waving vs gesture

waving and gesture both are nouns.

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

As nouns, gesture is a hypernym of waving; that is, gesture is a word with a broader meaning than waving:

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

Other hypernyms of waving include motion.

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

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