I know that the one-dimensional wave equation can be written as $$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial ^2 u}{\partial x^2}$$ and has solutions of the form $$ u = F(x+ct) + G(x-ct)$$
I'm having trouble developing a proper intuition about the meaning of the solution, though. I superficially understand that it's the sum of two functions "travelling" in different directions with time, but that doesn't help me be able to really visualize what solutions look like. More specifically, I'd like to develop an intuitive or visual understanding of what solutions to the wave equation have in common, and what separates them from functions that aren't solutions.
In the hope an animation is worth a thousand words: The blue wave $F$ travels left, the red wave $G$ travels right. Their sum is magenta. Both $F$ and $G$ were chosen to be spatially periodic so the loop would be smooth.