In old western movies, cowboys often would put their ear to a train track to listen for an oncoming train. What benefit would this method provide over listening for a train by listening for sound through the air?

Because the sound travels faster in solids

Explanation:

Sound waves are mechanical waves, produced by the oscillations of the particles in a medium. The particles vibrate back and forth along the direction of propagation of the wave (longitudinal waves).

The speed of a sound wave depends on several factors, such as the density of the medium. In particular, the higher the density, the higher the speed of the wave, because in denser mediums the particles are more close to each other, so the transmission of the wave is more efficient since it can be transmitted better through the colliding particles.

For this reason, a sound wave travels faster in solids, and slower in gases.

Therefore in this problem, the sound of the oncoming train travels faster through the rail track (which consists of solids), rather than the air (which is a gas), so by putting the ear next to the track, it is possible to hear the sound of the train coming before the sound is heard in air.