The bridge's tendency to sway by up to 75 mm in each direction, has been reduced to a few millimetres by fitting dampers at a cost of £5m.
Secret tests, witnessed by Building, were held last week to check that the dampers had cured the problem (see below). These involved 700 Arup staff crossing the bridge. More tests were due to be held in public on Wednesday.
Arup chairman Bob Emmerson said: "The bridge performed exactly as we expected it to perform. We believe it has been fixed." He said the bridge, which cost £18.2m to build, could reopen in a few weeks.
Emmerson added that hundreds of bridges could be affected by similar wobbles. He said the longer the bridge, the more likely it was to be affected, and noted that any type of structure, including suspension, truss and arch bridges, was potentially vulnerable.
Roger Ridsdill-Smith, structural engineer with Arup, agreed with this assessment. He said: "Other bridges would do the same thing if they had enough people on them." He described the diagnosis of the problem as a watershed, and added that in future all long-span footbridges would need dampening.
Safety guidelines may also have to be rewritten. Warnings have already been added to building codes in Britain and Canada, and European codes are likely to be redrafted.
Arup confirmed that it had been asked to advise on several other structures in this country, but would not reveal which ones.
Emmerson said that since Arup had begun its investigation into why the bridge was wobbling, it had discovered examples of the same effect throughout the world.
One bridge in Canada, which was 100 years old, wobbled for the first time when a large crowd gathered on it. In New Zealand, the Auckland harbour road bridge on North Island, started swaying in 1975 when it was used for a protest march.
The problem occurs only when an unusually large number of people use a bridge at the same time.
About 100,000 people crossed the Millennium Bridge when it opened on 10 June 2000, causing the sway. "We were lucky nobody got hurt," Emmerson said. "That could have happened."
Engineers believe the sequence of events that causes the wobble is as follows: the bridge sways slightly, causing large numbers of pedestrians to exert a synchronised sideways force in response. This in turn leads to the bridge swaying more violently, and so on.
Ridsdill-Smith agreed that the problems with the Millennium Bridge, designed by Arup with Foster and Partners and sculptor Sir Anthony Caro, had been a "painful process," but claimed the Arup brand had not been damaged.
To correct the sway, 37 viscous dampers and 54 tuned mass dampers have been installed beneath the deck. Arup has contributed to the £5m cost.