The ground shakes, under your feet

Complete? Thanks.

One afternoon earlier this year, I used a pedestrian bridge to walk across a wide river. Unlike Morbi, nothing really happened when I walked. But after Morbi, I have had the opportunity to recall the nickname the pedestrians gave the bridge.

Morbi bridge was popularly called “Julto bridge” or “swinging bridge”. The one I crossed last March is popularly called the “Wobbly Bridge”.

A strange synchronicity in those names, isn’t it? The official name of Wobbly Bridge is Millennium Bridge, and it spans the River Thames in the center of London. It was opened on 10 June 2000. It was closed on 12 June 2000—yes, exactly two days later—and did not reopen for almost two years, until 22 February 2002. In that time, it underwent significant structural changes.

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

With suspension bridges such as the Julto Bridge and Wobbly Bridge, we have long known of a certain specific phenomenon. When a group of soldiers climb over it, the bridge may begin to shake. This is because soldiers move step by step, and this regular strike of dozens or even hundreds of feet causes the bridge to oscillate synchronously. This can quickly become dangerous. In 1850, a battalion of soldiers marched on the Basse-Chain bridge in Angers, France. The bridge was already overflowing with thunder, and with the crossing of these soldiers, it swelled even more. The cables holding it broke, and its collapse killed more than 200 soldiers.

This is why soldiers on the march are called upon, when they come across a bridge like this, to break the formation and walk either way. The more unorganized, the better.

The day the Wobbly Bridge opened, about 90,000 people walked on it. At any given time, it had about 2,000. A theory about what happened then follows.

While walking, his natural gentle swaying motion caused the bridge to sway slightly from side to side. This causes the occupants of the bridge to fall spontaneously, consciously or otherwise, with the rhythm of the bridge.

That is, synchronously with the speed of the bridge. This shook the bridge even more – the amplitude of its oscillations increased – which affected pedestrians even more… and as the whole swaying phenomenon was steadily strengthened, it soon became apparent That the bridge was dangerous.

Fortunately, it was closed before a major disaster struck.

In December 2000, in an attempt to understand how the bridge had behaved, engineers conducted a “diagnostic wobbling test”. He sent pedestrians to the bridge only a few at a time, gradually increasing their number to about 200. As they walked and as the count increased, the engineers measured the “wave amplitude,” which means the distance to the bridge. Simultaneously, they also calculated the “order parameter,” a measure of how synchronous the pedestrians were in their walk. This measure goes from 0, which means completely asynchronous, to 1, which means in complete lockstep.

He found something really interesting. “For small crowds, the walkers are synchronized” – and thus the order parameter becomes close to zero. The shaking of the bridge is also minimal. Almost certainly, the people on the bridge didn’t notice any movement. But “at a significant crowd size, the bridge begins to vibrate and the congestion begins to synchronize, with each process pumping the other into a positive feedback loop.” This is the result of each walker “import[ing] force for an alternate sideways bridge”. In turn, the speed of the bridge “changes”[s] the movement of each pedestrian”.

That grim size is about 175 people.

The graphs plotting these two measures are eye-opening. At 175 people, the amplitude of the waver and the order parameter suddenly begin to increase, finding itself in lockstep. The former grows to more than 5 cm, the latter reaches about 0.7. (Numbers and quotes from Crowd Synchronize on the Millennium Bridge, Steven Strogets et al., Nature, 2 November 2005).

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

As always with mathematics, however, there are doubts about this synchronization explanation for what happened with the Millennium Bridge.

A more recent paper shows that “no synchronization of pedestrian foot placement is likely to cause instability.” There is little evidence that pedestrians synchronize their steps; In fact, the bridge was moving over time with the speed of the bridge at only 20% of them. ,The rise of London Millennium Bridge instability without synchronization, Igor Belykh et al., Nature, 10 December 2021,

This is essentially what Brian Josephson, the Nobel laureate for physics, said in a letter to The Guardian just days after the bridge opened and closed in 2000.

He said that the behavior of the bridge has nothing to do with people following in their footsteps. Instead, it’s tied to what people do as they try to maintain balance if the surface they’re on starts to shake, and it’s the same thing as in a small boat. Many people can stand in time.

Josephson’s point of view comes from a famous video of a crowd at the Millennium Bridge, opened in 2000.

The structure is clearly moving forward, yes. But the way people are proceeding is certainly strange. Think about how you can walk if the ground beneath your feet is shaking. That’s what you see, in those pedestrians. I don’t know if such an analysis could have been done for the Julto bridge, or if it could have saved those lives. But why not do the same for other pedestrian bridges now?

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

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