pde

I've always known the wave equation since even before taking a proper PDE lecture, from browsing the internet and watching videos, but I never saw how it came to be derived. Like, why does this describe a wave? $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ I decided to read Stein and Shakarchi's book on Fourier Analysis (as it is part one of a tetralogy, and I feel like I really need more insight on this topic), and a nice derivation is found right in the beginning.