Wave chess is a variant of chess meant to be played over multiple games.
Each player starts the match with x1 and x2 'waves' respectively. They start the match by first playing a game of standard, orthodox Queen's Chess. In each game after the first, the loser starts with all their pieces and all their clock time, but with one fewer wave in reserve. The winner of the previous game plays with the pieces and time they had remaining after the previous game, but the winner does not lose a wave.
Each remaining winner's piece can be placed in any space that it could legally start at. For example, a player with one remaining rook can choose either corner to start that rook in. They can also place their remaining pawns anywhere in the front rank.
A player is considered to have lost or won a previous game regardless of whether they have lost by forfeit or checkmate. The loser of the previous game becomes the white player.
In a draw, both players lose a wave and begin their next game with all pieces and time, and the previous game's white player becomes the black player.
This variant extends the idea of material odds handicapping. In a single game, one common handicap is to forfeit one or more pieces at the beginning of the game. In wave chess, smaller incremental handicaps are possible by forfeiting pieces only for a player's first wave. Also, larger handicaps are possible by giving one player more waves.
The wave mechanic also reduces the incentive of forfeiting. A player from a lost position may still want to continue a game in order to reduce their opponent's material for the next game.
After the first non-draw game, subsequent draws become less likely
because one player will start with a material advantage. I would expect that matches between strong, evenly matched opponents would start with one side beating the other with only a few pieces in the end game, followed by alternations between short games where the winner of the previous game manages to take a few pieces before being mated, and normal length games where the previous winner starts only a few pieces behind.
I would also expect that matches between badly mismatched players would result in the weaker player throwing wave after wave at the stronger player, which could make the game more exciting for both players. How many waves does it a take for a 1400-rated player to take down a single wave of a world class chess AI like Stockfish? How many waves of novice players, or randomly selected moves, can you withstand against?
One possible compromise between Queen's Chess and Wave Chess is to allow a winning player to regain x points of material between games.
As there are thousands of variants out there, it's plausible this has
been done before. If someone reading this knows of a similar concept, I
would love to know so I can send a link to it and give credit.