What are archaic words?I ask this question about the effect of the archaic words in the context of William Cullen Bryant' work in "To a Waterfowl."

Your question is very open ended. However, since you ask about William Cullen Bryant, who was an American poet that lived in the 19th century, I will assume that you mean archaic words that are related to the English language. In this case, it is almost certain that archaic language refers to the classical world. Keep in mind that classical learning was still very much a part of American culture then. People could even still speak Latin and dissertations in universities were written in Latin! Also, people were excellent in ancient Greek. For example, Alexander Pope, a British poet, whom Bryant admired, translated the Iliad. Also it is good to keep in mind that the title of one of Bryant's poems, Thanatopsis, was from two Greek words, meaning "mediations on death". In addition, Bryant’s poem, “To the Waterfowl” was written to be a didactic poem. He probably got this idea from the Greeks (Hesiod) and Romans (Ovid), who were fond of didactic poems.

Hence, in the light of these points, archaisms refer to Bryant's use of classical culture, language and literary conventions. It is also my experience that many of these types of writers make extraordinarily learned allusions to classical culture. So, if you read carefully, you will undoubtedly hear many echoes from Greece and Rome.