The wave equation with a boundary condition that turns on at $t=0$

Consider the wave equation in 1 spatial dimension (in units where $c=1$)
$$
frac{partial^2 u(t,x)}{partial t^2} – frac{partial^2 u(t,x)}{partial x^2} = 0 .
$$
Now suppose that the solution is $u(t,x)=0$ for all $t<0$. Then suddenly at $t=0$ a boundary condition gets enforced at $x=x_0$
$$
u(x_0, t) = Theta(t) b(t,x_0) ,
$$
for some function $b$.

Can progress be made towards a solution for such a boundary condition? For concreteness, I would like to consider the behaviour of the solution $u(t,x)$ for $t>0$ and $x>x_0$ (only to the right of the boundary condition).

For $t>0$ and $x>x_0$ the solution should have the standard form $u(t,x) = F(x – t) + G(x + t)$ for some functions $F$ and $G$.

My guess is that $G=0$ should be true for this set-up (ie. no incoming wave for $x>x_0$), since the disturbance caused by the boundary condition should probably cause only an outgoing wave (described just by $F$, a function of the retarded time $t-x$).

My second guess is that $F$ is probably proportional to a $Theta(t – x)$ for $x>x_0$, since the outgoing wave needs to take some time to arrive at the point $x >x_0$

Is this true? and How to see this?