Concepts: speed of travelling waves

3.1 Speed on a stretched string

A transverse wave on a string travels because each part of the string pulls on its neighbour. The tension T provides the restoring force, and the mass per unit length mu provides the inertia. A tighter string restores its shape faster, while a heavier string responds more slowly, so the speed rises with the square root of the tension and falls with the square root of the linear mass density.

The relation is valid for small transverse displacements and for a uniform string. In a real string, a larger amplitude can change the effective tension, but the standard treatment assumes small oscillations. Notice that the speed does not depend on the frequency: the frequency is set by the source, and the wavelength then adjusts so that the product of frequency and wavelength equals the fixed speed.

A typical numerical use is to find the time for a pulse to travel along a string. The time is the length divided by the speed, and the speed comes from the tension and the linear density. A second common use is the reverse: given a measured speed and a known tension, find the mass of a given length of wire.

3.2 Speed in fluids and solids

In a fluid the disturbance is a pressure change, and the restoring force comes from the bulk modulus B. The inertia again comes from the density. A stiffer fluid therefore carries sound faster, and a denser fluid carries it more slowly.

In a solid rod the relevant elastic modulus is Young's modulus, because a longitudinal wave stretches and compresses the rod along its length. For a bulk solid or a liquid, the bulk modulus replaces Young's modulus. This is why sound travels faster in steel, with a large elastic modulus, than in water, and faster in water than in air.

3.3 Newton's calculation and Laplace's correction

Newton treated the propagation of sound as an isothermal process and obtained a speed equal to the square root of pressure divided by density. For air at ordinary pressure, this predicts about 280 metres per second, which is well below the measured value of about 332 metres per second at zero degrees Celsius.

Laplace pointed out that the compressions and rarefactions happen so quickly that heat cannot flow between the compressed and rarefied regions. The process is therefore adiabatic, not isothermal, and the pressure change is governed by the adiabatic relation. This multiplies the squared speed by the ratio gamma of the specific heats, which is 1.4 for air, and the predicted speed becomes about 331 metres per second, in good agreement with experiment.

For an ideal gas the pressure divided by density equals R T divided by the molar mass M, so the speed depends only on the temperature and on the gas, not on the pressure. Air at higher pressure has proportionally higher density, so the ratio stays the same and the speed stays the same.

3.4 Temperature, pressure and humidity

The speed of sound in a gas is proportional to the square root of the absolute temperature. Heating the air therefore raises the speed, and the standard practical formula for air near room temperature is the zero-degree value plus 0.61 metres per second for each degree Celsius rise.

Pressure by itself does not change the speed of sound in a gas, because pressure and density change together and their ratio in the formula stays constant. Humidity does change it: water vapour is lighter than the same volume of dry air, so moist air has a lower average density and sound travels slightly faster in it.

In a solid or a liquid, temperature affects the elastic modulus and the density, and for most solids the modulus falls faster, so sound slows down slightly as the solid is heated. In general, the effect of temperature on speed depends on how temperature changes the elastic modulus and the density of the medium.

The string and sound formulas look different but have the same structure: an elastic property divided by an inertial property, with a square root. In a string the elastic property is the tension and the inertial property is the mass per unit length. In a fluid the elastic property is the bulk modulus and the inertial property is the density. Recognising this pattern makes the formulas easier to recall and harder to mix up.

3.5 A worked example

A steel wire of length 12.0 metres has a mass of 2.10 kilograms. The linear density is 2.10 divided by 12.0, which is 0.175 kilogram per metre. Suppose the wave on the wire should move as fast as sound travels in dry air at 20 degrees Celsius, namely 343 metres per second. Then the tension must satisfy T = mu v squared. This gives 0.175 times 343 squared, which is about 2.06 times ten to the fourth newtons. A tension of this size is quite large for a wire, which is why tuning a stringed instrument needs a strong frame.