If we consider the wave equation $u_{tt}-c^2u_{xx}=0$ on the one-dimensional half space (i.e. $x>0$), and suppose the solution for $t<0$ takes the form of a left-moving wave $u=f(x+ct)$, how does the solution evolve for $t\geq0$?
Two cases are to be considered: Dirichlet, and Neumann boundary conditions at $x=0$.
Could anyone tell me how I would arrive at a solution to this problem? I don't understand how anything can be said about the solution with the information given, it seems insufficient to me.
Here's the Dirichlet case.
Note that the Dirichlet condition implies $f(s) = 0$ for $s < 0$. Then take $u(x,t) = f(x + ct) - f(-x + ct)$.