heat (n.)youdaoicibaDictYouDict[heat 词源字典]
Old English hætu, hæto "heat, warmth, quality of being hot; fervor, ardor," from Proto-Germanic *haita- "heat" (cognates: Old Saxon hittia, Old Norse hiti, Old Frisian hete, German hitze "heat," Gothic heito "fever"), from PIE *kaid-, from root *kai- "heat." The same root is the source of Old English hat "hot" and hæða "hot weather."

Meaning "a single course in a race," especially a horse race, is from 1660s, perhaps from earlier figurative sense of "violent action; a single intense effort" (late 14c.), or the meaning "run given to a horse to prepare for a race" (1570s). The latter word over time was extended to "division of a race or contest when there are too many contestants to run at once," the winners of each heat then competing in a final race.

Meaning "sexual excitement in animals" is from 1768, especially of females, corresponding to rut in males. Meaning "trouble with the police" attested by 1920. Heat wave "period of excessive hot weather" first attested 1890; earlier in reference to solar cycles. Heat-stroke is from 1874. Heat-seeking (adj.) of missiles, etc., is from 1956. Red heat, white heat are in reference to the color of heated metals, especially iron.[heat etymology, heat origin, 英语词源]
wave (n.)youdaoicibaDictYouDict
"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" (cognates: Old Saxon, Old High German wag, Old Frisian weg, Old Norse vagr "water in motion, wave, billow," Gothic wegs "tempest;" see wag (v.)). 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.