For the problem:
$u_{tt}-c^2u_{xx}=\phi(x,t),\quad a<x<b,\quad t>0$
$u(x,0)=f(x),\ u_t(x,0)=g(x),\quad a \leq x \leq b$
$u(a,t)=\alpha(t),\ u(b,t)=\beta(t), \quad t \geq 0$
To have a solution we require the compatibility mode, which is the different constraints to fit one with each. Naturally it means $f(a)=\alpha(0),\ f(b)=\beta(0)$ and $g(a)=\alpha'(0),\ g(b)=\beta'(0)$, but according to my notes it means also $\alpha''(0)=c^2 f''(a),\ \beta''(0)=c^2 f''(b)$. It seems to originate from the homogeneous wave equation, but I don't see why? Why do we assume for the edges $u_{tt}-c^2u_{xx}=0$ instead of $u_{tt}-c^2u_{xx}=\lim_{t\rightarrow 0^+,\ x\rightarrow a^+ / b^-} \phi(x,t)$?