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]
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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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brain-wave (n.)
"apparent telepathic vibration transferring a thought from one person to another without any other medium, 1869, from brain (n.) + wave (n.).
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short-wave (adj.)
in reference to radio wavelength less than c.100 meters, 1907, from short (adj.) + wave (n.).
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wavelet (n.)
1808, mainly in poetry, from wave (n.) + diminutive suffix -let.
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wavy (adj.)
1580s, from wave (n.) + -y (2). Related: Waviness.
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wavelength (n.)
also wave-length, 1850, "distance between peaks of a wave," from wave (n.) + length. Originally of spectra; radio sense is attested by 1925. Figurative sense of "mental harmony" is recorded from 1927, on analogy of radio waves.
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waver (v.)
late 13c., weyveren, "to show indecision," probably related to Old English wæfre "restless, wavering," from Proto-Germanic *wæbraz (source also of Middle High German wabern "to waver," Old Norse vafra "to hover about"), a frequentative form from the root of wave (v.). Related: Wavered; wavering.
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microwave (n.)

type of electromagnetic wave, 1931, coined in English from micro- + wave (n.). First record of microwave oven is from 1961 (micro-oven also was used); microwave as short for this is attested from 1974; as a verb, from 1976. For a verb, "Better Homes and Gardens: magazine tried micro-cook (1976).

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