Do you know how at football matches the opposing sides have songs and organised cheers? The Romans did this, particularly in chariot racing, when the factions were well-organised. It is also possible that they had cheerleaders. There were people attached to the teams called iubilatores. As far as I know there is no consensus on what exactly they did, but one theory is that they led the crowd in exhortations, cheers and songs. Some carvings of them give this impression.

I don't know about the wave, but the crowd did all rise together at specific moments, such as welcoming the emperor and other luminaries when they arrived at the grounds. Coincidentally, at the games was one of the few times the people could directly petition the emperor as a body and express their wishes. So there was a political aspect to this, and it isn't impossible these 'spontaneous' cries to the emperor were organised somehow.

Late edit: I just realised that the crowd rising for the emperor could be compared to people rising today for national anthems. In a way the same thing is happening, in that the people at the games are respecting the civil society in which they live.

Sources: Eckart Köhne's Gladiators and Caesars: The Power of Spectacle in Ancient Rome; Gregory S. Aldrete's Daily Life in the Roman City: Rome, Pompeii and Ostia; Plutarch Galba