normal modes

Normal modes are the independent oscillation patterns of a system with more than one degree of freedom. In each normal mode, every part of the system oscillates at the same frequency and passes through equilibrium at the same time. Any motion of the system, no matter how complicated, can be written as a sum of its normal modes.

The simplest example is two identical pendulums connected by a spring. The system has two normal modes. In one, both pendulums swing together in the same direction at the same frequency — the spring never stretches. In the other, they swing in opposite directions, stretching and compressing the spring, at a higher frequency. If you start one pendulum swinging and hold the other still, the energy sloshes back and forth between them — a phenomenon called beating — because you have excited both normal modes simultaneously and they drift in and out of phase.

Normal-mode analysis extends to any number of coupled oscillators: molecules, crystal lattices, vibrating strings, drumheads. It is the bridge between the physics of one oscillator and the physics of waves.