If a character with flight is hit by pulse wave and the roll is doubles and they would be knocked off elevated does the characters flight continue to be ignored because of pulse wave and they would then take knockback damage from falling off elevated?
Or is pulse wave considered to be resolved after damage is applied and then the character would be knocked back but take no damage from falling off elevated ?

Pulse Wave ignores the game effects of the characters in the AoE until the action resolves. Knockback will happen during the action, however I don't think knockback will ignore any characters effects so if a flyer is knocked off of elevation I do not believe that they will take knockback from the fall. If you are curious, I have attempted an order of turn layout that to the best of my knowledge covers the order of the steps accurately. Using that and the wording of PW is what led me to my conclusion.

Flight will be ignored when the knock back happens, so the character will be dealt 2 damage.

If VP disagrees than I am likely wrong.

Yeah it ignores all powers and abilities possessed by hit characters so it would abolutely take two damage.