Wave the Swallow

Wave the Swallow is one of the Babylon Rogues, with Jet and Storm. She is the most level-headed of the three, taking charge whenever Jet cannot get things done. She does not get along well with Storm, and is rival to Tails (who is friendlier towards her than she is towards him).

Sonic Riders
Jet, Wave, and Storm entered a Grand Prix on Dr. Eggman's behalf to get the Chaos Emeralds and go to Babylon Garden for its treasure. Wave quickly became Tails's rival when she insulted his Extreme Gear, but secretly acknowledged that it was powerful and planted a bomb on it. Wave won her race against Amy, but lost to Tails. When Storm lost to Knuckles, Wave said that his skills suck. When Jet lost interest in the treasure of Babylon Garden, learning that it was a form of technology instead of a gem, Wave convinced him to continue his search since it could still be of value if a genius like Eggman was interested in it.

When Sonic nearly won his race against Jet, Wave detonated the bomb and blew up his Gear, causing Jet to win. The Babylon Rogues went to Babylon Garden, where Wave and Storm nearly got into a fight with Tails and Knuckles, but a fight was averted when E-10000R ran in with his army, forcing them to retreat (though E-10000R was killed by Tails as he took off, despite the scene being followed by a race where the player controls Wave and must defeat E-10000R along with Tails, Knuckles, Storm, and E-10000G, which may just be a representation of that scene).

Eventually, Wave raced alongside Jet, Storm, Sonic, Tails, Knuckles, and Amy against Babylon Guardian, destroying him and obtaining the treasure, a flying carpet. The Babylon Rogues went their separate ways from Sonic's team, acknowledging each other as worthy opponents.

Sonic Riders: Zero Gravity
Wave, Jet, and Storm were more rivals to Sonic than villains at this point, racing Sonic's team to get to various locations as they sought to obtain the five Arks of the Cosmos. Wave held onto them once the Babylon Rogues got all five of them, but SCR-HD took them from Wave, injuring her in the process (to which Tails tried to help her up). When SCR-HD transformed into Master Core: ABIS and opened the black hole at Babylon Garden, Wave raced alongside Jet, Storm, Sonic, Tails, Knuckles, and Amy to destroy ABIS and save the world.

Sonic Free Riders
By this point, the Babylon Rogues were simply rivals to Sonic, competing in a Grand Prix, ending with the four teams racing against Eggman.

If the player controls Team Babylon, however, Wave still cheats to win, as she sabotages Team Heroes' Gears without Jet's knowledge or approval, prompting Jet to seek a rematch to avoid his reputation being ruined. If the player controls Team Heroes, however, Wave does no such thing, and Jet's gear has mechanical trouble, causing Sonic to seek a rematch to be fair.

Trivia

 * Wave is one of three female villains in the Sonic series, other two being Rouge the Bat and Zeena. Like Rouge, Wave is more of a rival or anti-hero than a true villain, and turns good for the most part, unlike the pure evil Zeena, who is presumably destroyed.