Alliteration is used here and there throughout "An Absolutely Ordinary Rainbow." Alliteration can draw readers' attention to various words and phrases, reinforce thematic connections between words, and simply make the poem sound more interesting.
In the first stanza, for example, alliteration works alongside consonance to reflect the city dwellers' chattering, evoking the sound of gossip and intrigue. In line 1, "round Repins" uses a shared /r/ to signify the way in which everyone is talking about the same thing—the crying man. The "Stock Exchange scribblers" phrase has a similar effect, with the added whisper of sibilance evoking that scribbling.
Another striking moment of alliteration is in line 15, in the poem's third stanza:
and does not declaim it, nor beat his breast, nor even
The speaker describes the dignity of the man's weeping, explaining that the man doesn't "sob very loudly." The forceful alliteration is thus somewhat ironic, mimicking the way that it would sound if the man were to "declaim" (shout) it or beat his chest. At the same time, the strength and insistence of this alliteration reflects the crying man's powerful, firm dignity.
In the first line of the following stanza, the poem uses many /h/ sounds:
holds us back from his space, the hollow he makes about him
The breathy /h/ sound evokes the "hollow" space being described. These alliterative sounds also lend a sense of cohesion to the line, making the line itself like a halo or force field.