The ground moves, beneath your feet

Done? Thank you.

One afternoon earlier this year, I used a pedestrian bridge to walk across a wide river. Unlike in Morbi, nothing really happened as I walked. But after Morbi, I’ve had occasion to remember the nickname pedestrians gave that bridge. 

The Morbi bridge was popularly called “Julto Pul”, or “Swinging Bridge”. The one I crossed last March is popularly called “Wobbly Bridge”.

An odd synchronicity in those names, right? The Wobbly Bridge’s official name is the Millennium Bridge, and it spans the River Thames in the heart of London. It was opened on 10 June 2000. It was closed on 12 June 2000—yes, just two days later—and did not reopen for nearly two years, till 22 February 2002. In that time, it underwent significant structural changes.

Why were modifications necessary on a brand new bridge? Let me return to that.

With suspension bridges, like Julto Pul and Wobbly Bridge, we’ve known for a long time of a certain characteristic phenomenon. When a group of soldiers march over it, the bridge can start swaying. This is because the soldiers march in step, and this regular thumping of dozens or hundreds of feet sets the bridge oscillating in synchrony. This can get quickly dangerous. In 1850, a battalion of soldiers marched onto the Basse-Chaine bridge in Angers, France. The bridge was already swaying with a thunderstorm, and with these soldiers crossing, it swayed even more. The cables holding it up snapped, and over 200 soldiers were killed as it collapsed.

This is why soldiers on the march are told, when they come to a bridge like this, to break formation and walk across any way they can. The more disorganized, the better.

The day the Wobbly Bridge opened, about 90,000 people walked on it. At any given moment, it carried about 2,000. One theory about what happened then goes like this. 

While walking, their natural gentle swaying motion caused the bridge to sway slightly from side to side. This caused those on the bridge, consciously or otherwise, to spontaneously fall in with the rhythm of the bridge. 

That is, synchronously with the bridge’s movement. This made the bridge sway even more—the amplitude of its oscillations increased—which in turn made the pedestrians sway more, too … and as the whole swaying phenomenon was steadily reinforced, it soon was clear that the bridge was dangerous.

Luckily, it was closed before a horrific disaster ensued.

In December of 2000, in an effort to understand how the bridge had behaved, engineers carried out a “diagnostic wobble test”. They sent pedestrians onto the bridge a few at a time, slowly increasing their numbers until nearly 200. As they walked and as the count rose, the engineers measured the “wobble amplitude”, meaning the distance the bridge sways. Simultaneously, they also calculated the “order parameter”, a measure of how synchronized the pedestrians were in their walk. This measure goes from 0, meaning completely asynchronous, to 1, meaning in perfect lockstep.

They found something interesting indeed. “For small crowds, walkers are desynchronized” — and thus the order parameter hovers close to zero. The swaying of the bridge, too, is minimal. Almost certainly, the people on the bridge did not notice any swaying. But “at a critical crowd size, the bridge starts to sway and the crowd starts to synchronize, with each process pumping the other in a positive feedback loop.” That’s a result of each walker “impart[ing] an alternating sideways force to the bridge”. In turn, the movement of the bridge “alter[s] each pedestrian’s gait”.

That critical size is about 175 people.

The graphs that plot these two measures are eye-opening. At 175 people, the wobble amplitude and the order parameter suddenly begin rising, themselves in seeming lockstep. The former rises to over 5cm, the latter reaches about 0.7. (Numbers and quotes from Crowd synchrony on the Millennium Bridge, Steven Strogatz et al., Nature, 2 November 2005).

What’s also interesting about this model of the bridge’s behaviour is that it takes ideas about synchrony from biology. They describe, for example, how individual fireflies manage to synchronize how they glow.

As ever with mathematics, though, there are doubts about this synchronization explanation for what happened to the Millennium Bridge.

A more recent paper suggests that “any synchronization of pedestrians’ foot placement is a consequence of, not a cause of the instability”. For there’s very little evidence that the pedestrians synchronized their footsteps; in fact, at most only 20% of those on the bridge were striding along in time with the movements of the bridge. (Emergence of the London Millennium Bridge instability without synchronization, Igor Belykh et al., Nature, 10 December 2021).

The Physics Nobel Prize winner Brian Josephson made essentially this point in a letter he wrote to The Guardian only days after the bridge was opened and closed in 2000. 

The behaviour of the bridge, he pointed out, had nothing to do with people walking in step. Instead, it is “connected with what people do as they try to maintain balance if the surface on which they are walking starts to move, and is similar to what can happen if a number of people stand up at the same time in a small boat”.

Josephson’s point is borne out by a famous video of the crowds on the Millennium Bridge when it opened in 2000. 

The structure is visibly moving, yes. But the way the people are moving is decidedly awkward. Think of how you might walk if the ground below your feet is moving. That’s what you see, in those pedestrians. I have no idea if any such analysis could have been done for Julto Pul, or if it would have saved those lives. But why not do it now, for other pedestrian bridges?

Why not do it, in memory of the 140 we lost?

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.