Two masses connected to walls and to each other by springs exhibit normal modes — collective oscillations in which every part of the system moves at a single frequency. The symmetric mode (ω₋) has both masses moving together; the antisymmetric mode (ω₊) has them moving in opposite directions. When the coupling spring κ is small, the two frequencies are close and energy sloshes back and forth, producing the characteristic beating envelope.
Click Symmetric to excite the low-frequency mode (both masses displace the same direction). Click Antisymmetric for the high-frequency mode. Click Beating to start with only mass 1 displaced — watch the energy transfer completely to mass 2 and back. Reduce coupling κ to lengthen the beat period. Increase damping γ to see energy dissipate.
The physics of coupled oscillators underpins vast areas of science. Phonons in crystals are quantised normal modes of coupled atoms. Neutrino oscillations arise from quantum mixing of two mass eigenstates — a quantum analogue of the beating pattern. Even the CO₂ greenhouse effect depends on the antisymmetric bending mode of the molecule, which carries an oscillating dipole moment that absorbs infrared radiation.