Born to entitled parents she was forced to grow up fast and make the decision of whether or not to take care of her siblings... Or to abandon them to their circumstances like she had been. Cursed with the ability to see the immutable future she still chose to step up and take care of them, at the of her own life and the sacrifice of her childhood. After that the only thing that was her's in her life was track and field.
It was around the time that her brother's health was inexplicably deteriorating that she meet the person who had Bound Pyre, gave Seafoam his transformation broach and introduced Wave to the world of Dark Magic. As she was increasingly exposed to Pyre's Magic her powers began to grow in response, deliberately fed and shielded by him. He was able to speak to her and teach her how to see other probabilities. The more Wave reached for better futures the more her power burgeoned and extended outward, allowing her to unwittingly make reality more fluid and to make the immutable futures mutable again.
Eventually, she realized what was going on, who the actual villains were and managed to rescue herself and her siblings from the situation. Unwilling to accept defeat the villians went after Wave and her siblings relentlessly until Ripple killed them and temporarily crippled the Ancient One who had killed Wave. Impressed with Ripple (and somewhat hopeful about her) that Ancient One told Ripple about Pyre, though not what he had done to save Wave, nor about Pyre's connection to Wave. They did this to see how she would respond. When Midnight showed the (temporarily) dead Wave all of this she asked what the conclusion about her sister was. Midnight simply told her that Ripple had passed an unreasonable test to see if she could upend the situation.
Wave continues to keep Midnight company until her life can be restored, unwilling to divulge that her current death is her own doing; she had traded her Fate for her sister's in order to save Ripple. It's a choice that still devours her heart because it meant also sacrificing Seafoam. Pyre did not make her choose in order to be cruel... He simply did not have enough power by that point to save all three siblings, though he wished that he could. She currently still cannot face her sister yet.
It remains unknown why she has no animal emblem, though it may have something to do with Midnight.