I am aware that the wave equation $$u_{xx}=u_{tt}$$ can be analytically solved on a finite spatial domain even if it has a non-smooth initial condition. For instance, one such problem is analytically solved here. However, there are a couple of things that are not clear to me.
Let $x \in [0, 1]$. If we have one smooth and one non-smooth initial condition in the form of for example:
$$ u(0,x)=0 \tag 1$$ $$ {\partial u \over \partial t}\Bigg |_{t=0}=0 \ \ \forall \ \ x > 0 \ \ \tag 2$$ $$ {\partial u \over \partial t}\Bigg |_{t=0,x=0}=V \tag 3$$
where $V$ is a non-zero constant, is it possible for the wave function $u(t,x)$ to be a smooth function? My guess is no, but I'm not $100 \ \%$ sure. Also, when we have these types of initial conditions, the usual approach is to solve it piece-wise (assuming that the wave function has only one discontinuity), meaning we are solving two wave equations and would somehow have to determine one more boundary condition at the interface. However, for these instances of initial conditions, we should still solve only one wave equation for every $x > 0$, while the boundary condition at $x = 0$ would define the wave function at $x = 0$, right? Meaning, that one more boundary condition at the interface would not be needed?