"moving billow of water," 1520s (Tyndale), alteration (by influence of wave (v.)) of Middle English waw, waue, 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 noun and the verb have both been somewhat confused with waive. The usual Old English word for "moving billow of water" was yð.
The meaning "motion to and from with the hand or a held object" is recorded from 1680s. The sense of "surface in the form of a ridge and trough" is by 1540s. The meaning "undulating line" is by 1660s.
Figuratively, in reference to a flood, rush, or influx of anything, by 1580s. Of forward movements among people in masses, it is recorded by 1852; in reference to public feeling, opinion, etc., by 1851.
Specific use in physics is from 1832, in reference to sound. Wave-front, "continuous line including all particles in the same phase" is by 1867.
In meteorology, in reference to an advancing movement of atmosphere (heat wave, etc.), from 1843. Of earthquakes by 1760. The figurative phrase make waves "cause trouble" is attested by 1959, probably from boating.
Our culture is a nice flat-bed boat in the middle of a serene lake of complacency; anyone who splashes or makes waves is immediately pinioned by the state, the church, and the indignant populace. [Daily Tar Heel, April 4, 1959]
"I'm in one canoe, you're in another, the captain's in a canoe, the colonel ... You know what you do when you complain to somebody or report somebody or request something? (Irvin shakes his head). You make waves." [Ronald D. Scofield column in Santa Barbara News-Press, Dec. 6, 1959]
The crowd stunt making a wave-like effect 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.