Alliteration appears throughout "Winter Swans." In general, it helps to amplify the imagery and emotional heft of certain phrases. The first example is in lines 3 and 4. At this point, the poem is setting its scene, using pathetic fallacy to indicate that the speaker and his partner have been arguing and that the mood is still tense: it's been raining, and the earth is sodden with water. The couple is trudging over the damp ground, hinting at the way that they are also trudging through their relationship. Alliteration is used skillfully to show this:
in which we walked,
the waterlogged earth
These relentless /w/ sounds extend across the stanza break, conveying the persistent difficulties of the couple's relationship.
The alliteration that occurs across lines 6, 7, and 8 is specifically sibilance, so it's covered more extensively in that section of the guide. In brief, though, the whispering /s/ sounds here in "skirted," "silent," "swans," and "stopped" help build a sense of the hushed, tense atmosphere).
Another key example of alliteration is in lines 11-12:
... paused before returning again
like boats righting in rough weather.
Here, the poem is describing the graceful, almost ritualistic, movement of the swans. In "unison," they dunk their heads under the water, rotating their bodies downwards. The repeated /r/ sounds add a sense of sure-footedness to the lines, conveying the swans' easy ability to stabilize themselves after doing so. This process of "righting" themselves also works as a kind of symbol for the couple, subtly inspiring them to reconcile their relationship; the use of alliteration, then, helps focus readers' attention on a particularly important moment in the poem.