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5

There's nothing wrong with the first order wave equation mathematically, but it's just a little boring. If you want to use this equation to describe waves, it basically amounts to having a 1d solid with speed of sound $v$ for left moving waves (say) and speed of sound $0$ for right moving waves. It wouldn't surprise me if such a thing could be constructed ...

7

There are a few reasons I can think of: (1) The second order system is that it is time-reversible. If you let $t\to-t$, you get $$\frac{\partial^2f}{\partial(-t)^2}=\frac{\partial^2f}{\partial t^2}=v^2\frac{\partial^2f}{\partial x^2}$$ whereas the first order system has  \frac{\partial f}{\partial(-t)}=-\frac{\partial f}{\partial t}=\pm v\frac{\partial ...

3

Eh. It's like saying that Maxwell's equations are incomplete because they allow similar solutions. If you start spacetime out with radiation, that radiation will propogate. Just like if you have a universe containing nothing but a light pulse, that pulse will propogate.

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