# An explicit formula for the solution of half-infinite string wave equation

Find an explicit formula for the solution of half-infinite string wave equation with boundary and initial conditions:

That is the system is $u_{tt}-c^2u_{xx}=0$ for $x>0,t>0$

$u(0,t)=h(t)$ for $t>0$ and $u(x,0)=f(x), u_t(x,0)=g(x)$ for $x\geq 0$ where $f,g,h$ are $C^2$ with compatibility conditions $h(0)=f(0),h'(0)=g(0),h''(0)=c^2f''(0)$.

For the full string we know that $u(x,t)=F(x+ct)+G(x-ct)$ for some $F,G$. But here we have half string, so can I still write $u(x,t)$ as above? Also, the question asks for an explicit formula. I have seen such solutions that involves $\sin$ and $\cos$ functions, but how do we derive such a solution for this case?

• Do you know Duhamel's principle? – Uskebasi Mar 8 '17 at 9:35