What does the poet mean by eternal note of silence?

I think you are referring here to the end of the first stanza, where Arnold describes how the waves "bring / The eternal note of sadness in."
This poem is generally considered to be a lament by Arnold for a time when England had "certitude" based upon its near-universal faith—a time when "the Sea of Faith" was "full" and curled around the island like "a bright girdle." Arnold's poem is a reflection of the mid-Victorian anxiety that sprang from the crisis of religious faith that swept the country, as geological discoveries and Darwin's work, among other things, began to generate "confusion." As such, although he does not identify specifically what this "eternal note of sadness" is, we can interpret that it is painted in contrast to the "Sea of Faith."
Arnold states that Sophocles, too, once heard the eternal note of sadness, and that it brought "human misery" into his mind. This "human misery" is connected strongly, in Arnold's poem, to the "confusion" which has led the "ignorant armies" he speaks of—the public—to be "stranded on a darkling plain," unsure of where to head. The "sadness" is the human fear of the unknown and the distress engendered by an ebbing-away of faith, which leaves us confused and without direction.