Why doesn't pressure affect the speed of sound in air? I keep getting the answer "because density increases when pressure increases" but that doesn't really make sense to me since in denser materials - like water - sound travels faster. And if air molecules are closer together, wouldn't they interact more often. I thought it might have something to do with the elasticity of an object - i.e. sound travels rigid an object, like steel, much faster than and elastic medium, like rubber - but water is more elastic than air so that doesn't make sense either.
My understanding of physics is very basic so I may just not be able to understand this issue with my current knowledge, but if there is a simple answer it would be appreciated. Thank you.
 A: (a) the general formula for the speed of sound in a gas or liquid is
$$v=\sqrt \frac {K_s}{\rho}$$
$\rho$ is the density; $K_s$ is the adiabatic bulk modulus, which is a measure of how hard it is to squash the fluid into a smaller volume.
• " in denser materials - like water - sound travels faster." Sound certainly does travel faster in water than in air. But water is much, much harder to squash than air. More precisely,
$$\frac {K_s \text{for water}}{K_s \text{for air}}>\frac {\rho \text{ for water}}{\rho \text{ for air}}$$
• "Why doesn't pressure affect the speed of sound in air? We can show that for air, $K_s=1.4 p$ in which $p$    is the pressure. But we can also show that
$$\frac p{\rho}=\text{constant} \times T$$
The ratio of pressure to density, and therefore the speed of sound in air, depends just on the temperature. If you double the pressure you put on a sample of air (as long as you don't let its temperature change), you double its density and therefore leave the speed of sound unchanged.
Note that this applies to any one gas. It isn't the case that the ratio $p/\rho$ is the same for different gases even at the same temperature.
(b) There's a simple relationship that applies for diatomic gases like oxygen and nitrogen...
$$v=0.68\ c_{rms}.$$
$v$ is the speed of sound and $c_{rms}$ is the root mean square speed of the molecules, a sort of average speed of the molecules as they zip about randomly, colliding frequently with each other – whether or not sound is being propagated. The higher the temperature, the higher $c_{rms}$ and therefore the higher $v$.
This doesn't contradict the formula in (a), indeed it is a consequence of it. But it does give a new insight into the way sound travels in a gas. A sound source, such as a vibrating loudspeaker cone, superimposes an extra velocities on molecules in contact with it. These extra velocities encode the sound. They are much smaller than the molecules' ordinary random speeds. When molecules collide with each other they pass on the extra velocities, and the speed at which the sound travels through the gas is, not surprisingly, quite close to the random speeds of the molecules.
A: The important thing when discussing the speed of a wave in a medium is the ratio of something to do with elasticity and something to do with mass.
Increasing the elasticity means that the restoring forces are larger and hence the accelerations are larger, hence the speed of the wave is larger whereas increasing the mass means the accelerations are smaller, hence the speed of the wave is smaller.
Now you have compared steel and air.
It might well be that the density of steel is greater than the density of air but that is outweighed by the greater difference in the elasticity of steel as compared with the elasticity of air.
