Waveriders first appearance was Armageddon 2001 # 1.
Haunted by blurred memories of the past, where an vague hero save him from the debris of an building collapse, Matthew Ryder lived his life as a scientist from a future Earth ruled by Monarch, an armored despot who had eliminated all heroes. The most tragic element it was than Monarch itself was one of the heroes than Ryder admired so much. But who exactly is something than nobody knew.
Despite live in a clean, ordered relatively pacific world, Matthew felt than something was wrong with the world as the monarch stormtroppers made arbitrary detentions and executions and Monarch itself destroyed randomly problematic communities.
Knowing than Monarch was working in a time travel device to prevent somebody else would try to stop him, Ryder convinced the despot to let him be a guinea pig in his own device. The tyrant agreed, believing Matthew Ryder, like several initial test subjects, would die in the process.
Matthew Ryder however survived, merging with the timestream and transforming into Waverider, a golden super-powered being. He arrived ten years before Monarch would size the control of the world. Capable of read the potential, ever changing future of everyone he could touch, Waverider started a frantic quest among earths heroes to discover the identity of Monarch and stop him/her before he/she betray the world superheroes.
However, unknown to him, this was anticipated by Monarch: the actions of Waverider were the ones which lead to the premature birth of Monarch ten years before in his time, who was revealed to be Hank Hall, formerly the hero known as Hawk, when his future self killed Dove, his adventure companion.
Joining forces with Earth's Greatest Heroes, including the Justice League of America, the Teen Titans and Captain Atom to defeat the twisted Hawk, Waverider had the chance to rescue a kid trapped under the debris of a collapsed building. Finally he got the answers he has everyday of his life.
Waveraider would join soon later the Linear Men, a group devoted to maintaining the timestream integrity and which included an alternate version of himself, a Matthew Ryder than has lived with his parents all his childhood; a older version of veteran time traveler Rip Hunter; a mysterious woman called Liri Lee. Eventually, Waverider would join the team.
Return of the JSA and Zero Hour
After defeat Monarch Waverider discovered the plans of Lord Abraxis to take over the earth dimension through a time rift (Armageddon: Inferno). To stop Abraxis followers Waverider recruited various heroes from different time periods. Was then than the Spectre, also fighting Abraxis, was capable to free the JSA from the Ragnarok-like limbo where they were prisioners.
Waverider had a relevant role during Zero Hour, a Crisis in time managed by Hal Jordan (later revealed possesed by Parallax then) and Hank Hall (in his Extant identity). Extant apparently killed Waverider and absorbed his powers, but thanks to time travels mechanics, Waverider reappered to help the surviving heroes to reset time itself.
There has been also several apperances in adventures involving time travel. Mainly in the Superman titles, as their fight with Dominus and the trial of Superman after the defeat of Imperiex and Brainiac 13, when, along the Linear Men, acted as prosecutor against the man of Steel, accusing him of crimes against time. The Quintessence, who acted as jugde, instead ruled in behalf of Superman and then proceed to dissolve the Linear men, stripping them of their powers. But that don't stoped Waverider to still travel through time.
Death and post death appearences.
Waverider found a more permanent end during the 52 maxi-series. He was discovered first by the sons of Sivana, to whom he revealed than there was somebody hunting time trevelers. The culpruit was later discovered as Skeets, Booster Gold's robotic valet. Using his technolgy Skeets started to torture Waverider, but first he told his victim than his exterior shell was built using Waverider's skin. Off-panel, Skeet proceeded to tear off Waverider's skin.
His remains would appear again during the Time Masters: Vanishing Point mini-series. The remaining energy in his corpse was used to transform Liri Lee in the Linear woman.
There has been no new appearances of Waverider post-Flashpoint.
Thanks to being composed of tachyons, Waverider has mastered the art of time-travelling. Able to move through time forward and backward, he can fly, being invisible, he can project quantum-powered force blasts. When he touch someone he can access a person's aura and predict the future the most probable for them; he also can read the past of the person.
Appearances in Other Media
Waverider appears in the TV Series known as Justice League Unlimited. He is never truly a "spotilighted" character, but more as an object placed in the background, to add more depth to the team. Thus, we cannot discuss if his character remains true to the comics or not.