"The Second Coming" uses alliteration sparingly throughout. It is first used in the opening three lines, with repetition of the /t/ and /f/ sounds:
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart;
This opening image is focused on disorientation, and the alliteration furthers the idea of repeated, confused movement. The falcon and the falconer have been separated, neither able to locate the other, and the scattered /f/ sound shows that they can't bridge the gap that separates them.
The next notable example of alliteration comes in line 13, as the speaker describes the vision that came from the Spiritus Mundi. The speaker starts by setting the scene, and that's where the alliteration comes in:
Troubles my sight: somewhere in sands of the desert
The alliterative /s/ sounds have a sibilant, whispering kind of sound that conjures up the howling winds of an inhospitable desert. This sense of atmosphere makes the beast that turns out to live in the desert seem all the more ominous.
The poem then dials up the alliteration as it draws to its conclusion:
Were vexed to nightmare by a rocking cradle,
And what rough beast, its hour come round at last,
Slouches towards Bethlehem to be born?
The /r/ sounds link "rocking," "rough," and "round" together, creating an atmosphere of threat and potential violence. The four /b/ sounds in the last two lines build the sense of something taking to shape, as though the beast is growing in strength as the poem comes to its conclusion.