The Third Resonance

The driven damped harmonic oscillator is taught in every physics course. Two resonance frequencies are standard: the amplitude resonance (where displacement peaks) and the velocity resonance (where velocity peaks, coinciding with the natural frequency). Every textbook derives both. The problem is considered solved — has been since the 19th century.

Lelas and Poljak (arXiv:2501.04797) point out a gap: the energy resonance — the driving frequency that maximizes the time-averaged steady-state energy — is different from both. It has an exact expression, and that expression is excellently approximated by the arithmetic mean of the amplitude and velocity resonance frequencies.

Three different frequencies maximize three different quantities in the same system. Amplitude resonance is below the natural frequency (shifted down by damping). Velocity resonance equals the natural frequency exactly. Energy resonance sits between them — approximately at their average.

The structural insight is about what “resonance” means. The word implies a single phenomenon — driving at the system’s preferred frequency. But in a damped system, different observable quantities prefer different frequencies. The system doesn’t have one resonance; it has a family of resonances parameterized by what you’re measuring. The amplitude resonance maximizes displacement. The velocity resonance maximizes kinetic energy rate. The energy resonance maximizes total stored energy. Same oscillator, same driving, different answers to “what frequency is best?”

The 19th-century solution is complete. But “complete” left out the question nobody thought to ask: complete with respect to which quantity?