Charles Ellicott Commentary Numbers 8:11

"and Aaron shall offer the Levites before Jehovah for a wave-offering, on the behalf of the children of Israel, that it may be theirs to do the service of Jehovah." — Numbers 8:11 (ASV)

And Aaron shall offer the Levites before the Lord for an offering. —Literally, and Aaron shall wave the Levites as a wave-offering before the Lord. The manner in which the Levites were thus set apart to the Lord is not expressed. It may have been done by leading them backwards and forwards in front of the Tabernacle and in the presence of the people, or by the waving of Aaron’s hands.

The same word is used elsewhere in reference to offerings of different kinds—as, for example, of gold in Exodus 35:22. (Numbers 8:15; Numbers 8:21 of this chapter.) The symbolical meaning of the ceremony is obvious from the concluding words of the verse, and is further explained in Numbers 8:13-14. (Compare to Leviticus 7:30 and Note.)