Why is Sophocles mentioned in "Dover Beach"?

Sophocles is mentioned in "Dover Beach" because his plays show that tragedy is an eternal element in human life. His work is also relevant to Arnold in its use of tidal metaphors and its portrayals of losses of faith.

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In this poem, the speaker's view from his window as he looks out over the beach at Dover at night causes him to dwell on past ages. He thinks of Sophocles, an ancient Greek playwright. Talking of the "eternal note of sadness" that the tide brings in, the speaker says,

Sophocles long ago
Heard it on the Ægean, and it brought
Into his mind the turbid ebb and flow
Of human misery.
The Aegean Sea is the part of the Mediterranean sea near Greece. Here, the speaker is saying that Sophocles, too, heard the eternal sadness of the tide and that it made him think of humanity's suffering.
The simplest reason that Sophocles is mentioned in "Dover Beach" is to show that the "ebb and flow / Of human misery" that the speaker is reflecting on has persisted across centuries, going back at least to ancient Greece. Sophocles's plays exemplify the eternally tragic aspects of human life and remain relevant in Arnold's day as well as our own.
An additional reason for the reference to Sophocles may be that he sometimes used imagery of the sea in his tragedies. Take, for example, his tragic play Antigone. Antigone is the daughter of an incestuous union between her father, Oedipus, and Oedipus's mother, Jocasta. In the play, Sophocles likens the ways the tides of the sea roll in and out to the way fate brings tragedy back like an incoming tide to haunt Antigone. This fits well with the speaker's sad and melancholy mood as he watches the sea. Nature is not a source of solace for him but a source of despair.
A final reason that Sophocles is mentioned is that his plays depict losses of faith. In Sophocles's plays, loss of faith in the gods leads to tragedy. Oedipus, for instance, believes he can defy what the gods have ordained, only to come to a tragic end. Antigone dies because she battles Creon's defiance of the gods when he does not allow her brother a burial. Likewise, the modern age in Victorian England has, according to the speaker, lost its religious faith.
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