In modern physics, a quantity is said to behave like a wave if it satisfies a wave equation. For a contemporary instance of this type of reasoning, see e.g. this. With some quick googling, it is quite simple to find out in what context the wave equation was first derived and applied (e.g. here).
Furthermore, it is known that throughout the 18th and 19th century, the wave equation kept on popping up everywhere and that, by the time Maxwell put the final touch on what are now known as Maxwell's equations, there was some kind of consensus that his demonstration that the electromagnetic fields satisfy a wave equation is enough to proclaim the discovery that certain kinds of radiation (most notably visible light; pun intended) should be interpreted as electromagnetic waves.
This means that, somewhere in this period, there must have been a certain shift in the understanding of waves. I would imagine that initially, one would observe wavelike behavior and, a posteriori, derive a wave equation that turned out to describe the phenomenon. However, by the time Maxwell did his most famous work, the approach had apparently reversed its order: Finding a wave equation now implied wavelike behavior!
My question is simple: Is it known when people started defining a wave via the wave equation?