Speed of a travelling wave

Speed of a wave on a string

Narrated video. Transcript:

Put a marker on one crest of a travelling wave and follow it. A crest is a point of fixed phase, so as it moves, k x minus ω t stays constant.

As t increases, x must increase to keep the phase fixed. Differentiate: k times dx by dt minus ω equals zero. So the crest moves at v equals ω over k.

Put in ω equals 2πν and k equals 2π over λ. The 2π cancels, and v equals λ ν. It is the same as λ over T: in one period, the wave moves forward exactly one wavelength.

The speed is fixed by the medium, not by your hand. Shake faster and the frequency goes up, but the wavelength gets shorter, so λν, the speed, stays the same.

On a stretched string, the restoring force comes from the tension T. The inertia comes from the mass per unit length, μ.

Let’s use dimensions. Tension is a force, M L T to the minus 2. μ is M L to the minus 1. Their ratio T over μ has dimensions L squared T to the minus 2: a speed squared.

So v equals C times the square root of T over μ. A detailed analysis shows that the constant C is exactly 1. The speed is the square root of T over μ.

Tighten a sitar string, and waves run along it faster. A thicker, heavier string carries them more slowly. The frequency does not appear at all.

Example: a steel wire 0.72 metres long has a mass of 5.0 grams and is under a tension of 60 newtons. μ is 0.005 divided by 0.72, about 6.9 × 10⁻³ kilograms per metre.

Then v equals the square root of 60 divided by 6.9 × 10⁻³, which is about 93 metres per second.

The speed of sound

Narrated video. Transcript:

Sound is a longitudinal wave of compressions and rarefactions. What sets its speed? Again, a restoring property and an inertia property.

The restoring property is the bulk modulus B: the extra pressure needed to squeeze the medium, per fractional change in its volume. The inertia is the density ρ.

B has the dimensions of pressure, M L to the minus 1 T to the minus 2. ρ is M L to the minus 3. So B over ρ is again a speed squared, and v equals the square root of B over ρ.

Water is hard to compress, so sound travels through it at about 1500 metres per second. In a long steel rod, where v equals the square root of Y over ρ, it is several kilometres per second, more than fifteen times faster than in air.

Now air. Newton assumed the temperature of the air stays constant as it is squeezed. Then P V is constant, the bulk modulus equals the pressure P, and v equals the square root of P over ρ.

At normal temperature and pressure, P is 1.01 × 10⁵ pascals and ρ is 1.29 kilograms per cubic metre. Newton’s formula gives about 280 metres per second.

But the measured speed is 331 metres per second. Newton’s answer is about 15 percent too low.

Laplace found the mistake. The compressions and rarefactions come and go so quickly that heat has no time to flow in or out. The process is adiabatic, so P V to the power γ is constant.

Then the bulk modulus becomes γ P, and v equals the square root of γ P over ρ. For air, γ is 1.4. Multiply 280 by the square root of 1.4, and you get about 331 metres per second. A perfect match.

Changing the pressure alone does not change the speed, because at a fixed temperature P over ρ stays the same. But warm air carries sound faster than cold air.

Try it: waves on a stretched string

Interactive simulation. Things to try:

• The speed v = √(T/μ) depends only on the string. Change the frequency: λ changes, v does not.
• Make the tension four times bigger: the speed doubles.
• Check v = λν with the readouts at any setting.
• The motion is shown slowed down; real string waves vibrate tens to hundreds of times a second.

Shake it faster

You double the frequency of the waves you send along a stretched string, keeping the tension the same. What happens?

1. The speed doubles and the wavelength stays the same
2. The speed stays the same and the wavelength halves
3. Both the speed and the wavelength double
4. The speed halves and the wavelength stays the same

Wavelength on a guitar string

A string 50 cm long has a mass of 0.5 g and is stretched with a tension of 10 N. A 200 Hz wave travels along it. What is its wavelength?

1. 0.5 m
2. 0.71 m
3. 2 m
4. 0.016 m

Summary

• Speed of any travelling wave: v = ω/k = λν = λ/T
• Transverse wave on a string: v = √(T/μ)
• Sound in a fluid: v = √(B/ρ)
• Newton (isothermal) → Laplace (adiabatic): v = √(P/ρ) → v = √(γP/ρ)

• The wave speed is set by the medium (elasticity and inertia). The source sets the frequency; then λ = v/ν.
• On a string, higher tension means faster waves; heavier string (larger μ) means slower waves.
• Newton’s formula gives about 280 m/s for air at STP. Laplace’s adiabatic correction, with γ = 1.4, gives 331 m/s, matching experiment.
• At a fixed temperature, changing the air pressure does not change the speed of sound; raising the temperature increases it.