To solve the Wave Equation
$ u_{tt} - c^2 u_{xx} = 0$
One method is to start with operator factorization
$ u_{tt} - c^2 u_{xx} = \bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) \bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg) u = 0 $
Then it is claimed the solution is
$u(x,t) = f(x+ct) + g(x-ct)$
where $f$ ve $g$ are arbitrary functions of single variable.
The proof goes by letting $v = u_t + cu_x$
then
$ v_t - cv_x = 0 $
has to be true. Then simultaneously the two equations are solved
$ v_t - cv_x = 0 $
$ u_t + cu_x = v $
We know the solution for the top equation above is
$ v(x,t) = h(x+ ct) $
where $h$ is an arbitrary function. Now wave equation is
$ u_t + cu_x = h(x + ct) $
Here is what I do not understand. At this point that a solution is guessed as $u(x,t) = f(x+ct)$ and the book says "it is easy to check by differentiation"
f'(s) = h(s) / 2c.
What do I do with $h(s) / 2c$? Is the proof complete with this conclusion? When I plug in $f(x+ct)$ for $u(x,t)$ yes, I do get this equality but how does this help?
Also, after proving $f(x+ct)$ is a solution, then magically a $g(x-ct)$ is added, where does this come from? I do understand linear independence, but why didnt we add $g(x+2ct)$ for example?
Thanks,