I am trying to understand when solutions of the wave equation could have issues, so I made a continuous function which some extreme behaviors: has compact-support given by a smooth bump function term, and also has issues in a zero-measure point where it becomes somehow "smoothly"/"softly" into a non-differentiable function.
Since the 1D wave equation $f_{xx}=\dfrac{1}{c^2}f_{tt}$ could indeed stand some distributions as solutions (at least as is said vaguely in Wikipedia), I want to know if the following function could be a solution of the 1D wave equation, and if not, if there is some ways it could make sense (like in weak solutions): for $z = x - ct$, the function I made is:$$f(z) = \begin{cases} 0,\quad z=0 \\ 0, \quad |z|\geq 1\\ z\ln(z^2)\ \exp\left(\dfrac{z^2}{z^2-1}\right),\,\text{otherwise}\end{cases}$$
From its plot it looks quite innocent and similar to a common wave (plot at $t=0$):
But just in the simple scenario of $t=0$ their derivatives blown-up, so I got lost in trying to figure out if it could fulfill the 1D wave equation or not. Hope you could explain why is a so (if it fulfill it or not).
But if it do solves the solution, hope you could comment what is happening near $z=0$, since my intuition tells me that near that point the wave-points would be moving faster than the wave speed $c$: Would that violate causality? or at least the restrictions the same wave equation is supposedly fulfilling? or the wave speed don't restrict at all the speed the wave-points could have?