← Wave Systems

5 Resonance & damping

Resonance is what happens when a system is driven at its own natural frequency: each push arrives in phase with the existing motion, so energy is added cumulatively and the amplitude grows far beyond what a single push could produce.

What determines the natural frequency

  • For a string, air column or any standing-wave system, the natural (resonant) frequencies are exactly the harmonic frequencies from the previous section: fn = nv/2L (both ends alike) or fn = nv/4L (one end closed).
  • Driving at any OTHER frequency still makes the system vibrate, but with small, sustained amplitude — the pushes are not synchronised with the motion, so they sometimes add energy and sometimes remove it.
  • The closer the driving frequency is to a natural frequency, the larger the resulting amplitude.
Damping is energy loss from a vibrating system, usually to friction, air resistance or internal heating. It limits how large a resonant amplitude can grow, and it is why a system does not oscillate forever once the driving force stops.

Damping and resonance interact

  • Light damping ⟹ energy leaks away slowly, so resonant amplitude can build to very large, potentially destructive values.
  • Heavy damping ⟹ energy leaks away quickly, so even driving exactly at the natural frequency produces only a modest amplitude.
  • This is why real structures are deliberately damped (rubber mounts, dashpots, tuned mass dampers): removing the “light damping” condition is what stops resonance from being dangerous.
Worked example Explaining a resonance disaster A pedestrian bridge starts to sway alarmingly when a large crowd crosses it in step. Explain, in terms of resonance and damping, why this happens and how engineers usually fix it. Identify the driving frequency: pedestrians walking in step apply a small periodic push at their footfall frequency. Compare it with the natural frequency: if the footfall frequency happens to match one of the bridge's own natural (resonant) frequencies, each footstep adds energy in phase with the bridge's existing motion. Explain the buildup: because the pushes are synchronised with the motion, energy accumulates cumulatively rather than cancelling out, so amplitude grows well beyond what one person's footstep could cause alone. State the fix in terms of damping: adding dampers (devices that convert the bridge's kinetic energy to heat) increases the rate of energy loss, so even if the driving frequency matches a natural frequency, the amplitude cannot build to a dangerous level. ✓ The crowd's footfalls drive the bridge at (or near) one of its natural frequencies, and because pushes and motion stay in phase the amplitude builds cumulatively; installing dampers increases energy loss per cycle so the same driving can no longer sustain a large amplitude. ✕

Resonance is not just "vibrating a lot" — it needs a matching frequency

A common wrong answer describes resonance as simply "the system vibrating with a large amplitude," with no mention of frequency at all. That misses the actual physics mark: resonance specifically requires the driving frequency to match a natural frequency of the system. Any answer explaining resonance, or an increase/decrease in resonant amplitude, must name BOTH frequencies and say they are equal (or moving toward/away from being equal) — not just describe the size of the wobble.

Practise just this