I am trying to solve the following 1D wave equation problem with initial and boundary conditions.
$u_{tt} - c^2 u_{xx} = 0$
Initial and boundary conditions:
$u=f(x)\;, u_t=g(x) \;\;\; \text{for} \; t=0, x>0$
$u_t = \alpha u_x \;(\alpha \neq -c) \;\;\; \text{for} \; t>0, x=0$
I am also given that f and g are differentiable twice ($ \in C^2$) and that they vanish near x=0.
Here is what I have/know so far:
We know that u must be of the form:
$u = F(x+ct) + G(x-ct)$ where F and G are some functions 8we take that as a given ... it can be derived from e.g. characteristics). Taking that, we can conclude from the conditions above that:
$u_t(0,t) = cF'(ct)-cG'(ct) =\alpha(F'(ct) + G'(ct)) = \alpha u_x(0,t) \\ u(x,0) = F(x) + G(x) = f(x) \implies f'(x) = F'(x) + G'(x) \\ u_t(x,0) = cF'(x) - cG'(x) = g(x)$
I wrote those down because lines 2 and 3 are basic steps on the way to deriving d'Alembert's formula when we only have initial conditions at t=0 and solve in the upper xt - plane (i.e. for t > 0).
However, with this additional boundary condition, I am a bit lost ... I am lacking the idea of how to proceed and incorporate that bound condition.
Of course, I am not asking you to solve it for me, I would be happy about any hint/suggestion on how to proceed from here. Thanks very much in advance!