I'm currently trying to solve a PDE, the problem is as follows: Solve the equation $\frac{\partial ^{2} u}{\partial t^{2} } = c^{2}\frac{\partial ^{2} u}{\partial x^{2} }$ on the semi-infinite interval $0<x<L$ and with the conditions $$u_{x} + \alpha u \bigg |_{x=0} = 0 $$ $$f(x) = u \bigg |_{t=0} = \delta(x- x_{0}) \qquad and \qquad g(x) = u _{t}\bigg |_{t=0} = c\delta'(x- x_{0})$$
As far as I know, d'alembert solution should work to satisfy the initial conditions f and g, but I don't know how to deal with the boundary conditions.
If I have a general form for u, how would I apply this kind of boundary condition?