Behavior of wave equation when one of the initial conditions = 0 I have gathered in my reading that there is a difference between the behavior of the wave equation depending on whether the initial position or initial velocity is zero, but my impression is not precise enough to be very clear to me. 
If we have the one dimensional wave equation $u_{tt} = u_{xx}$ with $x \in \mathbb{R}$ and $t \geq 0$, with initial conditions
\begin{equation}
u(x,0) = f(x) 
\\
u_t(x,0) = g(x)
\end{equation}
As I understand it, if $f(x) = 0$ for all $x$ (and $g(x)$ not zero), then the wave will propagate in one direction, whereas if $g(x) = 0$ (and $f(x)$ not zero), then the disturbance splits in half and goes both directions. This latter case can be seen by looking at d'Alembert's Formula. Though I don't have a good reference for the former case.
Can somebody help make this more precise, confirm my understanding, or point to a reference where this particular matter is treated?  
 A: This is not an answer, but an unsuccessful attempt to reconcile the intuition of d'Alembert's formula with the explanation you've given, as well as an iteration of a cry for help since in my humble opinion there are same gaps in the intuition, which your question calls upon.
The general solution is
$$2u(x,t)=f(x-t)+f(x+t)+\int_{x-t}^{x+t}g(\tau)d\tau.$$
As you pointed out, if $g\equiv0$ then the initial wave $f$ splits in two and the solution goes in opposite directions (I imagine a string stretched and let go from rest). This case makes sense.
For the other situation, if $f\equiv0$, then for any $x$, $u(x,t)$ approaches the integral of $g$ as $t$ goes to infinity. This makes no intuitive sense, since straight from the formula we know then that if say, $g$ is any bump function with area 1, then for any $x$, $u(x,t)$ is eventually $1$, which is like saying "if you pluck a string with unit velocity, the entire string will eventually sit one unit in the direction that you've plucked it". This statement of course makes no sense physically...
A: Update: see
https://www.researchgate.net/publication/340085346
This paper shows that f can be any (reasonable) function.  The if g = -df/dx there will be no backward wave--the wave will propagate in one direction only.
In particular see appendix C.
