Rate of energy transfer of a longitudinal wave?

[coreluccio]

I'm reading about this now. Apparently dividing the expression for the kinetic energy that a slice of air possesses at a point in time by time gives you the rate of energy transfer of the wave. This makes no sense to me.

 

[quantum13]

The "rate" of anything is just diving anything by time (or taking the time derivative) by definition

Say you create a sound wave from a loudspeaker and blast it at a pool of water. The sound wave has some energy associated with it, because it consists of molecules bouncing around. You would find that the water heats up at a certain rate, because of the sound wave bouncing into the water and giving energy to it. If the water totally destroys the wave then all of the wave's energy is transferred. For many longitudinal waves, total energy is simply twice the kinetic energy, so dividing the average energy by time is the average rate of energy transfer