Question: The prayer for rain is recited annually during the Musaf Amidah (Ashkenazi rite) on Shemini Atzeret. Has there been any recent academic research on this prayer?
Answer: The prayer for rain is treated briefly in the "Af Beri" article in The Encyclopedia of Jewish Prayer by Macy Nulman (1993), and in the 1993 translation (Jewish Liturgy: A Comprehensive History) of Ismar Elbogen's earlier work.
Brigitte Kern-Ulmer discusses rain as theological redemption and regeneration, as understood in early rabbinic texts: "Consistency and Change in Rabbinic Literature as Reflected in the Terms Rain and Dew" in Journal for the Study of Judaism in the Persian, Hellenistic and Roman Period v 26, no 1 (April 1995), p. 55-75. The article focuses on the general prayer for rain, not specifically the Shemini Atzeret prayer.
Micahel Rand looks at the Shemini Atzeret prayer from a completely different point of view. He analyses paytan Eliezer Kallir's attempt to provide a rational meteorological explanation for the production of rain, based on Kallir's understanding of earlier rabbinic midrashim. The article is "Clouds, Rain and the Upper Waters: From Bereshit Rabbah to the Piyyutim of Eleazar be-Rabbi Qullir" in Aleph v 9, no. 1 (2009) p. 13-39.