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When were waves discovered/treated mathematically? Ex: basic sine wave

I searched google for any information regarding the invention/establishment of a mathematical description of waves. I did not find any results. I know that there was an understanding of SHM in springs at around 1700 but I do not know if there exists a time period before then in which waves were abstractly constructed.

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    $\begingroup$ (per your comment to JgL:) You seem to misunderstand your responsibility here. You need to show us that you have done some prior research, and you have not done so. Please show us what you have tried. $\endgroup$ Oct 20 '16 at 0:28
  • $\begingroup$ @rory Sorry, I did not know I had to show research effort. I will edit the question. $\endgroup$
    – John T.
    Oct 20 '16 at 1:35
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    $\begingroup$ This is answered in Did wave optics anticipate quantum mechanics? One of the earliest mathematical treatments of waves was in Huygens's wave optics around 1670. $\endgroup$
    – Conifold
    Oct 20 '16 at 17:25
  • $\begingroup$ @conifold Is it the earliest time period, though? The first time waves were thought of, in an abstract sense, was in a description of light? I doubt that because pendulums,springs, water waves, sound , etc were all much more easily observed and understood. $\endgroup$
    – John T.
    Oct 20 '16 at 18:20
  • $\begingroup$ Talk of waves can be read into some passages of Aristotle, although it is believed that Leonardo was first to clearly articulate the concept. Galileo talked about acoustic waves, and suggested that pitch is related to their frequency c.1600, I suppose that was mathematical-ish. But I am not aware of a mathematical physics theory involving waves before Huygens, see What is the philosophical origin of waves? $\endgroup$
    – Conifold
    Oct 21 '16 at 0:20

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