Wave
Old English wafian (verb), from the Germanic base of waver; the noun by alteration (influenced by the verb) of Middle English wawe ‘(sea) wave’.
wiktionary
From Middle English waven, from Old English wafian(“to wave, fluctuate, waver in mind, wonder”), from Proto-Germanic *wabōną, *wabjaną(“to wander, sway”), from Proto-Indo-European *webʰ-(“to move to and from, wander”). Cognate with Middle High German waben(“to wave”), German wabern(“to waft”), Icelandic váfa(“to fluctuate, waver, doubt”). See also waver.
From Middle English *wave, partially from waven(“to fluctuate, wave”) (see above) and partially from Middle English wawe, waghe(“wave”), from Old English wǣg(“a wave, billow, motion, water, flood, sea”), from Proto-Germanic *wēgaz(“motion, storm, wave”), from Proto-Indo-European *weǵʰ-(“to drag, carry”). Cognate with North Frisian weage(“wave, flood, sea”), German Woge(“wave”), French vague(“wave”) (from Germanic), Gothic 𐍅𐌴𐌲𐍃( wēgs, “a wave”). See also waw.
See waive.
etymonline
wave (v.)
"move back and forth," Old English wafian "to wave, fluctuate" (related to wæfre "wavering, restless, unstable"), from Proto-Germanic *wab- (source also of Old Norse vafra "to hover about," Middle High German waben "to wave, undulate"), possibly from PIE root *(h)uebh- "to move to and fro; to weave" (see weave (v.)). Transitive sense is from mid-15c.; meaning "to make a sign by a wave of the hand" is from 1510s. Related: Waved; waving.
I was much further out than you thought
And not waving but drowning.
[Stevie Smith]
wave (n.)
"moving billow of water," 1520s, alteration (by influence of wave (v.)) of Middle English waw, which is from Old English wagian "to move to and fro," from Proto-Germanic *wag- (source also of Old Saxon, Old High German wag, Old Frisian weg, Old Norse vagr "water in motion, wave, billow," Gothic wegs "tempest"), probably from PIE root *wegh- "to go, move." The usual Old English word for "moving billow of water" was yð.
The "hand motion" meaning is recorded from 1680s; meaning "undulating line" is recorded from 1660s. Of people in masses, first recorded 1852; in physics, from 1832. Sense in heat wave is from 1843. The crowd stunt in stadiums is attested under this name from 1984, the thing itself said to have been done first Oct. 15, 1981, at the Yankees-A's AL championship series game in the Oakland Coliseum; soon picked up and popularized at University of Washington. To make waves "cause trouble" is attested from 1962.
