How does a wave of energy disperse into a medium? Where does the energy go?

Answer 1

Think about dropping a pebble into a lake.

There's lots of energy near the spot where the pebble hit - you can see this because the ripples are larger.

They shrink as you get farther from the center of the impact because the energy that operates on a straight line, in wave form, from the center of impact has to push more water around to make those ripples. Basically - that's where the energy goes: the energy has to push more stuff around in order to make itself felt, and it gets weaker and weaker because it loses more of itself into that more stuff.

Energy (or in the case of light, intensity) decreases an an exponential rate as you get farther from the source.

Answer 2

A wave is not made of "energy" in the way you're describing. A wave is made of oscillating motion. Energy is simply a property that moving things possess.

If you were to disturb the surface of a still body of water, you would see ripples (waves) propagating outward from the disturbance. The energy contained in the waves would be dissipated by friction (turned into heat) due to friction between water molecules rubbing together.

In general, all energy eventually becomes heat.

For example, the chemical energy in gasoline is turned into kinetic energy by a car's engine. To stop the car, you convert the kinetic energy into heat using the brakes. The aerodynamic drag also imparts heat to the air.