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To signal with one's hand for someone or something to continue or proceed with their current course. A noun or pronoun can be used between "wave" and "on." The police officer waved us on as soon as the other cars had passed through the intersection. We need a team of volunteers to stand at certain parts of the race and wave on competitors so they don't get lost.
wave someone or something on
to make a signal with the hand for someone or something to move on or keep moving. The traffic cop waved us on. The cop waved on the hordes of pedestrians.
To encourage or signal someone or something to proceed by or as if by waving the hand or arm: The police officer waved the pedestrians on. The crowd waved on the runners.