Wave Speed in Strings & Media

Three numbers describe a periodic travelling wave, and they come from different places. Drive the end of a string at f oscillations per second and every point along it must repeat at f: the source owns the frequency, and nothing downstream can change it. The string owns the speed. Send a pulse of any shape along it and the pulse advances at one fixed v, which the next section gets from tension and mass per unit length alone. That leaves the wavelength as the only quantity free to adjust, so λ = v/f. Written as v = fλ the relation looks symmetric, but it is an identity between three numbers, not a statement that a higher frequency makes a faster wave. Drive the guitar's low E string at 220 Hz and the wavelength is 106.8/220 = 0.486 m; at 440 Hz it is 0.243 m; the speed is 106.8 m s⁻¹ both times. The graphs say the same thing: a snapshot measures λ along the x-axis, a history graph measures T at one fixed point, and v = λ/T converts between them.

Take a pulse of any shape moving at speed v along a string under tension F_T, and watch it from a frame moving with it. In that frame the shape stands still and the string material streams backwards through it at speed v. Look at a short arc at the top of the pulse, where the string curves with radius R and the arc subtends a small angle Δθ, so its length is RΔθ and its mass is μRΔθ. The tension pulls tangentially at each end of the arc; those two pulls are not parallel, and their resultant points towards the centre of curvature with magnitude 2F_T sin(Δθ/2) ≈ F_T Δθ. That resultant is the centripetal force on material rounding a bend of radius R at speed v, which is (μRΔθ)v²/R = μv²Δθ. Setting the two equal cancels Δθ and R together: F_T = μv², so v = √(F_T/μ). Nothing about the pulse survived the cancellation — not its height, not its width, not its shape. The dimensions check: F_T/μ is (M L T⁻²)/(M L⁻¹) = L² T⁻², whose root is a speed.

Frequency is absent, so every note on a given string travels at one speed. Amplitude is absent, so a hard pluck and a soft one arrive together. Both surviving dependences are square roots, and that sets the practical scale. Raising a string's pitch one semitone needs the speed up by 2¹⁄¹² = 1.0595, so the tension must rise by 1.0595² = 1.1225 — 12.2%, from 71.9 N to 80.7 N. An octave needs four times the tension, which is why a string near its breaking stress cannot be tuned much higher. The other route is μ. A guitar's high E sounds two octaves above its low E at the same 0.648 m length, so it needs four times the speed and therefore one sixteenth the mass per unit length: 3.94 × 10⁻⁴ against 6.30 × 10⁻³ kg m⁻¹. In plain steel that is a diameter ratio of four, 0.25 mm against 1.01 mm, and 1 mm steel is far too stiff to bend around a nut. Bass strings are wound instead: a thin core takes the tension while a heavy wrap supplies μ.

The string result is one instance of a form every mechanical wave speed shares: v = √(elastic property / inertial property). The numerator says how hard the medium pushes back when deformed, the denominator how much mass has to be accelerated. A long thin solid rod carrying a longitudinal wave gives v = √(Y/ρ) with Y the Young modulus; steel at Y = 2.0 × 10¹¹ Pa and ρ = 7850 kg m⁻³ gives 5.0 × 10³ m s⁻¹. A fluid has no shear stiffness, so only bulk compression is available and v = √(B/ρ); water at B = 2.2 × 10⁹ Pa and ρ = 998 kg m⁻³ gives 1.48 × 10³ m s⁻¹, and air near 20 °C gives about 344 m s⁻¹. The ordering looks wrong at first — steel is 6500 times denser than air and still carries sound about 15 times faster — until you compare numerators: steel's stiffness beats air's by a factor near 1.4 × 10⁶, far more than enough to outweigh the density.

Every step so far assumed one tension and one μ along the whole string, and a heavy rope hanging freely from a ceiling has neither. At height y above the free lower end the rope must support the weight below it, so F_T = μgy, and μ cancels out of the speed: v(y) = √(gy). A transverse pulse therefore starts almost at rest at the bottom and accelerates as it climbs. Integrating dt = dy/v gives the travel time, t = ∫₀^L dy/√(gy) = 2√(L/g), which for a 12.0 m rope with g = 9.81 m s⁻² is 2.21 s, arriving at the top at √(9.81 × 12.0) = 10.9 m s⁻¹. The lesson generalises: v = √(F_T/μ) is a local statement, true at each point, and it collapses into a single travel time only when F_T and μ really are constant. It also settles what happens where two strings are joined — the speed changes, the driving frequency cannot, so the wavelength must.

Three idealisations remain. The derivation held the tension magnitude fixed, which needs small slopes: the neglected term is of order (∂y/∂x)², so a sinusoid with A/λ = 0.05 already carries a maximum slope of 2π(0.05) = 0.31 and about a 5% error. Drive it harder and the crest genuinely stretches the string, raising the local tension and speeding the steep parts up until the shape distorts. Second, a real wire resists bending as well as stretching, which adds a term that makes speed grow with wave number: v(k) = √(F_T/μ) √(1 + EIk²/F_T), with I = πr⁴/4. For that high E — E = 2.0 × 10¹¹ Pa, r = 0.126 mm, so EI = 4.0 × 10⁻⁵ N m² — the correction moves the fundamental by under one part in 10⁵ but pushes the twentieth partial 0.26% sharp, 4.5 cents. That is dispersion: wavelengths travelling at different speeds. Third, an ideal string never loses energy; a real one damps, so the wave arriving is quieter than the wave sent.