The equation provided is:

$u_{tt} - c^2 u_{xx} = 0$

This is the one-dimensional wave equation, where $c$ is the wave speed, and $u = u(x, t)$ is the unknown function of space $x$ and time $t$.

Solution

The general solution to the wave equation in one dimension can be found using the method of d'Alembert. The general solution is:

$u(x, t) = f(x - ct) + g(x + ct)$

where $f$ and $g$ are arbitrary functions determined by the initial conditions of the problem. This form represents two waves traveling in opposite directions with speed $c$: one wave $f(x - ct)$ traveling to the right, and the other wave $g(x + ct)$ traveling to the left.

This solution arises because the wave equation can be decomposed into characteristic lines along which the solution remains constant.

Would you like a more detailed derivation of this solution or examples of initial conditions?

Here are some follow-up questions to deepen understanding:

1. How would the solution change if we introduced a source term in the wave equation?
2. What are some physical interpretations of the functions $f$ and $g$ in the context of waves?
3. How does the wave speed $c$ affect the behavior of the solution?
4. What are the implications of different boundary conditions, such as fixed or free ends?
5. How would you find a specific solution given initial conditions for $u(x, 0)$ and $u_t(x, 0)$?

Tip: When solving partial differential equations like the wave equation, check if characteristic lines or coordinate transformations simplify the problem.