gesture vs waving

gesture and waving both are nouns.

Word	Noun	Adjective	Verb	Adverb
gesture	Yes	No	Yes	No
waving	Yes	No	No	No

As nouns, waving is a hyponym of gesture; that is, waving is a word with a more specific, narrower meaning than gesture:

gesture: the use of movements (especially of the hands) to communicate familiar or prearranged signals
waving: the act of signaling by a movement of the hand

gesture (noun)	waving (noun)
motion of hands or body to emphasize or help to express a thought or feeling	the act of signaling by a movement of the hand
something done as an indication of intention
the use of movements (especially of the hands) to communicate familiar or prearranged signals

gesture (verb)	waving (verb)
show, express or direct through movement