Another equivalent analogy is a pendulum. Hold a pendulum with mass m at height h, so its potential energy is mgh. Let it go, and after a long time of swinging back and forth the pendulum eventually stops at the bottom (h=0). Where'd the energy go? It's all lost to heat through friction from the air and tension on the string. The rate of energy lost is just a lot less than the water barrel example. The general name for this kind of system is a harmonic oscillator.

Now I want to try the water barrel thing myself and see how many times the water goes back and forth before it finally stops...