The Millennium Bridge wobble and how crowds fall into step
Why London's Millennium Bridge swayed on opening day in 2000, what Strogatz and colleagues (2005) showed about crowd synchrony, and where the idea stops.
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The first people onto the Millennium Bridge on 10 June 2000 were there for the view: St Paul’s behind them, the Thames brown and wide underneath, a new steel ribbon thinner than anything that size had a right to be. Then the deck began to move sideways, a slow lean one way and back, and walkers who had been strolling at their own paces began to walk in time with it. By the time the lean was obvious, almost everyone on the span was doing it together.
The bridge wobbled because a large crowd, walking naturally, ended up pushing it sideways in time with its own sway. Strogatz, Abrams, McRobie, Eckhardt and Ott (Nature, 2005) showed that the wobbling and the crowd’s synchrony arise together, as one instability that switches on once the crowd passes a critical size.
What actually happened on opening day?
About 90,000 people crossed on opening day, with up to 2,000 on the bridge at once. The sway reached roughly 70 millimetres, and the bridge was shut on 12 June. Arup, the engineers, named the effect synchronous lateral excitation. Designers had long planned for vertical bounce and for wind. Nobody had planned for the small sideways force of a step, because a single walker produces almost nothing.
The sequence runs like a feedback loop. A walker’s body sways a little with each stride, and a crowd of walkers, each out of phase, mostly cancels out. A slight bridge movement then nudges people to adjust their balance, and the adjustment happens to push in the direction that feeds the movement. More people adjust, the push grows, the movement grows. Arup judged that stiffening the bridge was not feasible because it would have changed its appearance greatly. It added dampers instead, and the bridge reopened in February 2002.
How do coupled oscillators make a crowd synchronize?
Strogatz and colleagues borrowed from a line of mathematics that began with Yoshiki Kuramoto in 1975: a population of oscillators, each with its own preferred rhythm, each weakly tugged by the others. When the tugging is feeble, every oscillator keeps to itself. Raise it, and nothing much changes for a while. Then, at a particular strength, a cluster locks together, and the cluster pulls in more. In the simplest versions of the model, nothing locks below a critical coupling strength, and above it a locked cluster grows.
In the bridge version, the walkers do not look at each other. They are coupled through the deck, which every foot pushes and which pushes back. Nobody on the bridge chose the rhythm, nobody led it, and the synchrony did not exist at all below a certain crowd size. The same maths has been used for fireflies. John and Elisabeth Buck described synchronous flashing in Thai fireflies in 1968, and the working explanation is that each insect nudges the timing of its next flash when it sees a neighbour’s.
Our own clocks are oscillators too, which is why circadian entrainment sits so close to this subject: a rhythm held in step by a shared outside cue.
Where does the bridge story stop?
It stops sooner than the tidy versions suggest. The crowd-size cliff comes from a model, and the explanation has had critics: a letter in Physics Today from Feldman argued it is at best partial, pointing to the bridge’s lateral modes near 0.5 and 1.0 Hz and to conflicting results between watching walkers and testing people on moving platforms. I would hold the model loosely. It describes one structure, with one set of masses and stiffnesses, and says nothing about people choosing anything.
A small, tentative analogy
What follows is an analogy, and it is not evidence. Imagine five friends who each wake at their own hour. Coupling between them is nearly zero: they see each other on weekends, if then. Add a shared cue that every member feels (what chronobiologists call a zeitgeber), such as a group chat where the morning is visible, and the coupling rises. A task the whole group needs done may couple people harder still, which is the idea behind Sherif’s Robbers Cave camp, though that study has serious flaws. The bridge story hints that there might be a size and a strength below which the group’s rhythm does nothing, and above which it starts to feel like nobody’s decision. Maybe five is too few. Maybe the cue matters more than the headcount. I do not know, and neither does the bridge.
The opening-day crowd was thousands of people who each believed they were simply walking. What would it take for the rhythm of your mornings to belong to a few other people, and would you notice if it did?
Frequently asked questions
How did Arup fix the Millennium Bridge?
Arup chose damping over stiffening. The retrofit added 37 fluid viscous dampers for sideways motion and 52 tuned mass dampers for vertical motion. Repairs ran from May 2001 to January 2002 and cost about £5 million, and the bridge reopened on 22 February 2002 after a test with 700 Arup staff.
Do other bridges sway like this?
Yes. Bridges with a sideways natural frequency below about 1.3 Hz and little mass can show the same behaviour, and the Auckland Harbour Bridge (0.67 Hz) and the Birmingham NEC Link bridge (0.7 Hz) are the usual examples. Strogatz and colleagues proposed their model partly as a way to estimate the damping other crowded footbridges need.
What did Buck and Buck find about fireflies?
John and Elisabeth Buck’s 1968 paper in Science (volume 159) studied Pteroptyx malaccae fireflies along a tidal river in Thailand. They released males in a darkened hotel room and concluded that each firefly shifts the timing of its next flash in response to its neighbours’ flashes.
Who designed the Millennium Bridge?
Foster and Partners with the sculptor Anthony Caro designed it, and Arup were the engineers. Construction cost £18.2 million, £2.2 million over budget, before the £5 million retrofit.