# Sound waves

Sound waves are longitudinal waves propagated through a material by the transfer of kinetic energy from one molecule to another
(a) in solids and liquids by intermolecular forces and collisions,
(b) in gases by intermolecular collisions alone.
Hence the velocity of sound is greater in solids and liquids than in gases (the intermolecular forces in a gas are very small or zero).
The velocity of sound in a gas will thus be slightly less than the root mean square velocity of the gas molecules themselves. It will increase with increasing temperature since this will give a larger molecular velocity.

The table below gives the velocity of sound in a number of materials.

 Material Velocity of sound (ms-1) Material Velocity of sound (ms-1) Air (273 K) 330 Aluminium 5100 Water (298 K) 1430 Copper 3650 Steel 5060 Iron 5130 Vulcanised rubber 54 Glass 4000 - 5500 Granite (293 K) 6000 Pine 3313 Hydrogen (273 K) 1286 Oak 3837 Lead 1230 Elm 4108

The velocity of sound v in a solid in the form of a rod or wire may be found from the formula:

Velocity of sound (v) = [E/ρ]1/2

where E is the Young modulus for the material and ρ is its density

For an ideal gas the formula becomes:

Velocity of sound (v) in a gas = [γP/ρ]1/2

where P is the gas pressure and γ the ratio of the principal specific heat capacities of the gas. Substituting for P from PV = RT we have:

Velocity of sound (v) in a gas = [γRT/M]1/2

where M (=rV) is the molar mass of the gas.
This last equation shows that the velocity of sound in an ideal gas is:
(a) independent of the gas pressure,
(b) directly proportional to the square root of the absolute temperature,
directly proportional to the square root of g
(d) inversely proportional to the square root of the molar mass of the gas.

It is for this reason that the velocity of sound at high altitude is low since the air there is cooler. The speed of an aircraft relative to the speed of sound (its Mach number) is therefore greater when it flies at high altitude even though its actual speed may be the same as that before it began to climb. The change in the velocity of sound with temperature also explains why an instrument such as a flute becomes sharp when taken into a warm concert hall, the frequency change being greater than any effects due to the expansion of the instrument.

## Humidity and the speed of sound

The presence of water vapour in the atmosphere, known as humidity, causes a slight increase in the velocity of sound as the humidity rises.
The speed of sound in a gas in inversely proportional to the square root of the molecular weight of the gas through which the sound is passing the speed of sound will be greater in moist air. This is because the molecular weight of water is less than that of dry air.

At 20oC 100 kPa and 0% humidity the speed of sound is 343.4 m/s.
At 20oC 100 kPa and 50% humidity the speed of sound is 344 m/s.
At 20oC 100 kPa and 90% humidity the speed of sound is 344.5 m/s.
At 20oC 100 kPa and 100% humidity the speed of sound is 344.6 m/s.

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