How is "Dover Beach" by Matthew Arnold a dramatic monologue?
A dramatic monologue is a poem in which the speech of a single character in a single scene makes up the entirety of the poem. The speaker may be addressing someone else, but only the person delivering the monologue speaks. The focus of a dramatic monologue is to reveal something about the speaker's inner character. It may also be used to express the speaker's point of view, which the poet may or may not share.
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I would agree with you. The poem is certainly a monologue, but not really a dramatic monologue, since it lacks the element of a speaker distinct from the author. But this is all in the definition, since some people don't regard the distinct speaker as part of the definition of a dramatic monologue. Terms in literary criticism don't necessarily have the precise and agreed meaning that terms in biology or math do.