When did physicists start defining waves as 'solutions of a wave equation'?

Question

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?

Answer 1

Let's sort out a few things first. It is true that originally people called things waves if they behaved like waves in the colloquial sense. This is still true today even in physics, waves do not have to be solutions to the wave equation, solutions to the Schrödinger or Korteweg–de Vries equations for example are also called waves, as is any "disturbance or oscillation (of a physical quantity), that travels through matter or space, accompanied by a transfer of energy", regardless of what equation it satisfies if any.

Now concerning light. The idea that light is a wave came up before the wave equation was written down. Huygens offered wave theory of light in 1678. However, his version of it was essentially equivalent to geometric optics, which could just as well be explained by the corpuscular theory of Newton. On Newton's authority, and due to difficulties with explaining polarization in wave theory, the corpuscular optics dominated in the 18-th century. Experience with surface waves however brought about the realization that there were effects that distinguished the wave theory, such as diffraction and interference. The wave nature of light was accepted after 1816 experiments of Fresnel showed that light exhibited diffraction and interference patterns, and in quantitative agreement with wave theory's predictions. The dark spot at the center of diffraction pattern, straight across the hole through which the light is shone, also looked very convincing. This started the theory of the "luminiferous ether", a hypothetical medium in which optical waves were supposed to propagate.

So Maxwell's contribution was not establishing the wave nature of light, it was establishing that light was a particular type of electromagnetic field. The deciding factor in proving it was not that light and electromagnetic waves both satisfied the wave equation, so did acoustic and elastic waves among others. It was that Maxwell showed in 1862 that the propagation speed of electromagnetic waves predicted by his equations matched the known speed of light. The formula was $c=\frac1{\sqrt{\varepsilon_0\mu_0}}$, where $c$ is the speed of light, and $\varepsilon_0$ and $\mu_0$ (electric permittivity and magnetic permeability of vacuum) are constants obtained from purely electric and magnetic measurements. It was quite a tell.