"Dover Beach" is a dramatic monologue. In a dramatic monologue the speaker is not necessarily the author, but most critics suggest that the speaker is a poet who does express some of Arnold's own beliefs. At least, we can say the speaker is (probably a poet) with his lover, looking over the straits of Dover toward the French coast while contemplating humanity, faith, and love.
In a dramatic monologue, we (readers) know who the speaker is addressing by clues in the speaker's monologue. In the first stanza, it is unclear whether the speaker is speaking to another person, to himself, or if he's thinking (as an internal soliloquy). It isn't until the fourth stanza that the speaker actually addresses someone:
Ah, love, let us be true
To one another! for the world, which seems
To lie before us like a land of dreams,
We might be inclined to suggest that the speaker is addressing a personified "Love" or the abstract quality of love. But since the speaker repeats the word "us," it is much more likely that he is speaking to a lover, his "love." So, although there are no clues in the first stanza, the clues in the fourth stanza imply that he's been speaking to his "love" throughout the entire poem.
The speaker of "Dover Beach" is thought to be a poet who acts as the voice of Matthew Arnold; apparently, he stands at an open window of an inn where he is afforded a clear view of the straits of Dover on the English Channel. He addresses his lover, a silent audience: "Come to the window, sweet is the night air!"
Robert Browning once expressed the role of the poet as a person who has been sent to remedy "the misapprehensiveness of his age." In "Dover Beach," written as a dramatic monologue, a poet expresses his thoughts on the beauty of the scene that reminds him of the disconcerting conditions of his time. His heart is drawn to the beauty of the scene, but his mind must heed the historical sounds of the surf, sounds that Sophocles heard on the Aegean, a sea that experienced many wars. Then, as he hears the water rub the beach, Arnold's speaker reflects upon the "ebb and flow/Of human misery."