Wave speed on a stretched string

Follow a point of fixed phase, such as a crest. For kx − ωt to stay constant as t grows, x must grow at the rate ω/k, so the wave speed is v = ω/k.

Using ω = 2πν and k = 2π/λ, v = λν = λ/T. In one period of any particle the pattern moves forward by one wavelength. This holds for every progressive wave.

The speed of a mechanical wave is set by the medium: an elastic property that supplies the restoring force, and an inertial property, the density.

On a stretched string the restoring force is the tension T and the inertia is the linear mass density µ = m/L, mass per unit length.

Dimensional analysis: T/µ has dimensions [MLT⁻²]/[ML⁻¹] = [L²T⁻²], a speed squared, so v = C√(T/µ). Dimensions cannot fix C; the exact derivation gives C = 1, so v = √(T/µ).

The speed depends only on T and µ, not on the wavelength or frequency. The source sets the frequency, and λ = v/ν then follows.

Worked example (NCERT): a steel wire 0.72 m long of mass 5.0 × 10⁻³ kg under a tension of 60 N. µ = 5.0 × 10⁻³/0.72 = 6.9 × 10⁻³ kg m⁻¹, so v = √(60/6.9 × 10⁻³) = 93 m s⁻¹.

Tighten a string and waves on it speed up; use a heavier string and they slow down.