I don't really understand where the fundamental or general wave equation $$\frac{\partial^2y}{\partial t^2} = v^2\frac{\partial^2y}{\partial x^2}$$ comes from. I understood the derivation of wave equation for a string which shows that $v = \sqrt{T/\mu}$ . But I don't understand where the original equation is derivied from or is it a way to define ways? How would this equation be discovered or derived given fundamental properties of waves?
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2$\begingroup$ Does this answer your question? Derivation of 1D wave equation $\endgroup$– Vincent ThackerCommented May 21 at 5:50
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The $\frac{\partial ^2 y}{\partial t^2}$ term is an acceleration. $\frac{\partial ^2 y}{\partial x^2}$ is a term which sometimes shows up in the equations for force, such as the force due to tension of a string.
Over time, as we looked at more and more systems, we found this particular term kept cropping up. It shows up in strings, and in blown pipes, and in electrical transmission lines and in quantum mechanics. We keep finding that there are systems where if we define y and x carefully, this equation shows up.
So really we arrived at it the way we arrive at most abstractions. We explored a bunch of cases and noticed there was a useful common pattern to be had between them.
Other systems do not follow this pattern. Thermodynamics, for example, typically does not have this $\frac{\partial ^2 y}{\partial x^2}$. We simply don't think of such systems as waves. But enough systems do indeed have wave like behaviors that it's a useful pattern.
