Rabbi Yohanan said: He moves them to and fro to dedicate them to He Whom the four directions are His. He raises and lowers them to He Whom the heavens and earth are His. In the West, Eretz Yisrael, they taught it as follows. Rabbi Hama bar Ukva said that Rabbi Yosei, son of Rabbi Hanina, said: He moves them to and fro in order to request a halt to harmful winds, storms and tempests that come from all directions; he raises and lowers them in order to halt harmful dews and rains that come from above.
Perhaps the simplest explanation of the na’anu’im is given by the Me’iri, who describes them as a show of joy appropriate for the Sukkot holiday. This idea is developed by Rabbeinu Mano’ah who says that the shaking must be done with strength and vigor to fulfill the passage that commands that a person must praise God with his entire being (Tehillim 35:10), and by the Rosh who explains that it is to show particular love for the mitzva.
The explanation that appears in the Jerusalem Talmud is that waving the lulav is an act of defense – an attempt to ward off the prosecuting angel. Moreover, the Mishna instructs this shaking or waving of the lulav at specific points during Hallel – the prayer of thanksgiving. Based on the passage in Tehillim (96:12) that the trees of the forest sing out in praise of God, we are commanded to shake the symbolic trees as we praise God with our recitation of the Hodu prayer (Tehillim 118:1) and the plea Hoshi’a na (Tehillim 118:25).
The actual definition of ni’anu’ah is a matter of dispute. The Shulhan Arukh (Orah Hayyim 681:9) rules that it is a thrusting of the lulav back and forth in all directions, which is the tradition kept by the Sefardic communities. Most Ashkenazim follow the opinion of the Rema that the lulav must be shaken, as well.