Matthew Arnold projects his anguished viewpoint of the world upon the beautiful scene from his window, and he does it primarily through the connections he makes with what he sees to the melancholy that he feels about the world. To him, the sound of the ocean waves on the beach is not a calm, soothing, serene sound as many people often find it; rather, he thinks that it brings in "the eternal note of sadness" and the "turbid ebb and flow of human misery." So whereas other people might connect that peaceful scene to a calming, happy feeling, Arnold connects it to sadness and misery. This is him projecting his own worldview on the scene before him.
Also, he picks his words carefully as he describes the scene, using words that reflect his own emotional state about the world. Note the harsh, negative word choices: "grating roar," "fling," "tremulous," "vast edges," "naked shingles," "darkling plain," "confused alarms" and "clash". All of these words reflect Arnold's reality of anguish and despair that he feels over the state of the world, which taints even a serene view from a window. He feels that any beauty and goodness have disappeared from the world--in the last stanza, he goes through an entire list of things he feels are no longer in the world. He feels that all of those things, and faith, have departed, leaving only violence, darkness, exposure and misery behind.
The only possible hope he sees is if he and his love are "true to one another." By forging meaningful, loving and honest relationships, he feels that he can at least protect himself from the awful darkness descending on the world. It's not much hope, but a little tiny flare of it. I hope that those thoughts helped; good luck!
Arnold's poem, Dover Beach begins with a graphic, naturalistic description of the Dover nightscape: the "calm" sea, the tide being "full"; the moon being personified as lying fair on the straits; the gleaming light on the French coast disappearing; "the long line of spray" lapping on to the "moon-blanched" shore.
The images of sights and sounds are all objective with no subjective projection of the lyrical self. But then occurs the switch from the objective to the subjective, from the descriptive to the reflective. The poet asks his companion to listen to the "grating roar of pebbles" in the continuous to and fro movement of the waves. The waves draw the pebbles from the shore only to fling them back on to the sands, and this frictional sound in the waves picks up "a tremulous cadence slow" to bring into the listener's mind "an eternal note of sadness."
The poet remembers the ancient Greek master, Sophocles, who must have heard the same melancholy music many centuries ago at the Aegean sea, a music that subjectively connotes "the turbid ebb and flow of human misery."
Further, the poet refers to a symbolic sea--"the Sea of Faith"--which was full to the brim in the past. That Sea now retreats from the shores of human habitation, leaving a waste land of "naked shingles." This is all subjective and imaginative, meant to underpin the poet's melancholy regret and anguish at the fact of the crumbling away of faith in the world of man.
Arnold's melancholy in this brief and yet expansive poem is due to the loss of faith and the increasing hold of doubts and disputes on the mind of man in the modern Industrial age. The only ray of hope in this darkening world of faithlessness and blind hostility is love, as the poet urges his beloved--"...let us be true to one another."