I'm going through the solution of the boundary problem of the wave equation with the parallelogram theorem, and it states that for the problem:
$u_{tt}-c^2u_{xx}=\phi(x,t),\quad a < x < b,\quad t>0$
$u(x,0) = f(x),\ u_t(x,0) = g(x),\quad a \leq x \leq b$
$u(a,t)=\alpha(t),\ u(b,t)=\beta(t),\quad t > 0$
the compatibility conditions are:
$\alpha(0)=f(a),\ \beta(0)=f(b)$
$\alpha'(0)=g(a),\ \beta'(0)=g(b)$
which are clear, and finally:
$\alpha''(0)=c^2 f''(a),\ \beta''(0)=c^2 f''(b)$
which are equivalent for demanding that on the boundaries, the solution will obey to the homogeneous equation. Why is that a demand?