Coupled and driven oscillators

Push a swing at random moments and little happens; push it once per cycle, at the right point in the cycle, and it climbs higher every time. That is resonance, and the important part is that the size of the push is almost irrelevant. What matters is timing. When the driving frequency matches the oscillator's natural frequency, every push arrives in phase with the motion and adds energy, and a tiny force can build an enormous amplitude — limited only by how fast damping bleeds energy away.

Off resonance, the oscillator does not simply respond less; it responds late. Drive it slowly and it follows the force almost exactly. Drive it fast and it lags by half a cycle, moving one way while the force pushes the other. Right at resonance the lag is a quarter cycle — which is precisely what makes the force push hardest when the oscillator is moving fastest, and so pump in the most energy per cycle.

Coupling two oscillators does something related. Linked by a spring or a shared support, they stop having independent frequencies and acquire shared patterns of motion — normal modes — in which the whole system oscillates at one frequency. Two identical pendulums on a common bar have a mode where they swing together and one where they swing oppositely, at slightly different frequencies. Start one alone and you have excited both modes at once; they drift in and out of step, and the visible result is energy sloshing from one pendulum to the other and back.