What does the solution to a wave equation represent?

What does the solution of a wave equation represent? When one finds a solution to a wave equation, how can one interpret this solution? Does the solution show the behavior of a wave?

Answer

For the usual wave equation,

$$\frac{\partial^2 u}{\partial t^2} - v^2 \nabla^2 u = 0$$

$u = u(x,t)$ is taken to be some property of the wave, at position $x$ and time $t$. For example, for the electromagnetic wave equation, $E(x,t)$ would be the force per unit charge at $(x,t)$.

If we had a rope, and a wave $u(x,t)$ propagated along the rope, then we could define $u(x,t)$ as the displacement of the infinitesimal piece of rope along it at position $x$, at the instant $t$.