Gemara

33b - Mishnah: Tzaftzafah is Pasul for Aravah. An Aravah of Ba'al (a field that need not be irrigated) is Kosher.

Beraisa: "Arvei Nachal" - they grow by a river. Also, this teaches that its leaves are long like a river.

Beraisa - Question: "Arvei Nachal" connotes Aravos that grow by a river. What is the source to include Aravos of Ba'al and of mountains?

Answer: "Arvei (plural) Nachal" - in any case.

Aba Sha'ul says, "Arvei" alludes to two Aravos - one for the four Minim, and one taken in the Mikdash (on Hoshanah Rabah).

Chachamim learn about the Aravah in the Mikdash from a tradition from Sinai.

Beraisa: "Arvei Nachal" excludes Tzaftzafah, which grows between mountains.

R. Zeira: We learn from "Kach Al Mayim Rabim Tzaftzafah Shemo";

R. Avahu: Hash-m wanted Yisrael to be like what grows by water, i.e. Aravah, but they made themselves like mountain Tzaftzafah.

Beraisa #1: Aravah has red branches and its leaves are long with smooth edges. Tzaftzafah has white branches and its leaves are round with (jagged) edges like a sickle.

Contradiction (Beraisa #2): (If the edges of the leaves are) like a sickle (it) is Kosher. (If it is) like a saw is Pasul.

Answer (Abaye): Beraisa #2 discusses Chilfa Gila (a species of Aravah; its leaves are like a sickle). This teaches that Chilfa Gila is Kosher for Aravah. One might have thought that it is Pasul because it has a Shem Levai (an accompanying name). "Arvei Nachal" includes it.

34b - Mishnah - R. Yishmael: We take three Hadasim, two Aravos...

R. Akiva says, we take one of each.

44a - Question (Abaye): Why do we take the Lulav (i.e. all four Minim together) all seven days to commemorate the Mikdash, but not the Aravah (by itself)?

Answer: Rava taught that since Lulav is mid'Oraisa, we take it all seven days to commemorate the Mikdash. Aravah is mid'Rabanan, so we do not take it all seven days.

Objection: This is unlike Aba Sha'ul, who says that "Arvei" alludes to two Aravos - one with the Lulav, and one taken in the Mikdash. It is unlike Chachamim, who have a tradition from Sinai for Aravah in the Mikdash!

Correction: Rather, since there is a source in (written) Torah for Lulav, we take it all seven days. Aravah has no source in Torah, so we do not take it all seven days.

Rishonim

Rif: "Arvei Nachal" teaches that they grow by a river and that its leaves are long like a river. It also excludes Tzaftzafah, which grows between mountains.

Rambam (Hilchos Lulav 7:3): "Arvei Nachal" is not anything that grows by a river. It is a particular species of this name. Its leaves are long with smooth edges and the branch is red. This is called Aravah. Most of this species grows by a river, therefore it says "Arvei Nachal". Even if it grows in the wilderness or mountains it is Kosher.

Tosfos (34a DH v'Rabanan): We can expound only one matter from the plural "Arvei Nachal". Aba Sha'ul and the first Tana use it to teach about the Aravah in the Mikdash and to include Aravos of Ba'al and of mountains, respectively. They hold like R. Akiva, who says that only one Aravah is taken. Since the Halachah does not follow R. Akiva, we have no source to include Aravos of Ba'al and of mountains. Therefore, one must be careful to use only 'river' Aravos. It seems that the Rif agrees, for he did not bring the Beraisa that permits them. Rashi says that river Aravos are l'Chatchilah. B'Di'eved, those of Ba'al and of mountains are Kosher, for "Arvei Nachal" includes them.

Rebuttal #1 (Ran DH v'Chosvu): Tosfos rules against our Stam Mishnah because there is an argument in a later Mishnah (in such case, the Halachah need not follow the Stam). He holds that the Halachah does not follow R. Akiva, but this is not necessarily so. Abaye is Basra, and he includes Chilfa Gila from "Arvei Nachal". He does not say that it teaches about two Aravos and nothing else!

Rebuttal #2 (Rosh 3:13): People are not careful to take river Aravos. I did not see my Rebbi'im strive to use them. The first Beraisa expounds "Arvei Nachal" - they grow by a river. It seems to me that this means the species that grows primarily by rivers, It excludes Tzaftzafah, which grows on mountains. This Tana does not need a Ribuy for Aravos (of the species that usually grows by rivers) that grew in the wilderness or mountains. This is the opinion of Chachamim, who require two Aravos. Also the Rambam holds like this.

Rebuttal (of Rosh - Re'em, cited in Kesef Mishneh): The Rambam allows Aravos that grew in the wilderness or mountains from the Ribuy "Arvei". The Rosh's text of the Rambam said, 'therefore it is called Arvei Nachal' 'in place of 'therefore it says "Arvei Nachal"'. This misled him.

Defense (of Rosh - Kesef Mishneh): The Gemara clearly says that we can include only one thing from the plural "Arvei". (This forced it to say that Chachamim learn about the Aravah in the Mikdash from a tradition! Aravos that grew in the wilderness or mountains are like one thing, because there is no reason to favor one over the other.) The Rambam requires two Aravos. This is from the plural, so he must hold that "Arvei Nachal" connotes even those that grow in the wilderness or mountains

Rebuttal (Lechem Mishneh): The Rosh himself brought this proof from the Gemara. The Re'em properly challenges the Rosh's claim to additional support for himself from the Rambam.

R. Mano'ach: It says "Arvei Nachal" instead of 'Arvei Mayim' to include Aravos that did not grow by water. Nachal can also mean valley, e.g. "Va'Yachperu Avdei Yitzchak ba'Nachal", and "Nachal Eisan" according to Targum Onkelus.

Poskim

Shulchan Aruch (OC 647:1): "Arvei Nachal" is a particular species of this name. Its leaves are long with smooth edges and the branch is red (even if it is still green it is Kosher).

Kaf ha'Chayim (4): It need not be red. Anything that is not white is called red, since the sun will redden it.

Shulchan Aruch (ibid.): Most of this species grows by a river, therefore it is called "Arvei Nachal".

Question: The Rosh says that most Poskim require river Aravos; he gave an alternative explanation. Why didn't his Rebbi'im fulfill the majority opinion?

Answer (Taz 2): They wanted to show that the Halachah does not require river Aravos.

Mishnah Berurah (3): Some say that l'Chatchilah one should take river Aravos. The Taz disagrees.

Kaf ha'Chayim (6): Letter of the law, Aravos that grow in the wilderness or mountains are Kosher. It is good to take river Aravos to fulfill the opinion of the other Poskim.

Mishnah Berurah (6): Often, children who know nothing pick Aravos. A buyer must beware.