The halakhic significance of נענועים should not be minimized. While the details of how and when to wave the lulav vary by community, the act of waving itself is considered much more than a mere “custom” in the halakhic literature. Technically, the halakhah requires only lifting or holding the four species to fulfill the mitzvah. But early sources view na’anu’im as the essential, even defining, aspect of arba minim. Two halakhot recorded in the Mishnah, regarding a lulav’s minimum size and the appropriate age to begin performing the mitzvah, make this clear:
לולב שיש בו שלשה טפחים כדי לנענע בו, כשר (משנה סוכה ג:א)
קטן היודע לנענע, חייב בלולב (משנה סוכה ג:טו)
היה משכים לצאת לדרך נוטל שופר ותוקע, לולב ומנענע, מגילה וקורא בהן ומתפלל. ולכשיגיע זמן קריאת שמע קורא.
It appears that נענועים is more closely linked to the four species obligation than any other action (see, however, the Talmudic passage below where Raba implies that waving is considered only שירי מצוה - ancillary to the primary mitzvah).The Talmud offers two reasons for waving the lulav, which it compares to the biblical תנופה, the waving of bread and sheep offerings on Shavuot (Sukka 37b, with modified Soncino translation below):
אמר רבי יוחנן: מוליך ומביא ־ למי שהארבע רוחות שלו, מעלה ומוריד ־ למי שהשמים והארץ שלו. במערבא מתנו הכי, אמר רבי חמא בר עוקבא אמר רבי יוסי ברבי חנינא: מוליך ומביא ־ כדי לעצור רוחות רעות, מעלה ומוריד ־ כדי לעצור טללים רעים. אמר רבי יוסי בר אבין, ואיתימא רבי יוסי בר זבילא: זאת אומרת ,שירי מצוה מעכבין את הפורענות. שהרי תנופה שירי מצוה היא, ועוצרת רוחות וטללים רעים. ואמר רבא: וכן בלולב. רב אחא בר יעקב ממטי ליה ומייתי ליה, אמר: דין גירא בעיניה דסטנא. ולאו מלתא היא, משום דאתי לאיגרויי ביה.
R. Johanan explained, One waves them to and fro in honor of Him to Whom the four directions belong, and up and down in acknowledgment of Him to Whom are Heaven and Earth. In the Land of Israel they taught us thus: R. Hama b. ‘Ukba stated in the name of R. Jose son of R. Hanina, He waves them to and fro in order to restrain harmful winds; up and down, in order to restrain harmful dews. R. Jose b. Abin, or, as some say, R. Jose b. Zebila, observed, This implies that even the ancillary parts of a commandment prevent calamities; for the waving is obviously ancillary to the commandment, and yet it shuts out harmful winds and harmful dews. In connection with this Raba remarked, And so with the lulav. R. Aha b. Jacob used to wave it to and fro, saying, ‘This is an arrow in the eye of Satan’.* This, however, is not proper, since Satan might in consequence be provoked against him.
The reasons proposed here by the Sages - omni-directional waving of the four species acknowledges God's omnipresence; waving restrains harmful weather - reflect two major themes of Sukkot: Thanksgiving for the completed harvest, and prayer for the upcoming rainy season. As Sukkot is the holiday on which we pray for the upcoming rains, we can readily understand how shaking the lulav could be linked with a desire to “restrain harmful winds and dews.” It is also very reasonable that on the biblical Thanksgiving festival, we grasp the four harvest species and ritually wave them in all directions, to signify God’s omnipresence and his control over the agricultural cycle.
Medieval Talmudists added to the Talmud's reasons. For example, Tosafot (Sukka 37b, s.v. Be-Hodu) cites the juxtaposition of verses in I Chron. (16:33-35) to account for the Mishnah's requirement to wave the lulav while reciting specific verses within the Hallel prayer:
אָז יְרַנְּנוּ עֲצֵי הַיָּעַר מִלִּפְנֵי ה’ כִּי-בָא לִשְׁפּוֹט אֶת-הָאָרֶץ: הוֹדוּ לַה’ כִּי טוֹב כִּי לְעוֹלָם חַסְדּוֹ: וְאִמְרוּ הוֹשִׁיעֵנוּ אֱ-לֹהֵי יִשְׁעֵנוּ וְקַבְּצֵנוּ וְהַצִּילֵנוּ מִן-הַגּוֹיִם לְהֹדוֹת לְשֵׁם קָדְשֶׁךָ לְהִשְׁתַּבֵּחַ בִּתְהִלָּתֶךָ:
In this view, shaking the lulav mimics treetops blowing around in the wind, a sign of nature’s joy and delight in God's presence. By waving the lulav, we demonstrate that our human songs of praise for God are in harmony with nature's songs.
There are several biblical poems, especially in Psalms and Isaiah, where trees, mountains, and rivers "sing" (יְרַנְּנוּ) and "clap" (יִמְחֲאוּ-כָף) for God or for Israel:
יַעֲלֹ֣ז שָׂ֭דַי וְכָל־אֲשֶׁר־בּ֑וֹ אָ֥ז יְ֝רַנְּנ֗וּ כָּל־עֲצֵי־יָֽעַר׃ (תהלים צו:יב)
הָרִיעוּ לַה' כָּל-הָאָרֶץ פִּצְחוּ וְרַנְּנוּ וְזַמֵּרוּ: זַמְּרוּ לַה' בְּכִנּוֹר בְּכִנּוֹר וְקוֹל זִמְרָה . . . נְהָרוֹת יִמְחֲאוּ-כָף יַחַד הָרִים יְרַנֵּנו
(הֶהָרִים וְהַגְּבָעוֹת יִפְצְחוּ לִפְנֵיכֶם רִנָּה וְכָל-עֲצֵי הַשָּׂדֶה יִמְחֲאוּ-כָף (ישעיה נה:י
Each of these natural phenomena generate impressive sounds and, in the poet's imagination, joyful song; mountains also seem to sing as the wind rushes through their trees.
In one of its rabbinic usages, the word נענוע itself connotes song and music; e.g., Tosefta Sanhedrin 12:5:
רבי עקיבא אומר המנענע קולו בשיר השירים בבית המשתה ועושה אותו כמין זמר אין לו חלק לעולם
A similar musical usage is employed by the author of שמיני אשפוך, a piyyut for Maariv on Shemini Azeret, when he refers to playing a new eighth string -- there were previously seven -- on the harp of the future Temple:
שמיני לעודף שמחה לראות / לנענע שמיני בעשותו נוראות
I recently came across an intriguing and novel approach to the origin of na’anu’im. Moshe Zeev Sole suggests that waving the four species derives from an ancient desire to create a sort of artificial wind, by which to remind the angel appointed on the wind and rain to fulfill his annual duties.** He does not present any evidence for this claim. Leaving that aside, this theurgical interpretation – one which sees the ritual as an attempt to influence the behavior of a divinity – may make na’anu’im seem primitive and obsolete. Today, it is laughable to imagine that anyone would shake a lulav to awaken a wind spirit.
To be fair, a theurgical element also appears to underlie the latter part of the Talmudic passage above, which refers to "restraining harmful winds and dews” and to shooting “an arrow in the eye of Satan” by means of waving the lulav (see Maharsha's commentary, ad loc.). Even if that were true, however, in its conclusion, the Talmud is uncomfortable with the idea of using the lulav to fight Satan. In an attempt, perhaps, to suppress overt theurgy, the Talmud rejects R. Aha's practice of invoking Satan’s name explicitly during na’anu’im.
In fact, we are already familiar with the "angel of rain" from another context in the Sukkot Machzor. He is mentioned in the opening line of Tefillat Geshem, recited on Shemini Azeret: אף ברי אותת שם שר מטר – “The prince of rain has been named Af Beri.” Even if not addressed directly, the שר מטר was clearly seen by Eleazar Ha-Kalir, the prayer's circa-seventh-century composer, as a prominent force in bringing rain.
When considering such theories regarding the origins of religious practices, it is important not to confuse origin with meaning. Whatever the historical origin (real or presumed) of a practice, its meaning is far more significant - at least to those observing it. Meaning is highly subjective and can come from a variety of sources. For some, meaning and origin are intertwined: Insight into the origin of a practice may indeed enrich its meaning. But historical origin need not exclude other variables from the determination of meaning. Over the centuries, ta'ame ha-mitzvot and ta'ame ha-minhagim have been highly dynamic and open to widely varying creative interpretation precisely because they deal not so much with origin as with meaning, which changes with time and place.
As modern people and as heirs to the rationalist tradition within Judaism, we tend – for good reason – to minimize the role of angels and demons in our tradition. It is understandable, then, that the “wind angel” theory of na’anu’im causes us much discomfort. On the other hand, even as moderns we may allow ourselves to draw inspiration from the idea of waving the lulav in order to generate a musical wind – taken by itself, perhaps, and with apologies to any meteorological angel who may feel left out. The idea is not too distant, after all, from Tosafot’s suggestion that na’anu’im represent the joining of nature’s personified song with human paeans of joy and thanksgiving following the harvest; of אָז יְרַנְּנוּ עֲצֵי הַיָּעַר in concert with הוֹדוּ לַה כִּי-טוֹב.
* Shooting an arrow into Satan's eye was one of many images -- literary and visual -- used in the ancient war on the evil eye. See, for example, this second century Roman mosaic from Antioch.
**Ha-Moadim Ve-Hamikra, (Jerusalem: Mabat, 1985), pp. 83ff. The author (1908-1994) was trained at the Hildesheimer Rabbinical Seminary in Berlin and wrote several books on various topics in Philosophy and Jewish Studies. A brief biography in David Tidhar's recently digitized Entziklopedia Le-Halutze Ha-Yishuv U-Vonav is available here.