Resonance

Why can a series of small pushes destroy a structure that a single hard shove cannot?

Lean on a footbridge with all your weight and nothing happens. Walk across it in step with a few hundred others and it can sway alarmingly. The force in the second case is not obviously larger — it is merely repeated. So the destructive ingredient cannot simply be force. What does repetition supply that a single shove, however hard, does not?

A single shove delivers one fixed packet of energy, absorbed at once through bending, friction, and air. If the packet is too small to break anything, the story ends there.

So what would it take to make a small packet grow? You would have to add another before the first has drained away — and here is the constraint that does all the work. A structure does not accept energy at any moment you please. Left alone after a disturbance it swings back and forth at a rate set by its own stiffness and mass: its natural frequency. Push it while it is already moving away from you and you add energy; push it while it is coming back toward you and you take energy out.

The ideal timing, then, is once per cycle, always in the direction the thing is already going. Each push adds to a store that has not had time to leak away, and the amplitude climbs — not because any push got stronger, but because they are stacking.

Does it climb forever? Damping — internal friction, air, joints rubbing — drains energy at a rate that grows with how hard the structure is moving, so there is a size of swing at which the leak matches the supply, and growth stops there. A well-damped structure settles at a modest swing; a lightly damped one must grow enormous before its losses catch up, and may break before it does.

Notice what this predicts: the effect should be fussy about frequency. Drive slightly off the natural rate and your pushes drift out of step, spending part of each cycle undoing the last one's work. Which is why a wine glass shatters at one sung pitch and ignores its neighbours, and why armies break step on a bridge — a marching cadence is exactly the clean, repeated, single-frequency input you would choose if you wanted to find a structure's weak note.

Two corrections. First, resonance is not a property of the push or of the structure alone but of the relationship between them — the same bridge is indifferent to one rhythm and dangerous under another. Second, most famous "resonance disasters" are not resonance. A bridge failing in wind is usually suffering flutter, where the motion itself alters the airflow and so generates its own driving force — a self-excited loop, not an outside rhythm that happens to match. The 1940 Tacoma Narrows collapse, the picture everyone reaches for, is now attributed to that aeroelastic mechanism rather than to forced resonance.

London's Millennium Bridge in 2000 is the cleaner case, and stranger, because there the driving rhythm was not fixed either: a slight lateral sway made walkers unconsciously adjust their gait to match it, which fed the sway, which recruited more walkers. It closed within two days and was later fitted with dampers — damping being the lever engineers actually pull, since you cannot always avoid a natural frequency but you can make energy leak out faster than it arrives.

• Why a struck object rings at several natural frequencies at once, and what decides their pitches.
• How tuned mass dampers — a heavy pendulum deliberately mistuned against the building — drain a skyscraper's sway.
• Flutter and other self-excited oscillations, where the structure supplies its own driving rhythm.

Every term the collection defines is gathered in the glossary.