in 8/31.6 ≈ 0.253 s per traverse and completes five bounces in about 2.53 s before the simulation halts. Each bounce is marked by a flash at the relevant endpoint, and the Bounces readout increments in real time.
The v(T) curve on the right panel shows wave speed as a function of tension at whatever linear density the slider currently holds. Its concave-down shape is the visual signature of the square-root law: equal steps in T produce smaller and smaller gains in v as tension rises. The live operating point, the filled blue circle on the curve, moves right when the tension slider increases and up or down as linear density changes. Comparing the operating point's position to the curve's shape makes the diminishing-return effect immediate: at μ = 0.020 kg/m, moving from T = 20 N to T = 40 N shifts the readout from 31.6 m/s to sqrt(40/0.020) = sqrt(2000) ≈ 44.7 m/s, a gain of 13.1 m/s. Moving from T = 40 N to T = 60 N gains only sqrt(60/0.020) − 44.7 = sqrt(3000) − 44.7 ≈ 54.8 − 44.7 = 10.1 m/s for the same 20 N increase.
Linear density acts as the inertial brake. A heavier string per unit length accelerates more slowly under the same restoring tension force, lowering v. At T = 20 N, increasing μ from 0.020 kg/m to 0.080 kg/m drops the readout from 31.6 m/s to sqrt(20/0.080) = sqrt(250) ≈ 15.8 m/s, roughly half the original speed, because the density quadrupled and the square root of four is two.
Key Equations
With T = 20 N and μ = 0.020 kg/m, this gives v = sqrt(20 / 0.020) = sqrt(1000) ≈ 31.6 m/s. The Wave Speed readout in the simulator reports 31.6 m/s at those slider positions, confirming the formula to readout precision. Doubling T to 40 N gives sqrt(40 / 0.020) = sqrt(2000) ≈ 44.7 m/s, a factor of sqrt(2) ≈ 1.41 above the baseline, not a factor of 2.
Speed doubles only when tension quadruples. At μ = 0.020 kg/m, T = 20 N produces v ≈ 31.6 m/s and T = 80 N produces v = sqrt(80 / 0.020) = sqrt(4000) ≈ 63.2 m/s. Setting the tension slider to 80 N with linear density at 0.020 kg/m puts the Wave Speed readout at 63.2 m/s, exactly twice the default value. No intermediate tension achieves this doubling.
Quadrupling linear density halves wave speed. At T = 20 N, μ = 0.020 kg/m gives 31.6 m/s and μ = 0.080 kg/m gives sqrt(20 / 0.080) = sqrt(250) ≈ 15.8 m/s, half the original. The simulator confirms this: moving the linear density slider from 0.020 to 0.080 kg/m drops the readout from 31.6 m/s to 15.8 m/s in the pre-run state, with no simulation needed to verify the static formula.
Each one-way traverse covers the string length L = 8 m. At the default settings (T = 20 N, μ = 0.020 kg/m, v ≈ 31.6 m/s), five bounces require the pulse to travel 5 × 2 × 8 = 80 m total, giving a stop time of 80 / 31.6 ≈ 2.53 s. The Time readout reaches approximately 2.53 s when the Bounces readout hits 5, matching this prediction.
Key Variables
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| v | Wave speed | m/s | Speed of a transverse pulse along the string |
| T | Tension | N | Force pulling the string taut along its length |
| μ | Linear density | kg/m | Mass per unit length of the string |
| L | String length | m | Fixed at 8 m in the simulator |
| t | Time | s | Elapsed simulation time shown on the Time readout |
| n | Bounce count | , | Number of endpoint reflections, shown on the Bounces readout |
Real World Examples
Why do guitar players tighten strings to raise pitch?
Pitch depends on the frequency of standing waves on the string, and frequency is proportional to wave speed divided by string length. Tightening a string raises the tension T, which raises v = sqrt(T/μ), which raises every resonant frequency by the same factor. A guitarist tuning the high E string from T = 54 N to T = 64 N raises wave speed from sqrt(54/0.000385) ≈ 374.9 m/s to sqrt(64/0.000385) ≈ 407.8 m/s, a ratio of about 1.088, which corresponds to just under a semitone in the equal-tempered scale. The same mechanism works in reverse: slack strings lower wave speed and flatten pitch.
Setting T = 20 N and μ = 0.020 kg/m in the simulator puts the Wave Speed readout at 31.6 m/s. Raising T to 80 N at the same linear density brings the readout to 63.2 m/s, exactly doubling the speed and therefore doubling every resonant frequency on that string. In practice a guitar string cannot sustain a fourfold tension increase without snapping, so luthiers also use different gauges, varying μ across strings to spread the required frequencies without extreme tension.
How do engineers choose wire gauges for piano strings?
A piano spans over seven octaves, requiring an enormous range of fundamental frequencies. Engineers cannot achieve the full range by tension variation alone, because strings would snap at the high end or go too slack to vibrate cleanly at the low end. Instead, linear density μ is varied systematically across the keyboard: bass strings are wound with copper to increase μ without excessive length, while treble strings are thin, high-tension steel. Increasing μ lowers wave speed at constant tension, lowering the fundamental frequency by the factor 1/sqrt(μ).
A bass string with μ = 0.080 kg/m at T = 20 N reaches v = sqrt(20/0.080) = sqrt(250) ≈ 15.8 m/s. A treble string at the same tension but μ = 0.008 kg/m reaches v = sqrt(20/0.008) = sqrt(2500) = 50.0 m/s, roughly three times faster, a ratio of sqrt(0.080/0.008) = sqrt(10) ≈ 3.16. This roughly three-to-one speed ratio at equal tension is directly readable from the simulator: setting μ to 0.080 kg/m shows 15.8 m/s; changing to μ = 0.008 kg/m shows 50.0 m/s. Piano designers exploit this continuous control over μ to cover seven octaves while keeping string tensions within the safe operating range of the soundboard and frame.
Why does doubling tension not double wave speed on a transmission line?
Overhead transmission cables behave mechanically like strings under tension, and engineers who first encounter the wave-speed formula sometimes expect that doubling tension doubles the speed of transverse disturbances, which would halve the travel time for a fault signal to propagate down the line. The square-root law prevents this: doubling T multiplies v by only sqrt(2) ≈ 1.41, a 41% increase rather than 100%. To double wave speed, tension must be quadrupled.
The simulator makes the nonlinearity concrete. With μ = 0.020 kg/m, T = 20 N gives Wave Speed 31.6 m/s. Doubling tension to T = 40 N gives sqrt(40/0.020) = sqrt(2000) ≈ 44.7 m/s, a factor of sqrt(2) above 31.6 m/s, not a factor of 2. Reaching 63.2 m/s requires T = 80 N, confirming that speed doubles only when tension quadruples. Cable designers who need faster transverse-wave communication must account for this diminishing return when specifying sag and tension in long-span lines.
Further Reading
- Simple pendulum: another mechanical system whose restoring force and inertia combine under a square root to set oscillation frequency, with direct parallels to the tension-density balance in string waves.
- Spring-mass oscillator: the analogous one-dimensional system where stiffness and mass determine the natural frequency through the same square-root structure as v = sqrt(T/μ).
- Damped spring: extends the spring-mass picture to include energy loss, showing how the oscillation frequency shifts when damping is added, a precursor to wave attenuation on real strings.