Etymology
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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 .

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.

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Definitions of wave
1
wave (n.)
one of a series of ridges that moves across the surface of a liquid (especially across a large body of water);
Synonyms: moving ridge
wave (n.)
a movement like that of a sudden occurrence or increase in a specified phenomenon;
troops advancing in waves
a wave of settlers
wave (n.)
(physics) a movement up and down or back and forth;
Synonyms: undulation
wave (n.)
something that rises rapidly;
a wave of conservatism in the country led by the hard right
a wave of emotion swept over him
there was a sudden wave of buying before the market closed
wave (n.)
the act of signaling by a movement of the hand;
Synonyms: waving / wafture
wave (n.)
a hairdo that creates undulations in the hair;
wave (n.)
an undulating curve;
Synonyms: undulation
wave (n.)
a persistent and widespread unusual weather condition (especially of unusual temperatures);
a heat wave
2
wave (v.)
signal with the hands or nod;
She waved to her friends
He waved his hand hospitably
Synonyms: beckon
wave (v.)
move or swing back and forth;
She waved her gun
Synonyms: brandish / flourish
wave (v.)
move in a wavy pattern or with a rising and falling motion;
the waves rolled towards the beach
Synonyms: roll / undulate / flap
wave (v.)
twist or roll into coils or ringlets;
Synonyms: curl
wave (v.)
set waves in;
she asked the hairdresser to wave her hair
3
Wave (n.)
a member of the women's reserve of the United States Navy; originally organized during World War II but now no longer a separate branch;
From wordnet.princeton.edu