The waving of the Arba Minim (the Four Species) -- the Na'anu'im-- serves as a focal point of the t'filot of Chag HaSukkot. Interestingly, though, one can fulfill his minimal obligation by merely picking of the Arba Minim (See Sukka 42a). However, the clear implication of many Rishonim is that Na'anu'im are a l'chat'chila d'Oraisa - a Biblically mandated ideal enhancement to the performance of the mitzva (see Rambam (Hilchot Lulav (7:9); also Tosfot and Ritva to 37b who rule stringently in case of doubt, presumably following the rule S'feika D'Oraisa L'Chumra). Indeed, the Mishna in Masechet Sukka (29b see Rashi there) indicates that unlike the Hoshana (myrtle) and Arava (willow) whose minimum size is three t'fachim (handbreadths), the Lulav requires 4 t'fachim. The extra tefach, as the Mishna explains, is necessary "k'dei l'na'nai'a bo" -- in order to wave it. Although the Ba'al Ha'Itur and Me'iri maintain that this requirement for the fourth tefach is only Mi'd'rabbanan, the overwhelming majority of Rishonim and subsequent Poskim maintain that 4 t'fachim is an absolute requirement which would seem to indicate its Biblical origin. Even though the Na'anu'im are only l'chat'chila, the lulav must at least be fit for waving.
What is signified by this Biblically mandated waving? Furthermore, it is clear from the Mishna (37b) that the Four Minim are to be waved at two places in the Hallel -- once at "Hodu Lashem Ki Tov" -- "Give praise to G-d for He is good" and once at "Ana Hashem Hoshi'a Na" -- "Please G-d, save us now!". These expressions contain drastically different themes. "Hodu" is a call to praise Hashem for his everlasting kindness. "Ana" is a heartfelt prayer for salvation. How does the waving of the Arba Minim enhance these diverse prayers?
What clearly emerges is that the waving of the Arba Minim serves as both a t'fila of bakasha (request) and a song of hoda'ah (praise). But how does the lulav accomplish both tasks? The Mishna in Rosh Hashana (16a) relates that "Be'chag nidonin al ha'mayim" -- on Sukkot the world is judged concerning the amount of rainfall for that year. The four species all require additional watering in addition to rainfall to survive. The Torah even refers to Aravot as Arvei Nachal -- willows that grow by a river. There is even a minority position in Tosfot (34a) that the Aravot must grow by a river to be valid. Even though we do not follow this position (see Shulchan Aruch (647:1) and Mishna B'rura (3)), certainly the Torah highlights the role of water in the Arba Minim. Indeed, one of the textual proofs that the Biblical "Pri eitz hadar" (Leviticus 23:40) is a citron is that the word "hadar" is etymologically related to the Greek "hydro" meaning water (Sukka 35a). Hence, the Torah refers to a fruit that needs much additional watering to survive. The waving of the species then, is a form of t'fila for rain. Besides the t'fila for rain on Shmini Atzeret, we pray every day of Sukkot for rain in the Hallel by waving the Arba Minim at "Ana".
But the waving of the Arba Minim symbolizes more than bakasha alone. The Torah refers to Sukkot as the "Chag Ha'asif" (Exodus 23:16) -- the festival of ingathering. The beginning of the fall ushers in the period of gathering the produce from the field into the silos and warehouses. The long, arduous task of carefully sowing, watering, harvesting, and drying is over. The Jewish farmer can finally partake of his bounty. Herein lies the danger of the farmer attributing his material success to his own efforts alone. "Kochi v'otzem yadi asah li es hachayil hazeh" (Deutoronomy 8:17) -- "My strength alone brought about this abundance of wealth." By waving the Arba Minim -- the more prominent and beautiful of the crops -- in the Hallel at "Hodu", the farmer turns heavenward and declares to the Mashgiach Al Hakol (Provider for All): "I thank You, Hashem, for your kindness to me. I am fully cognizant that it is you "Hanosein koach la'asos chayil" -- that crowns my efforts with success, and without Whom there would be no crops to celebrate.
This then can be the underlying meaning of the statement by R. Yochanan (Sukka 37b) that we wave back and forth to the Creator of the four corners of the world, and we wave up and down to the Creator of Heaven and Earth. R. Yochanan underscores the role of the Na'anu'im as an expression of praise and thanksgiving. R. Yose b'Rabbi Chanina states (ibid.) that we wave back and forth to stop harmful winds and up and down to prevent harmful precipitation. R. Yose emphasizes the bakasha nature of the waving.
These underlying themes of prayer and thanksgiving through mitzvot other than oral t'fila itself appears in other mitzvot as well. Rav Joseph B. Soloveitchik ZT"L, based on earlier sources, explains the blowing of the shofar as a form of prayer without words -- a cry symbolically emanating from the heart of the mispallel (petitioner) (see Y'Mei Zikaron). Similarly, the Gemara in Megilla (4a) derives the requirement of the dual reading of the Megilla -- once at night and once during the day -- from two sources. One is the passage in Tehilim (30) "Lma'an y'zamercha chavod v'lo yidom Hashem Elokai l'olam odeka" -- "in order that my soul sing to you and not be silent, Hashem, my G-d, forever I will thank you." The second is from T'hilim (22) "Elokai ekra yomam v'lo ta'aneh v'layla lo dumiya li" -- "My G-d, I call to you by day and you do not answer, and at night and I am not silent." Rav Betzalel Zolti ZT"L points out that the two sources apparently indicate that Mikra Megila contains the same dual nature as we find regarding the waving of the Lulav. First, it serves as a paean to the Ribono Shel Olam who foiled Haman's plot to decimate the Jewish people. Indeed, the Gemara (Megilla 14a) even suggests that Hallel need not be recited on Purim since the Megila itself serves as the Hallel. However, the reading of the Megilla also serves as a t'fila of bakasha. By reading the narrative of how Hashem has saved us in the past, we implicitly beseech Him that he redeem us from our present exile as well.
The theme of prayer through the performance of mitzvot ma'asiyot -- active commandments -- highlights the recurrent them of the Seifer Hachinuch. "Acharay hape'ulot nimshachim ha'l'va'vot" -- the mind follows the actions. Praying to Hashem and praising him through words alone does not suffice. We must make use of the Kol Shofar, the Na'anuim, and Mikra Megilla to enhance our communication with the Shomai'a T'filos. Together with the words of our t'fillot these mitzvot help stir our emotions in a way that oral prayer alone cannot do. Ultimately though, the goal is the "Avoda She'b'laiv -- Zo T'fila" (The service of the heart which is prayer). The Mispallel with the Shofar, the Lulav and Megilla is urged to elicit in himself an outpouring of his heart and soul to the Source of all coupled with the recognition of his utter dependence on his Creator who his only true source of help. May all of our various forms of t'filot this Sukkot and throughout the whole year be answered L'Tova.