Etymology 1 
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”), Icelandic váfa (“to fluctuate, waver, doubt”). See also waver.
- (intransitive) To move back and forth repeatedly.
- The flag waved in the gentle breeze.
- 2011 October 1, Tom Fordyce, “Rugby World Cup 2011: England 16-12 Scotland”, BBC Sport:
- But the World Cup winning veteran's left boot was awry again, the attempt sliced horribly wide of the left upright, and the saltires were waving aloft again a moment later when a long pass in the England midfield was picked off to almost offer up a breakaway try.
- (intransitive) To wave one’s hand in greeting or departure.
- I waved goodbye from across the room.
- (intransitive) To have an undulating or wavy form.
- (transitive) To produce waves to the hair.
- (intransitive, baseball) To swing and miss at a pitch.
- Jones waves at strike one.
- (transitive) To cause to move back and forth repeatedly.
- The starter waved the flag to begin the race.
- (transitive) To signal (someone or something) with a waving movement.
Derived terms 
Etymology 2 
From Middle English *wave, wawe, waghe (“wave”), partially from waven (“to fluctuate, wave”) (see above) and partially from Old English wǣg (“a wave, billow, motion, water, flood, sea”), from Proto-Germanic *wēgaz (“motion, storm, wave”), from Proto-Indo-European *weǵhe- (“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.
wave (plural waves)
- A moving disturbance in the level of a body of water; an undulation.
- The wave traveled from the center of the lake before breaking on the shore.
- (physics) A moving disturbance in the energy level of a field.
- A shape which alternatingly curves in opposite directions.
- Her hair had a nice wave to it.
- sine wave
- (figuratively) A sudden unusually large amount of something that is temporarily experienced.
- A wave of shoppers stampeded through the door when the store opened for its Christmas discount special.
- A wave of retirees began moving to the coastal area.
- A wave of emotion overcame her when she thought about her son who was killed in battle.
- 2011 January 11, Jonathan Stevenson, “West Ham 2 - 1 Birmingham”, BBC:
- Foster had been left unsighted by Scott Dann's positioning at his post, but the goalkeeper was about to prove his worth to Birmingham by keeping them in the game with a series of stunning saves as West Ham produced waves after wave of attack in their bid to find a crucial second goal.
- A sideway movement of the hand(s).
- With a wave of the hand.
- A group activity in a crowd imitating a wave going through water, where people in successive parts of the crowd stand and stretch upward, then sit. Usually referred to as "the wave"
Derived terms 
- (an undulation): und (obsolete, rare)
Etymology 3 
- Obsolete spelling of waive.