Speed and Frequency of Waves on a Stretched String

The speed of a transverse wave on a stretched string depends on how tight the string is and how heavy it is:v = √(T / μ)where T is the tension in newtons and μ is the mass per unit length of the string in kg m^-1. You find μ by dividing the mass of a piece of string by its length.A greater tension gives a greater restoring force, so the wave travels faster. A heavier string has more inertia, so the wave travels slower. Because of the square root, quadrupling the tension only doubles the speed.Worked example 1: a string has T = 40 N and μ = 1.0 × 10^-3 kg m^-1. Then v = √(40 / 1.0 × 10^-3) = √(40 000) = 200 m s^-1.Once you know the speed, the frequency of any stationary wave pattern on the string comes from f = v / λ, using the wavelength of that harmonic (λ = 2l/n for a string of length l).Worked example 2: the same string is 0.50 m long and vibrates in its second harmonic, so λ = 0.50 m. The frequency is f = 200 / 0.50 = 400 Hz.Worked example 3: a 2.0 m length of wire has a mass of 3.0 g. μ = 0.0030 / 2.0 = 1.5 × 10^-3 kg m^-1.Remember that v here is the speed of waves along the string, not the speed of the sound the string makes in the air.