A wave offering was an offering that the priest would “wave” in front of the Tabernacle. For instance, a priest might wave his own food in front of the Tabernacle, but then eat it for his own use. In Numbers 6:20, we read, “Then the priest shall wave them for a wave offering before the Lord. It is holy for the priest, together with the breast offered by waving and the thigh offered by lifting up; and afterward the Nazirite may drink wine.” Therefore, the wave offering did involve literal waving. Ronald Allen writes,
The notion of ‘wave offering’ is somewhat mysterious to us. We have some concept of the burning of sacrifices, of the pouring of libations, and of the presenting of grain offerings. But the ‘wave’ offering is the most obscure to us. The idea in the wave offering was to hold an object, usually the part of the offering that would be the food for the priests, before the Lord, to wave it back and forth, and then to keep it for one’s own use.
The symbolism here was that the meat (or in Numbers 8, the Levites) were being kept from being literally sacrificed. But they were consciously being given over to God. In the NT, we are “living and holy sacrifices” to God (Rom. 12:1).
 Allen, Ronald. Numbers. In F. E. Gaebelein (Ed.), The Expositor’s Bible Commentary, Volume 2: Genesis, Exodus, Leviticus, Numbers (F. E. Gaebelein, Ed.). Grand Rapids, MI: Zondervan Publishing House. 1990. 767.