What Arnold has in common with Sophocles is that they both have heard ‘’the eternal note of sadness;’’ Arnold at Dover Beach, Sophocles on the Aegean Sea. He is reminded of Sophocles with the ‘’grating roar of pebbles’’ as the waves ebb and flow. Such a sound brought to his mind what dawned on Sophocles’, ‘’the turbid ebb and flow of human misery.’’
Just as Arnold listened to "an eternal note of sadness" in the waves of the Dover sea, he imagined Sophocles to have heard in the Aegean sea a similar melancholy note, "the turbid ebb and flow of human misery". Arnold, an English poet of the Victorian age, was substantially grounded in ancient Greek literature, and discovered himself in line with the Greek master, Sophocles.
Yes, Dover Beach can be read as a dramatic monologue spoken by the poet at a time of moral-spritual crisis, a monologue addressed to his mistress on a moon-lit night at the beach of the Dover sea.
Don't listen to kc4u. He misunderstands the definition of a dramatic monologue:
"Type of poem in which a character other than the author is the speaker of the poem."
Matthew Arnold is clearly the speaker of this poem, addressing his wife, so I would say that it is NOT a dramatic monologue.
I know this question is old, but just for the people who might come here for help. The answer, IMO, is no.