TITLE: Method of reflections for the wave equation.
QUESTION [6 upvotes]: For the Wave Equation:

$ \left\{\begin{matrix}
u_{tt}-c^2u_{xx}=0 & x>0, t>0\\ 
u(x,0)=f(x) & x>0 &  \hspace{0.5cm} (1) \\ 
u_{t}(x,0)=g(x) & x>0 &  \hspace{0.5cm} (1)  \\
u(0,t)=h(t) & t>0  \\ 
\end{matrix}\right. $

The general solution is:

$ u(x,t) = \frac{f(x+ct)-f(ct-x)}{2}+ \frac{1}{2c}\int_{x+ct}^{ct-x}g(s)ds + h(t-\frac{x}{c})  $

My notes explain how to obtain it. But then they say that when $h\equiv0$, we can use the method of reflections to solve the problem for $ x\in{\mathbb{R}} $ with the restrictions (1).

Can someone post an example using method of reflections? (*)
Also I don't get why we need to solve it for $x\in{\mathbb{R}}$... Isn't the problem defined for $x>0$ ?

(*) I have been searching for it but I found nothing. Maybe it is a problem with the name, I have translated the name method of reflections from spanish método de reflexiones.

REPLY [2 votes]: Here is your complete answer. Please look at the link pdf. It deals both Dirichlet and Neumann boundary conditions
http://web.math.ucsb.edu/~grigoryan/124A/lecs/lec13.pdf