Whenever one is studying poetry, they need to be sure to understand the devices of a poem. Eugene Field's poem "The Duel" contains many different poetic devices.

Alliteration: Alliteration is the repetition of a consonant sound within a line of poetry. Alliteration is, typically, used to create a "sing-song" melody. The use of alliteration within the line adds to the sing-song effect seen throughout.Alliteration appears in the second line with the repetition of the "s" sound.

Side by side on the table sat;

Assonance: Assonance is similar to alliteration, but a vowel sound is repeated. An example of assonance appears in line three (the vowel sound in "do" and "you" is repeated).

'T was half-past twelve, and (what do you think!)

Personification: Personification is the giving of human characteristics/qualities to non-human/non-living things. The last five lines of stanza one contain many different examples of personification.

The old Dutch clock and the Chinese plate
Appeared to know as sure as fate
There was going to be a terrible spat.
(I was n't there; I simply state
What was told to me by the Chinese plate!)

Onomatopoeia: Onomatopoeia is, according to eNOtes,

the formation and use of words that suggest, by their sounds, the object or idea being named or the imitation of natural sounds by words such as “bang” or “buzz.” It is a figure of speech and is especially useful for rhetorical effect.

Therefore, the use of both Bow-wow-wow and Mee-ow are examples of onomatopoeia.

The gingham dog went "Bow-wow-wow!"
And the calico cat replied "Mee-ow!"