Solve, using the methods of characteristics: $$\begin{cases}u_{tt}(x,t)=u_{xx}(x,t)\\ IC: u(x,0)=u_t(x,0)=0\\ BC: u(0,t)=h(t),\ u(L,t)=0\end{cases}$$
I tried to use the substitution $u(x,t)=v(x,t)+h(t)$ Which leads to: $$\begin{cases}v_{tt}(x,t)+h''(t)=v_{xx}(x,t)\\ IC: v(x,0)=-h(0),\ v_t(x,0)=-h'(0)\\ BC: v(0,t)=0,\ u(L,t)=-h(t)\end{cases}$$
I am not quite sure how to continue. The formula of d'Alembert would only work if for $u(L,t)=0$. Could anyone give me a hint on how to continue?